De Gruyter Expositions in Mathematics 50 Editors Victor P. Maslov, Academy of Sciences, Moscow Walter D. Neumann, Colum...

Author:
Yevhen Zelenyuk

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De Gruyter Expositions in Mathematics 50 Editors Victor P. Maslov, Academy of Sciences, Moscow Walter D. Neumann, Columbia University, New York, NY Markus J. Pflaum, University of Colorado at Boulder, Boulder, CO Dirk Schleicher, Jacobs University, Bremen Raymond O. Wells, Jr., Jacobs University, Bremen

Yevhen G. Zelenyuk

Ultrafilters and Topologies on Groups

De Gruyter

Mathematical Subject Classification 2010: 22A05, 22A15, 54D80, 54G05.

ISBN 978-3-11-020422-3 e-ISBN 978-3-11-021322-5 ISSN 0938-6572 Library of Congress Cataloging-in-Publication Data Zelenyuk, Yevhen G. Ultrafilters and topologies on groups / by Yevhen G. Zelenyuk. p. cm. ⫺ (De Gruyter expositions in mathematics ; 50) Includes bibliographical references and index. ISBN 978-3-11-020422-3 (alk. paper) 1. Topological group theory. 2. Ultrafilters (Mathematics) I. Title. QA166.195.Z45 2011 5121.55⫺dc22 2010050782

Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de. ” 2011 Walter de Gruyter GmbH & Co. KG, Berlin/New York Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ⬁ Printed on acid-free paper Printed in Germany www.degruyter.com

Preface

This book presents the relationship between ultraﬁlters and topologies on groups. It shows how ultraﬁlters are used in constructing topologies on groups with extremal properties and how topologies on groups serve in deriving algebraic results about ultraﬁlters. The contents of the book fall naturally into three parts. The ﬁrst, comprising Chapters 1 through 5, introduces to topological groups and ultraﬁlters insofar as the semigroup operation on ultraﬁlters is not required. Constructions of some important topological groups are given. In particular, that of an extremally disconnected topological group based on a Ramsey ultraﬁlter. Also one shows that every inﬁnite group admits a nondiscrete zero-dimensional topology in which all translations and the inversion are continuous. ˇ In the second part, Chapters 6 through 9, the Stone–Cech compactiﬁcation ˇG of a discrete group G is studied. For this, a special technique based on the concepts of a local left group and a local homomorphism is developed. One proves that if G is a countable torsion free group, then ˇG contains no nontrivial ﬁnite groups. Also the ideal structure of ˇG is investigated. In particular, one shows that for every inﬁnite jGj Abelian group G, ˇG contains 22 minimal right ideals. In the third part, using the semigroup ˇG, almost maximal topological and left topological groups are constructed and their ultraﬁlter semigroups are examined. Projectives in the category of ﬁnite semigroups are characterized. Also one shows that every inﬁnite Abelian group with ﬁnitely many elements of order 2 is absolutely !resolvable, and consequently, can be partitioned into ! subsets such that every coset modulo inﬁnite subgroup meets each subset of the partition. The book concludes with a list of open problems in the ﬁeld. Some familiarity with set theory, algebra and topology is presupposed. But in general, the book is almost self-contained. It is aimed at graduate students and researchers working in topological algebra and adjacent areas. Johannesburg, November 2010

Yevhen Zelenyuk

Contents

Preface 1

v

Topological Groups 1.1 The Notion of a Topological Group . . . . . . 1.2 The Neighborhood Filter of the Identity . . . 1.3 The Topology T .F / . . . . . . . . . . . . . 1.4 Topologizing a Group . . . . . . . . . . . . . 1.5 Metrizable Reﬁnements . . . . . . . . . . . . 1.6 Topologizability of a Countably Inﬁnite Ring

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1 1 4 7 10 14 18

Ultraﬁlters 2.1 The Notion of an Ultraﬁlter . . . 2.2 The Space ˇD . . . . . . . . . . 2.3 Martin’s Axiom . . . . . . . . . 2.4 Ramsey Ultraﬁlters and P -points 2.5 Measurable Cardinals . . . . . .

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26 26 29 34 36 39

3

Topological Spaces with Extremal Properties 3.1 Filters and Ultraﬁlters on Topological Spaces . . . . . . . . . . . . . 3.2 Spaces with Extremal Properties . . . . . . . . . . . . . . . . . . . . 3.3 Irresolvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 42 46

4

Left Invariant Topologies and Strongly Discrete Filters 4.1 Left Topological Semigroups . . . . . . . . . . . . . 4.2 The Topology T ŒF . . . . . . . . . . . . . . . . . . 4.3 Strongly Discrete Filters . . . . . . . . . . . . . . . 4.4 Invariant Topologies . . . . . . . . . . . . . . . . .

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52 52 54 56 63

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68 68 71 72 76

The Semigroup ˇS 6.1 Extending the Operation to ˇS . . . . . . . . . . . . . . . . . . . . . 6.2 Compact Right Topological Semigroups . . . . . . . . . . . . . . . .

82 82 86

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Topological Groups with Extremal Properties 5.1 Extremally Disconnected Topological Groups 5.2 Maximal Topological Groups . . . . . . . . . 5.3 Nodec Topological Groups . . . . . . . . . . 5.4 P -point Theorems . . . . . . . . . . . . . .

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viii

Contents

6.3 6.4

Hindman’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . Ultraﬁlters from K.ˇS/ . . . . . . . . . . . . . . . . . . . . . . . . .

91 94

7

Ultraﬁlter Semigroups 97 7.1 The Semigroup Ult.T / . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.2 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.3 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

8

Finite Groups in ˇG 8.1 Local Left Groups and Local Homomorphisms 8.2 Triviality of Finite Groups in ˇZ . . . . . . . . 8.3 Local Automorphisms of Finite Order . . . . . 8.4 Finite Groups in G . . . . . . . . . . . . . . .

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110 110 117 120 128

Ideal Structure of ˇG 9.1 Left Ideals . . . . . . . . . . . 9.2 Right Ideals . . . . . . . . . . 9.3 The Structure Group of K.ˇG/ 9.4 K.ˇG/ is not Closed . . . . .

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130 130 136 140 144

10 Almost Maximal Topological Groups 10.1 Construction . . . . . . . . . . . . . . . . . . . . . . . 10.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Semilattice Decompositions and Burnside Semigroups 10.4 Projectives . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Topological Invariantness of Ult.T / . . . . . . . . . .

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147 147 150 154 158 165

11 Almost Maximal Spaces 11.1 Right Maximal Idempotents in H 11.2 Projectivity of Ult.T / . . . . . . . 11.3 The Semigroup C.p/ . . . . . . . 11.4 Local Monomorphisms . . . . . .

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170 170 174 177 181

12 Resolvability 12.1 Regular Homeomorphisms of Finite Order . . . . . . . . . . . . . . . 12.2 Resolvability of Topological Groups . . . . . . . . . . . . . . . . . . 12.3 Absolute Resolvability . . . . . . . . . . . . . . . . . . . . . . . . .

188 188 194 198

13 Open Problems

208

Bibliography

211

Index

217

9

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Chapter 1

Topological Groups

In this chapter some basic concepts and results about topological groups are presented. The largest group topology in which a given ﬁlter converges to the identity is described. As an application Markov’s Criterion of topologizability of a countable group is derived. Another application is computing the minimum character of a nondiscrete group topology on a countable group which cannot be reﬁned to a nondiscrete metrizable group topology. We conclude by proving Arnautov’s Theorem on topologizability of a countably inﬁnite ring.

1.1

The Notion of a Topological Group

Deﬁnition 1.1. A group G endowed with a topology is a topological group if the multiplication W G G 3 .x; y/ 7! xy 2 G and the inversion

W G 3 x 7! x 1 2 G

are continuous mappings. A topology which makes a group into a topological group is called a group topology. The continuity of the multiplication and the inversion is equivalent to the continuity of the function 0 W G G 3 .x; y/ 7! xy 1 2 G: Indeed, 0 .x; y/ D .x; .y//, .x/ D 0 .1; x/ and .x; y/ D 0 .x; .y//. The continuity of 0 means that whenever a; b 2 G and U is a neighborhood of ab, there are neighborhoods V and W of a and b, respectively, such that V W 1 U: It follows that whenever a1 ; : : : ; an 2 G, k1 ; : : : ; kn 2 Z and U is a neighborhood of a1k1 : : : ankn 2 G, there are neighborhoods V1 ; : : : ; Vn of a1 ; : : : ; an , respectively, such that V1k1 Vnkn U: Another immediate property of a topological group G is that the translations and the inversion of G are homeomorphisms. Indeed, for each a 2 G, the left translation a W G 3 x 7! ax 2 G

2

Chapter 1 Topological Groups

and the right translation a W G 3 x 7! xa 2 G are continuous mappings, being restrictions of the multiplication. The inversion is continuous by the deﬁnition. Since we have also that .a /1 D a1 , .a /1 D a1 and 1 D , all of them are homeomorphisms. A topological space X is called homogeneous if for every a; b 2 X , there is a homeomorphism f W X ! X such that f .a/ D b. If G is a topological group and a; b 2 G, then ba1 W G ! G is a homeomorphism and ba1 .a/ D ba1 a D b. Thus, we have that Lemma 1.2. The space of a topological group is homogeneous. Now we establish some separation properties of topological groups. Lemma 1.3. Every topological group satisfying the T0 separation axiom is regular and hence Hausdorff. In this book, by a regular space one means a T3 -space. Proof. Let G be a T0 topological group. We ﬁrst show that G is a T1 -space. Since G is homogeneous, it sufﬁces to show that for every x 2 G n¹1º, there is a neighborhood U of 1 not containing x. By T0 , there is an open set U containing exactly one of two points 1; x. If 1 2 U , we are done. Otherwise xU 1 is a neighborhood of 1 not containing x. Now we show that for every neighborhood U of 1, there is a closed neighborhood of 1 contained in U . Choose a neighborhood V of 1 such that V V 1 U . Then for every x 2 G n U , one has xV \ V D ;. Indeed, otherwise xa D b for some a; b 2 V , which gives us that x D ba1 2 V V 1 U , a contradiction. Hence cl V U . In fact, the following stronger statement holds. Theorem 1.4. Every Hausdorff topological group is completely regular. Proof. See [55, Theorem 10]. Theorem 1.4 is the best possible general separation result. However, for countable topological groups, it can be improved. A space is zero-dimensional if it has a base of clopen (D both closed and open) sets. Note that if a T0 -space is zero-dimensional, then it is completely regular. Proposition 1.5. Every countable regular space is normal and zero-dimensional.

3

Section 1.1 The Notion of a Topological Group

Proof. Let X be a countable regular space and let A and B be disjoint closed subsets of X . Enumerate A and B as A D ¹an W n < !º

and

B D ¹bn W n < !º:

Inductively, for each n < !, choose neighborhoods Un and Vn of an and bn respectively such that (a) cl Un \ B D ; and A \ cl Vn D ;, S S (b) Un \ . i

[ Ui \ Vi D ;:

in

It follows that U D

[ n

in

Un

and

V D

[

Vn

n

are disjoint neighborhoods of A and B, respectively. Now, having established that X is normal, let U be an open neighborhood of a point x 2 X . Without loss of generality one may suppose that U ¤ X . Then by Urysohn’s Lemma, there is a continuous function f W X ! Œ0; 1 such that f .x/ D 0 and f .X n U / D ¹1º. Since X is countable, there is r 2 Œ0; 1 n f .X /. Then f 1 .Œ0; r// D f 1 .Œ0; r/ is a clopen neighborhood of x contained in U . It follows from Lemma 1.3 and Proposition 1.5 that Corollary 1.6. Every countable Hausdorff topological group is normal and zerodimensional. Note that every ﬁrst countable Hausdorff topological group is also normal. (A space is ﬁrst countable if every point has a countable neighborhood base.) This is immediate from the fact that every metric space is normal and the following result. Theorem 1.7. A Hausdorff topological group is metrizable if and only if it is ﬁrst countable. In this case, the metric can be taken to be left invariant. Proof. See [34, Theorem 8.3]. Starting from Chapter 5, all topological groups are assumed to be Hausdorff.

4

Chapter 1 Topological Groups

1.2

The Neighborhood Filter of the Identity

For every set X , P .X / D ¹Y W Y X º: Deﬁnition 1.8. Let X be a nonempty set. A ﬁlter on X is a family F P .X / with the following properties: (1) X 2 F and ; … F , (2) if A; B 2 F , then A \ B 2 F , and (3) if A 2 F and A B X , then B 2 F . In other words, a ﬁlter on X is a nonempty family of nonempty subsets of X closed under ﬁnite intersections and supersets. A classic example of a ﬁlter is the set Nx of all neighborhoods of a point x in a topological space X called the neighborhood ﬁlter of x. By a neighborhood of x one means any set whose interior contains x. The system ¹Nx W x 2 X º of all neighborhood ﬁlters on X is called the neighborhood system of X . Theorem 1.9. Let X be a space and let ¹Nx W x 2 X º be the neighborhood system of X . Then (i) for every x 2 X and U 2 Nx , x 2 U , and (ii) for every x 2 X and U 2 Nx , ¹y 2 X W U 2 Ny º 2 Nx . Conversely, given a set X and a system ¹Nx W x 2 X º of ﬁlters on X satisfying conditions (i)–(ii), there is a unique topology T on X for which ¹Nx W x 2 X º is the neighborhood system. Proof. That the neighborhood system ¹Nx W x 2 X º of a space X satisﬁes (i)–(ii) is obvious. We need to prove the converse. Deﬁne the operator int on the subsets of X by putting for every A X int A D ¹x 2 X W A 2 Nx º: We claim that it satisﬁes the following conditions: (a) int X D X , (b) int A A, (c) int .int A/ D int A, and (d) int .A \ B/ D .int A/ \ .int B/.

Section 1.2 The Neighborhood Filter of the Identity

5

Indeed, for every x 2 X , X 2 Nx , consequently x 2 int X , and so (a) is satisﬁed. For (b), if x 2 int A, then A 2 Nx , and so by (i), x 2 A. To check (c), let x 2 int A. Then A 2 Nx . Applying (ii) we obtain that int A 2 Nx . It follows that x 2 int .int A/. Hence int A int .int A/. The converse inclusion follows from (b). To check (d), let x 2 .int A/\.int B/. Then A 2 Nx and B 2 Nx , so A\B 2 Nx . It follows that x 2 int .A \ B/. Hence .int A/ \ .int B/ int .A \ B/. Conversely, let x 2 int .A\B/. Then A\B 2 Nx , consequently A 2 Nx and B 2 Nx . It follows that x 2 .int A/ \ .int B/. Hence int .A \ B/ .int A/ \ .int B/. It follows from (a)–(d) that there is a unique topology T on X such that int is the interior operator for .X; T /. We have that a subset U X is a neighborhood of a point x 2 X in T if and only if x 2 int U , and so if and only if U 2 Nx . Hence, ¹Nx W x 2 X º is the neighborhood system for .X; T /. In a topological group, the neighborhood system is completely determined by the neighborhood ﬁlter of the identity. Lemma 1.10. Let G be a topological group and let N be the neighborhood ﬁlter of 1. Then for every a 2 G, aN D N a is the neighborhood ﬁlter of a. Here, aN D ¹aB W B 2 N º

and

N a D ¹Ba W B 2 N º:

Proof. Since both a and a are homeomorphisms and a .1/ D a .1/ D a, aN D a .N / D a .N / D N a is the neighborhood ﬁlter of a. The next theorem characterizes the neighborhood ﬁlter of the identity of a topological group. Theorem 1.11. Let .G; T / be a topological group and let N be the neighborhood ﬁlter of 1. Then (1) for every U 2 N , there is V 2 N such that V V U , (2) for every U 2 N , U 1 2 N , and (3) for every U 2 N and x 2 G, xUx 1 2 N . Conversely, given a group G and a ﬁlter N on G satisfying conditions (1)–(3), there is a unique group topology T on G for which N is the neighborhood ﬁlter of 1. The topology T is Hausdorff if and only if T (4) N D ¹1º.

6

Chapter 1 Topological Groups

Note that conditions (2) and (3) in Theorem 1.11 are equivalent, respectively, to (20 ) N 1 D N , and (30 ) for every x 2 G, xN x 1 D N , where N 1 D ¹A1 W A 2 N º and xN x 1 D ¹xAx 1 W A 2 N º. Proof. That the neighborhood ﬁlter of 1 satisﬁes (1)–(3) follows from the continuity of the multiplication .x; y/ at .1; 1/ and the mappings .x/ and x .x 1 .y// at 1. To prove the converse, consider the system ¹xN W x 2 Gº. We claim that it satisﬁes the conditions of Theorem 1.9. To check (i), let x 2 G and U 2 N . It follows from (1)–(2) that there is V 2 N such that V V 1 U . Then x 2 xV V 1 xU . To check (ii), let x 2 G and U 2 N . It follows from (1) that there is V 2 N such that V V U . For every y 2 xV , yV xV V xU , consequently xV ¹y 2 G W xU 2 yN º; and so ¹y 2 G W xU 2 yN º 2 xN : Now by Theorem 1.9, there is a unique topology T on G such that for each x 2 G, xN is the neighborhood ﬁlter of x, that is, the neighborhoods of x are of the form xU , where U is a neighborhood of 1. To see that T is a group topology, let a; b 2 G be given and let U be a neighborhood of 1. Using conditions (1)–(3) choose a neighborhood V of 1 such that bV V 1 b 1 U . Then aV .bV /1 D aV V 1 b 1 D ab 1 bV V 1 b 1 ab 1 U: Since T is a group T topology, it is Hausdorff if and only if it is a T1 -topology, and so if and only if N D ¹1º. The notion of a ﬁlter is closely related to that of a ﬁlter base. Deﬁnition 1.12. Let X be a nonempty set. A ﬁlter base on X is a nonempty family B P .X / with the following properties: (1) ; … B, and (2) for every A; B 2 B there is C 2 B such that C A \ B. Equivalently, B P .X / is a ﬁlter base if F D ¹A X W A B for some B 2 Bº is a ﬁlter, and in this case we say that B is a base for F . Note that if F is a ﬁlter, then B F is a base for F if and only if for every A 2 F there is B 2 B such that B A.

Section 1.3 The Topology T .F /

7

If X is a topological space and x 2 X , then a base for the neighborhood ﬁlter of x is called a neighborhood base at x. As a consequence we obtain from Theorem 1.11 the following. Corollary 1.13. Let B be a ﬁlter base on G satisfying the following conditions: (1) for every U 2 B, there is V 2 B such that V V U , (2) for every U 2 B, U 1 2 B, and (3) for every U 2 B and x 2 G, xUx 1 2 B. Then there is a unique group topology T on G for which B is a neighborhood base at 1. The topology T is Hausdorff if and only if T (4) B D ¹1º.

1.3

The Topology T .F /

Deﬁnition 1.14. For every ﬁlter F on a group G, let T .F / denote the largest group topology on G in which F converges to 1. Deﬁnition 1.14 is justiﬁed by the factW that for every family ¹Ti W i 2 I º of group topologies on G, the least upper bound i2I Ti taken in the lattice of all topologies on G is a group topology. Deﬁnition 1.15. For every ﬁlter F on a group G, let FQ denote the ﬁlter with a base consisting of subsets of the form [ 1 x.Ax [ A1 ; x [ ¹1º/x x2G

where for each x 2 G, Ax 2 F . Lemma 1.16. For every ﬁlter F on a group G, FQ is the largest ﬁlter contained in F such that T (i) 1 2 FQ , (ii) FQ 1 D FQ , and (iii) for every x 2 G, x FQ x 1 D FQ . Proof. That FQ satisﬁes (i) is obvious. To check (ii) and (iii), let Ax 2 F for each x 2 G. Then [ 1 [ 1 1 x.Ax [ A1 D x.Ax [ A1 : x [ ¹1º/x x [ ¹1º/x x2G

x2G

8

Chapter 1 Topological Groups

Consequently, FQ 1 D FQ . Next, for every y 2 G, [ [ 1 1 y x.Ax [ A1 yx.Ax [ A1 y 1 D x [ ¹1º/x x [ ¹1º/.yx/ x2G

x2G

D

[

1 x.Ay 1 x [ Ay1 1 x [ ¹1º/x

x2G

D

[

x.Bx [ Bx1 [ ¹1º/x 1 ;

x2G

where Bx D Ay 1 x for each x 2 G. It follows that y FQ y 1 D FQ . To see that FQ is the largest ﬁlter on G contained in F and satisfying (i)–(iii), let G be any such ﬁlter and let A 2 G . Then 1 2 A and for each x 2 G, there is Ax 2 G 1 A. Since G F , A 2 F for each x 2 G. Deﬁne such that x.Ax [ A1 x x /x Q B 2 F by [ 1 BD x.Ax [ A1 : x [ ¹1º/x x2G

Then B A, and so A 2 FQ . For every n 2 N, let Sn denote the group of all permutations on ¹1; : : : ; nº. The next theorem describes the topology T .F /. Theorem 1.17. For every ﬁlter F on a group G, the neighborhood ﬁlter of 1 in T .F / has a base consisting of subsets of the form 1 [ Y n [

B.i/ ;

nD1 2Sn iD1

Q where .Bn /1 nD1 is a sequence of members of F . Proof. It is clear that these subsets form a ﬁlter base on G. In order to show that this is the neighborhood ﬁlter of 1 in a group topology, it sufﬁces to check conditions Q (1)–(3) of Corollary 1.13. Let .Bn /1 nD1 be any sequence of members of F . 1 Q To check (1), deﬁne the sequence .Cn /nD1 in F by Cn D B2n \ B2n1 . Then for for every n 2 N and ; 2 Sn , n Y iD1

C.i/

n Y

C.i/

iD1

where 2 S2n is deﬁned by .j / D

n Y

B2.i/1

iD1

´

2.j / 1 2.j n/

n Y

B2.i/ D

iD1

if j n if j > n:

2n Y j D1

B.j /

Section 1.3 The Topology T .F /

It follows that 1 [ Y n [

C.i/

nD1 2Sn iD1

9

1 [ Y n [

1 [ Y n [ C.i/ B.i/ ;

nD1 2Sn iD1

nD1 2Sn iD1

1 Q To check (2), deﬁne the sequence .Cn /1 nD1 in F by Cn D Bn (Lemma 1.16). Then for every n 2 N and 2 Sn , n Y

B.i/

1

D

iD1

n Y

1 B.i/

D

iD1

n Y

C.i/

iD1

where 2 Sn is deﬁned by .i / D .n C 1 i /. Consequently, 1 [ Y n [

B.i/

1

nD1 2Sn iD1

D

1 [ Y n [

C.i/ :

nD1 2Sn iD1

1 Q To check (3), let x 2 G. Deﬁne the sequence .Cn /1 nD1 in F by Cn D xBn x (Lemma 1.16). Then for every n 2 N and 2 Sn ,

x

n Y

n n Y Y B.i/ x 1 D xB.i/ x 1 D C.i/ :

iD1

iD1

iD1

Consequently, x

1 [ Y n [

1 [ Y n [ B.i/ x 1 D C.i/ :

nD1 2Sn iD1

nD1 2Sn iD1

Now let G be endowed with any group topology in which F converges to 1 and let U be a neighborhood of 1. Note that every neighborhood of 1 is a member of FQ (Lemma 1.16). Choose inductively a sequence .Vn /1 nD0 of neighborhoods of 1 such that V0 D U and for every n, VnC1 VnC1 VnC1 Vn : Then whenever n1 ; : : : ; nk are distinct numbers in N, one has Vn1 Vnk Vn ; where n D min¹n1 ; : : : ; nk º 1. (To see this, pick i 2 ¹1; : : : ; kº such that ni D min¹n1 ; : : : ; nk º and write Vn1 Vnk as .Vn1 Vni1 /Vni .VniC1 Vnk /.) It follows that 1 [ Y n [ V.i/ U; nD1 2Sn iD1

and so U is a neighborhood of 1 in T .F /.

10

1.4

Chapter 1 Topological Groups

Topologizing a Group

Deﬁnition 1.18. Let G be a countably inﬁnite group. Enumerate G as ¹gn W n < !º without repetitions and with g0 D 1. 1 (i) For every inﬁnite sequence .an /1 nD1 in G, deﬁne U..an /nD1 / G by U..an /1 nD1 / D

1 [ Y n [

B.i/ ;

nD1 2Sn iD1

S ˙1 ; a˙1 1 where Bi D j1D0 gj ¹1; aiCj iCj C1 ; : : :ºgj . (ii) For every ﬁnite sequence a1 ; : : : ; an in G, deﬁne U.a1 ; : : : ; an / G by n [ Y

U.a1 ; : : : ; an / D

n B.i/ ;

2Sn iD1

Sni

˙1 ; a˙1 ˙1 1 where Bin D j D0 gj ¹1; aiCj iCj C1 ; : : : ; an ºgj . That is, U.a1 ; : : : ; an / consists of all elements of the form

gjn cn gj1 ; gj1 c1 gj1 n 1 ˙1 ; : : : ; an˙1 º for each i D 1; : : : ; n, where ji 2 ¹0; : : : ; n .i /º and ci 2 ¹1; a.i/Cj i and 2 Sn . In particular, U.a1 / D ¹1; a1˙1 º. Also put U.;/ D ¹1º. (iii) For every ﬁnite sequence a1 ; : : : ; an1 in G, let T .a1 ; : : : ; an1 ; x/ denote the set of group words f .x/ in the alphabet G [¹xº in which variable x occurs and which have the form f .x/ D gj1 c1 gj1 gjn cn gj1 ; n 1 ˙1 ˙1 ; x ˙1 º for each i D ; : : : ; an1 where ji 2 ¹0; : : : ; n .i /º and ci 2 ¹1; a.i/Cj i 1; : : : ; n, and 2 Sn . In particular, T .x/ consists of two group words x and x 1 .

Of course, in the case where G is Abelian, all these deﬁnitions look simpler. In particular, Bin D ¹0; ˙ai ; : : : ; ˙an º and

U.a1 ; : : : ; an / D

n X

Bin :

iD1

Theorem 1.19. For every sequence .an /1 nD1 in G, the following statements hold: 1 (1) U..an /1 nD1 / is a neighborhood of 1 in T ..an /nD1 /, S 1 (2) U..an /1 nD1 U.a1 ; : : : ; an /, nD1 / D

(3) U.a1 ; : : : ; an / D U.a1 ; : : : ; an1 / [ ¹f .an / W f .x/ 2 T .a1 ; : : : ; an1 ; x/º for every n 2 N, and (4) for every n 2 N and f .x/ 2 T .a1 ; : : : ; an1 ; x/, f .1/ 2 U.a1 ; : : : ; an1 /.

11

Section 1.4 Topologizing a Group

Proof. (1) follows from Theorem 1.17, and (2)–(4) from Deﬁnition 1.18. Theorem 1.19 gives us a method of topologizing a countable group. We illustrate it by proving Markov’s Criterion. We say that a group is topologizable if it admits a nondiscrete Hausdorff group topology. An inequality over a group G is any expression of the form f .x/ ¤ b, where f .x/ is a group word in the alphabet G [ ¹xº and b 2 G. Theorem 1.20 (Markov’s Criterion). A countable group G is topologizable if and only if every ﬁnite system of inequalities over G having a solution has also another solution. Proof. Necessity is obvious. Indeed, let T be a nondiscrete Hausdorff group topology on G. Consider any ﬁnite system of inequalities over G, say fi .x/ ¤ bi , where i D 1; : : : ; n, having a solution, say a 2 G, that is, fi .a/ ¤ bi for each i D 1; : : : ; n. Since T is a Hausdorff group topology, there is a neighborhood U of a in T such that bi … fi .U / for each i D 1; : : : ; n. Then every element of U is a solution of the system, and since T is nondiscrete, U n ¹aº ¤ ;. The proof of sufﬁciency is based on Theorem 1.19. It is enough to construct a sequence .an /1 nD1 in G n ¹1º such that gi … U.ai ; aiC1 ; : : : ; an / for each n 2 N and i D 1; : : : ; n. This implies that gi … U..an /1 nDi / for each i 2 N. Then the topology T ..an /1 nD1 / would be nondiscrete and Hausdorff. At the ﬁrst step pick any a1 2 G n ¹1; g1˙1 º. Then g1 … U.a1 / D ¹1; a1˙1 º. Now ﬁx n 2 N and suppose that elements a1 ; : : : ; an 2 G have already been chosen so that gi … U.ai ; : : : ; an / for each i D 1; : : : ; n. We need to ﬁnd anC1 2 G n ¹1º such that gi … U.ai ; : : : ; anC1 / for each i D 1; : : : ; n C 1. Since U.ai ; : : : ; anC1 / D U.ai ; : : : ; an / [ ¹f .anC1 / W f .x/ 2 T .ai ; : : : ; an ; x/º; it follows that gi … U.ai ; : : : ; anC1 / for each i D 1; : : : ; n C 1 if and only if anC1 is a solution of the system of inequalities f .x/ ¤ gi ; where i D 1; : : : ; n C 1 and f .x/ 2 T .ai ; : : : ; an ; x/. Since f .1/ 2 U.ai ; : : : ; an / for each i D 1; : : : ; n C 1 and f .x/ 2 T .ai ; : : : ; an ; x/, 1 is a solution of this system. Hence, there is a solution anC1 ¤ 1.

12

Chapter 1 Topological Groups

In connection with Theorem 1.20, A. Markov asked whether every inﬁnite group is topologizable. The next two theorems show that for inﬁnite Abelian groups and for free groups the answer is “Yes”. Theorem 1.21. For every Abelian group G and for every a 2 G n ¹0º, there is a homomorphism f W G ! T such that f .a/ ¤ 1. Proof. We ﬁrst deﬁne a homomorphism f0 W hai ! T such that f0 .a/ ¤ 1. If a has ﬁnite order, say n, deﬁne 2 ik f .ka/ D e n for every k D 1; : : : ; n. If a has inﬁnite order, pick any x 2 T n ¹1º and put f .ka/ D x k for every k 2 Z. Now consider the set of all pairs .H; g/, where H is a subgroup of G containing hai and g W H ! T is a homomorphism extending f0 , ordered by .H1 ; g1 / .H2 ; g2 / if and only if

H1 H2 and g2 jH1 D g1 :

By Zorn’s Lemma, there is a maximal pair .H; f /. We claim that H D G. Indeed, assume on the contrary that there is c 2 G n H . To derive a contradiction, let H 0 D H C hci and deﬁne a homomorphism f 0 W H 0 ! T extending f . If there is n 2 N with nc 2 H , choose the smallest such n and then z 2 T such that z n D f .nc/, and if there is no such n, choose arbitrary z 2 T . Deﬁne f 0 .b C kc/ D f .b/ z k for every b 2 H and k 2 Z. A topological group G is called totally bounded if it is Hausdorff and for every nonempty open U G there is a ﬁnite F G such that F U D G. A topological group is totally bounded if and only if it can be topologically and algebraically embedded into a compact group (see [4, Corollary 3.7.17]). By a compact group one means a compact Hausdorff topological group. Corollary 1.22. Every inﬁnite Abelian group admits a totally bounded group topology. Proof. Let G be an inﬁnite Abelian group. By Theorem 1.21, for everyQ a 2 G n ¹0º, there is a homomorphism fa W G ! T with fa .a/ ¤ 1. Let K D a2Gn¹0º Ta where Ta D T . Deﬁne f W G ! K by .f .x//a D fa .x/. Clearly f is an injective homomorphism. Being a subgroup of a compact group, f .G/ is totally bounded. Hence, the topology on G consisting of subsets f 1 .U /, where U ranges over open subsets of f .G/, is as required.

Section 1.4 Topologizing a Group

13

Theorem 1.23. Let X be a set and let F be the free group generated by X . Then for every w 2 F n ¹;º, there exist n 2 N and a homomorphism f W F ! SnC1 such that f .w/ ¤ , where is the identity permutation. Proof. Write w as x1"1 x2"2 : : : xn"n where each "i D ˙1 and "i D "iC1 whenever xi D xiC1 . It sufﬁces to deﬁne a mapping X 3 x 7! x 2 SnC1 such that x"11 ı x"22 ı ı x"nn ¤ : Given x 2 X and " D ˙1, let D" .x/ D ¹i 2 ¹1; : : : ; nº W xi D x and "i D "º: Note that D1 .x/ \ .D1 .x/ C 1/ D ; and .D1 .x/ C 1/ \ D1 .x/ D ;, so .D1 .x/ [ .D1 .x/ C 1// \ .D1 .x/ [ .D1 .x/ C 1// D ;: First deﬁne x on D1 .x/ [ .D1 .x/ C 1/ by ´ i C 1 if i 2 D1 .x/ x .i / D i 1 if i 2 D1 .x/ C 1: Then extend it in any way to a member x 2 SnC1 . Now we claim that for each i D 1; : : : ; n, x"ii .i C 1/ D i . Indeed, if "i D 1, then x"ii .i C 1/ D xi .i C 1/ D i . If "i D 1, then xi .i / D i C 1, and consequently, x"ii .i C 1/ D i . It follows that x"11 ıx"22 ı ıx"nn .nC1/ D 1 and hence x"11 ıx"22 ı ıx"nn ¤ . Corollary 1.24. Every free group admits a totally bounded group topology. Markov’s question had been open for a long time. However, eventually it was solved in the negative. Example 1.25. Let m; n 2 N, m 2, n 665 and n is odd. Consider the Adian group A.m; n/. This is a torsion free m-generated group whose center is an inﬁnite cyclic group hci and the quotient A.m; n/=hci is an inﬁnite group of period n. More precisely, A.m; n/=hci is the Burnside group B.m; n/, the largest group on m generators satisfying the identity x n D 1. Let x 2 A.m; n/ n hci. It is clear that x n 2 hci, because A.m; n/=hci has period n. We claim that x n … hc n i. To see this, assume the contrary. So x n D .c n /k D .c k /n for some k 2 Z. Let z D xc k . Then z … hci and z n D x n c k n D 1, since hci is the center. But this contradicts that A.m; n/ is torsion free.

14

Chapter 1 Topological Groups

Now let G D A.m; n/=hc n i and let D D hci=hc n i. Then G is inﬁnite, D D ¹1; d1 ; : : : ; dn1 º G and for every x 2 G n D, one has x n 2 ¹d1 ; : : : ; dn1 º. It follows that for every T1 -topology on G in which 1 is not an isolated point, the mapping G 3 x 7! x n 2 G is discontinuous at 1. Hence G is nontopologizable.

1.5

Metrizable Reﬁnements

The character of a space is the minimum cardinal such that every point has a neighborhood base of cardinality . Deﬁnition 1.26. For every countable topologizable group G, let pG denote the minimum character of a nondiscrete Hausdorff group topology on G which cannot be reﬁned to a nondiscrete metrizable group topology. Equivalently, pG is the supremum of all cardinals such that every nondiscrete Hausdorff group topology on G of character < has a nondiscrete metrizable reﬁnement. In this section we show that the cardinal pG is equal to a well-known cardinal invariant of the continuum. We say that a family T F P .X / has the strong ﬁnite intersection property if for every ﬁnite H F , H is inﬁnite. A subset A X is a pseudo-intersection of a family F P .X / if A n B is ﬁnite for all B 2 F . Deﬁnition 1.27. The pseudo-intersection number p is the minimum cardinality of a family F P .!/ having the strong ﬁnite intersection property but no inﬁnite pseudo-intersection. We show that Theorem 1.28. For every countable topologizable group G, pG D p. Before proving Theorem 1.28 we establish several auxiliary statements. Lemma 1.29. Let T be a nondiscrete group topology on G, let .Vn /1 nD1 be a sequence of neighborhoods of 1, and let A G be such that 1 2 clT A n A. Then there is a sequence .an /1 nD1 in A such that U..an /1 nDi / Vi for every i 2 N.

15

Section 1.5 Metrizable Reﬁnements

Proof. Construct inductively a sequence .an /1 nD1 in A such that U.ai ; : : : ; an / Vi for every n 2 N and i D 1; : : : ; n. Without loss of generality one may assume that all Vn are open. Pick a1 2 A \ V1 \ V11 . Then U.a1 / V1 . Now ﬁx n > 1 and suppose that elements a1 ; : : : ; an1 2 A have already been chosen so that U.ai ; : : : ; an1 / Vi for every i D 1; : : : ; n 1. Choose a neighborhood Wn of 1 such that for every i D 1; : : : ; n 1 and f .x/ 2 T .ai ; : : : ; an1 ; x/, one has f .Wn / Vi . This can be done because f .1/ 2 U.ai ; : : : ; an1 /. Pick an 2 A \ Vn \ Vn1 \ Wn . Then U.an / Vn and for every i D 1; : : : ; n 1, U.ai ; : : : ; an / D U.ai ; : : : ; an1 / [ ¹f .an / W f .x/ 2 T .ai ; : : : ; an1 ; x/º Vi : Lemma 1.30. Let T be a nondiscrete Hausdorff group topology on G and let A G be such that 1 2 clT A n A. Then there is a sequence .an /1 nD1 in A such that (i) an … U.a1 ; : : : ; an1 / for every n 2 N, and (ii) whenever k 2 N and 1 n1 < < nk < !, U.an1 ; : : : ; ank / n U.an1 ; : : : ; ank1 / U.a1 ; a2 ; : : : ; ank / n U.a1 ; a2 ; : : : ; ank 1 /: Proof. Construct inductively a sequence .an /1 nD1 in A such that for every n 2 N and f .x/ 2 T .a1 ; : : : ; an1 ; x/, either f .an / D f .1/ or f .an / … U.a1 ; : : : ; an1 /. Then (i) is satisﬁed because x 2 T .a1 ; : : : ; an1 ; x/ and 1 … A. To check (ii), let g 2 U.an1 ; : : : ; ank / n U.an1 ; : : : ; ank1 /: Then g D f .ank / for some f .x/ 2 T .an1 ; : : : ; ank1 ; x/. Since T .an1 ; : : : ; ank1 ; x/ T .a1 ; : : : ; ank 1 ; x/; f .x/ 2 T .a1 ; : : : ; ank 1 ; x/, and since g … U.an1 ; : : : ; ank1 /, f .ank / ¤ f .1/. Hence by the construction, g D f .ank / … U.a1 ; : : : ; ank 1 /. Lemma 1.31. Let T be a nondiscrete Hausdorff group topology on G and let U T . Then there is a Hausdorff group topology T 0 on G such that U T 0 T and the character of T 0 does not exceed max¹!; jUjº.

16

Chapter 1 Topological Groups

Proof. It sufﬁces to show that for every U 2 U, there is a HausdorffW metrizable group topology TU on G such that U 2 TU T . Then the topology T 0 D U 2U TU would be as required. Let V D ¹Ux 1 W x 2 U º. Enumerate V and G n ¹1º as ¹Vn W n 2 Nº and ¹xn W n 2 Nº, respectively. Construct inductively a sequence .Wn /1 nD0 of neighborhoods of 1 in T such that W0 D G and for every n 2 N the following conditions are satisﬁed: (a) Wn Vn , (b) xn … Wn , (c) Wn Wn 1 Wn1 , and (d) xk Wn xk 1 Wn1 for all k D 1; : : : ; n. It follows from (b)–(d) that there is a group topology TU on G for which ¹Wn W n 2 Nº is a neighborhood base at 1. Then TU is metrizable, TU T and by (a), U 2 TU . For every group topology T0 on G, let T04 denote the lattice of all group topologies T such that that T0 T . Let F0 denote the Fréchet ﬁlter on N and F04 the lattice of all ﬁlters F such that F0 F . Theorem 1.32. Let T 0 be a nondiscrete metrizable group topology on G. Then there is a metrizable group topology T0 T 0 and a mapping h W G ! N such that T04 3 T 7! h.N .T // 2 F04 is a surjective order preserving mapping. Here, N .T / denotes the neighborhood ﬁlter of 1 in T , and h.N .T // the ﬁlter on N with a base consisting of subsets h.U / where U 2 N .T /. 0 Proof. Pick any sequence .bn /1 nD1 in G n ¹1º converging to 1 in T and let A D ¹bn W n 2 Nº. By Lemma 1.30, there is a sequence .an /1 nD1 in A such that

(i) an … U.a1 ; : : : ; an1 / for every n 2 N, and (ii) whenever k 2 N and 1 n1 < < nk < !, U.an1 ; : : : ; ank / n U.an1 ; : : : ; ank1 / U.a1 ; a2 ; : : : ; ank / n U.a1 ; a2 ; : : : ; ank 1 /: 0 Clearly, .an /1 nD1 converges to 1 in T . By Lemma 1.31, there is a metrizable group topology T0 on G such that

T 0 T0 T ..an /1 nD1 /: Deﬁne a function h W G ! N by the condition that for every n 2 N and x 2 U.a1 ; : : : ; an / n U.a1 ; : : : ; an1 /, one has h.x/ D n.

17

Section 1.5 Metrizable Reﬁnements

It is clear that the mapping T04 3 T 7! h.N .T // 2 F04 is order preserving. We need to show that it is surjective. Let I be an inﬁnite subset of N. Write I as ¹nk W k 2 Nº, where .nk /1 is an kD1 increasing sequence in N. Deﬁne the group topology TI on G by TI D T ..ank /1 kD1 /: Then h.N .TI // is the ﬁlter on N consisting of all subsets J N with ﬁnite I n J . Indeed, for every x 2 U.an1 ; : : : ; ank / n U.an1 ; : : : ; ank1 /, one has h.x/ D nk , and ank 2 U.an1 ; : : : ; ank / n U.an1 ; : : : ; ank1 /. Now let F be any ﬁlter on N containing F0 . Deﬁne the group topology T on G by T D

_

TI :

I 2F

Then h.N .T // D F . Now we are in a position to prove Theorem 1.28. Proof of Theorem 1.28. To prove that pG p, let T 0 be a nondiscrete Hausdorff group topology on G of character < p. We need to ﬁnd a metrizable group topology T0 T 0 . Let ¹V˛ W ˛ < º be a neighborhood base at 1 in T 0 . Since < p, there is an inﬁnite pseudo-intersection of the neighborhood base, and consequently, a oneto-one sequence .an /1 nD1 in G converging to 1. Using Lemma 1.29 and that < p, 1 inductively for each ˛ , construct a subsequence .an˛ /1 nD1 of .an /nD1 such that (i) for each ˛ < , U..an˛ /1 nD1 / V˛ , and

(ii) for each ˛ and < ˛, ¹an˛ W n 2 Nº n ¹an W n 2 Nº is ﬁnite. 1 Then .an /1 nD1 is a subsequence of .an /nD1 with the following property: for each ˛ < , there is i.˛/ 2 N such that U..an /1 / V˛ . By Lemma 1.31, the nDi.˛/ topology T ..an /1 / can be weakened to a metrizable group topology T0 in which for nD1 each i 2 N, U..an /1 / remains a neighborhood of 1. It then follows that T 0 T0 . nDi To prove that p pG , let A be a family of subsets of N having the strong ﬁnite intersection property and with jAj D < pG . We need to ﬁnd an inﬁnite pseudointersection of A. Deﬁne the ﬁlter F on N by

\ ® ¯ F D ANWA B for some ﬁnite B A : Without loss of generality one may suppose that F contains the Fréchet ﬁlter F0 . Since G is topologizable, there is a nondiscrete Hausdorff group topology T 0 on G,

18

Chapter 1 Topological Groups

and by Lemma 1.31, T 0 can be chosen to be metrizable. By Theorem 1.28, there is a metrizable group topology T0 T 0 and a mapping h W G ! N such that T04 3 T 7! h.N .T // 2 F04 is a surjective order preserving mapping. Pick T 2 T04 such that h.N .T // D F . Choosing a family U of open neighborhoods of 1 in T such that jUj D and ¹h.U / W U 2 Uº is a base for F and applying Lemma 1.31, one may suppose that T has character . Since < pG , there is a nondiscrete metrizable group topology T1 T . Then the ﬁlter F1 D h.N .T1 // has a countable base, and consequently, an inﬁnite pseudo-intersection A N. Since F1 F A, A is also a pseudo-intersection of A.

1.6

Topologizability of a Countably Inﬁnite Ring

A ring is topologizable if it admits a nondiscrete Hausdorff ring topology, that is, a topology in which the addition, the additive inversion, and the multiplication are continuous. In this section we show that Theorem 1.33 (Arnautov’s Theorem). Every countably inﬁnite ring is topologizable. Note that the ring in Theorem 1.33 is not assumed to be associative. The proof of Theorem 1.33 is based on two auxiliary results. First is the ring version of Theorem 1.20. Proposition 1.34. A countable ring R is topologizable if and only if every ﬁnite system of inequalities over R having a solution has also another solution. By an inequality over R one means any expression of the form f .x/ ¤ b, where f .x/ is a ring word in the alphabet R [ ¹xº and b 2 R. The proof of Proposition 1.34 is similar to that of Theorem 1.20. First of all we need the following lemma. Lemma 1.35. A ﬁlter N on a ring R is the neighborhood ﬁlter of 0 in a ring topology if and only if the following conditions are satisﬁed: (1) for every U 2 N , there is V 2 N such that V C V U , (2) for every U 2 N , U 2 N , (3) for every U 2 N , there is V 2 N such that V V U , (4) for every U 2 N and x 2 R, there is V 2 N such that xV U and V x U .

Section 1.6 Topologizability of a Countably Inﬁnite Ring

19

Proof. Necessity is obvious. We need to check sufﬁciency. It follows from (1), (2) and Theorem 1.11 that there is a topology T on R in which, for each x 2 R, x C N is the neighborhood ﬁlter of x, and the addition and the additive inversion are continuous mappings. To see that the multiplication is continuous, let x; y 2 R and let Uxy be a neighborhood of xy 2 R. Put U D xy C Uxy . Then U is a neighborhood of 0 and Uxy D xy C U . Choose a neighborhood V of 0 such that V C V C V U . Applying (3) and (4) we obtain that there is a neighborhood W of 0 such that W W V , xW V and W y V . Then Wx D x C W is a neighborhood of x, Wy D y C W is a neighborhood of y and Wx Wy D .x C W /.y C W / D xy C xW C W y C W W xy C V C V C V xy C U D Uxy : Proof of Proposition 1.34. Necessity is obvious. We have to prove sufﬁciency. Assume that every ﬁnite system of inequalities over R having a solution has also another solution. Enumerate R without repetitions as ¹bn W n < !º with b0 D 0. Deﬁne inductively a set P of ring words in the alphabet ¹x1 ; x2 ; : : :º and to each f D f .x1 ; : : : ; xk / 2 P assign a vector rEf D .r1 ; : : : ; rk / 2 N k as follows: (i) x1 2 P and rEx1 D .1/, (ii) if f 2 P and rEf D .r1 ; : : : ; rk /, then g D g.x1 ; : : : ; xk / D f .x1 ; : : : ; xk / 2 P and rEg D .r1 ; : : : ; rk /, (iii) if f; g 2 P , rEf D .r1 ; : : : ; rk / and rEg D .s1 ; : : : ; sl /, then h D h.x1 ; : : : ; xkCl / D f .x1 ; : : : ; xk / C g.xkC1 ; : : : ; xkCl / 2 P; t D t .x1 ; : : : ; xkCl / D f .x1 ; : : : ; xk / g.xkC1 ; : : : ; xkCl / 2 P; and rEh D rEt D .2r1 ; : : : ; 2rk ; 2s1 ; : : : ; 2sl /, (iv) if f 2 P and rEf D .r1 ; : : : ; rk /, then for each m 2 N, h D h.x1 ; : : : ; xk / D bm f .x1 ; : : : ; xk / 2 P; t D t .x1 ; : : : ; xk / D f .x1 ; : : : ; xk / bm 2 P; and rEh D rEt D .r1 C m; : : : ; rk C m/. Lemma 1.36. For every ﬁlter F on R, the subsets [ f .Ar1 ; : : : ; Ark /; f 2P

where .r1 ; : : : ; rk / D rEf and .An /1 nD1 is a sequence of members of F , form a neighborhood base at 0 in the largest ring topology on R in which F converges to 0.

20

Chapter 1 Topological Groups

Proof. Let .An /1 nD1 be any sequence of members of F and let U D

[

f .Ar1 ; : : : ; Ark /:

f 2P

By (ii), U D U . Deﬁne the sequence .Bn /1 nD1 of members of F by Bn D A2n and let [ V D f .Br1 ; : : : ; Brk /: f 2P

Then by (iii), V C V U . Indeed, f .Br1 ; : : : ; Brk / C g.Bs1 ; : : : ; Bsl / D f .A2r1 ; : : : ; A2rk / C g.A2s1 ; : : : ; A2sl / D h.A2r1 ; : : : ; A2rk ; A2s1 ; : : : ; A2sl /: Similarly, V V U . Next, for each m 2 N, deﬁne the sequence .Cn /1 nD1 of members of F by Cn D AnCm and let W D

[

f .Cr1 ; : : : ; Crk /:

f 2P

Then by (iv), bm W U and W bm U . Applying Lemma 1.35, we obtain that there is a ring topology T on R in which the above subsets form a neighborhood base at 0, and so F converges to 0. It remains to check that T is the largest such topology. Let T 0 be any ring topology on R in which F converges to 0 and let U be any neighborhood of 0 in T 0 . We need to show that U is a neighborhood of 0 in T . 0 Choose a sequence .Un /1 nD0 of neighborhoods of 0 in T with U0 D U0 U such that for each n 2 N and m D 1; : : : ; n, Un D Un ; Un C Un Un1 ; Un Un Un1 ; bm Un Un1 and Un bm Un1 : We show that for every f 2 P with rE.f / D .r1 ; : : : ; rk / and for every j D 0; 1; : : : , one has f .Ur1 Cj ; : : : ; Urk Cj / Uj ; in particular, f .Ur1 ; : : : ; Urk / U0 . Suppose that the statement holds for some f; g 2 P with rEf D .r1 ; : : : ; rk / and rEg D .s1 ; : : : ; sl /. Consider the words h D h.x1 ; : : : ; xkCl / D f .x1 ; : : : ; xk / C g.xkC1 ; : : : ; xkCl / 2 P; t D t .x1 ; : : : ; xk / D bm f .x1 ; : : : ; xk / 2 P

Section 1.6 Topologizability of a Countably Inﬁnite Ring

21

with rEh D .2r1 ; : : : ; 2rk ; 2s1 ; : : : ; 2sl / and rEt D .r1 C m; : : : ; rk C m/. We have that h.U2r1 Cj ; : : : ; U2sl Cj / D f .U2r1 Cj ; : : : ; U2rk Cj / C g.U2s1 Cj ; : : : ; U2sl Cj / f .Ur1 Cj C1 ; : : : ; Urk Cj C1 / C g.Us1 Cj C1 ; : : : ; Usl Cj C1 / Uj C1 C Uj C1 Uj and t .Ur1 CmCj ; : : : ; Urk CmCj / D bm f .Ur1 CmCj ; : : : ; Urk CmCj / bm UmCj UmCj 1 Uj : Considering the words f .x1 ; : : : ; xk /g.xkC1 ; : : : ; xkCl / and f .x1 ; : : : ; xk /bm is similar, and the case of the word f .x1 ; : : : ; xk / is trivial. Now for every n 2 N, denote by Pn the subset of P consisting of all f 2 P such that max¹r1 ; : : : ; rk º n, where .r1 ; : : : ; rk / D rEf , in particular, P1 D ¹x1 ; x1 º. Note that Pn is ﬁnite. For every ﬁnite sequence a1 ; : : : ; an in R, deﬁne the subset U.a1 ; : : : ; an / R by [ U.a1 ; : : : ; an / D ¹0º [ f .¹ar1 ; : : : ; an º; : : : ; ¹ark ; : : : ; an º/; f 2Pn

where .r1 ; : : : ; rk / D rEf . We then obtain that for every inﬁnite sequence .an /1 nD1 in R, 1 [ [ U.a1 ; : : : ; an / D f .Ar1 ; : : : ; Ark /; f 2P

nD1

where Ai D ¹ai ; aiC1 ; : : :º for all i 2 N. Consequently by Lemma 1.36, 1 [

U.a1 ; : : : ; an /

nD1

is a neighborhood of 0 in the largest ring topology on R in which .an /1 nD1 converges to 0. Hence, in order to prove that R is topologizable, it sufﬁces to construct a sequence .an /1 nD1 in R such that an ¤ 0 and bi … U.ai ; aiC1 ; : : : ; an / for all n 2 N and i D 1; : : : ; n. To this end, for every ﬁnite sequence a1 ; : : : ; an1 in R, denote by T .a1 ; : : : ; an1 ; x/ the set of ring words in the alphabet ¹a1 ; : : : ; an1 ; xº in which variable x occurs and which are obtained from words f 2 Pn , say f D f .x1 ; : : : ; xk / with rEf D .r1 ; : : : ; rk /, by substituting xi 2 ¹ari ; : : : ; an1 ; xº into f .x1 ; : : : ; xk / for each i D 1; : : : ; k, in particular, T .x/ D ¹x; xº. Then for every ﬁnite sequence a1 ; : : : ; an in R,

22

Chapter 1 Topological Groups

(a) U.a1 ; : : : ; an / D U.a1 ; : : : ; an1 / [ ¹f .an / W f .x/ 2 T .a1 ; : : : ; an1 ; x/º and U.a1 / D ¹0º [ ¹f .a1 / W f .x/ 2 T .x/º, (b) for every f .x/ 2 T .a1 ; : : : ; an1 ; x/, f .0/ 2 U.a1 ; : : : ; an1 /, and for f .x/ 2 T .x/, f .0/ D 0. Statement (a) is obvious and (b) follows from the next lemma. Lemma 1.37. Let f 2 P and let rEf D .r1 ; : : : ; rk /. Then (1) f .0; : : : ; 0/ D 0, and (2) whenever ; ¤ ¹i1 ; : : : ; il º ¹1; : : : ; kº and g.xi1 ; : : : ; xil / is the word obtained from f .x1 ; : : : ; xk / by substituting xi D 0 for each i 2 ¹1; : : : ; kº n ¹i1 ; : : : ; ik º, there exists h 2 P with rEh D .s1 ; : : : ; sl / such that sj rij for all j D 1; : : : ; l and g.c1 ; : : : ; cl / D h.c1 ; : : : ; cl / for all c1 ; : : : ; cl 2 R. Proof. Suppose that the lemma holds for some f1 ; f2 2 P with rEf1 D .r10 ; : : : ; rk0 1 / and rEf2 D .r100 ; : : : ; rk002 /. Let k D k1 C k2 and consider the word f D f .x1 ; : : : ; xk / D f1 .x1 ; : : : ; xk1 / C f2 .xk1 C1 ; : : : ; xk / 2 P with rEf D .2r10 ; : : : ; 2rk0 1 ; 2r100 ; : : : ; 2rk002 /. Clearly f .0; : : : ; 0/ D f1 .0; : : : ; 0/ C f2 .0; : : : ; 0/ D 0 C 0 D 0: Let ; ¤ ¹i1 ; : : : ; il º ¹1; : : : ; kº and let g.xi1 ; : : : ; xil / be the word obtained from f .x1 ; : : : ; xk / by substituting xi D 0 for each i 2 ¹1; : : : ; kº n ¹i1 ; : : : ; ik º. Without loss of generality one may assume that ¹i1 ; : : : ; il º \ ¹1; : : : ; k1 º ¤ ;. Let ¹i1 ; : : : ; il º \ ¹1; : : : ; k1 º D ¹i1 ; : : : ; il1 º and let g1 .xi1 ; : : : ; xil1 / be the word obtained from f1 .x1 ; : : : ; xk1 / by substituting xi D 0 for each i 2 ¹1; : : : ; k1 º n ¹i1 ; : : : ; il1 º. By the hypothesis, there exists h1 2 P with rEh1 D .s10 ; : : : ; sl01 / such that sj0 ri0j for all j D 1; : : : ; l1 and g1 .c1 ; : : : ; cl1 / D h1 .c1 ; : : : ; cl1 / for all c1 ; : : : ; cl1 2 R. Suppose ﬁrst that l1 < l. Let ¹i1 ; : : : ; il º \ ¹k1 C 1; : : : ; kº D ¹k1 C q1 ; : : : ; k1 C ql2 º and let g2 .xq1 ; : : : ; xql2 / be the term obtained from f2 .x1 ; : : : ; xk2 / by substituting xq D 0 for each q 2 ¹1; : : : ; k2 º n ¹q1 ; : : : ; ql2 º. By the hypothesis, there exists h2 2 P with rEh2 D .s100 ; : : : ; sl002 / such that sj00 rq00j for all j D 1; : : : ; l2 and g2 .c1 ; : : : ; cl2 / D h2 .c1 ; : : : ; cl2 / for all c1 ; : : : ; cl2 2 R. Then the word h D h.x1 ; : : : ; xl / D h1 .x1 ; : : : ; xl1 / C h2 .xl1 C1 ; : : : ; xl / 2 P

Section 1.6 Topologizability of a Countably Inﬁnite Ring

23

with rEh D .2s10 ; : : : ; 2sl01 ; 2s100 ; : : : ; 2sl002 / is as required. Indeed, whenever c1 ; : : : ; cl 2 R, we have that g.c1 ; : : : ; cl / D g1 .c1 ; : : : ; cl1 / C g2 .cl1 C1 ; : : : ; cl / D h1 .c1 ; : : : ; cl1 / C h2 .cl1 C1 ; : : : ; cl / D h.c1 ; : : : ; cl / and 2sj0 2ri0j for all j D 1; : : : ; l1 and 2sj00 2rq00j for all j D 1; : : : ; l2 . Suppose now that l1 D l. Then the word h D h.x1 ; : : : ; xl / D h1 .x1 ; : : : ; xl1 / 2 P with rEh D .s10 ; : : : ; 2sl01 / is as required. Indeed, whenever c1 ; : : : ; cl 2 R, we have that g.c1 ; : : : ; cl / D g1 .c1 ; : : : ; cl1 / C f2 .0; : : : ; 0/ D h1 .c1 ; : : : ; cl1 / C 0 D h.c1 ; : : : ; cl / and sj0 ri0j for all j D 1; : : : ; l1 . Considering the words f1 .x1 ; : : : ; xk1 /f2 .xk1 C1 ; : : : ; xk /, bm f1 .x1 ; : : : ; xk1 / and f1 .x1 ; : : : ; xk1 /bm is similar, and the case of the word f .x1 ; : : : ; xk / is trivial. Using (a), (b) and our assumption, the required sequence .an /1 nD1 can be constructed in the same way as in the proof of Theorem 1.20. Another result that we need is Hindman’s Theorem. Deﬁnition 1.38. Given a set X , Pf .X / is the set of ﬁnite nonempty subsets of X . Deﬁnition 1.39. Let S be a semigroup. Given an inﬁnite sequence .xn /1 nD1 in S, the set of ﬁnite products of the sequence is deﬁned by °Y ± FP..xn /1 / D x W F 2 P .N/ : n f nD1 n2F

Given a ﬁnite sequence .xn /m nD1 in S, °Y ± / D x W F 2 P .¹1; : : : ; mº/ : FP..xn /m n f nD1 n2F

If S is an additive semigroup, we write FS instead of FP and say ﬁnite sums instead of ﬁnite products.

24

Chapter 1 Topological Groups

We state Hindman’s Theorem in the form involving product subsystem. Deﬁnition 1.40. Let S be a semigroup and let .xn /1 nD1 be a sequence in S. A se1 1 quence .yn /nD1 is a product subsystem of .xn /nD1Q if there is a sequence .Hn /1 nD1 in Pf .N/ such that max Hn < min HnC1 and yn D i2Hn xi for each n 2 N. If S is an additive semigroup, we say sum subsystem instead of product subsystem. Theorem 1.41. Let S be a semigroup and let .xn /1 nD1 be a sequence in S. Whenever FP..xn /1 / is partitioned into ﬁnitely many subsets, there exists a product subsystem nD1 1 such that FP..y /1 / is contained in one subset of the partition. .yn /1 of .x / n n nD1 nD1 nD1 We shall prove Theorem 1.41 in Chapter 6. Now we need the following. Corollary 1.42. Let S be an inﬁnite cancellative semigroup and let A S with jAj < jS j. Whenever S n A is partitioned into ﬁnitely many subsets, there exists a 1 sequence .yn /1 nD1 such that FP..yn /nD1 / is contained in one subset of the partition. 1 Proof. Construct inductively a sequence .xn /1 nD1 in S such that FP..xn /nD1 / \ A D ;. Then apply Theorem 1.41.

Given a ring R, let RŒx denote the set of all ring words in the alphabet R [ ¹xº. Using ring identities every f .x/ 2 RŒx can be rewritten as a noncommutative polynomial. We denote by deg f .x/ the degree of that polynomial. Lemma 1.43. For every f .x/ 2 RŒx with deg f .x/ > 0, there is g.x/ 2 RŒx such that deg g.x/ < deg f .x/ and g.a/ D f .a C b/ f .a/ f .b/ for all a; b 2 R. Proof. Rewriting f .x C b/ as a polynomial of x we obtain that f .x C b/ D f .x/ C f .b/ C g.x/; where deg g.x/ < deg f .x/. Now we are in a position to prove Theorem 1.33. Proof of Theorem 1.33. Let R be a countably inﬁnite ring and assume on the contrary that R is not topologizable. By Proposition 1.34, there is a ﬁnite sequence f1 .x/; : : : ; fm .x/

Section 1.6 Topologizability of a Countably Inﬁnite Ring

25

in RŒx with the following properties: (i) fi .0/ ¤ 0 for each i D 1; : : : ; m, and (ii) for every a 2 R n ¹0º, there is i D 1; : : : ; m such that fi .a/ D 0. Let k D max¹deg fi .x/ W i D 1; : : : ; mº. By Theorem 1.41, applied to the additive group of R, there exist a sequence .an /kC1 nD1 in R n ¹0º and i D 1; : : : ; m such that fi .a/ D 0 for every a 2 FS..an /kC1 nD1 /: Let f .x/ D fi .x/. Then by Lemma 1.43, there is g.x/ 2 RŒx with deg g.x/ < deg f .x/ such that g.a/ D f .a C b/ f .a/ f .b/ for all a; b 2 R. It follows that g.0/ D f .0 C 0/ f .0/ f .0/ D f .0/ ¤ 0 and for every a 2 FS..an /knD1 /, g.a/ D f .a C akC1 / f .a/ f .akC1 / D 0: After at most k such reductions we obtain a ring word h.x/ with deg h.x/ D 0 such that h.0/ D ˙f .0/ ¤ 0 and h.a1 / D 0, which is a contradiction.

References The standard references for topological groups are [55] and [34]. A great deal of information about topological groups can be found also in [7], [11], and [4]. Theorem 1.17 is from [62], a result of collaboration with I. Protasov. Its Abelian case was proved in [83]. Theorem 1.20 is due to A. Markov [48]. The ﬁrst example of a nontopologizable group was produced by S. Shelah [68]. It was an uncountable group and its construction used the Continuum Hypothesis CH. G. Hesse [32] showed that CH can be dropped in Shelah’s construction. Example 1.25 is due to A. Ol’šanski˘ı [53]. For the Adian group see [1]. That the Burnside group B.m; n/ is inﬁnite for m 2 and for n sufﬁciently large and odd was proved by P. Novikov and S. Adian [51]. Theorem 1.28 is from [110]. For more information about p and other cardinal invariants of the continuum see [79]. Theorem 1.33 is due to V. Arnautov [5]. Theorem 1.41 is due to N. Hindman [35]. Our proof of Theorem 1.33, as well as Theorem 1.20, is based on the treatment in [62].

Chapter 2

Ultraﬁlters

ˇ This chapter contains some basic facts about ultraﬁlters and the Stone–Cech compactiﬁcation of a discrete space. We also discuss Ramsey ultraﬁlters, P -points, and countably complete ultraﬁlters.

2.1

The Notion of an Ultraﬁlter

Let D be a nonempty set. Recall that a ﬁlter on D is a family F P .D/ with the following properties: (1) D 2 F and ; … F , (2) if A; B 2 F , then A \ B 2 F , and (3) if A 2 F and A B D, then B 2 F . Deﬁnition 2.1. An ultraﬁlter on D is a ﬁlter on D which is not properly contained in any other ﬁlter on D. In other words, an ultraﬁlter is a maximal ﬁlter. Deﬁnition 2.2. A family A P .D/ has the ﬁnite intersection property if for every ﬁnite B A.

T

B¤;

It is clear that every ﬁlter has the ﬁnite intersection property. Conversely, every family A P .D/ with the ﬁnite intersection property generates a ﬁlter ﬂt.A/ on D by \ ﬂt.A/ D ¹A D W A B for some ﬁnite B Aº: Proposition 2.3. Let A P .D/. Then the following statements are equivalent: (1) A is an ultraﬁlter, (2) A is a maximal family with the ﬁnite intersection property, (3) A is a ﬁlter and for every A D, either A 2 A or D n A 2 A. Proof. .1/ ) .3/ Consider two cases. Case 1: there is B 2 A such that B \ A D ;. Then B D n A and, since A is closed under supersets, D n A 2 A.

Section 2.1 The Notion of an Ultraﬁlter

27

Case 2: for every B 2 A, B \ A ¤ ;. Then ¹B \ A W B 2 Aº is a family of nonempty sets closed under ﬁnite intersections, because A is closed under ﬁnite intersections. Consequently, F D ¹C D W C B \ A for some B 2 Aº is a ﬁlter. Clearly A F and A 2 F . Since A is an ultraﬁlter, A D F . Hence A 2 A. .3/ ) .2/ Let A D and A … A. Then D n A 2 A. Since A \ .D n A/ D ;, A [ ¹Aº has no ﬁnite intersection property. .2/ ) .1/ Since A has the ﬁnite intersection property, so does ﬂt.A/, and since A is maximal, ﬂt.A/ D A. It follows that A is a ﬁlter, and consequently, an ultraﬁlter. It is obvious that for every a 2 D, ¹A D W a 2 Aº is an ultraﬁlter. Such ultraﬁlters are called principal. Ultraﬁlters which are not principal are called nonprincipal. Lemma 2.4. Let U be an ultraﬁlter on D. Then the following statements are equivalent: (1) U is a nonprincipal ultraﬁlter, T (2) U D ;, (3) for every A 2 U, jAj !. T Proof. .1/ ) .2/ If a 2 U, then U ¹A D W a 2 Aº, and since U is a maximal ﬁlter, U D ¹A D W a 2 Aº. .2/ ) .3/ Suppose that some A 2 U is ﬁnite. Then, applying Proposition 2.3, we obtain that there is a 2 A such that ¹aº 2 U. It follows that U D ¹A D W a 2 Aº. .3/ ) .1/ is obvious. The existence of nonprincipal ultraﬁlters involves the Axiom of Choice. Proposition 2.5 (Ultraﬁlter Theorem). Every ﬁlter on D can be extended to an ultraﬁlter on D. Proof. Let F be a ﬁlter on D and let P D ¹G P .D/ W F G and G is a ﬁlter on Dº: S S Given any chain C in P , C is a ﬁlter, and so C is an upper bound of C . Hence, by Zorn’s Lemma, P has a maximal element U. Clearly, U is a maximal ﬁlter.

28

Chapter 2 Ultraﬁlters

An ultraﬁlter U on D is uniform if for every A 2 U, jAj D jDj. Corollary 2.6. There are uniform ultraﬁlters on any inﬁnite set. Proof. Let D be any inﬁnite set and let F D ¹A D W jD n Aj < jDjº: By Proposition 2.3, there is an ultraﬁlter U on D containing F . If A D and jAj < jDj, then D n A 2 F U, so A … U. Hence U is uniform. Deﬁnition 2.7. Let F be a ﬁlter on D. If C D and C \ A ¤ ; for every A 2 F , then F jC D ¹C \ A W A 2 F º is a ﬁlter on C called the trace of F on C . If f W D ! E, then f .F / D ¹f .A/ W A 2 F º is a ﬁlter base on E called the image of F with respect to f . Note that if f is surjective, then f .F / is a ﬁlter. It is clear that if F is an ultraﬁlter, so is F jC . Lemma 2.8. If F is an ultraﬁlter, f .F / is an ultraﬁlter base. Proof. Let B E and let A D f 1 .B/. Since F is an ultraﬁlter, either A 2 F or D n A 2 F . Then either B f .A/ 2 f .F / or E n B f .D n A/ 2 f .F /. Recall that a space X is called compact if every open cover of X has a ﬁnite subcover. Equivalently, X is compact if every family of closed subsets of X with the ﬁnite intersection property has a nonempty intersection. A ﬁlter base B on a space X converges to a point x 2 X if for every neighborhood U of x 2 X , there is A 2 B such that A U . Note that in the case where B is a ﬁlter, B converges to x if and only if B contains the neighborhood ﬁlter of x. Proposition 2.9. A space X is compact if and only if every ultraﬁlter on X is convergent. Proof. Let X be a compact space and let U be an ultraﬁlter on X . Assume on the contrary that for every point x 2 X , there is a neighborhood Ux of x such that Ux … U. Clearly one can choose Ux to be open. Then, since U is an ultraﬁlter, X nUx 2 U. The open sets Ux , where x 2 X , cover X . Since X is compact, there is a ﬁnite T subcover ¹Uxi W i < nº of ¹Ux W x 2 X º. But then ; D i

29

Section 2.2 The Space ˇD

Conversely, suppose that every ultraﬁlter on X is convergent and let A be a family of closed T subsets of X with the ﬁnite intersection property. Assume on the contrary that A D ;. By Proposition 2.5, there is an ultraﬁlter U on XTsuch that A U. We claim that U is not convergent. Indeed, let x 2 X . Since A D ;, there is Fx 2 A such that x … Fx . Then Ux D X n Fx is a neighborhood of x and Ux … U. Hence U is not convergent, a contradiction.

2.2

The Space ˇD

Deﬁnition 2.10. Let D be a nonempty set and let ˇD denote the set of all ultraﬁlters on D. For every A D, deﬁne A ˇD by A D ¹p 2 ˇD W A 2 pº: Thus, for every A D and p 2 ˇD, p 2 A if and only if A 2 p. We have that A\B DA\B for all A; B D, so the family ¹A W A 2 P .D/º is closed under ﬁnite intersections. Consequently, this family is a base for a topology on ˇD. For every pair of different p; q 2 ˇD, there are A 2 p and B 2 q such that A \ B D ;, and so A \ B D ;. Hence, this topology is Hausdorff. Note also that the set of principal ultraﬁlters on D is a discrete open dense subset of ˇD with respect to this topology. Deﬁnition 2.11. Given a discrete space D, ˇD is the space of ultraﬁlters on D with the topology generated by taking as a base the subsets A, where A 2 P .D/. The principal ultraﬁlters are being identiﬁed with the points of D. For every A D, A D A n A. We have that D n A D ˇD n A for every A D. Consequently, the subset A ˇD is closed as well as open. Since also A is dense in A, it follows that clˇD A D A: ˇ Deﬁnition 2.12. The Stone–Cech compactiﬁcation of a discrete space D is a compact Hausdorff space Y containing D as a dense subspace such that every mapping f W D ! Z from D into any compact Hausdorff space Z can be extended to a continuous mapping f W Y ! Z. ˇ It is easy to see that the Stone–Cech compactiﬁcation of D is unique up to a homeomorphism leaving D pointwise ﬁxed.

30

Chapter 2 Ultraﬁlters

ˇ Theorem 2.13. ˇD is the Stone–Cech compactiﬁcation of D. Before proving Theorem 2.13, let us consider the following general situation. Suppose that X is a dense subset of a space Y and f W X ! Z is a continuous mapping from X into a Hausdorff space Z. For every p 2 Y , denote by Fp the trace of the neighborhood ﬁlter of p 2 Y on X and let limX3x!p f .x/ denote the limit of the ﬁlter base f .Fp / in Z, if exists. It is clear that if f has a continuous extension f W Y ! Z, then it is unique and f .p/ D

lim f .x/

X3x!p

for every p 2 Y . Lemma 2.14. Suppose that Z is regular and for every p 2 Y, there is limX3x!p f .x/. Deﬁne f W Y ! Z by f .p/ D lim f .x/: X3x!p

Then f is the continuous extension of f . Proof. It follows from the deﬁnition of f that for every A X , f .clY A/ clZ f .A/: To see this, let q 2 clY A and let W be a neighborhood of f .q/ 2 Z. Since f .q/ D limx!q f .x/, there is B 2 Fq such that f .B/ W . We have that B \ A ¤ ; and f .B \ A/ W . Hence, f .q/ 2 clZ f .A/. Now to see that f is continuous, let p 2 Y and let U be a neighborhood of f .p/ 2 Z. Since Z is regular, one may suppose that U is closed. Choose A 2 Fp such that f .A/ U . Put V D clY A. Then V is a neighborhood of p 2 Y and f .V / U . Now we are ready to prove Theorem 2.13. Proof of Theorem 2.13. Since D n A D ˇD n A, the sets A form also a base for the closed sets (that is, every closed set in ˇD is the intersection of some sets A). Consequently, in order to show that ˇD is compact, it sufﬁces to show that every family A of sets of the form A with the ﬁnite intersection property has a nonempty intersection. Let B D ¹A D W A 2 Aº Clearly B has the ﬁnite intersectionT property. Then by Proposition 2.5, there is p 2 ˇD such that B p. But then p 2 A. Now let f W D ! Z be any mapping from D into any compact Hausdorff space Z and let p 2 ˇD. The trace of the neighborhood ﬁlter of p 2 ˇD on D is the ultraﬁlter

31

Section 2.2 The Space ˇD

p. Since Z is compact, the ultraﬁlter base f .p/ converges, so limD3x!p f .x/ exists. Hence by Lemma 2.14, f W ˇD 3 p 7!

lim

D3x!p

f .x/ 2 Z

is the continuous extension of f . Deﬁnition 2.15. For every mapping f W D ! Z of a discrete space D into a compact Hausdorff space Z, let f W ˇD ! Z denote the continuous extension of f . Note that if f W D ! E ˇE and p 2 ˇD, then f .p/ is the ultraﬁlter on E with a base f .p/. Theorem 2.16. Let f W D ! D be a mapping with no ﬁxed points. Then there is a 3-partition ¹Ai W i < 3º of D such that Ai \ f .Ai / D ; for each i < 3. Proof. Consider the set G of all functions g such that (i) dom.g/ D and ran.g/ ¹0; 1; 2º, (ii) f .dom.g// dom.g/, and (iii) g.a/ ¤ g.f .a// for each a 2 dom.g/,

S ordered by . Note that G is nonempty, and if C is a chain in G , then C 2 G . Consequently by Zorn’s Lemma, there is a maximal element g 2 G . We claim that dom.g/ D D. Let Dg D dom.g/ and assume on the contrary that there is b 2 D n Dg . To obtain a contradiction, it sufﬁces to deﬁne an extension h of g such that (a) dom.h/ D Dg [ ¹f n .b/ W n < !º, where f 0 .b/ D b and f nC1 .b/ D f .f n .b//, and (b) g.f n .b// ¤ g.f nC1 .b// for each n < !. We distinguish between three cases. Case 1: there is n < ! such that f k .b/ … Dg for all k n and f nC1 .b/ 2 Dg . Let i D g.f nC1 .b// and pick j < 2 different from i . For each k n, put ´ j if k is even h.f nk .b// D i otherwise: Case 2: all elements f n .b/, where n < !, are different and do not belong to Dg . For each n < !, put ´ 0 if n is even h.f n .b// D 1 otherwise:

32

Chapter 2 Ultraﬁlters

Case 3: there are m < n < ! such that all elements f k .b/, where k n, are different, do not belong to Dg , and f nC1 .b/ D f m .b/. If n m is odd, for each k n, put ´ 0 if k is even k h.f .b// D 1 otherwise: If n m is even, we correct the latter deﬁnition at f n .b/ by putting h.f n .b// D 2. Having established that dom.g/ D D, let Ai D g 1 .i / for each i < 3. Corollary 2.17. If f W D ! D has no ﬁxed points, neither does f W ˇD ! ˇD. Proof. By Theorem 2.16, there is a partition ¹Ai W i < 3º of D such that Ai \f .Ai / D ; for each i < 3. Let p 2 ˇD. Then Aj 2 p for some j < 3. Consequently f .Aj / 2 f .p/. We have that p 2 Aj , f .p/ 2 f .Aj / and Aj \ f .Aj / D ;. Hence f .p/ ¤ p. Corollary 2.18. Let f W D ! D and let p 2 ˇD. Then f .p/ D p if and only if ¹a 2 D W f .a/ D aº 2 p: Proof. Denote F D ¹a 2 D W f .a/ D aº and A D D n F . Let f .p/ D p and assume on the contrary that F … p. Then A 2 p. Pick any b 2 A and deﬁne g W D ! D by ´ f .x/ if x 2 A g.x/ D b if x 2 F: Then g has no ﬁxed points and gjA D f jA , so gjA D f jA . Since p 2 A, we obtain that g.p/ D f .p/ D p. But by Corollary 2.17, g.p/ ¤ p, a contradiction. Conversely, let F 2 p. Then p 2 F and f .x/ D x for all x 2 F . Hence f .p/ D p. Deﬁnition 2.19. A space is extremally disconnected if the closure of an open set is open. Equivalently, a space is extremally disconnected if the closures of disjoint open sets are disjoint. Lemma 2.20. ˇD is extremally disconnected. Proof. Let U be an open subset of ˇD. Put A D U \ D. Then U clˇD A. Hence clˇD U D A. Remark 2.21. Although ˇD is extremally disconnected, D is not (see [22, Example 6.2.31]).

33

Section 2.2 The Space ˇD

A space is called -compact if it can be represented as a countable union of compact subsets. Theorem 2.22. Let X be a regular extremally disconnected space and let A and B be -compact subsets of X . If .cl A/ \ B D A \ .cl B/ D ;, then .cl A/ \ .cl B/ D ;. S S Proof. Write A D n

It follows that U D

[ n

in

Un

and

V D

[

Vn

n

are disjoint open neighborhoods of A and B respectively. Since X is extremally disconnected, .cl U / \ .cl V / D ; and so .cl A/ \ .cl B/ D ;. Corollary 2.23. Let A and B be countable subsets of ˇD. If .cl A/ \ B D A \ .cl B/ D ;, then .cl A/ \ .cl B/ D ;. S S Corollary 2.24. Let X D n

Theorem 2.26. Let D be an inﬁnite set of cardinality . Then jU.D/j D jˇDj D 22 .

Proof. Since every ultraﬁlter is a member of P .P .D//, jˇDj 22 . Let U D U.D/. In order to show that jU j 22 , it sufﬁces to construct a mapping of U onto a set of cardinality 22 .

34

Chapter 2 Ultraﬁlters

Let Z be the product of 2 copies of the discrete space ¹0; 1º. Then jZj D 22 . By the Hewitt–Marczewski–Pondiczery Theorem (see [22, Theorem 2.3.15]), Z has a dense subset E of cardinality . Enumerate E as ¹q˛ W ˛ < º. Now let ¹A˛ W ˛ < º be a partition of D into subsets of cardinality . For each ˛ < , there is p˛ 2 U with A˛ 2 p. Deﬁne f W D ! E by f .A˛ / D ¹q˛ º. Then f .p˛ / D q˛ . Since U ˇD is closed, it follows that f .U / is a compact subset of Z containing E. Hence f .U / D Z. We conclude this section by establishing a one-to-one correspondence between nonempty closed subsets of ˇD and ﬁlters on D. Deﬁnition 2.27. Given a family A P .D/, deﬁne A ˇD by \ AD A: A2A

Lemma 2.28. For every ﬁlter F on D, F is a nonempty closed subset of ˇD consisting of all p 2 ˇD such that F p. Conversely, for every nonempty closed subset X ˇD, the intersection of all ultraﬁlters from X is a ﬁlter F on D such that F D X. Proof. The ﬁrst part of the lemma is obvious, so it sufﬁces to prove the second. It is clear that F is a ﬁlter and X F . To see the reverse inclusion, let p 2 F . Assume on the contrary that p … X . Then there is A 2 p such that A \ X D ;. It follows that for each q 2 X , D n A 2 q. Hence D n A 2 F p, a contradiction.

2.3

Martin’s Axiom

Let P D .P; / be a partially ordered set. A ﬁlter in P is a nonempty subset G P such that (i) for every a; b 2 G, there is c 2 G such that c a and c b, and (ii) for every a 2 G and b 2 P , a b implies b 2 G. Elements a; b 2 P are incompatible if there is no c 2 P such that c a and c b. An antichain in P is a subset of pairwise incompatible elements. P has the countable chain condition if every antichain in P is countable. A subset D P is dense if for every a 2 P there is b 2 D such that b a. Deﬁnition 2.29. MA is the following assertion: whenever .P; / is a partially ordered set with the countable chain condition and D is a family of < 2! dense subsets of P , there is a ﬁlter G in P such that G \ D ¤ ; for all D 2 D. We ﬁrst show that MA follows from the Continuum Hypothesis CH.

35

Section 2.3 Martin’s Axiom

Lemma 2.30. CH implies MA. Proof. Let .P; / be a partially ordered set with the countable chain condition and let D be a family of < 2! dense subsets of P . Then by CH, D is countable. Enumerate D as ¹Dn W n < !º. Construct inductively a sequence .an /n

if and only if

F F 0 ; H H 0 and F 0 n F

\

H:

It is routine to verify that is a partial order. Next notice that if .F; H / and .F 0 ; H 0 / are incompatible elements of P , then F ¤ F 0 . Indeed, otherwise .F; H [ H 0 / .F; H / and .F; H [ H 0 / .F; H 0 /. Since the set of ﬁnite subsets of ! is countable, it follows that P has the countable chain condition. For each B 2 F , let DB D ¹.F; H / 2 P W B 2 H º: Given any .F; H / 2 P , one has .F; H [ ¹Bº/ 2 DB and .F; H [ ¹Bº/ .F; H /. So DB is dense. Also for each n < !, let Dn D ¹.F; H / 2 P W max F nº: T Given any .F; H / 2 P , there is m 2 H such that m n. Then .F [¹mº; H / 2 Dn and .F [ ¹mº; H / .F; H /. So Dn is dense. Now by MA, there is a ﬁlter G in P such that G \ DB ¤ ; for each B 2 F and G \ Dn ¤ ; for each n < !. Let AD

[ .F ;H /2G

F:

36

Chapter 2 Ultraﬁlters

Since G \ Dn ¤ ; for each n < !, it follows that A is inﬁnite. To show that A n B is ﬁnite for each B 2 F , pick .F; H / 2 G \ DB . We claim that A n B F . Indeed, let x 2 A n B. Then there is .F 0 ; H 0 / 2 G such that x 2 F 0 . Since G is a ﬁlter, there 00 ; H 00 / .F; H / and .F 00 ; H 00 / .F 0 ; H 0 /. We have is .F 00 ; H 00 / 2 G such that .F T that x 2 F 0 F 00 , F 00 n F H and B 2 H . Hence x 2 F .

2.4

Ramsey Ultraﬁlters and P-points

Given a set X and a cardinal k, ŒX k D ¹Y X W jY j D kº: Ramsey’s Theorem says that whenever k; r 2 N and Œ!k is r-colored, there exists an inﬁnite A ! such that ŒAk is monochrome (see [30]). Deﬁnition 2.32. An ultraﬁlter p on ! is Ramsey if whenever k; r 2 N and Œ!k is r-colored, there exists A 2 p such that ŒAk is monochrome. Theorem 2.33. Let p be an ultraﬁlter on !. Then the following statements are equivalent: (1) p is Ramsey, (2) for every 2-coloring of Œ!2 , there exists A 2 p such that ŒA2 is monochrome, and (3) for every partition ¹An W n < !º of !, either An 2 p for some n < ! or there exists A 2 p such that jA \ An j 1 for all n < !. Proof. .1/ ) .2/ is obvious. .2/ ) .3/ Deﬁne W Œ!2 ! ¹0; 1º by ´ 0 if ¹i; j º An for some n < ! .i; j / D 1 otherwise: By (2), there is A 2 p be such that ŒA2 is monochrome. Suppose that An … p for all n < !. It then follows that .ŒA2 / D ¹1º, and consequently, jA \ An j 1 for all n < !. .3/ ) .1/ We prove (1) for ﬁxed r by induction on k. For k D 1, it is obvious. Now assume (1) holds for some k and let W Œ!kC1 ! ¹0; 1; : : : ; r 1º be given. For each i < !, deﬁne i W Œ!k ! ¹0; 1; : : : ; r 1º by ´ .¹i º [ x/ if min x > i i .x/ D 0 otherwise:

37

Section 2.4 Ramsey Ultraﬁlters and P -points

By the inductive assumption, for each i < !, there are Bi 2 p and li < r such that i .ŒBi k / D ¹li º. Clearly one may suppose that min Bi > i and the sequence .Bi /i

n

There is s < 2 such that ¹bi W i 2 Is º 2 p. Then whenever i; j 2 Is and i < j , there is n < ! such that i < in < inC1 j , and so j bi . Finally, there is l < r, such that ¹bi W i 2 Is and lbi D lº 2 p: Denote this member of p by A. Then .ŒAkC1 / D ¹lº. Theorem 2.34. Assume p D c. Then there exists a nonprincipal Ramsey ultraﬁlter on !. Proof. Let ¹X˛ W ˛ < 2! º be an enumeration of P .!/ and let ¹˛ W ˛ < 2! º be an enumeration of 2-colorings Œ!2 ! ¹0; 1º. We shall construct inductively a 2! sequence .A˛ /˛<2! of inﬁnite subsets of ! such that for every ˛ < 2! , the following conditions are satisﬁed: (i) for every ˇ < ˛, jA˛ n Aˇ j < !, (ii) ŒA˛ 2 is monochrome with respect to ˛ , and (iii) either A˛ X˛ or A˛ ! n X˛ . By Ramsey’s Theorem, there is an inﬁnite C0 ! such that ŒC0 2 is monochrome with respect to 0 . If jC0 \ X0 j D !, put A0 D C0 \ X0 . Otherwise put A0 D C0 \ .! n X0 /. Now ﬁx < 2! and suppose that we have already constructed a sequence .A˛ /˛< satisfying conditions (i)–(iii) for all ˛ < . Since p D c, there is B ! such that jB n A˛ j < ! for all ˛ < . Choose an inﬁnite C B such that ŒC 2 is monochrome with respect to and put ´ if jC \ X j D ! C \ X A D C \ .! n X / otherwise:

38

Chapter 2 Ultraﬁlters

Having constructed .A˛ /˛<2! , let p D ¹A ! W jA˛ n Aj < ! for some ˛ < 2! º: Then p is a nonprincipal Ramsey ultraﬁlter. A point x in a topological space X is called a P -point if the intersection of countably many neighborhoods of x is again a neighborhood of x. Deﬁnition 2.35. We say that a nonprincipal ultraﬁlter p on ! is a P -point if p is a P -point in ! . Lemma 2.36. Let p be a nonprincipal ultraﬁlter on !. Then p is a P -point if and only if whenever ¹An W n < !º is a partition of ! and An … p for all n < !, there exists A 2 p such that A \ An is ﬁnite for all n < !. Proof. Necessity. Let ¹An W n < !º be a partition of ! and AS n … p for all n < !. Deﬁne the sequence ¹B W n < !º of members of p by B D n n n*
*

Section 2.5 Measurable Cardinals

2.5

39

Measurable Cardinals

Deﬁnition 2.39. Let and be inﬁnite cardinals. A ﬁlter F on is -complete if T G 2 F whenever G F and jG j < . Every ﬁlter is !-complete. An ! C -complete ﬁlter is called countably complete. Lemma 2.40. An ultraﬁlter p on is -complete if and only if whenever < and ¹A˛ W ˛ < º is a partition of , there is ˛ < such that A˛ 2 p. Proof. Necessity. Assume on the contrary T that for every ˛ < , one T has A˛ … p, so nA˛ 2 p. Then by -completeness of p, ˛< nA˛ 2 p. But ˛< nA˛ D ;, a contradiction. Sufﬁciency. Assume on the contrary that p is not -complete. Choose < and T ¹C˛ T W ˛ < º p such that ˛< C˛ … p and T is as small as possible. Taking C˛ n ˛< C˛ instead of C˛ , one may suppose that ˛< T C˛ D ;. Using minimality of , deﬁne ¹B˛ W ˛ < º p by B0 D and B˛ D ˇ <˛ C˛ if ˛ > 0. Then the sets A˛ D B˛ n B˛C1 , ˛ < , form a partition of and A˛ … p for every ˛ < , a contradiction. Corollary 2.41. An ultraﬁlter p on is countably complete if and only if whenever ¹An W ˛ < !º is a partition of , there is n < ! such that An 2 p. Deﬁnition 2.42. A cardinal is measurable (Ulam-measurable) if there is a -complete (countably complete) nonprincipal ultraﬁlter on . Note that ! is a measurable cardinal, but not Ulam-measurable. Theorem 2.43. A cardinal is Ulam-measurable if and only if it is greater than or equal to the ﬁrst uncountable measurable cardinal. Proof. It is clear that every uncountable measurable cardinal is Ulam-measurable, and if a cardinal is Ulam-measurable, then any greater cardinal is also Ulam-measurable. Therefore, it sufﬁces to prove that the ﬁrst Ulam-measurable cardinal is measurable. Let be the ﬁrst Ulam-measurable cardinal and let p be a nonprincipal countably complete ultraﬁlter on . We show that p is -complete. Assume the contrary. Then by Lemma 2.40, there are < and a partition ¹A W < º of such that A … p for every < . Deﬁne f W ! by f .x/ D if x 2 A . Now let q D f .p/. It is easy to see that q is a nonprincipal countably complete ultraﬁlter on , contradicting the choice of . A cardinal is called strongly inaccessible if is regular and 2 < whenever < . Note that ! is a strongly inaccessible cardinal. Theorem 2.44. A measurable cardinal is strongly inaccessible.

40

Chapter 2 Ultraﬁlters

Proof. Let be a measurable cardinal and let p be a nonprincipal -complete ultraﬁlter on . We ﬁrst show that is regular. Let D cf. / and assume on the contrary that < . Pick an increasing coﬁnal T -sequence . /< of cardinals in . If n 2 p for every < , then ; D < n 2 p, since p is -complete. Consequently, 2 p for Tsome < . But then, since p is nonprincipal and -complete, we obtain that ; D ˛< n ¹˛º 2 p, a contradiction. Now let < and assume on the contrary that 2 . Pick an injective function F W ! 2. For every < and i < 2, let Xi D ¹˛ < W F .˛/. / D i º: g./

Note that ¹Xi W i < 2º is a partition of . Deﬁne a function g W ! 2 by X T g./ and let X D < X . Then X 2 p, by -completeness of p, and

2 p,

X D ¹˛ < W F .˛/. / D g. / for all < º D ¹˛ < W F .˛/ D gº D F 1 .g/: Since F is injective, it follows that X is a singleton. But this contradicts the assumption that p is nonprincipal. Theorem 2.45. It is consistent with ZFC that there is no uncountable strongly inaccessible cardinal. Proof. See [43, Corollary IV 6.9].

References Ultraﬁlters were introduced by F. Riesz [64] and S. Ulam [76]. The Ultraﬁlter Theorem is equivalent to the Prime Ideal Theorem, a well-known weaker form of the Axiom of Choice AC (see [41, Section 2.3]). The existence of a nonprincipal ultraﬁlter ˇ on ! cannot be established in ZF [41, Problem 5.24]. The Stone–Cech compactiﬁˇ cation was produced independently by M. Stone [72] and E. Cech [82]. A great deal ˇ of information about ultraﬁlters and the Stone–Cech compactiﬁcation can be found in [12]. Concerning ˇ! see also [81]. Theorem 2.16 goes back to N. de Bruijn and P. Erd˝os [16]. Theorem 2.22 is apparently due to Z. Frolík [26]. Theorem 2.31 is a result of D. Booth [6]. For more information about MA see [43] and [25]. Our proof of Theorem 2.33 is based on the treatment in [10]. P -points were invented by W. Rudin [66] who used them to show that under CH, ! is not homogeneous. Theorem 2.38 is due to S. Shelah [69]. Measurable cardinals were introduced by S. Ulam [77].

Chapter 3

Topological Spaces with Extremal Properties

In this chapter we discuss extremally disconnected, irresolvable, maximal and other spaces with extremal properties.

3.1

Filters and Ultraﬁlters on Topological Spaces

Deﬁnition 3.1. Let F be a ﬁlter on a topological space. (a) F is open (closed) if it has a base of open (closed) sets. (b) F is nowhere dense if there is A 2 F such that int cl A D ;. (a) F is dense if for every A 2 F , int cl A 2 F . Lemma 3.2. Every ultraﬁlter on a space is either dense or nowhere dense. Proof. Let U be an ultraﬁlter on a space X . Suppose that U is not dense. Then there is a closed A 2 F such that int A … U. Let B D A n int A. It follows that B 2 U and int cl B D ;. Hence, U is nowhere dense. Corollary 3.3. An ultraﬁlter U on a space is dense if and only if for every A 2 U, int cl A ¤ ;. Note that a ﬁlter F on a space is nowhere dense if and only if every ultraﬁlter U F is nowhere dense. Lemma 3.4. Let F be a ﬁlter on a space X and suppose that F is not nowhere dense. Deﬁne the ﬁlter G on X by taking as a base the subsets of the form U nY where U 2 F and Y ranges over nowhere dense subsets of X . Then for every ultraﬁlter U F , U is dense if and only if U G . If F is open, G is open as well. Proof. It follows from the deﬁnition of G that the ultraﬁlters containing G are precisely those of ultraﬁlters containing F which are not nowhere dense, and so by Lemma 3.2, these are all dense ultraﬁlters containing F . If F is open, then G has a base consisting of subsets of the form U n Y where U 2 F is open and Y X is closed nowhere dense, so G is open as well. We say that ﬁlters F and G are incompatible if A \ B D ; for some A 2 F and B 2 G.

42

Chapter 3 Topological Spaces with Extremal Properties

Lemma 3.5. Every open ﬁlter is contained in a maximal open ﬁlter. Different maximal open ﬁlters are pairwise incompatible. Proof. The ﬁrst statement is immediate from Zorn’s Lemma. To see the second, let F and G be maximal open ﬁlters on a space X . Suppose that U \ V ¤ ; for all U 2 F and V 2 G . Then the sets U \ V ¤ ;, where U 2 F and V 2 G , form a base for a ﬁlter H . Since H is open and contains F and G , it follows that F D G . Lemma 3.6. Let F be a maximal open ﬁlter on a space X and let A X be such that A \ U ¤ ; for all U 2 F . Then int cl A 2 F . Proof. Since F is open, it sufﬁces to show that cl A 2 F . Assume the contrary. Then U n cl A ¤ ; for all U 2 F . Consequently, the sets U n cl A, where U 2 F , form a base for an open ﬁlter G F . Since F is maximal open, F D G . It follows that .cl A/ \ U D ; for some U 2 F , and so A \ U D ;, a contradiction. Corollary 3.7. Every ultraﬁlter containing a maximal open ﬁlter is dense. Lemma 3.8. Let F be an open ﬁlter on a space and let U be a dense ultraﬁlter containing F . Then there is a maximal open ﬁlter G such that U G F . Proof. For every A 2 U, int cl A 2 U. Consequently, there is a maximal open ﬁlter G such that int cl A 2 G for all A 2 U. Let V 2 G be open and let A 2 U. Then V \ .int cl A/ ¤ ;, and so V \ A ¤ ;. It follows that G U. Summarizing, we obtain the following. Proposition 3.9. Let F be an open ﬁlter on a space X , let G be the ﬁlter with a base consisting of subsets U n Y where U 2 F and Y ranges over nowhere dense subsets of X , and let ¹Gi W i 2 I º be the family of all maximal open ﬁlters containing F . Then (1) for every ultraﬁlter U F , U is dense if and only if U G , T (2) the ﬁlters Gi , where i 2 I , are pairwise incompatible and i2I Gi D G , and (3) for every i 2 I , ultraﬁlter U Gi and A 2 U, one has int cl A 2 Gi .

3.2

Spaces with Extremal Properties

Deﬁnition 3.10. A topological space X is nodec if every nowhere dense subset of X is closed. Equivalently, a space is nodec if every nowhere dense subset is discrete.

43

Section 3.2 Spaces with Extremal Properties

Proposition 3.11. A T1 -space X is nodec if and only if every nonprincipal converging ultraﬁlter on X is dense. Proof. A T1 -space X contains no nonclosed nowhere dense set if and only if there is no nonprincipal converging nowhere dense ultraﬁlter on X , consequently by Lemma 3.2, if and only if every nonprincipal converging ultraﬁlter on X is dense. Recall that a space is extremally disconnected if the closures of disjoint open sets are disjoint. Proposition 3.12. A space X is extremally disconnected if and only if for every x 2 X , there is exactly one maximal open ﬁlter on X converging to x. Proof. Suppose that X is not extremally disconnected. Then there are disjoint open U; V X and x 2 X such that x 2 .cl U / \ .cl V /. Let F be the neighborhood ﬁlter of x and let FU D ﬂt.F [ ¹U º/

and

FV D ﬂt.F [ ¹V º/:

Then FU and FV are incompatible open ﬁlters on X converging to x. Hence, maximal open ﬁlters extending FU and FV (Lemma 3.5) are different. Conversely, suppose that there are at least two different maximal open ﬁlters on X , say G and H , converging to some x 2 X . Since they are incompatible (Lemma 3.5), there are disjoint open U 2 G and V 2 H . Clearly x 2 .cl U / \ .cl V /. Hence, X is not extremally disconnected. An important property of extremally disconnected spaces is contained in the following theorem. Theorem 3.13. Let X be an extremally disconnected Hausdorff space and let f W X ! X be a homeomorphism. Then the set M D ¹x 2 X W h.x/ D xº of all ﬁxed points of f is clopen. Proof. By Zorn’s Lemma, there is a maximal open set U X such that f .U / \ U D ;. Since X is extremally disconnected, U is clopen. Let F D U [ f .U / [ f 1 .U /: Then F is clopen and M \ F D ;. We claim that M D X n F . Indeed, assume on the contrary that f .x/ ¤ x for some x 2 X n F . Choose an open neighborhood V of x such that V \ F D ; and f .V / \ V D ;. It follows that f .V / \ U D ;, since V \ f 1 .U / D ;. But then f .U [ V / \ .U [ V / D ;, which contradicts maximality of U . Deﬁnition 3.14. A space X is strongly extremally disconnected if for every open nonclosed U X , there exists x 2 X n U such that U [ ¹xº is open.

44

Chapter 3 Topological Spaces with Extremal Properties

Proposition 3.15. A space X is strongly extremally disconnected if and only if it is extremally disconnected and every nonempty nowhere dense subset of X has an isolated point. Proof. Necessity. To see that X is extremally disconnected, let U be an open subset of X and let V D int cl U . We claim that V D cl U . Indeed, otherwise V is nonclosed and so there is x 2 cl V n V D cl U n V such that V [ ¹xº is open, which contradicts V D int cl U . Now let Y be a nonempty nowhere dense subset of X . One may suppose that Y is closed. Then U D X n Y is open and nonclosed. Consequently, there is x 2 Y such that U [ ¹xº is open. It follows that x is an isolated point of Y . Sufﬁciency. Let U be an open nonclosed subset of X . Then Y D cl U n U is a nonempty nowhere dense subset. Consequently, there is an isolated point x 2 Y . Also we have that cl U is open. It follows that U [ ¹xº is open. Note that to say that every nonempty nowhere dense subset of a space has an isolated point is the same as saying that every nonempty perfect subset has a nonempty interior. A subset of a space is called perfect if it is closed and dense in itself. Corollary 3.16. A dense in itself space X is strongly extremally disconnected if and only if every perfect subset of X is open. Proof. Necessity. Let F be a perfect subset of X and let U D int F . We have to show that U D F . Assume on the contrary that F n U ¤ ;. Then U is nonclosed. Indeed, otherwise F n U is perfect, so by Proposition 3.15, int .F n U / ¤ ;, which contradicts U D int F . Since U is nonclosed, there is x 2 F n U such that U [ ¹xº is open, a contradiction with U D int F . Sufﬁciency. By Proposition 3.15, we have to check only that X is extremally disconnected. Let U be a nonempty open subset of X . Since X is dense in itself, so is U . Consequently, cl U is perfect, and so it is open. Deﬁnition 3.17. A space is resolvable or irresolvable depending on whether or not it can be partitioned into two dense subsets. A space X is open-hereditarily irresolvable if every nonempty open subset of X is irresolvable. Proposition 3.18. A space X is open-hereditarily irresolvable if and only if every (converging) maximal open ﬁlter on X is an ultraﬁlter. Proof. Let F be a maximal open ﬁlter on X and suppose that F is not an ultraﬁlter. Then there are two different ultraﬁlters U and V on X containing F . Pick pairwise disjoint A 2 U and B 2 V . By Corollary 3.7, both U and V are dense, so int cl A 2 F and int cl B 2 F . Deﬁne an open U 2 F by U D .int cl A/ \ .int cl B/:

Section 3.2 Spaces with Extremal Properties

45

Then both A \ U and B \ U are dense in U , so U is resolvable. Hence, X is not open-hereditarily irresolvable. Conversely, suppose that every converging maximal open ﬁlter on X is an ultraﬁlter. Let U be any nonempty open subset of X and let ¹A0 ; A1 º be a partition of U . Pick any converging maximal open ﬁlter F on X containing U . By the assumption F is an ultraﬁlter, so Ai 2 F for some i < 2, say A0 2 F . Since F is open, int A0 ¤ ;. Consequently, A1 is not dense in U . Hence, X is not open-hereditarily irresolvable.

Deﬁnition 3.19. A space is submaximal if every dense subset is open. Proposition 3.20. A space is submaximal if and only if it is open-hereditarily irresolvable and nodec. Proof. Suppose that X is submaximal. To see that X is open-hereditarily irresolvable, let U be a nonempty open subset of X and let A be a dense subset of U . Then .X nU /[A is dense in X , and consequently, open. It follows that U n A is is not dense in U . Hence, U is irresolvable. To see that X is nodec, let Y be a nowhere dense subset of X . Then X n Y is dense, and consequently, open. So Y is closed. Now suppose that X is open-hereditarily irresolvable and nodec. To show that X is submaximal, let A be a dense subset of X . Since X is open-hereditarily irresolvable, int A is dense in X . Consequently, X n int A is nowhere dense, and so discrete, since X is nodec. It follows from this that int A D A. Deﬁnition 3.21. A space .X; T / is maximal if T is maximal among all dense in itself topologies on X . Equivalently, a space is maximal if it has no isolated point but it does have an isolated point in any stronger topology. Theorem 3.22. For a dense in itself Hausdorff space X , the following statements are equivalent: (1) X is maximal, (2) X is extremally disconnected, open-hereditarily irresolvable and nodec, (3) for each x 2 X , there is exactly one nonprincipal ultraﬁlter on X converging to x. Proof. Let X D .X; T /. .1/ ) .2/ By Proposition 3.20, it sufﬁces to show that that X is extremally disconnected and submaximal.

46

Chapter 3 Topological Spaces with Extremal Properties

Assume that X is not extremally disconnected. Then there is an open subset V X and x 2 cl V n V such that V [ ¹xº is not open. Deﬁne the topology on X by taking as a base T [ ¹U \ .V [ ¹xº/ W U 2 T º: This topology is dense in itself and stronger than T . Hence, T is not maximal, a contradiction. Now assume that X is not submaximal. So there is a dense nonopen subset A X . Deﬁne the topology on X by taking as a base T [ ¹U \ A W U 2 T º: Clearly, this topology is dense in itself and stronger than T , a contradiction. .2/ ) .3/ It follows from extremal disconnectedness of X that there is exactly one maximal open ﬁlter F on X converging to x (Proposition 3.12). Since X is nodec, every nonprincipal ultraﬁlter on X converging to x is dense (Proposition 3.11) and so contains F (Proposition 3.9). Finally, open-hereditary irresolvability of X implies that F is an ultraﬁlter (Proposition 3.18). .3/ ) .1/ Let T1 be any topology on X stronger than T . Then there is x 2 X and a nonprincipal ultraﬁlter U on X converging to x in T but not converging to x in T1 . It follows that x is an isolated point in T1 . Hence, T is maximal. The equivalence .1/ , .3/ in Theorem 3.22 justiﬁes the following deﬁnition. Deﬁnition 3.23. We say that a space X is almost maximal if it is dense in itself and for every x 2 X there are only ﬁnitely many ultraﬁlters on X converging to x.

3.3

Irresolvability

The notion of irresolvability (Deﬁnition 3.17) naturally generalizes as follows. Deﬁnition 3.24. Given a cardinal 2, a space is -resolvable or -irresolvable depending on whether or not it can be partitioned into -many dense subsets. A space X is hereditarily -irresolvable (open-hereditarily -irresolvable) if every nonempty subset of X (every nonempty open subset of X ) is -irresolvable. The next lemma contains the main simple properties of -resolvability. Lemma 3.25. For every 2, the following statements hold: (1) an open subset of a -resolvable space is -resolvable, (2) the closure of a -resolvable subset of a space is -resolvable, and (3) the union of a family of -resolvable subsets of a space is -resolvable.

47

Section 3.3 Irresolvability

Proof. (1) and (2) are obvious, so it sufﬁces to S show (3). Let R be a family of resolvable subsets of a space X and let Y D R. We have to show that Y is -resolvable. Consider the set P of all -sequences A S D ¹A˛ W ˛ < º consisting of pairwise disjoint nonempty subsets A˛ Y dense in A. Deﬁne the order on P by AB

if and only if

A˛ B˛ for all ˛ < :

Every chain .Ai /i2I in P , where Ai D ¹Ai˛ W ˛ < º, has an upper bound ± °[ Ai˛ W ˛ < : i2I

ConsequentlySby Zorn’s Lemma, there is a maximal element A D ¹A˛ W ˛ < º in P . Let Z D ˛< A˛ . Obviously, Z is -resolvable. It is clear also that Z is closed. Indeed, otherwise the element B D ¹B˛ W ˛ < º in P deﬁned by ´ A0 [ cl Z n Z if ˛ D 0 B˛ D A˛ otherwise is greater than A. We claim that Z D Y . To see this, assume the contrary. Then there is R 2 R such that R n Z ¤ ;. Since Z is closed, S R n Z 2 R as well. It follows that there is C D ¹C˛ W ˛ < º in P such that C \ Z D ;. Then the element D D ¹D˛ W ˛ < º in P deﬁned by D˛ D A˛ [ C˛ is greater than A, a contradiction. Lemma 3.25 leads to the following construction. Lemma 3.26. Given a space X and 2, let R .X / denote the union of all resolvable subsets of X . Then (i) R .X / is the largest -resolvable subset of X , (ii) R .X / is closed, (iii) X is -resolvable if and only if R .X / D X , and (iv) if X is -irresolvable, then X n R .X / is hereditarily -irresolvable. Proof. It is immediate from Lemma 3.25. Corollary 3.27. A homogeneous space is -irresolvable if and only if it is hereditarily -irresolvable.

48

Chapter 3 Topological Spaces with Extremal Properties

Proof. Let X be a homogeneous -irresolvable space. Assume on the contrary that X is not hereditarily -irresolvable. Then R .X / is a proper subset of X . Pick x 2 R .X / and y 2 X n R .X /. Let f W X ! X be a homeomorphism with f .x/ D y. Then f .R .X // is a -resolvable subset of X and f .R .X // n R .X / ¤ ;, which is a contradiction. Of special interest are ﬁnitely and !-irresolvable spaces. Recall that for every ﬁlter F on a set X , F denotes the set of all ultraﬁlters on X containing F . Proposition 3.28. A space X is open-hereditarily n-irresolvable if and only if for every (converging) maximal open ﬁlter F on X , jF j < n. Proof. The same as that of Proposition 3.18. Lemma 3.29. Let X be a space and suppose that there is an open ﬁlter F on X such that jF j < n for some n < !. Then X is n-irresolvable. Proof. Assume on the contrary that there is aSpartition of X into n dense sets Ai , where i < n. Then there is j < n such that j ¤i

49

Section 3.3 Irresolvability

Lemma 3.34. Every countable discrete subset of a regular space is strongly discrete. Proof. Let X be a regular space and let D be a countably inﬁnite discrete subset of X . Enumerate D without repetitions as ¹xn W n < !º. For each n < !, pick a closed neighborhood Un of xn 2 X such that Un \ D D ¹xn º. For each n < !, deﬁne a neighborhood Vn of xn 2 X by [ Vn D Un n Ui : i

Then the subsets Vn , where n < !, are pairwise disjoint. Proof of Theorem 3.33. Let D be a countable discrete nonclosed subset of X and let a 2 cl D n D. For each x 2 X , there is a homeomorphism fx W X ! X with fx .a/ D x. Construct inductively a sequence .Dn /1 nD1 of discrete subsets of X such that Dn cl DnC1 n DnC1 : Put D1 D D. Fix k 2 N and suppose that the sequence .Dn /knD1 has already been constructed satisfying that condition. Denote ´S k1 if k > 1 nD1 cl Dn Yk1 D ; otherwise: It follows from the condition that Dk \ Yk1 D ;. By Lemma 3.34, for each x 2 Dk , there is an open neighborhood Ux of x 2 X n Yk1 such that the sets Ux , where x 2 Dk , are pairwise disjoint. Then the set [ DkC1 D fx .D \ fx1 .Ux // x2Dk

is as required. Now deﬁne Z X by ZD

1 [

Dn :

nD1

We claim that Z is !-resolvable. Indeed, let ¹In W n 2 Nº be any partition of N into inﬁnite sets. Deﬁne the partition ¹An W n 2 Nº of Z by [ An D Di : i2In

Then each An is dense in Z. Finally, since X contains a nonempty !-resolvable subset, X itself is !-resolvable by Corollary 3.27.

50

Chapter 3 Topological Spaces with Extremal Properties

We conclude this section by showing that !-irresolvable spaces are in fact ﬁnitely irresolvable. Theorem 3.35. If a space is n-resolvable for every n < !, then it is !-resolvable. To prove Theorem 3.35, we need two lemmas. Lemma 3.36. Given a space X , let W D W .X / denote the union of all open sets in X containing a dense hereditarily irresolvable subset. Then (i) W is the largest open set in X containing a dense open-hereditarily irresolvable subset, and (ii) every dense subset of X n cl W is resolvable, and consequently, !-resolvable. Proof. By Zorn’s Lemma, there is a maximal family U of pairwise S disjoint open sets in X containing a dense hereditarily irresolvable subset. Note that U is dense in W .SFor each U 2 U, pick a dense hereditarily irresolvable DU U . Then D D U 2U DU is a dense open-hereditarily irresolvable subset of W . Now let A be a dense subset of X n cl W . Assume on the contrary that A is irresolvable. It then follows from Lemma 3.26 that there is a nonempty open set V in X n cl W such that V \ A is hereditarily irresolvable. Clearly, V \ A is also dense in V . But this contradicts maximality of U. Lemma 3.37. Let D be an open-hereditarily irresolvable subset of a space X and suppose that X is .n C 1/-resolvable for some n. Then X n D is dense in X and n-resolvable. Proof. By Lemma 3.26, it sufﬁces to show that for every nonempty open set U X , U n D contains a nonempty n-resolvable subset. One may suppose that D \ U ¤ ;. Partition X into dense subsets A1 ; : : : ; AnC1 . Then D \ U is partitioned into subsets A1 \ D \ U; : : : ; AnC1 \ D \ U . Since no space is the union of ﬁnitely many nowhere dense sets, at least one of those sets, say AnC1 \ D \ U , is not nowhere dense in D \ U . So there is an open U0 U such that ; ¤ D \ U0 clD\U .AnC1 \ D \ U /: It follows that AnC1 \ D \ U0 is dense in D \ U0 . Then, since D \ U0 is irresolvable, .D \ U0 / n AnC1 is not dense in D \ U0 . Consequently, there is an open U1 U0 such that ; ¤ D \ U1 AnC1 \ D \ U0 , and so ; ¤ D \ U1 AnC1 \ U1 . It follows that A1 \ U1 ; : : : ; An \ U1 are pairwise disjoint dense subsets of U1 n D. Proof of Theorem 3.35. By Lemma 3.26, it sufﬁces to show that every nonempty open set X0 X contains a nonempty !-resolvable subset. Being open subset of X , X0 is

Section 3.3 Irresolvability

51

n-resolvable for each n. Consider the subset W .X0 / X0 given by Lemma 3.36. If it is not dense, then X0 n cl W .X0 / is !-resolvable and we are done. Otherwise pick a dense open-hereditarily irresolvable subset D1 W .X0 / and put X1 D W .X0 / n D1 . Being an open subset of X0 , W .X0 / is n-resolvable for each n, and then by Lemma 3.37, X1 is n-resolvable for each n. If W .X1 / X1 is not dense, then X1 n cl W .X1 / is !-resolvable. Otherwise pick a dense open-hereditarily irresolvable subset D2 W .X1 / and put X2 D W .X1 / n D2 , and so on. If at some m, W .Xm / Xm is not dense, then Xm n cl W .Xm / is the required nonempty !-resolvable subset of X0 . Otherwise we construct an inﬁnite sequence .Dm /1 mD1 of pairwise disjoint dense subsets of X0 , and then X0 itself is !-resolvable.

References The study of maximal and irresolvable spaces was initiated by E. Hewitt [33]. Theorem 3.13 is due to Z. Frolík [27]. Theorem 3.22 is from [99]. Another characterizations of maximal spaces can be found in [80]. Corollary 3.32 is a result of A. El’kin [18]. Theorem 3.35 is due to A. Illanes [40].

Chapter 4

Left Invariant Topologies and Strongly Discrete Filters

In this chapter left invariant topologies on semigroups are studied. We describe the largest left invariant topology in which a given ﬁlter converges to the identity. A special attention is paid to the case where the ﬁlter is strongly discrete, then the topology possesses certain important properties. We conclude by showing that every inﬁnite group admits a nondiscrete zero-dimensional Hausdorff invariant topology.

4.1

Left Topological Semigroups

Deﬁnition 4.1. A semigroup (group) S endowed with a topology is a left topological semigroup (group) and the topology itself a left invariant topology if for every a 2 S, the left translation a W S 3 x 7! ax 2 S is continuous. A topology T on a semigroup S is left invariant if and only if for every a 2 S and U 2 T , a1 U 2 T where a1 U D 1 a .U / D ¹x 2 S W ax 2 U º: Note that in a left topological group left translations are homeomorphisms. Consequently, if S is a group, a left invariant topology on S is completely determined by the neighborhood ﬁlter N of 1: for every a 2 S, the neighborhood ﬁlter of a is aN D ¹aU W U 2 N º. Topologies with this property on semigroups are characterized by the following lemma. Lemma 4.2. Let S be a semigroup with identity, let T be a topology on S, and let N be the neighborhood ﬁlter of 1 in T . Then the following statements are equivalent: (a) for every a 2 S, aN is a neighborhood base at a, (b) for every a 2 S, the left translation a W S 3 x 7! ax 2 S is continuous and open, and (c) for every a 2 S and U 2 T , both a1 U 2 T and aU 2 T .

Section 4.1 Left Topological Semigroups

53

Proof. (a) ) (b) To see that a is continuous, let b 2 S and let U be a neighborhood of a .b/ D ab. Pick V 2 N such that abV U . Then bV is a neighborhood of b and a .bV / D abV U . To see that a is open, let b 2 S and let U be a neighborhood of b. Pick V 2 N such that bV U . Then abV is a neighborhood of a .b/ and a .U / a .bV / D abV . (b) ) (c) Since a is continuous, a1 U D 1 a .U / 2 T , and since a is open, aU D a .U / 2 T . (c) ) (a) If U is an open neighborhood of 1, then a 2 aU 2 T , so aU is an open neighborhood of a. Conversely, let V be an open neighborhood of a and let U D a1 V . Then U is an open neighborhood of 1 and aU V . The next theorem characterizes the neighborhood ﬁlter of the identity of a left topological semigroup. Theorem 4.3. Let S be a left topological semigroup with identity and let N be the neighborhood ﬁlter of 1. Then (1) for every U 2 N , 1 2 U , and (2) for every U 2 N , ¹x 2 S W x 1 U 2 N º 2 N . Conversely, given a semigroup S with identity and a ﬁlter N on S satisfying conditions (1)–(2), there is a left invariant topology on S in which for each a 2 S, aN is a neighborhood base at a. Note that condition (2) in Theorem 4.3 is equivalent to (20 ) for every U 2 N , there are V 2 N and V 3 x 7! Wx 2 N such that xWx U . Proof. Suppose that S is a left topological semigroup with identity and let U 2 N . Clearly 1 2 U . Putting V D int U , we obtain that V 2 N and for every x 2 V , x 1 U 2 N . Conversely, suppose that N is a ﬁlter on S satisfying conditions (1)–(2). For every x 2 S , let Nx be the ﬁlter with a base xN . We show that, whenever x 2 S and U 2 Nx , one has (i) x 2 U , and (ii) ¹y 2 S W U 2 Ny º 2 Nx . (i) follows from (1) and deﬁnition of Nx . To show (ii), let U0 D x 1 U , V0 D ¹z 2 S W U0 2 Nz º; and V D xV0 . Clearly U0 2 N , so by (2), V0 2 N , and consequently V 2 Nx . We claim that for every y 2 V , U 2 Ny . To see this, write y D xz for some z 2 V0 . Then x 1 U D U0 2 Nz D zN , and so U 2 xzN D Ny . Now by Theorem 1.9, there is a topology T on S for which ¹Nx W x 2 Sº is the neighborhood system. By Lemma 4.2, T is left invariant.

54

Chapter 4 Left Invariant Topologies and Strongly Discrete Filters

A semigroup is called cancellative (left cancellative, right cancellative) if all translations (left translations, right translations) are injective. T T Note that whenever f W XT! Y and T B P .X /, one has f . B/ f .B/, and if f is injective, then f . B/ D f .B/. From this and Theorem 4.3 we obtain as a consequence the following. Corollary 4.4. Let S be a semigroup with identity and let B be a ﬁlter base on S satisfying the following conditions: (1) for every U 2 B, 1 2 U , and (2) for every U 2 B and a 2 U , there is V 2 B such that aV U . Then there is a left invariant topology T on S in which for each a 2 S, aB is an open neighborhood base at a. Furthermore, if S is left cancellative, then T is a T1 -topology if and only if T (3) B D ¹1º. Theorem 4.3 gives us also one more characterization of topologies from Lemma 4.2. Corollary 4.5. Let S be a semigroup with identity, let T be a topology on S, and let N be the neighborhood ﬁlter of 1 in T . Then the following statements are equivalent: (a) for every a 2 S , aN is a neighborhood base at a, and (d) T is the largest left invariant topology on S in which N is the neighborhood ﬁlter of 1. Proof. (a) ) (d) That T is left invariant follows from Lemma 4.2. To see that it is the largest one, let T 0 be any left invariant topology on S in which N is the neighborhood ﬁlter of 1. Then whenever a 2 S and U is a neighborhood of a in T 0 , there is V 2 N such that aV U . Hence T 0 T . (d) ) (a) By Theorem 4.3, there is a left invariant topology T 0 on S in which for each a 2 S, aN is a neighborhood base at a. Applying (a))(d) to T 0 , we obtain that T 0 is the largest left invariant topology on S in which N is the neighborhood ﬁlter of 1. Hence T 0 D T .

4.2

The Topology T ŒF

Deﬁnition 4.6. Let S be a semigroup with identity. For every ﬁlter F on S, let T ŒF denote the largest left invariant topology on S in which F converges to 1. Note that by Corollary 4.5, if N is the neighborhood ﬁlter of 1 in T ŒF , then for every a 2 S , aN is a neighborhood base at a, and by Lemma 4.2, the left translations in T ŒF are continuous and open.

Section 4.2 The Topology T ŒF

55

Deﬁnition 4.7. For every mapping M W S ! P .S / and a 2 S, deﬁne ŒM a 2 P .S / by ŒM a D ¹x0 x1 xn W n < !; x0 D a and xiC1 2 M.x0 xi / for each i < nº: The next theorem describes the topology T ŒF . Theorem 4.8. Let S be a semigroup with identity and let F be a ﬁlter on S. Then for every a 2 S, the subsets ŒM a , where M W S ! F , form an open neighborhood base at a in T ŒF . Proof. The subsets ŒM a , where a 2 S and M W S ! F , possess the following properties: (i) for every a 2 S and M; N W S ! F , there is K W S ! F such that ŒKa ŒM a \ ŒN a , (ii) for every a 2 S and M W S ! F , a 2 ŒM a , (iii) for every a 2 S, M W S ! F and x 2 ŒM a , one has ŒM x ŒM a , and (iv) for every a 2 S and M W S ! F , there is N W S ! F such that ŒM a D aŒN 1 . For (i), deﬁne K W S ! F by K.x/ D M.x/ \ N.x/. (ii) and (iii) are obvious. To see (iv), deﬁne N W S ! F by N.x/ D M.ax/. Then ŒM a D ¹x0 x1 xn W n < !; x0 D a and xiC1 2 M.x0 xi / for each i < nº D a¹x0 x1 xn W n < !; x0 D 1 and xiC1 2 M.ax0 xi / for each i < nº D a¹x0 x1 xn W n < !; x0 D 1 and xiC1 2 N.x0 xi / for each i < nº D aŒN 1 : By (i), the subsets ŒM 1 , where M W S ! F , form a ﬁlter base B on S , and by (ii), 1 2 U for all U 2 B. (iii) and (iv) give us that for every U 2 B and a 2 U , there is V 2 B such that aV U . Hence by Corollary 4.4, there is a left invariant topology T on S in which for every a 2 S, aB is an open neighborhood base at a. (iii) and (iv) give us also that for every a 2 S and M W S ! F , ŒM a is an open neighborhood of a in T . On the other hand, let U be an open neighborhood of a 2 S in any left invariant topology on S in which F converges to 1. Deﬁne M W S ! F by ´ x 1 U if x 2 U M.x/ D S otherwise. Then ŒM a U . It follows that the subsets ŒM a , where M W S ! F , form an open neighborhood base at a in T and T D T ŒF .

56

Chapter 4 Left Invariant Topologies and Strongly Discrete Filters

Corollary S 4.9. IfA 2 F and for every x 2 A, Ux is a neighborhood of 1 in T ŒF , then x2A xUx [ ¹1º is a neighborhood of 1 in T ŒF . Proof. Without loss of generality one may assume that for every x 2 A, Ux is open, S so U D x2A xUx is open. Deﬁne M W S ! F by 8 ˆ

if x D 1 if x 2 U n ¹1º otherwise.

Then ŒM 1 U [ ¹1º. Corollary 4.10. If S is left cancellative and

T

F D ;, then T ŒF is a T1 -topology.

Proof. Let 1 ¤ a 2 S . Since S is left cancellative and F such that a … xM.x/ for all x 2 S. Then a … ŒM 1 .

T

F D ;, there is M W S !

Of special importance is the case where F is an ultraﬁlter. Theorem 4.11. For every nonprincipal ultraﬁlter F on S, T ŒF is strongly extremally disconnected. Proof. Let U be an open nonclosed subset of .S; T ŒF / and let C D S n U . Consider two cases. Case 1: there is a 2 C such that U 2 aF . Choose M W S ! F such that aM.a/ U and for every x 2 U , xM.x/ U . Then ŒM a U [ ¹aº. Hence, U [ ¹aº is open. Case 2: for every x 2 C , C 2 xF . Choose M W S ! F such that for every x 2 C , xM.x/ C . Then ŒM b C for every b 2 C . Consequently, C is open, and so U is closed. Hence, this case is impossible.

4.3

Strongly Discrete Filters

Deﬁnition 4.12. Let S be a semigroup with identity. Given a mapping M W S ! P .S /, an M -product is any product of elements of S of the form x0 xn where xiC1 2 M.x0 xi / for each i < n, and an M -decomposition of an element x 2 S is a decomposition x D x0 xn , where x0 xn is an M -product. An M -decomposition x D x0 xn is trivial if n D 0 (and then x0 D x). A mapping M W S ! P .S / is basic if every x 2 S has a unique M -decomposition x D x0 xn with x0 D 1.

Section 4.3 Strongly Discrete Filters

57

Lemma 4.13. Let M W S ! P .S / be a basic mapping. Then (1) there is no nontrivial M -decomposition of 1, (2) the subsets xM.x/, where x 2 S, are pairwise disjoint, and (3) for each x 2 S, the mapping M.x/ 3 y 7! xy 2 xM.x/ is injective. Proof. (1) It follows from the deﬁnition of a basic mapping that there is no nontrivial M -decomposition 1 D x0 xn with x0 D 1. So suppose that 1 D x0 xn is an M decomposition and x0 ¤ 1. Then there is an M -decomposition x0 D y0 ym with y0 D 1 and m > 0. But then 1 D y0 ym x1 xn is a nontrivial M - decomposition with y0 D 1, which is a contradiction. (2) Suppose that xu D yv D z for some x; y; z 2 S, u 2 M.x/ and v 2 M.y/. Let x D x0 xn and y D y0 ym be M -decompositions with x0 D y0 D 1. Then z D x0 xn u and z D y0 ym v are M -decompositions with x0 D y0 D 1. It follows that u D v, n D m and xi D yi for each i n, and so x D y. (3) Suppose that xu D xv for some x 2 S and u; v 2 M.x/. Let x D x0 xn be M -decomposition with x0 D 1. Then x0 xn u and x0 xn v are M -products and x0 xn u D x0 xn v. Hence u D v. Lemma 4.14. Let M W S ! P .S / and suppose that the subsets xM.x/, where x 2 S , are pairwise disjoint and for each x 2 S, M.x/ 3 y 7! xy 2 xM.x/ is injective. Let x0 xn and y0 ym be M -products and let x0 xn D y0 ym . Then either (1) x0 xnm D y0 and xnmCj D yj , 1 j m, if n m, or (2) x0 D y0 ymn and xi D ymnCi , 1 i n, if n < m. Proof. We proceed by induction on min¹n; mº. It is trivial for min¹n; mº D 0. Let min¹n; mº > 0. We claim that x0 xn1 D y0 ym1 and then xn D ym , since M.x0 xn1 / 3 y 7! x0 xn1 y 2 x0 xn1 M.x0 xn1 / is injective. Indeed, if x0 xn1 ¤ y0 ym1 , then x0 xn ¤ y0 ym , since x0 xn1 M.x0 xn1 / \ y0 ym1 M.y0 ym1 / D ;: We then apply the inductive assumption. Recall that a subset D of a space X is called strongly discrete if for every x 2 D, there is a neighborhood Ux of x 2 X such that the subsets Ux , where x 2 D, are pairwise disjoint. The Fréchet ﬁlter on an inﬁnite set X consists of all subsets A X such that jX n Aj < jX j.

58

Chapter 4 Left Invariant Topologies and Strongly Discrete Filters

Deﬁnition 4.15. Let S be a semigroup with identity and let F be a ﬁlter on S. We say that F is strongly discrete if (1) F contains the Fréchet ﬁlter, and (2) there is M W S ! F such that the subsets xM.x/, where x 2 S, are pairwise disjoint, and for every x 2 S, M.x/ 3 y 7! xy 2 xM.x/ is injective. Theorem 4.16. For every strongly discrete ﬁlter F on S, there is a basic mapping M WS !F. Proof. Let D jSj and enumerate S as ¹s˛ W ˛ < º with s0 D 1. Choose M0 W S ! F such that (i) the subsets xM0 .x/, where x 2 S, are pairwise disjoint, and for every x 2 S, M0 .x/ 3 y 7! xy 2 xM0 .x/ is injective, and (ii) s˛ … sˇ M0 .sˇ / for all ˛ ˇ < . It follows from (ii) that 1 … xM0 .x/ for all x 2 S, which in turn implies that there is no nontrivial M0 -decomposition of 1. Construct inductively a -sequence .a /< with maximally possible by putting a0 D 1 and taking a , for > 0, to S be the ﬁrst element in the sequence ¹s˛ W ˛ < º such that a … X where X D ı< ŒM0 aı . We deﬁne M W S ! F by ´ M0 .1/ [ ¹a W 0 < < º M.x/ D M0 .x/

if x D 1 otherwise.

Clearly, for every x 2 S , M.x/ 3 y 7! xy 2 xM.x/ is injective. Let us check that the subsets xM.x/, where x 2 S, are pairwise disjoint. Assume the contrary. Then by the deﬁnition of M and (i), a 2 xM0 .x/ for some 1 ¤ x 2 S and 0 < < . We have also that a D s˛ and x D sˇ for some ˛; ˇ < , so s˛ 2 sˇ M0 .sˇ /. It follows from this and (ii) that ˛ > ˇ. But then by the choice of a D s˛ , one has sˇ 2 X . This implies that also every M0 -product x0 xn with x0 D sˇ belongs to X . Since s˛ 2 sˇ M0 .sˇ /, we obtain that s˛ D a 2 X , which is a contradiction. Now we claim that M is basic. Indeed, for every x 2 S , there is < such that x 2 ŒM0 a , so there is an M0 decomposition x D x0 xn with x0 D a . If D 0, this is M -decomposition with x0 D 1. Otherwise so is x D 1x0 xn . To see that every x has only one M -decomposition x D x0 xn with x0 D 1, it sufﬁces, by Lemma 4.14, to check that if 1 D x0 xn is an M -decomposition with x0 D 1, then n D 0. Assume on the contrary that n > 0. Then 1 D x1 xn is a nontrivial M0 -decomposition of 1, which is a contradiction.

Section 4.3 Strongly Discrete Filters

59

Deﬁnition 4.17. Let D be a discrete subset of a space X and let x be an accumulation point of D. We say that D is locally maximal with respect to x if for every discrete subset E X such that D \ U E \ U for some neighborhood U of x, there is a neighborhood V of x such that D \ V D E \ V . The next theorem gives us the main properties of topologies determined by strongly discrete ﬁlters. Theorem 4.18. Let F be a strongly discrete ﬁlter on S and let S D .S; T ŒF /. Then (1) S is zero-dimensional and Hausdorff, (2) there is D 2 F such that (i) D is a strongly discrete subset of S with exactly one accumulation point, and (ii) D is a locally maximal discrete subset of S with respect to 1. Furthermore, if in addition F is an ultraﬁlter, then (3) S is strongly extremally disconnected. Proof. (1) To see that T ŒF is a T1 -topology, let a; b be distinct elements of S. Pick N W S ! F such that b … xN.x/ for all x 2 S. Then b … ŒN a . Now to show that T ŒF is zero-dimensional, let a 2 S. By Theorem 4.16, there is a basic mapping M W S ! F , and let N W S ! F be any mapping such that N.x/ M.x/ for all x 2 S. We show that ŒN a is closed. To this end, pick K W S ! F such that for every x 2 S, K.x/ M.x/ and a … xK.x/. We claim that for every b 2 S n ŒN a , one has ŒKb \ ŒN a D ;. Indeed, assume the contrary. Then x0 xn D y0 ym for some n; m < ! and xi ; yj 2 S such that x0 D b, xiC1 2 K.x0 xi / for i < n, y0 D a, and yj C1 2 N.y0 yj / for j < m. Since K.x/; N.x/ M.x/ for all x 2 S, we obtain by Lemma 4.14, that either x0 xnm D y0 (if n m) or x0 D y0 ymn (if n < m). The ﬁrst possibilty gives us that a 2 ŒKb , a contradiction with a … xK.x/ for all x 2 S. And the second gives us that b 2 ŒN a , again a contradiction. (2) Pick a basic mapping M W S ! F and let D D M.1/. (i) We claim that the subsets ŒM a , where a 2 D, are pairwise disjoint. Indeed, let x1 ; y1 2 D and let ŒM x1 \ ŒM y1 ¤ ;. Then x1 xn D y1 ym for some n; m 1 and xi ; yj such that xiC1 2 M.x1 xi / and yj C1 2 M.y1 yj /. But then x0 x1 xn D y0 y1 ym where x0 D y0 D 1. Since x1 ; y1 2 M.1/, it follows from this and Lemma 4.14 that n D m and xi D yi for each i n, in particular, x1 D y1 . To see that D [ ¹1º is closed, let 1 ¤ x1 2 ŒM 1 and let ŒM x1 \ D ¤ ;. Then x1 xn D y1 for some y1 2 D, n 1 and xi such that xiC1 2 M.x1 xi /. But then x0 x1 xn D y0 y1 where x0 D y0 D 1. It follows from this that n D 1 and x1 D y1 .

60

Chapter 4 Left Invariant Topologies and Strongly Discrete Filters

(ii) Let E be a discrete subset of S such that D\U E\U for some neighborhood U of 1. One may suppose that U is open. For every x 2 D \ U S, choose a neighborhood Vx of 1 such that xVx U and xVx \E D ¹xº. Put V D x2D\U xVx [¹1º. By Corollary 4.9, V is a neighborhood of 1, and by the construction, D \V D E \V . (3) is immediate from Theorem 4.11. The next proposition shows how naturally strongly discrete ﬁlters arise on left topological groups. Proposition 4.19. Let .S; T / be a regular left topological group such that the intersection of < jSj open sets is open and let D be a discrete subset of .S; T / with exactly one accumulation point 1. Then any ﬁlter on S containing D and converging to 1 is strongly discrete. Proof. Let F be a ﬁlter on S containing D and converging to 1 and let D jSj. Enumerate S as ¹s˛ W ˛ < º. Construct inductively a sequence .M.s˛ //˛< such that M.s˛ / 2 F , M.s˛ / D and [ s˛ M.s˛ / \ sˇ M.sˇ / D ;: ˇ <˛

This can be done because [

0 sˇ M.sˇ / D ¹sˇ W ˇ < ˛º:

ˇ <˛

(For every subset Y of a space X , Y 0 denotes the set of accumulation points of Y X .) We now show that strongly discrete ﬁlters exist in profusion on any inﬁnite cancellative semigroup. Proposition 4.20. Let S be a cancellative semigroup with identity and let jSj D !. Enumerate S without repetitions as ¹s˛ W ˛ < º. Then there is a one-to-one sequence .x˛ /˛< in S such that the subsets s˛ X˛ , where X˛ D ¹xˇ W ˛ ˇ < º, are pairwise disjoint. Proof. Pick as x0 any element of S. Now let 0 < < and assume that we have constructed a one-to-one sequence .x˛ /˛< such that the subsets s˛ X˛; , where X˛; D ¹xˇ W ˛ ˇ < º, are pairwise disjoint. Choose x 2 S satisfying the condition s˛ x ¤ sˇ xı , where ˛ and ˇ ı < (this can be done because S is left cancellative). The condition s˛ x ¤ sˇ x , where ˛; ˇ and ˛ ¤ ˇ, is satisﬁed automatically (because S is right cancellative). Hence, the subsets s˛ X˛;C1 , where ˛ < C 1, are also pairwise disjoint.

61

Section 4.3 Strongly Discrete Filters

Let F and G be ﬁlters on sets X and Y , respectively. We say that F and G are isomorphic if there is a bijection f W X ! Y such that f .F / D G . Deﬁnition 4.21. Let F be a ﬁlter on an inﬁnite set X containing the Fréchet ﬁlter. We say that F is locally Fréchet if there is A 2 F such that F jA is the Fréchet ﬁlter on A. A locally Fréchet ﬁlter is proper if it is not Fréchet. Note that all proper locally Fréchet ﬁlters on X are isomorphic, and if F is not locally Fréchet, then for every A 2 F , F is isomorphic to F jA . To see the latter, pick B 2 F such that B A and jAnBj D jX j and take any bijection g W X nB ! AnB. Deﬁne f W X ! A by ´ x if x 2 B f .x/ D g.x/ otherwise. Then f is a bijection and f .F / D F jA . Now we obtain from Proposition 4.20 the following. Corollary 4.22. Let S be a cancellative semigroup with identity and let jSj D !. For every ﬁlter F on properly containing the Fréchet ﬁlter, there is a strongly discrete ﬁlter G on S isomorphic to F . Proof. Let .x˛ /˛< be a sequence in S guaranteed by Proposition 4.20, let A D ¹x˛ W ˛ < º, and let H be the ﬁlter on S such that A 2 H and H jA is the Fréchet ﬁlter. Then H is proper locally Fréchet and strongly discrete. Now let F be a ﬁlter on properly containing the Fréchet ﬁlter. If F is locally Fréchet, it is isomorphic to H . Otherwise take a ﬁlter G on S such that A 2 G and G jA is isomorphic to F . Then G is isomorphic to F and is strongly discrete. We conclude the section with a topological classiﬁcation of topologies determined by strongly discrete ﬁlters. Deﬁnition 4.23. Given a space X with exactly one accumulation point 1 2 X , deﬁne an increasing sequence .Xn /1 nD1 of extensions of X D X1 as follows. For each n 1, put [ XnC1 D Xn0 [ X.y/; y2Xn nXn0

where X.y/ is a copy of X with y D 1 (we suppose that X.y/ \ Xn D ¹yº and that X.y/ \ X.z/ D ; if y ¤ z), and topologize XnC1 by declaring a subset U XnC1 to be open if and only if U \ Xn is open in Xn and for every y 2 Xn n Xn0 , U \ X.y/ is open in X.y/. Finally, we deﬁne X! D lim Xn ; n!1

62

Chapter 4 Left Invariant Topologies and Strongly Discrete Filters

S that is, X! D 1 nD1 Xn and a subset U X is open if and only if U \ Xn is open in Xn for every n. Theorem 4.24. Let F be a strongly discrete ﬁlter on S and let X be a space with exactly one accumulation point 1 2 X such that the ﬁlter on X with a base consisting of subsets U n¹1º, where U runs over neighborhoods of 1, is isomorphic to F if F is not locally Fréchet and to the Fréchet ﬁlter otherwise. Then .S; T ŒF / is homeomorphic to X! . Proof. Suppose ﬁrst that F is not locally Fréchet. By Theorem 4.16, there is a basic M W S ! F . Deﬁne an increasing sequence .Yn /1 nD1 of subspaces of .S; T ŒF // by putting Y1 D M.1/ and [ .M.y/ [ ¹yº/: YnC1 D Yn0 [ y2Yn nYn0

Then .S; T ŒF / D lim Yn : n!1

For every y 2 S, M.y/ [ ¹yº is homeomorphic to X . It follows that for every n, Yn is homeomorphic to Xn , and consequently, .S; T ŒF / is homeomorphic to X! . Now suppose that F is locally Fréchet, so there is B 2 F such that F jB is the Fréchet ﬁlter. Choose a basic M W S ! F such that M.x/ B for all x 2 S. Then for every N W S ! F such that N.x/ M.x/ and for every a 2 S, ŒN a is homeomorphic to X! , in particular, ŒM 1 is homeomorphic to X! . We shall show that there are N W S ! F and A S with 1 2 A such that N.x/ M.x/, jAj D jS j D , and the subsets ŒN a , where a 2 A, form a partition of ŒM 1 . This implies that both ŒM 1 and .S; T ŒF / are homeomorphic to the sum of copies of X! , and consequently, .S; T ŒF / is homeomorphic to X! . For each x 2 M.1/, pick yx 2 M.x/, and let C D ¹xyx W x 2 M.1/º. Then jC j D , Sthe subsets ŒM a ŒM 1 , where a 2 C , are pairwise disjoint, and U D ŒM 1 n a2C ŒM a is an open neighborhood of 1 (see Theorem 4.18 (2)). Deﬁne N W S ! F by ´ .x 1 U / \ M.x/ if x 2 U N.x/ D M.x/ otherwise. The mapping N , in turn, determines D S such that the subsets ŒN a , where a 2 D, form a partition of S. Put A D D \ ŒM 1 . Remark 4.25. Let be an inﬁnite cardinal, let X be a space with jX j D and exactly one accumulation point 1X 2 X such that the ﬁlter on X with a base consisting of subsets U n ¹1X º, where U runs over neighborhoods of 1X , is isomorphic to the Fréchet ﬁlter on , and let Y be a space with jY j D and exactly one accumulation

Section 4.4 Invariant Topologies

63

point 1Y 2 Y such that the ﬁlter on Y with a base consisting of subsets U n ¹1Y º, where U runs over neighborhoods of 1Y , is isomorphic to the proper locally Fréchet ﬁlter on . It is not hard to see that the space Y! is homeomorphic to the sum of copies of X! , and so just to X! . Hence Theorem 4.24 can be stated also as follows: Let F be a strongly discrete ﬁlter on S and let X be a space with exactly one accumulation point 1 2 X such that the ﬁlter on X with a base consisting of subsets U n ¹1º, where U runs over neighborhoods of 1, is isomorphic to F . Then .S; T ŒF / is homeomorphic to X! . Corollary 4.26. Let F and G be strongly discrete ﬁlters on S. Then the topologies T ŒF and T ŒG are homeomorphic if and only if the ﬁlters F and G are isomorphic. Proof. Let f W .S; T ŒF / ! .S; T ŒG / be a homeomorphism. Without loss of generality one may suppose that f .1/ D 1. We claim that f .F / D G . Let DF 2 F and DG 2 G be sets satisfying Theorem 4.18 (2). It sufﬁces to show that for every A 2 F with A DF , f .A/ 2 G . Assume the contrary. Then B D DG n f .A/ is a discrete subset of .S; T ŒG / with cl B D B [ ¹1º. Consequently, C D A [ f 1 .B/ is a discrete subset of .S; T ŒF / such that A C and for every neighborhood U of 1, .C n A/ \ U ¤ ;. But this contradicts Theorem 4.18 (2). Conversely, let F and G be isomorphic. Then T ŒF and T ŒG are homeomorphic by Theorem 4.24.

4.4

Invariant Topologies

Deﬁnition 4.27. We say that a topology on a group is invariant if all translations and the inversion are continuous. In this section we show that Theorem 4.28. Every inﬁnite group admits a nondiscrete zero-dimensional Hausdorff invariant topology. By Theorem 4.18 (1), in order to prove Theorem 4.28, it sufﬁces to show that for every inﬁnite group G, there is a strongly discrete ﬁlter F on G such that the topology T ŒF is invariant. We ﬁrst derive a simple sufﬁcient condition for the topology T ŒF to be invariant. Lemma 4.29. Let T be a left invariant topology on G and let N be the neighborhood ﬁlter of 1 in T . Then T is invariant if and only if N 1 D N and xN x 1 D N for all x 2 G. Proof. Necessity is obvious. Sufﬁciency follows from the fact that for every U 2 N and x; y 2 G, one has .xU /y D xy.y 1 Uy/ and .xU /1 D x 1 .xU 1 x 1 /.

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Chapter 4 Left Invariant Topologies and Strongly Discrete Filters

Lemma 4.30. Let F be a ﬁlter on G such that F 1 D F and xF x 1 D F for all x 2 G. Then the topology T ŒF is invariant. Proof. By Lemma 4.29 and Theorem 4.8, it sufﬁces to show that (a) for every M W G ! F and a 2 G, there is N W G ! F such that aŒN 1 a1 ŒM 1 , and (b) for every M W G ! F , there is N W G ! F such that ŒN 1 1 ŒM 1 . To see (a), deﬁne N W G ! F by N.x/ D a1 M.axa1 /a: Let x 2 ŒN 1 and let x D x0 xn be N -decomposition with x0 D 1. Then axa1 D ax0 xn a1 D ax0 a1 axn a1 and for each i < n, one has axiC1 a1 2 aN.x0 xi /a1 D M.ax0 xi a1 / D M.ax0 a1 axi a1 /: To see (b), deﬁne N W G ! F by N.x/ D x 1 .M.x 1 //1 x: Let x 2 ŒN 1 and let x D x0 xn be N -decomposition with x0 D 1. Then x 1 D .x0 xn /1 D x01 x0 x11 x01 .x0 xn1 /xn1 .x0 xn1 /1 and for each i < n, one has 1 .x0 xi /xiC1 .x0 xi /1

2 .x0 xi /.N.x0 xi //1 .x0 xi /1 D M..x0 xi /1 / D M.x01 x0 x11 x01 .x0 xi1 /xi1 .x0 xi1 /1 /: In the case where F is a strongly discrete ﬁlter, the condition in Lemma 4.30 is also necessary for the topology T ŒF to be invariant. Proposition 4.31. Let F be a strongly discrete ﬁlter on G. Then the topology T ŒF is invariant if and only if F 1 D F and xF x 1 D F for all x 2 G.

Section 4.4 Invariant Topologies

65

Proof. Sufﬁciency is immediate from Lemma 4.30. Necessity follows from Theorem 4.18 (2). Thus, by Theorem 4.18 (1) and Lemma 4.30, in order to prove Theorem 4.28, it sufﬁces to show that Theorem 4.32. For every inﬁnite group G, there is a strongly discrete ﬁlter F on G such that F 1 D F and xF x 1 D F for all x 2 G. Proof. We will prove the assertion: For every inﬁnite subgroup H of G and for every subset F G with jF j < jGj, there is a ﬁlter F on G and a mapping M W H ! F such that (1) F 1 D F and xF x 1 D F for all x 2 H , (2) F jM.1/ contains the Fréchet ﬁlter on M.1/, jM.1/j D jH j and M.x/ M.1/ for all x 2 H , (3) the subsets xM.x/, where x 2 H , are pairwise disjoint and disjoint from F . Here, F jM.1/ is the trace of F on M.1/. We proceed by induction on D jH j. Let D !. Enumerate H as ¹xn W n < !º. It sufﬁces to construct a one-to-one sequence .yn /n

where n W ! ! !, and the mapping M W H ! F by M.xn / D Zn (see Lemma 1.16). Pick as y0 any element of G such that x0 y0 x01 … F . Fix 0 < k < ! and assume that we have constructed an one-to-one sequence .yn /n

66

Chapter 4 Left Invariant Topologies and Strongly Discrete Filters

Lemma 4.33. Every ﬁnite system of inequalities of the form ayb ¤ y " , where a; b 2 G, " D ˙1 and ab ¤ 1, has jGj-many solutions in G. Proof. Assume the contrary. Then there is a subset F G with jF j < jGj such that for every x 2 G n F , there exists an inequality ayb ¤ y " of the system with axb D x " . By Corollary 1.42, there is a 3-sequence .xn /n<3 in G and an inequality ayb ¤ y " of the system such that axb D x " for every x 2 FP..xn /n<3 /. Consider two cases. Case 1: " D 1. Then on the one hand ax1 x2 b D x1 x2 and on the other hand ax1 x2 b D ax1 b b 1 a1 ax2 b D x1 .ab/1 x2 ; so

x1 .ab/1 x2 D x1 x2

which contradicts ab ¤ 1. Case 2: " D 1. Then for any x 2 FP..xn /1n<3 /, on the one hand ax0 xb D .x0 x/1 D x 1 x01 and on the other hand ax0 xb D ax0 b b 1 a1 axb D x01 .ab/1 x 1 ; so and consequently Hence

x01 .ab/1 x 1 D x 1 x01 x01 .ab/1 x 1 x0 D x 1 : .x01 /x.abx0 / D x

and since x01 abx0 ¤ 1, we can apply Case 1. Fix > !, assume that the assertion has been proved for all inﬁnite < , and prove it for D . Choose S an increasing sequence .H˛ /˛< of inﬁnite subgroups of H with jH˛ j < and ˛< H˛ D H . For every ˛ < , construct a ﬁlter F˛ on G and a mapping M˛ W H˛ ! F˛ such that (1) F˛1 D F˛ and xF˛ x 1 D F˛ for all x 2 H˛ , (2) F jM˛ .1/ contains the Fréchet ﬁlter on M˛ .1/, jM˛ .1/j D jH˛ j and M˛.x/ M˛ .1/ for all x 2 H˛ , and

67

Section 4.4 Invariant Topologies

(3) the subsets xM˛ .x/, where x 2 H˛ , are pairwise disjoint and disjoint from F and all yM .y/, where < ˛ and y 2 H . Deﬁne the ﬁlter F on G by taking as a base the subsets of the form [ A˛ ; ˛<

where < and A˛ 2 F˛ , and the mapping M W H ! F by [ M.x/ D M˛ .x/; ˛<

where D min¹˛ W x 2 H˛ º. Then F is as required. jGj

Remark 4.34. In fact, the proof of Theorem 4.32 shows more: there are 22 nondiscrete zero-dimensional Hausdorff invariant topologies T˛ on G such that for any disjGj tinct ˛; < 22 , the topology T˛ _ T is discrete. (The latter means that there are neighborhoods U˛ , U of 1 in T˛ , T respectively with U˛ \ U D ¹1º.) To see this, let H D G and for every uniform ultraﬁlter p on , deﬁne the ﬁlter F .p/ on G by taking as a base the subsets of the form [ A˛ ; ˛2P

where P 2 p and A˛ 2 F˛ . It is clear that .F .p//1 D F .p/, xF .p/x 1 D F .p/ for all x 2 G, and for any distinct p and q, there exist A.p/ 2 F .p/ and A.q/ 2 F .q/ with A.p/ \ A.q/ D ;. That T .F .p// _ T .F .q// is discrete follows from Theorem 4.18 (2) and Corollary 4.9.

References The results about the topology T ŒF and strongly discrete ﬁlters in the case where S is a group were proved in [103]. That every inﬁnite group admits a nondiscrete zero-dimensional Hausdorff invariant topology was proved in [100].

Chapter 5

Topological Groups with Extremal Properties

In this chapter, assuming MA, we construct important topological groups with extremal properties. We also show that some of them cannot be constructed in ZFC. Recall that starting from this chapter, all topological groups are assumed to be Hausdorff.

5.1

Extremally Disconnected Topological Groups

Theorem 5.1.LFor every n < !, let Gn be a nontrivial ﬁnite group written additively, and let G D n

Section 5.1 Extremally Disconnected Topological Groups

69

Since p is Ramsey, there exists A 2 p such that A B0 and ŒA2 is monochrome. Observe that .ŒA2 / D ¹1º. We claim that HA ŒM 0 . To see this, let 0 ¤ x 2 HA and let supp.x/ D ¹n1 ; : : : ; nk º, where n1 < < nk . We have that n1 2 B0 and niC1 2 Bni for each i D 1; : : : ; k 1. Write x as x D x1 C C xk , where supp.xi / D ¹ni º. Then x1 2 DB0 M.0/ and xiC1 2 DBni M.x1 C C xi /: Hence x 2 ŒM 0 . .2/ ) .3/ Let E be a discrete subset of .G; T / such that D \ U E \ U for some neighborhood U of 0. One may suppose that U is open. For every x 2 D \ U , choose a neighborhood Vx of 0 such that x C Vx U and .x C Vx / \ E D ¹xº. Let [ V D .x C Vx / [ ¹0º: x2D\U

By Corollary 4.9, V is a neighborhood of 0 and by the construction, D \ V D E \ V . .3/ ) .1/ Let ¹An W n < !º be any partition of ! such that An … p for all n < !. Deﬁne E G by E D ¹x 2 G W supp.x/ An for some n < ! and jsupp.x/j D 2º: Notice that every point from E is isolated in D [ E. Since An … p for all n < !, it follows that also every point from D is isolated in D [ E. Consequently D [ E is discrete. Since D is a locally maximal discrete subset with respect to 0, there is a neighborhood U of 0 such that U \ E D ;. Choose A 2 p such that HA U . Then jA \ An j 1 forL all n < !. Now let G D ! Z2 . Then F is an ultraﬁlter. .2/ ) .4/ and .2/ ) .5/ follow from Theorem 4.11 and Proposition 3.15. To show .4/ ) .3/, let E be a discrete subset of G D .G; T / such that D \ U E \U for some neighborhood U of 0. For every x 2 E, choose an open neighborhood Ux of 0 such that the subsets x C Ux are pairwise disjoint. Put [ [ UD D .x C Ux / and UE nD D .x C Ux /: x2D

x2E nD

Then UD and UE nD are disjoint open subsets and, obviously, 0 2 cl UD . But then, since G is extremally disconnected, 0 … cl UE nD . Hence, V D U n UE nD is a neighborhood of 0 and D \ V D E \ V . To show .5/ ) .1/, suppose that p is not Ramsey. Then there is a partition ¹An W n < !º of ! such that An … p for all n < ! and for every A 2 p, jA \ An j 2 for some n < !. Deﬁne the subset F of .G; T / by F D ¹x 2 G W jsupp.x/ \ An j 1 for all n < !º: We claim that F is perfect and nowhere dense.

70

Chapter 5 Topological Groups with Extremal Properties

Clearly, F is closed. To see that F is dense in itself, let x 2 F . Pick n0 < ! such that max supp.x/ < min An for all n n0 and deﬁne A 2 p by AD

[

An :

n0 n

Then x C DA F and x is an accumulation point of x C DA . Finally, to see that F is nowhere dense, let x 2 G and let U be a neighborhood of 0 in T . Pick A 2 p such that max supp.x/ < min A and HA U . Then jA \ An j 2 for some n. Consequently, there is y 2 HA such that supp.y/ is a 2-element subset of An , and so x C y … F . Since F is closed, this shows that F is nowhere dense. Corollary 5.2. Assume p D c. Then there exists a countable nondiscrete extremally disconnected topological group. A group of exponent 2 is called Boolean. Note that if G is a Boolean group, then 1 1 1 for every x; y 2 G, one has xy D L.xy/ D y x D yx, so G is Abelian. It then follows that G is isomorphic to Z2 for some cardinal . For every group G, let B.G/ D ¹x 2 G W x 2 D 1º: Note that if G is Abelian, then B.G/ is the largest Boolean subgroup of G. Lemma 5.3. Let G be a topological group. If B.G/ is a neighborhood of 1, then G contains an open Boolean subgroup. Proof. Let B D B.G/. Since B is a neighborhood of 1, there is a neighborhood W of 1 such that W 2 B. Then for every x; y 2 W , xy 2 B and consequently xy D .xy/1 D y 1 x 1 D yx. It then follows that for every x1 ; : : : ; xn 2 W , .x1 xn /2 D x12 xn2 D 1 and so x1 xn 2 B. Hence, hW i is an open Boolean subgroup of G. Theorem 5.4. Every extremally disconnected topological group contains an open Boolean subgroup. Proof. Let G be an extremally disconnected topological group and let W G 3 x 7! x 1 2 G. Then by Theorem 3.13, ¹x 2 G W .x/ D xº is clopen. Since ¹x 2 G W .x/ D xº D B.G/, G contains an open Boolean subgroup by Lemma 5.3.

Section 5.2 Maximal Topological Groups

5.2

71

Maximal Topological Groups

We say that a topological group (or a group topology) is maximal if the underlying space is maximal. Theorem 5.5. Assume p D c. Then there exists a maximal topological group. L Proof. Let G D ! Z2 . Enumerate the subsets of G as ¹Z˛ W ˛ < cº with Z0 D G. For every ˛ < c, we shall construct a sequence .x˛;n /n 1Sand suppose that the statement holds for all positive integers less than n. Let X D A and for every x 2 X , let Ax D ¹A 2 A W x 2 Aº. Consider two cases.

72

Chapter 5 Topological Groups with Extremal Properties

Case 1: for every x 2 X , Ax is countable. By transﬁnite recursion on ˛ < !1 , pick so that A˛ \ A D ; for all < ˛. This can be done because S A˛ 2 A S ¹Ax W x 2 <˛ A º is countable. Let B D ¹A˛ W ˛ < !1 º. Then B is a

-system with root ;. Case 2: there is x 2 X with uncountable Ax . Let A0 D ¹A n ¹xº W A 2 Ax º. By the inductive hypothesis, there is an uncountable -system B 0 A0 . Let R denote the root of B 0 . Deﬁne B Ax by B D ¹A [ ¹xº W A 2 B 0 º. Then B is a -system with root R [ ¹xº. Theorem 5.7. Every maximal topological group contains a countable open Boolean subgroup. Proof. Let G be a maximal topological group. Then G is extremally disconnected. L Consequently by Theorem 5.4, one may suppose that G D Z2 for some inﬁnite cardinal . For every a 2 N, let 2 .a/ D max¹n < ! W 2n jaº: Deﬁne the partition ¹Ai W i < 2º of G n ¹0º by Ai D ¹x 2 G n ¹0º W 2 .jsupp.x/j/ i .mod 2/º: Since G is maximal, there exists a neighborhood U of 0 such that either U n ¹0º A0 or U n ¹0º A1 . Choose a neighborhood V of 0 such that V C V C V C V U: We claim that V is countable. Indeed, assume the contrary. Then applying the -system Lemma gives us an uncountable X V and a ﬁnite F such that, whenever x and y are different elements of X , one has supp.x/ \ supp.y/ D F and jsupp.x/j D jsupp.y/j. Pick different x0 ; x1 ; x2 ; x3 2 X and let x D x0 C x1 and y D x0 C x1 C x2 C x3 . Then x; y 2 U and jsupp.y/j D 2 jsupp.x/j, so x; y belong to different sets A0 ; A1 , a contradiction.

5.3

Nodec Topological Groups

Theorem 5.8. Assume p D c. Then every nondiscrete group topology on an Abelian group of character < c can be reﬁned to a nondiscrete nodec group topology. Before proving Theorem 5.8 we establish several auxiliary statements.

73

Section 5.3 Nodec Topological Groups

Lemma 5.9. Let Y be a topological space, let X be a nowhere dense subset of Y , and let F be a ﬁnite set of continuous open mappings f W Y ! Y . Then for every nonempty open set U Y , there is a nonempty open set V U such that f .V / \ X D ; for all f 2 F . Proof. Let f 2 F . Since f is open and X is nowhere dense, there is a nonempty open Wf f .U / such that Wf \ X D ;. Then, since f is continuous, there is a nonempty open Vf U such that f .Vf / Wf , and so f .Vf / \ X D ;. Now let F D ¹f1 ; : : : ; fn º. Construct inductively a decreasing sequence .Vi /niD1 of nonempty open subsets of U such that fi .Vi / \ X D ; for each i D 1; : : : ; n. Then the set V D Vn is as required. Lemma 5.10. Let G be a nondiscrete metrizable Abelian topological group of prime period and let X be a nowhere dense subset of G such that 0 … X . Then there is a nondiscrete subgroup H of G such that H \ X D ;. Proof. Let ¹Un W n 2 Nº be a neighborhood base at 0 2 G. It sufﬁces to construct a sequence .an /1 nD1 in G such that 0 ¤ an 2 Un and ha1 ; : : : ; an i \ X D ; for every n < !. Then the group H D han W n 2 Ni D

1 [

ha1 ; : : : ; an i

nD1

would be as required. Without loss of generality one may suppose that X [ ¹0º is closed. Let p be the period of G. Since p is prime, the mapping x 7! mx is a homeomorphism for each m D 1; : : : ; p 1. Then by Lemma 5.9, there is a1 2 U1 n¹0º such that ha1 i\X D ;. Now ﬁx n < ! and suppose that we have constructed a sequence .ai /niD1 in G such that ha1 ; : : : ; an i \ X D ;. Then there is a neighborhood V of 0 such that .ha1 ; : : : ; an i n ¹0º C V / \ X D ;: Choose a neighborhood W of 0 such that W C W… V: „ C ƒ‚ p1

By Lemma 5.9, there is anC1 2 .W \ UnC1 / n ¹0º such that hanC1 i \ X D ;. Since ha1 ; : : : ; anC1 i n ha1 ; : : : ; an i D .ha1 ; : : : ; an i n ¹0º/ C hanC1 i and hanC1 i V , it follows that ha1 ; : : : ; anC1 i \ X D ;. Lemma 5.11. Let .G; T / be an Abelian topological group with no open subgroup of ﬁnite period. Then T can be reﬁned to a nondiscrete group topology T 0 of the same character that T and such that for every m 2 N, the mapping x 7! mx is open in T 0 .

74

Chapter 5 Topological Groups with Extremal Properties

Proof. Deﬁne the group topology T 0 on G by taking as a neighborhood base at 0 the subsets of the form mU , where U runs over a neighborhood base at 0 in T and m 2 N. Lemma 5.12. Let .G; T / be a nondiscrete metrizable Abelian topological group such that for every m 2 N, the mapping x 7! mx is open, and let X be a nowhere dense subset of .G; T / such that 0 … X . Then the topology T can be reﬁned to a nondiscrete metrizable group topology T 0 such that 0 … clT 0 X . Proof. Let ¹Un W n 2 Nº be a neighborhood base at 0 in T . We shall construct a sequence .an /1 nD1 in G such that 0 ¤ an 2 Un and n X

Bin \ X D ;

iD1

for all n 2 N, where Bin D ¹0; ˙ai ; : : : ; ˙an º. Without loss of generality one may suppose that X [ ¹0º is closed. Clearly, there is a1 2 U1 n ¹0º such that ˙a1 … X . Fix n 2 N and suppose that we have constructed a sequence .ai /niD1 such that n X

Bin \ X D ;:

iD1

Then there is a neighborhood V of 0 such that n X

Bin n ¹0º C V \ X D ;:

iD1

Choose a neighborhood W of 0 such that W C W… V: „ C ƒ‚ nC1

By Lemma 5.9, there is anC1 2 W \ UnC1 n ¹0º such that ¹˙anC1 ; : : : ; ˙.n C 1/anC1 º \ X D ;: Then

nC1 X

BinC1 \ X D ;:

iD1

Now let T1 D T

..an /1 nD1 /

and let U D

1 X n [ nD1 iD1

Bin

Section 5.3 Nodec Topological Groups

75

By Theorem 1.19, U is a neighborhood of 0 in T1 , and by the construction, U \ X D ;. Since an 2 Un , .an /1 nD1 converges to 0 in T , and consequently, T T1 . By Lemma 1.31, T1 can be weakened to a metrizable group topology T 0 in which U remains a neighborhood of 0. Now we are ready to prove Theorem 5.8. Proof of Theorem 5.8. Let .G; T / be a nondiscrete Abelian topological group of character < c. Without loss of generality one may assume that G is countable. By Theorem 1.28, one may assume also that T is metrizable. Enumerate the subsets of G n ¹0º into a sequence .X /

76

Chapter 5 Topological Groups with Extremal Properties

Fix ˛ < c and suppose that for every < ˛, the topology T has already been deﬁned and the mappings x 7! mx, where m 2 N, are open in T . If ˛ is a limit ordinal, deﬁne T˛ by (ii). Then the mappings x 7! mx, where m 2 N, are open in T˛ as well. Let ˛ D ı C 1 for some ı < c. By Theorem 1.28 and Lemma 5.17, there is a nondiscrete metrizable group topology T˛0 on G such that Tı T˛0 and the mappings x 7! mx, where m 2 N, are open in T˛0 . Let X.˛/ be the ﬁrst set in the sequence .X /

from Œı; c/, X is a nowhere dense set in T with 0 2 clT X , then X is the ﬁrst such set and so one canStake C 1 as ˛. /. Now let T 0 D ˛

5.4

P-point Theorems

In this section we prove two theorems. The ﬁrst of them is concerned with extremally disconnected topological groups containing countable nonclosed discrete subsets. Lemma 5.13. Let X be an extremally disconnected space, let D be a strongly discrete subset of X , and let x 2 cl DnD. Then there is exactly one ultraﬁlter on X containing D and converging to x. Proof. Assume on the contrary that there are two different ultraﬁlters on G containing D and converging to x. Then there are disjoint subsets A and B of D with x 2 .cl A/\.cl B/. For every y 2 D, there is a neighborhood S Uy of y such that the S subsets Uy , where y 2 D, are pairwise disjoint. Let UA D y2A Uy and UB D y2B Uy . Then UA and UB are disjoint open subsets of X with x 2 .cl UA / \ .cl UB /, which is a contradiction. Theorem 5.14. Let G be an extremally disconnected topological group and let p be a nonprincipal converging ultraﬁlter on G. Suppose that p contains a countable

77

Section 5.4 P -point Theorems

discrete subset of G. Then there is a mapping f W G ! ! such that f .p/ is a P -point. Proof. Without loss of generality one may assume that G is Boolean and p converges to 0. Let D be a countable discrete subset of G such that D 2 p. Then by Lemma 3.34, D is strongly discrete. Enumerate D without repetitions as ¹xn W n < !º. Construct a decreasing sequence .Un /n

Then P and Q are disjoint open subsets of G, F Q, and 0 2 cl P . It follows that 0 … cl Q and so 0 … cl F . Choose neighborhoods V and W of 0 such that V \ F D ; and W C W V . Let E D D \ W and A D f .E/. We claim that for every m < !, A \ Am is ﬁnite. Indeed, assume the contrary. Pick any xn 2 E \ Dm . Then there exists xk 2 E \ Dm \ Un . But then, on the one hand, xn C xk 2 E C E W C W V and, on the other hand, xn C xk 2 xn C .Dm \ Un / F G n V; a contradiction.

78

Chapter 5 Topological Groups with Extremal Properties

It follows from Theorem 5.14 and Theorem 2.38 that Corollary 5.15. It is consistent with ZFC that there is no extremally disconnected topological group containing a countable discrete nonclosed subset. Our second theorem deals with countable topological Boolean groups containing no nonclosed discrete subset, and even more generally, containing no subset with exactly one accumulation point. A nonempty subset of a topological space is called a P-set if the intersection of any countable family of its neighborhoods is again its neighborhood. Note that every isolated point of a P -set is a P -point. In particular, every point of a ﬁnite P -set is a P -point. Deﬁnition 5.16. We say that a set of nonprincipal ultraﬁlters on ! is a P -set if it is a P -set in ! . Lemma 5.17. Let F be a ﬁlter on ! with are equivalent:

T

F D ;. Then the following statements

(1) F is a P-set, (2) for every decreasing sequence .An /n

and

.x/ D max supp.x/:

79

Section 5.4 P -point Theorems

L Theorem 5.19. Let T be a nondiscrete Q group topology on ! Z2 ﬁner than that induced by the product topology on ! Z2 and let F denote the neighborhood ﬁlter L of 0 in T . Suppose that . ! Z2 ; T / contains no subset with exactly one accumulation point. Then .F / is a P -set. In particular, if .F / is ﬁnite, then each of its points is a P -point. Proof. To show that .F / is a P -set, we use characterization (3) from Lemma 5.17. Let f W ! ! !. Denote S D F n ¹0º and for each p 2 S, pick Ap 2 p such that either f ..x// .x/ for all x 2 Ap or f ..x// < .x/ for all x 2 Ap . Consider two cases. Case 1: there is p 2 S such that f ..x// < .x/ for all x 2 Ap . Pick an accumulation point a ¤ 0 of Ap . Then there is q 2 S and Q 2 q such that a C Q Ap . Choose Q in addition so that for every x 2 Q, one has .a/ < .x/. Then for every x 2 Q, we have that f ..x// D f ..a C x// < .a C x/ D .a/: Hence f ..Q// is ﬁnite. Case 2: for every p 2 S, f ..x// .x/ for all x 2 Ap . Deﬁne the neighborhood V of 0 in T by [ V D Ap [ ¹0º p2S

and choose a neighborhood W of 0 such that W C W V . We claim that for every n < !, f 1 .n/ \ .W / is ﬁnite. Indeed, assume the contrary. Then there exist m < ! and a sequence .xn /n m. Hence f ..x // > m, which is a contradiction. We say that a topological group .G; T / is maximally nondiscrete if T is maximal among nondiscrete group topologies on G. By Zorn’s Lemma, every nondiscrete group topology can be reﬁned to a maximally nondiscrete group topology. Clearly, every maximal topological group is maximally nondiscrete. Lemma 5.20. Let .G; T0 / be a totally bounded topological group and let T1 be a maximally nondiscrete group topology on G. Then T0 T1 . Proof. Pick any nonprincipal ultraﬁlter U on G converging to 1 in T1 . Considering .G; T0 / as a subgroup of a compact group shows that the ﬁlter UU1 (with a base of subsets AA1 , where A 2 U) converges to 1 in T0 . Clearly UU1 converges to 1 in T1 as well. Since T1 is maximally nondiscrete, it follows that T1 D T .UU1 /. Hence T0 T1 .

80

Chapter 5 Topological Groups with Extremal Properties

Now as a consequence we obtain from from Theorem 5.19 that Corollary 5.21. The existence of a maximal topological group implies the existence of a P -point in ! . Proof. Let .G;L T / be a maximal topological group. By Theorem 5.7, one may suppose that G D ! Z2 . Let F be the nonprincipal ultraﬁlter on G converging Q to 0 in T and let T0 be the topology on G induced by the product topology on ! Z2 . By Lemma 5.20, T0 T . Hence by Theorem 5.19, .F / is a P -point. Combining Corollary 5.21 and Theorem 2.38 gives us that Corollary 5.22. It is consistent with ZFC that there is no maximal topological group. As another consequence we obtain from Theorem 5.19 the following result. Corollary 5.23. Let p be a nonprincipal ultraﬁlter on ! not being a P-point and let L T p be the group topology on G D ! Z2 with a neighborhood base at 0 consisting of subgroups HA D ¹x 2 G W supp.x/ Aº where A 2 p. Then for every nondiscrete group topology T on G ﬁner than T p , .G; T / has a discrete subset with exactly one accumulation point. Proof. Let F be the neighborhood ﬁlter of 0 in T . It is clear that .F / D p. Now assume on the contrary that .G; T / has a subset with exactly one accumulation point. Then by Theorem 5.19, .F / D p is a P -point, which is a contradiction. Lemma 5.24. There exists a nonprincipal ultraﬁlter on ! not being a P -point. Proof. Let ¹An W n < !º be a partition of ! into inﬁnite subsets. Deﬁne the ﬁlter F on ! by taking as a base the subsets of the form [

.Ai n Fi /

ni

where n < ! and Fi is a ﬁnite subset of Ai for each i . It then follows that every ultraﬁlter on ! containing F is not a P -point. Combining Corollary 5.23 and Lemma 5.24 gives us that Corollary 5.25. There exists a maximally nondiscrete group topology T on G D L Z such that .G; T / has a discrete subset with exactly one accumulation point. 2 !

Section 5.4 P -point Theorems

81

References The question whether there exists in ZFC a nondiscrete extremally disconnected topological group was raised by A. Arhangel’ski˘ı [3]. The ﬁrst consistent example of such a group L was constructed by S. Sirota [71], the implication .1/ ) .4/ in Theorem 5.1 for ! Z2L . The equivalence .1/ , .2/ and the fact that T ŒF is extremally disconnected for ! Z2 are due to A. Louveau [45]. Theorem 5.1, except for statement (5), is from [103]. Theorem 5.5 is a result of V. Malykhin [46]. Theorem 5.4 and Theorem 5.7 are also due to him [46, 47]. Theorem 5.8 was proved in [88]. Theorems 5.14 and 5.19 are from [97]. Corollary 5.21 is due to I. Protasov [56]. Corollary 5.25 is a partial case of a result from [65].

Chapter 6

The Semigroup ˇS

In this chapter we extend the operation of a discrete semigroup S to ˇS making ˇS a right topological semigroup with S contained in its topological center. We show that every compact Hausdorff right topological semigroup has an idempotent and, as a consequence, a smallest two sided ideal which is a completely simple semigroup. The structure of a completely simple semigroup is given by the Rees-Suschkewitsch Theorem. As a combinatorial application of the semigroup ˇS we prove Hindman’s Theorem. We conclude by characterizing ultraﬁlters from the smallest ideal of ˇS.

6.1

Extending the Operation to ˇS

Theorem 6.1. The operation of a discrete groupoid S extends uniquely to ˇS so that (1) for each a 2 S , the left translation a W ˇS 3 x 7! ax 2 ˇS is continuous, and (2) for each q 2 ˇS, the right translation q W ˇS 3 x 7! xq 2 ˇS is continuous. If S is a semigroup, the extended operation is associative.

Proof. For each a 2 S, the left translation la W S 3 x 7! ax 2 S extends uniquely to a continuous mapping la W ˇS ! ˇS. For each a 2 S and q 2 ˇS, deﬁne a q 2 ˇS by a q D la .q/: Next, for each q 2 ˇS, the mapping rq W S 3 x 7! x q 2 ˇS extends uniquely to a continuous mapping rq W ˇS ! ˇS. For each p 2 S and q 2 ˇS, deﬁne p q 2 ˇS by p q D rq .p/: Under the extended operation a D la and q D rq , consequently, conditions (1) and (2) are satisﬁed. Furthermore, whenever the operation of S extends to ˇS so that conditions (1) and (2) are satisﬁed, one has a D la and q D rq , hence such an extension is unique.

83

Section 6.1 Extending the Operation to ˇS

Finally, suppose that S is a semigroup and let p; q; r 2 ˇS. Then .pq/r D . lim xq/r x!p

because q is continuous

D . lim lim xy/r

because x is continuous

D lim lim .xy/r

because r is continuous

x!p y!q

x!p y!q

D lim lim lim .xy/z because xy is continuous x!p y!q z!r

D lim lim lim xyz x!p y!q z!r

and p.qr/ D lim .x.qr// x!p

D lim .x lim yr/ x!p

y!q

because qr is continuous because r is continuous

D lim .x lim lim yz/ because y is continuous x!p

y!q z!r

D lim lim lim x.yz/ because x is continuous x!p y!q z!r

D lim lim lim xyz; x!p y!q z!r

so .pq/r D p.qr/. Deﬁnition 6.2. A semigroup T endowed with a topology is a right topological semigroup if for each p 2 T , the right translation p W T 3 x 7! xp 2 T is continuous. The topological center of a right topological semigroup T , denoted ƒ.T /, consists of all a 2 T such that the left translation a W T 3 x 7! ax 2 T is continuous. Theorem 6.1 tells us that the operation of a discrete semigroup S extends uniquely to ˇS so that ˇS is a right topological semigroup with S ƒ.ˇS /. Lemma 6.3. Let R and T be Hausdorff right topological semigroups, let S be a dense subsemigroup of R such that S ƒ.R/, and let ' W R ! T be a continuous mapping such that '.S/ ƒ.T /. If 'jS is a homomorphism, so is '.

84

Chapter 6 The Semigroup ˇS

Proof. First let x 2 S and q 2 R. Then '.xq/ D '. lim xy/ S3y!q

because x is continuous

D lim '.xy/

because ' is continuous

D lim '.x/'.y/

because 'jS is a homomorphism

y!q y!q

D '.x/ lim '.y/ because '.x/ is continuous y!q

D '.x/'.q/: Now let p; q 2 R. Then '.pq/ D '. lim xq/ S3x!p

D lim '.xq/ x!p

because q is continuous because ' is continuous

D lim '.x/'.q/ x!p

D . lim '.x//'.q/ because '.q/ is continuous x!p

D '.p/'.q/: Corollary 6.4. Let S be a discrete semigroup and let ' W S ! T be any homomorphism of S into a compact Hausdorff right topological semigroup T such that '.S/ ƒ.T /. Then the continuous extension ' W ˇS ! T of ' is a homomorphism. Deﬁnition 6.5. Given a semigroup S endowed with a topology, a semigroup compactiﬁcation of S is a pair .'; T / where T is a compact right topological semigroup and ' W S ! T is a continuous homomorphism such that '.S / is dense in T and '.S/ ƒ.T /. ˇ Corollary 6.4 tells us that the Stone–Cech compactiﬁcation ˇS of a discrete semigroup S is the largest semigroup compactiﬁcation of S. ˇ From now on, for any semigroup S , ˇS denotes the Stone–Cech compactiﬁcation of the discrete semigroup S . The next lemma describes the operation of ˇS in terms of ultraﬁlters. That is, given ultraﬁlters p; q 2 ˇS, one characterizes the subsets of S which are members of the ultraﬁlter pq. Recall that given a semigroup S, A S and s 2 S, s 1 A D ¹x 2 S W sx 2 Aº D 1 s .A/:

Section 6.1 Extending the Operation to ˇS

85

Lemma 6.6. Let S be a semigroup, A S, s 2 S and p; q 2 ˇS. Then (1) A 2 sq if and only if s 1 A 2 q, and (2) A 2 pq if and only if ¹x 2 S W x 1 A 2 qº 2 p. Proof. (1) Let A 2 sq. Then A is a neighborhood of sq. Since s is continuous, there is Q 2 q such that sQ A. It follows that sQ A and so s 1 A 2 q. Conversely, let s 1 A 2 q. Assume on the contrary that A … sq. Consequently, S n A 2 sq. Then by the already established necessity, s 1 .S n A/ 2 q. But .s 1 A/ \ .s 1 .S n A// D ;; a contradiction. (2) Let A 2 pq. Since q is continuous, there is P 2 p such that P q A. Then for every x 2 P , A 2 xq and so by (1), x 1 A 2 q. Hence, ¹x 2 S W x 1 A 2 qº 2 p. Conversely, let ¹x 2 S W x 1 A 2 qº 2 p. Assume on the contrary that A … pq. Consequently, S n A 2 pq. Then by the already established necessity, ¹x 2 S W x 1 .S n A/ 2 qº 2 p. But .x 1 A/ \ .x 1 .S n A// D ; for each x 2 S. It follows that ¹x 2 S W x 1 A 2 qº \ ¹x 2 S W x 1 .S n A/ 2 qº D ;; a contradiction. Corollary 6.7. Let S be a semigroup, a 2 S and p; q 2 ˇS. Then (1) the ultraﬁlter aq has a base consisting of subsets of the form aQ where Q 2 q, and S (2) the ultraﬁlter pq has a base consisting of subsets of the form x2P xQx where P 2 p and Qx 2 q. We conclude this section by showing that if S is a cancellative semigroup, then S is a two-sided ideal of ˇS and the translations of ˇS are injective on S. A nonempty subset I of a semigroup S is a left ideal (two-sided ideal or just ideal) if SI I (both SI I and IS I ). Lemma 6.8. Let S be a cancellative (left cancellative, right cancellative) semigroup. Then S is an ideal (left ideal, right ideal) of ˇS. Proof. Suppose that S is left cancellative. Let p 2 ˇS and q 2 S . To see that pq 2 S , let A 2 pq. Then there exist x 2 S and B 2 q such that xB A. It follows that A is inﬁnite. Now suppose that S is right cancellative. Let p 2 S and q 2 ˇS. Assume on the contrary that pq D a 2 S. Then there exist A 2 p and, for each x 2 A, Bx 2 q such that xBx D ¹aº. Pick distinct x; y 2 A and any z 2 Bx \ By . Then xz D yz, a contradiction.

86

Chapter 6 The Semigroup ˇS

Lemma 6.9. Let S be a cancellative semigroup, let a and b be distinct elements of S , and let p 2 ˇS. Then ap ¤ bp and pa ¤ pb. Proof. Deﬁne f W S ! S by putting f .ax/ D bx for every x 2 S and f .y/ D a2 for every y 2 S n aS. Then f has no ﬁxed points and f .ap/ D bp. Hence, ap ¤ bp by Corollary 2.17. The proof that pa ¤ pb is similar. As usual, for every semigroup S, we use S 1 to denote the semigroup with identity obtained from S by adjoining one if necessary. Lemma 6.10. If S is a cancellative semigroup, so is S 1 . Proof. We ﬁrst show that if e is an idempotent in S, then e D 1. To see this, let x 2 S . Multiplying the equality e D ee by x from the left gives us xe D xee. Then cancellating the latter equality by e from the right we obtain x D xe. Similarly x D ex. Now assume on the contrary that S 1 is not cancellative. Then some translation in 1 S is not injective. One may suppose that this is a left translation. It follows that there exist a; b 2 S such that ab D a. But then ab 2 D ab, and so b 2 D b. Hence b D 1 2 S , a contradiction. Combining Lemma 6.9 and Lemma 6.10, we obtain the following. Corollary 6.11. Let S be a cancellative semigroup, let a 2 S, and let p 2 ˇS. If ap D p, then a D 1.

6.2

Compact Right Topological Semigroups

Recall that an element p of a semigroup is an idempotent if pp D p. Theorem 6.12. Every compact Hausdorff right topological semigroup has an idempotent. Proof. Let S be a compact Hausdorff right topological semigroup. Consider the set P of all closed subsemigroups of S partially ordered by the inclusion. Since S 2 P , T P ¤ ;. For every chain C in P , C 2 P . Hence by Zorn’s Lemma, C has a minimal element A. Pick x 2 A. We shall show that xx D x. (It will follow that A D ¹xº, but we do not need this.) We start by showing that Ax D A. Let B D Ax. Clearly B is nonempty. Since B D x .A/, B is closed. Also BB AxAx AAAx Ax D B, so B is a subsemigroup. Thus B 2 P . But then B D A, since B D Ax AA A and A is minimal.

Section 6.2 Compact Right Topological Semigroups

87

Now let C D ¹y 2 A W yx D xº. Since x 2 A D Ax, C is nonempty. And since C D x1 .x/, C is closed. Given any y; z 2 C , yz 2 AA A and yzx D yx D x, so C is a subsemigroup. Thus C 2 P . But then C D A, since C A and A is minimal. Hence x 2 C and so xx D x as required. A left ideal L of a semigroup S is minimal if S has no left ideal strictly contained in L. Corollary 6.13. Let S be a compact Hausdorff right topological semigroup. Then S contains a minimal left ideal. Every minimal left ideal of S is closed and has an idempotent. Proof. If L is any left ideal of S and x 2 L, then S x D x .S / is a closed left ideal contained in L. It follows that every minimal left ideal of S is closed and, by Theorem 6.12, has an idempotent. Thus we need only to show that S contains a minimal left ideal. To this end, consider the set P of all closed left ideals of S , partially ordered by the inclusion. Applying Zorn’s Lemma gives us a minimal element L 2 P . Since L is minimal among closed left ideals and every left ideal contains a closed left ideal, it follows that L is a minimal left ideal. A semigroup can have many minimal left (right) ideals. However, it can have at most one minimal two-sided ideal. Indeed, if K is a minimal ideal, then for any ideal I , KI is an ideal and KI K \ I , so K I . Deﬁnition 6.14. For every semigroup S, let K.S / denote the smallest ideal of S, if exists. Lemma 6.15. Let S be a semigroup and assume that there is a minimal left ideal L of S. Then (1) for every a 2 S , La is a minimal left ideal of S , (2) every left ideal of S contains a minimal left ideal, (3) different minimal left ideals of S are disjoint and their union is K.S /. Proof. (1) If M is a left ideal of S contained in La, then L0 D ¹x 2 L W xa 2 M º is a left ideal of S contained in L, so L0 D L and consequently M D La. (2) Let N be a left ideal of S. Pick a 2 N . Then La N and by (1), La is a minimal left ideal. (3) If M and N are minimal left ideals and L D M \ N ¤ ;, then L is a left ideal, so M D L D N . S Now let K D LS D a2S La. Clearly K is an ideal. If I is any ideal of S, then IL I \ L L, so IL D I \ L D L and consequently L I . Hence, K D LS IS I .

88

Chapter 6 The Semigroup ˇS

A semigroup is simple (left simple) if it has no proper ideal (left ideal). It is easy to see that a smallest ideal (a minimal left ideal) is a simple (left simple) semigroup. A semigroup S is completely simple if it is simple and there is a minimal left ideal of S which has an idempotent. Corollary 6.16. Every compact Hausdorff right topological semigroup has a smallest ideal which is a completely simple semigroup. Deﬁnition 6.17. Given a semigroup S, let E.S/ denote the set of idempotents of S. Lemma 6.18. Let S be a left simple semigroup with E.S / ¤ ; and let e 2 E.S /. Then (1) e is a right identity of S, and (2) eS is a group. Proof. (1) Let s 2 S. Since Se D S, there is t 2 S such that t e D s. Then se D t ee D t e D s. (2) Clearly, eS is a semigroup and e is a left identity of eS . By (1), e is also a right identity, so e is identity of eS. Now let x 2 eS . We have to ﬁnd y 2 eS such that yx D xy D e. Since Sx D S, there is s 2 S such that sx D e. Let y D es. Then yx D esx D ee D e. To see that xy D e, note that there is t 2 S such that ty D e, since Sy D S. It then follows that t D t e D tyx D ex D x. A semigroup satisfying the identity xy D x (xy D y) is called a left (right) zero semigroup. By Lemma 6.18 (1), if S is a left simple semigroup and E.S / ¤ ;, then E.S/ is a left zero semigroup. Proposition 6.19. Let S be a left simple semigroup with E.S / ¤ ;. Let e 2 E.S / and deﬁne f W E.S/ eS ! S by f .x; a/ D xa. Then f is an isomorphism. Proof. To see that f is a homomorphism, let .u1 ; a1 /; .u2 ; a2 / 2 E.S / eS. Then f ..u1 ; a1 /.u2 ; a2 // D f .u1 u2 ; a1 a2 / D u1 u2 a1 a2 D u1 a1 a2 D u1 a1 u2 a2 D f .u1 ; a1 /f .u2 ; a2 /: To see that f is surjective, let x 2 S. Since S is left simple, there is y 2 S such that yx D e. Then xyxy D xey D xy, so xy 2 E.S /. It follows that f .xy; ex/ D xyex D xyx D xe D x: To see that f is injective, let .u; a/ 2 E.S/ eS and let x D f .u; a/. We claim that a D ex and u D xy where y is the inverse of ex 2 eS. Indeed, a D ea D eua D ex and u D ue D uexy D uxy D xy, since ux D uua D ua D x.

Section 6.2 Compact Right Topological Semigroups

89

Lemma 6.20. Let S be a semigroup and let e 2 E.S /. If Se is a minimal left ideal of S , then eS is a minimal right ideal. Proof. Let a 2 S . We have to show that e 2 eaS . Since ae 2 Se and Se is a minimal left ideal, there is b 2 S such that aeb D u 2 E.Se/. Then e D eu D eaeb 2 eaS . Lemma 6.21. Let S be a semigroup and let R and L be minimal right and left ideals of S . Then R \ L D RL is a maximal subgroup of S . Proof. Let G D RL. Then for every a 2 G, aG D aRL D RL D G and Ga D RLa D RL D G, so G is a group. Clearly G R\L. To see the converse inclusion, let b 2 R \L and let e denote the identity of G. Since b 2 L and e 2 E.L/, it follows that b D be, and consequently b 2 RL. Finally, if G 0 is any subgroup of S with the identity e, then G 0 D eG 0 R and G 0 D G 0 e L, so G 0 R \ L D G. Deﬁnition 6.22. Let G be a group, let I; ƒ be nonempty sets, and let P D .p i / be a ƒ I matrix with entries in G. The Rees matrix semigroup over the group G with ƒ I sandwich matrix P , denoted M.GI I; ƒI P /, is the set I G ƒ with the operation deﬁned by .i; a; /.j; b; / D .i; ap j b; /: It is straightforward to check that M.GI I; ƒI P / is a completely simple semigroup, the minimal right ideals being the subsets ¹i º G ƒ, where i 2 I , the minimal left ideals the subsets I G ¹º, where 2 ƒ, and the maximal groups the subsets ¹i ºG ¹º, where i 2 I and 2 ƒ. The next theorem tells us that every completely simple semigroup is isomorphic to some Rees matrix semigroup. Theorem 6.23. Let S be a completely simple semigroup. Pick e 2 E.S / such that Se is a minimal left ideal. Let I D E.Se/, ƒ D E.eS /, G D eSe, and p i D i , and deﬁne f W M.GI I; ƒI P / ! S by f .i; a; / D i a. Then f is an isomorphism. Proof. Let .i; a; / and .j; b; / be arbitrary elements of M.GI I; ƒI P /. Then f ..i; a; /.j; b; // D f .i; ajb; / D i ajb D f .i; a; /f .j; b; /; so f is a homomorphism. Since E.Se/eSeE.eS/ D E.Se/eSeeSeE.eS / D SeeS D SeS; f is surjective. To see that f is injective, let i a D jb. Then a D eae D ei ae D ejbe D ebe D b; i D i e D i aa1 D i aea1 D jaea1 D jaa1 D je D j; D e D a1 a D a1 ei a D a1 eja D a1 a D e D :

90

Chapter 6 The Semigroup ˇS

If S is a completely simple semigroup and M.GI I; ƒI P / is a Rees matrix semigroup isomorphic to S , then every maximal subgroup of S is isomorphic to G. We call G the structure group of S. The next proposition and lemma are useful in identifying the smallest ideal of subsemigroups and homomorphic images. Proposition 6.24. Let S be a semigroup and let T be a subsemigroup S. Suppose that both S and T have a smallest ideal which is a completely simple semigroup. If K.S / \ T ¤ ;, then K.T / D K.S/ \ T . Proof. Clearly, K.S/ \ T is an ideal of T , so K.T / K.S / \ T . For the reverse inclusion, let x 2 K.S/ \ T . Then Sx is a minimal left ideal of S, x 2 S x, and T x is a left ideal of T . Since K.T / is completely simple, there is an idempotent e 2 K.T / \ T x. Then Se D Sx, so x 2 Se. It follows that x D xe 2 T e K.T /. Lemma 6.25. Let S and T be semigroups and let f W S ! T be a surjective homomorphism. If S has a smallest ideal, so does T and K.T / D f .K.S //. Proof. By surjectivity, f .K.S// is a two-sided ideal of T . If K 0 is any ideal of T , then f 1 .K 0 / is an ideal of S, so contains K.S / by minimality. Thus f .K.S // K 0 , whence f .K.S// is the smallest ideal of T . We conclude this section with discussing standard preorderings on the idempotents of a semigroup. Deﬁnition 6.26. Let S be a semigroup with E.S / ¤ ;. Deﬁne the relations L , R , and on E.S / by (a) e L f if and only if ef D e, (b) e R f if and only if f e D e, and (a) e f if and only if e L f and e R f , that is, ef D f e D e. Note that e L f if and only if Se Sf . Indeed, if ef D e, then Se D Sef Sf . Conversely, if Se Sf , then e D ee D sf for some s 2 S, and consequently ef D sff D sf D e. Similarly, e R f if and only if eS f S . It is easy to see that the relations L , R , and are reﬂexive and transitive, and , in addition, is antisymmetric. Thus, L and R are preorderings on E.S / and is an ordering. Given a preordering on a set E, an element e 2 E is minimal (maximal) if for every f 2 E, f e implies e f (e f implies f e). Lemma 6.27. Let S be a semigroup and let e 2 E.S /. Then the following statements are equivalent:

Section 6.3 Hindman’s Theorem

91

(a) e is minimal with respect to L , (b) e is minimal with respect to R , and (c) e is minimal with respect to . Proof. It sufﬁces to show the equivalence of (a) and (c). (a) ) (c) Assume that e is minimal with respect to L and let f e. Since e is L -minimal, it follows from f L e that e L f , that is, e D ef . But ef D f , since f R e. Hence, e D f . (c) ) (a) Assume that e is minimal with respect to and let f L e. Denote g D ef . Then gg D ef ef D eff D ef D g 2 E.S /. Also, ge D ef e D ef D g and eg D eef D ef D g, so g e. Consequently g D e. Thus ef D e and so e L f . Hence, e is L -minimal. We say that an idempotent e of a semigroup S is minimal if e is minimal with respect to any of the preorderings L , R , or on E.S /. Thus, to say that a semigroup S contains a minimal left ideal which has an idempotent is the same as saying that S has a minimal idempotent. So a completely simple semigroup may be deﬁned as a simple one having a minimal idempotent. It is clear also that if a semigroup S has a minimal idempotent, then the minimal idempotents of S are precisely the idempotents of K.S/. An idempotent e of a semigroup S is right (left) maximal if e is maximal in E.S / with respect to R (L ). Theorem 6.28. Let S be a compact Hausdorff right topological semigroup. Then for every idempotent e 2 S, there is a right maximal idempotent f 2 S with e R f . Proof. Let P D ¹x 2 E.S/ W e R xº. Clearly P ¤ ;. It sufﬁces to show that any chain C in P has an upper bound. Then by Zorn’s Lemma, C would have a maximal element. For each p 2 C , let Tp D ¹x 2 S W xp D pº. If x; y 2 Tp , then xyp D xp D p, so xy 2 Tp . Also Tp D p1 .p/. Thus, Tp is a closed subsemigroup of S. Furthermore, if p R q and x 2 Tq , then xp T D xqp D qp D p, so x 2 Tp and, consequently, Tp Tq . It follows that T D p2C Tp is a closed T subsemigroup of S. Now by Theorem 6.12, there is an idempotent q 2 T . Since p2C Tp D ¹x 2 S W xp D p for all p 2 C º, q is an upper bound of C .

6.3

Hindman’s Theorem

In this section we prove Hindman’s Theorem using the semigroup ˇS. Lemma 6.29. Let S be a semigroup and let p 2 ˇS. Then p is an idempotent if and only if for every A 2 p, one has ¹x 2 S W x 1 A 2 pº 2 p. Proof. It is immediate from Lemma 6.6.

92

Chapter 6 The Semigroup ˇS

Proposition 6.30. Let S be a semigroup and let p be an idempotent in S . Then there is a left invariant topology T on S 1 in which for each s 2 S 1 , ¹sA [ ¹sº W A 2 pº is a neighborhood base at s. Furthermore, if S is cancellative, then T is Hausdorff. Proof. Let N be the ﬁlter on S 1 with a base consisting of subsets A [ ¹1º where A 2 p. Then for every U 2 N , 1 2 U and and by Lemma 6.29, ¹x 2 S W x 1 U 2 N º 2 N . Hence by Theorem 4.3, there is a left invariant topology T on S 1 in which for each s 2 S 1 , sN is a neighborhood base at s. Now suppose that S is cancellative. Then (i) ap is a nonprincipal ultraﬁlter for every a 2 S 1 , and (ii) ap ¤ bp for all distinct a; b 2 S 1 . It follows that T is Hausdorff. Lemma 6.31. Let S be a left topological semigroup with identity and let U be an open subset of S such that 1 2 cl U . Then there is a sequence .xn /1 nD1 in S such that FP..xn /1 / U . nD1 Proof. Pick x1 2 U . Fix n 2 N and suppose we have chosen a sequence .xi /niD1 such that FP..xi /niD1 / U . Let F D FP..xi /niD1 /. Then there is a neighborhood V of 1 such that F V U . Pick xnC1 2 U \ V . Theorem 6.32. Let S be a semigroup, let p be an idempotent in ˇS, and let A 2 p. 1 Then there is a sequence .xn /1 nD1 in S such that FP..xn /nD1 / A. Proof. The statement is obvious if p 2 S . Let p 2 S . By Proposition 6.30, there is a left invariant topology T on S 1 in which for each s 2 S 1 , ¹sA [ ¹sº W A 2 pº is a neighborhood base at s. Let U D intT A. Note that U D A \ ¹x 2 S W x 1 A 2 pº. We have that U is an open subset of .S 1 ; T /, 1 2 clT U , and U A. Then apply Lemma 6.31. S Corollary 6.33. Let S be a semigroup, let r 2 N, and let S D riD1 Ai . Then there 1 exist i 2 ¹1; 2; : : : ; rº and a sequence .xn /1 nD1 in S such that FS..xn /nD1 / Ai . Proof. Pick by Theorem 6.12 an idempotent p 2 ˇS and pick i 2 ¹1; 2; : : : ; rº such that Ai 2 p. Then apply Theorem 6.32. As a special case of Corollary 6.33 we obtain Hindman’s Theorem. S Corollary 6.34 (Hindman’s Theorem). Let r 2 N and let N D riD1 Ai . Then there 1 exist i 2 ¹1; 2; : : : ; rº and a sequence .xn /1 nD1 in N such that FS..xn /nD1 / Ai . Corollary 6.34 is strong enough to derive all other versions of Hindman’s Theorem including Theorem 1.41.

93

Section 6.3 Hindman’s Theorem

Deﬁnition 6.35. For every x 2 N, deﬁne supp2 .x/ 2 Pf .!/ by X xD 2i i2supp2 .x/

and let 2 .x/ D min supp2 .x/

and

2 .x/ D max supp2 .x/:

In other words, supp2 .x/ is the set of indexes of nonzero digits in the binary expansion of x, and 2 .x/ and 2 .x/ are the indexes of the ﬁrst and the last nonzero digit, respectively. Equivalently, 2 .x/ D max¹i < ! W 2i jxº and

2 .x/ D max¹i < ! W 2i xº:

2 .x/ can be deﬁned also as n < ! such that x 2n

.mod 2nC1 /:

Note that we have already used the function 2 .x/ in the proof of Theorem 5.7. Lemma 6.36. Let .xn /1 nD1 be a sequence in N. Then there is a sum subsystem 1 .yn /1 of .x / such that 2 .yn / < 2 .ynC1 / for every n 2 N. n nD1 nD1 Proof. It sufﬁces to show that for every m; k < !, there is F 2 Pf .N/ such that P min F > m and 2k j n2F xn . We proceed by induction on k. If k D 0, put F D ¹m C 1º. Now assume the statement holds for some k. Then there are F1 ; F2 2 Pf .N/ such that m < min F1 , P P max F1 < min F2 , and 2k j n2Fi xn for each i 2 ¹1; 2º. If 2kC1 j n2Fi xn for P P some j 2 ¹1; 2º, put F D Fj . Otherwise 2kC1 j n2F1 xn C n2F2 xn , so put F D F1 [ F2 . Sr Corollary 6.37. Let r 2 N and let Pf .N/ D iD1 Ai . Then there exist i 2 ¹1; 2; : : : ; rº and a sequence .Gn /1 in P .N/ such that FU..Gn /1 f nD1 nD1 / Ai and for each n 2 N, max Gn < min GnC1 . Proof. Consider the bijection N 3 x 7! supp2 .x/ 2 Pf .!/ and apply Corollary 6.34 and Lemma 6.36. Corollary 6.37 gives us, in turn, the following. 1 Corollary 6.38. Sr Let S be a semigroup, let .xn /nD1 be a sequence in S, let r 2 N, and let S D iD1 Ai . Then there exist i 2 ¹1; 2; : : : ; rº and a product subsystem 1 1 .yn /1 nD1 of .xn /nD1 such that FP..yn /nD1 / Ai .

The next proposition tells us that the relationship between idempotents and ﬁnite products is even more intimate than indicated by Theorem 6.32.

94

Chapter 6 The Semigroup ˇS

Proposition 6.39. Let S be a semigroup and let .xn /1 nD1 be a sequence in S. Then T1 1 / is a closed subsemigroup of ˇS. Consequently, there is an idemFP..x / n nDm mD1 potent p 2 ˇS such that FP..xn /1 nDm / 2 p for every m 2 N. Before proving Proposition 6.39 we establish the following simple general fact. Lemma 6.40. Let S be a semigroup and let F be a ﬁlter on S . Suppose that for every A 2 F , ¹x 2 S W x 1 A 2 F º 2 F . Then F is a closed subsemigroup of ˇS. Proof. Clearly, F is a closed subset of ˇS. To see that it is a subsemigroup, let p; q 2 F and let A 2 F . We have to show that A 2 pq. Let B D ¹x 2 S W x 1 A 2 F º, and for every x 2 B, let Cx D x 1 A. Then B 2 F p, Cx 2 F q, and S x2B xCx A. Hence, A 2 pq. 1 Proof of Proposition 6.39. Let m Q2 N and let x 2 FP..xn /nDm /. Pick F 2 Pf .N/ with min F m such that x D n2F xn . Let k D max F C 1. Then 1 x FP..xn /1 nDk / FP..xn /nDm /:

T 1 Consequently, by Lemma 6.40, 1 mD1 FP..xn /nDm / is a closed subsemigroup of ˇS. For the second part, apply Theorem 6.12.

6.4

Ultraﬁlters from K.ˇS /

In this section we characterize ultraﬁlters from K.ˇS /. Deﬁnition 6.41. Let S be a semigroup. (a) A subset A S is syndetic if there is a ﬁnite F S such that F 1 A D S. (b) Let F and G be ﬁlters on S . A subset A S is .F ; G /-syndetic if for every V 2 F , there is a ﬁnite F V such that F 1 A 2 G . If F D G , we say F -syndetic instead of .F ; G /-syndetic. Note that if F G and A is either F -syndetic or G -syndetic, then A is also .F ; G /-syndetic. Lemma 6.42. Let T be a closed subsemigroup of ˇS, let L be a minimal left ideal of T , let F and G be the ﬁlters on S such that F D T and G D L, and let A S . Then the following statements are equivalent: (1) A \ L ¤ ;, (2) A is G -syndetic, and (3) A is .F ; G /-syndetic.

Section 6.4 Ultraﬁlters from K.ˇS/

95

Proof. .1/ ) .2/ Pick p 2 A \ L. Then for every q 2 L, one has p 2 Lq. Now to show that A is G -syndetic, let V 2 G . For every q 2 L, there is r 2 L such that p D rq, consequently, there is x 2 V such that xq 2 A, and so q 2 x 1 A. Thus, the sets of the form x 1 A, where x 2 V , cover the compact L. Hence, there is a ﬁnite F V such that L F 1 A, and so F 1 A 2 G . .2/ ) .3/ is obvious. .3/ ) .1/ Pick q 2 L. For every V 2 F , there is a ﬁnite F V such that F 1 A 2 G , and so F 1 A 2 q. It follows that for every V 2 F , there is xV 2 V such that A 2 xV q. Pick r 2 T \ c`ˇS ¹xV W V 2 G º. Then A 2 rq and rq 2 L. Hence, A \ L ¤ ;. The next theorem characterizes ultraﬁlters from K.T /. Theorem 6.43. Let T be a closed subsemigroup of ˇS , let F be the ﬁlter on S such that F D T , and let p 2 T . Then p 2 K.T / if and only if for every A 2 p, ¹x 2 S W x 1 A 2 pº is F -syndetic. Proof. Suppose that p 2 K.T /. Let L D Tp and let G be the ﬁlter on S such that G D L. Now let A 2 p, let B D ¹x 2 S W x 1 A 2 pº, and let V 2 F . By Lemma 6.42, A is G -syndetic, so there is F V such that F 1 A 2 G . Since L D Tp, there is W 2 F such that Wp F 1 A. We claim that W F 1 B. Indeed, let y 2 W . Then there is x 2 F such that yp 2 x 1 A. It follows that .xy/1 A 2 p. Hence, xy 2 B, and then y 2 x 1 B. Conversely, suppose that p … K.T /. Pick q 2 K.T /. Then p … T qp, since T qp K.T /. It follows that there is A 2 p such that A \ T qp D ;. Now let B D ¹x 2 S W x 1 A 2 pº. We claim that B is not F -syndetic. To show this, pick a minimal left ideal L of T contained in Tp and let G be the ﬁlter on S such that G D L. Assume on the contrary that B is F -syndetic. Then B is also .F ; G /syndetic. Hence by Lemma 6.42, B \ L ¤ ;. Consequently, B 2 rq for some r 2 T , and so ¹x 2 S W x 1 A 2 pº 2 rq. But then A 2 rqp, which is a contradiction. As a consequence we obtain the following characterization of ultraﬁlters from K.ˇS /. Corollary 6.44. Let p 2 ˇS. Then p 2 K.ˇS / if and only if for every A 2 p, ¹x 2 S W x 1 A 2 pº is syndetic.

References The fact that the operation of a discrete semigroup S can be extended to ˇS was implicitly established by M. Day [15] using a multiplication of the second conjugate of a Banach algebra, in this case l1 .S /, ﬁrst introduced by R. Arens [2] for arbitrary

96

Chapter 6 The Semigroup ˇS

Banach algebras. P. Civin and B. Yood [8] explicitly stated that if S is a discrete group, then the above operation produced an operation on ˇS, viewed as a subspace of that second dual. R. Ellis [21] carried out the extension in ˇS viewed as a space of ultraﬁlters, again assuming that S is a group. Theorem 6.12 was proved by K. Numakura [52] for topological semigroups and by R. Ellis [20] in the general case. Theorem 6.23 is a special case of the ReesSuschkewitsch Theorem proved by A. Suschkewitsch [73] for ﬁnite semigroups and by D. Rees [63] in the general case. Theorem 6.28 is due to W. Ruppert [67]. For more information about compact right topological semigroups, including references, see [67]. An introduction to this topic can be found also in [39]. Corollary 6.34 is Hindman’s Theorem [35] known also as the Finite Sums Theorem. The proof based on the semigroup ˇN is due to F. Galvin and S. Glazer. For other combinatorial applications of ˇS see [37]. The exposition of Section 6.4 is based on the treatment in [70].

Chapter 7

Ultraﬁlter Semigroups

Given a T1 left topological group .G; T /, the set Ult.T / of all nonprincipal ultraﬁlters on G converging to 1 in T is a closed subsemigroup of ˇG called the ultraﬁlter semigroup of T . In this chapter we study the relationship between algebraic properties of Ult.T / and topological properties of .G; T /. Not every closed subsemigroup of G is the ultraﬁlter semigroup of a left invariant topology on G. However, every ﬁnite subsemigroup is. A special attention is paid to the question when a closed subsemigroup of G is the ultraﬁlter semigroup of a regular left invariant topology. We conclude by showing how to construct homomorphisms of Ult.T /.

7.1

The Semigroup Ult.T /

Lemma 7.1. Let .S; T / be a left topological semigroup with identity and let N be the neighborhood ﬁlter of 1. Then (1) N is a closed subsemigroup of ˇS, (2) for every open subset U of .S; T /, U N U , and (3) if T satisﬁes the T1 separation axiom, then N n ¹1º D ¹p 2 S W p converges to 1 in T º is a closed subsemigroup. Proof. (1) follows from Theorem 4.3 and Lemma 6.40. (2) Let p 2 U and q 2S N . Since U is open, for every x 2 U , there is Vx 2 N such that xVx U . Then x2U xVx U . Since U 2 p and Vx 2 q, it follows that U 2 pq, so pq 2 U . T (3) Obviously, N n ¹1º is a closed subset of ˇS. Since T is a T1 -topology, N D ¹1º. It follows that N n¹1º S , and so N n¹1º D ¹p 2 S W p converges to 1 in T º. To see that N n ¹1º is a subsemigroup, let p; q 2 N n ¹1º. We have to show that for every U 2 N , pq 2 U n ¹1º. Clearly, one may suppose that U is open. Then U n ¹1º is also open, because T is a T1 -topology. Since p 2 U n ¹1º, we obtain by (2) that pq 2 U n ¹1º D U n ¹1º. Deﬁnition 7.2. Let .S; T / be a T1 left topological semigroup with identity. Deﬁne Ult.T / ˇS by Ult.T / D ¹p 2 S W p converges to 1 in T º:

98

Chapter 7 Ultraﬁlter Semigroups

By Lemma 7.1, Ult.T / is a closed subsemigroup of ˇS (if nonempty). We refer to Ult.T / as to the ultraﬁlter semigroup of T (or .S; T /). Recall that a ﬁlter on a space is called open (closed) if it has a base consisting of open (closed) sets. Lemma 7.3. Let .G; T / be a T1 left topological group, let F be a ﬁlter on G, and let Q D Ult.T /. (1) If F is open, then F Q F . (2) If F is closed, then pQ \ F D ; for every p 2 ˇG n F . In the case where Q is ﬁnite, the converses of the statements (1)–(2) also hold. Proof. (1) For every open U 2 F , F Q U Q, and by Lemma 7.1, U Q U , so F Q U . It follows that F Q F . Conversely, suppose that F Q F and Q is ﬁnite. To show that F is open, let U 2 F . For every p 2 F and q 2 Q, pick Ap;q 2 p such that Ap;q U and Ap;q q U . Let [ \ Ap;q : V D p2F q2Q

Then V 2 F , V U , and V Q U . It follows that V int U , and so int U 2 F . To see this, for every x 2 V and q 2 Q, pick Bx;q 2 q such that xBx;q U , and let [ Wx D ¹xº [ Bx;q : q2Q

Then Wx is a neighborhood of 1 and xWx U . (2) Since F is closed, for every p 2 ˇS n F , there is an open U 2 p such that U \ Q D ;. By Lemma 7.1, U Q U . It follows that pQ \ Q D ;. Conversely, suppose that pQ \ F D ; for every p 2 ˇG n F and that Q is ﬁnite. To show that F is closed, assume the contrary. Then there is U 2 F such that for every V 2 F , .cl V / n U ¤ ;. Since Q is ﬁnite, it follows that there is q 2 Q and, for every V 2 F , xV 2 G n U such that V 2 xV q. Let p be an ultraﬁlter on G extending the family of subsets XV D ¹xW W V W 2 F º where V 2 F . Then p 2 G n U and V 2 pq for every V 2 F . Consequently, p 2 ˇG n F and pq 2 F , a contradiction. Deﬁnition 7.4. For every ﬁlter F on a space X , denote by int F (respectively cl F ) the largest open (closed) ﬁlter on X , possibly improper, containing (contained in) F . Corollary 7.5. Let .G; T / be a T1 left topological group, let F be a ﬁlter on .G; T /, and let Q D Ult.T /. Suppose that Q is ﬁnite. Then

Section 7.1 The Semigroup Ult.T /

99

(1) int F D ¹p 2 F W pQ F º, and (2) cl F D F [ ¹p 2 ˇG W pQ \ F ¤ ;º. Proof. (1) By Lemma 7.3, int F Q int F F , so int F ¹p 2 F W pQ F º. To see the converse inclusion, note that ¹p 2 F W pQ F º is closed. Let G denote the ﬁlter on G such that G D ¹p 2 F W pQ F º. For every p 2 G and q 2 Q, one has pqQ pQ F , so G Q G . Now, since Q is ﬁnite, applying Lemma 7.3 gives us that G is open. Hence G int F . (2) If pQ \ F ¤ ; for some p 2 ˇG, then pQ \ cl F ¤ ;, and so p 2 cl F by Lemma 7.3. Consequently, F [ ¹p 2 ˇG W pQ \ F ¤ ;º cl F . To see the converse inclusion, note that ¹p 2 ˇG W pQ \ F ¤ ;º is closed, since Q is ﬁnite. Let G denote the ﬁlter on G such that G D F [ ¹p 2 ˇG W pQ \ F ¤ ;º. We claim for every p 2 ˇG n G , pQ \ G ¤ ;. Indeed, otherwise there exists q 2 Q such that pqQ \ F ¤ ;. It then follows that pQ \ F ¤ ;, and so p 2 G , which is a contradiction. Now, since Q is ﬁnite, applying Lemma 7.3 gives us that G is closed. Hence cl F G . Recall that an ultraﬁlter p on a space is dense if for every A 2 p, int cl A ¤ ;. Lemma 7.6. Let .G; T / be a T1 left topological group. Then the set of all dense ultraﬁlters on .G; T / converging to 1 is a closed ideal of Ult.T /. Proof. Let F be the ﬁlter on .G; T / with a base consisting of subsets of the form U n Y where U is a neighborhood of 1 and Y is a nowhere dense subset of .G; T /. Then by Proposition 3.9, F is the set of all dense ultraﬁlters on .G; T / converging to 1. Since F is open, F is a right ideal of Ult.T /, by Lemma 7.3. To see that F is a left ideal, let p 2 Ult.T /, q 2 F , and A 2 pq. Then there is x 2 G and B 2 q such that xB A. Since q is dense, int cl B ¤ ;. But x.int cl B/ int.x.cl B// and int.x.cl B// int cl.xB/. It follows that int cl A ¤ ;. Hence pq 2 F . The next proposition shows how the ultraﬁlter semigroup of a left topological group reﬂects topological properties of the group itself. Proposition 7.7. Let .G; T / be a T1 left topological group and let Q D Ult.T /. (1) If Q has only one minimal right ideal, then T is extremally disconnected. (2) If T is irresolvable, then K.Q/ is a left zero semigroup. (3) If T is n-irresolvable, then a minimal right ideal of Q consists of < n elements. (4) If p 2 K.Q/, then p is dense. (5) If Q D K.Q/, then T is nodec. In the case where Q is ﬁnite, the converses of the statements (1)–(5) also hold.

100

Chapter 7 Ultraﬁlter Semigroups

Proof. (1) Suppose that T is not extremally disconnected. Then there are two disjoint open subsets U and V of .G; T / such that 1 2 .cl U / \ .cl V /. But then by Lemma 7.1, U \ Q and V \ Q are two disjoint right ideals of Q. Conversely, suppose that there are two disjoint right ideals R and J of Q and that Q is ﬁnite. Let F and G denote ﬁlters on G such that F D R and G D J . Then, since Q is ﬁnite, both F and G are open by Lemma 7.3, so there are disjoint open U 2 F and V 2 G . But then 1 2 .cl U / \ .cl V /. Hence T is not extremally disconnected. (2) is a special case of (3). (3) Suppose that T is n-irresolvable. Then by Theorem 3.31, there is an open ﬁlter F on .G; T / converging to 1 with jF j < n. Then by Corollary 7.3, F n ¹1º is a right ideal of Q. Hence, a minimal right ideal of Q consists of jF j elements. Conversely, let R be a minimal right ideal of Q and suppose that jRj < n and Q is ﬁnite. Let F denote the ﬁlter on G such that F D R. Then by Lemma 7.3, F is open. Hence by Theorem 3.31, T is n-irresolvable. (4) By Lemma 7.6, the set of all dense ultraﬁlters from Q is an ideal of Q. It follows that every ultraﬁlter from K.Q/ is dense. Conversely, suppose that p … K.Q/ and Q is ﬁnite. Then by Corollary 7.5, cl p D ¹pº [ ¹q 2 ˇG W p 2 qQº and int cl p D ¹q 2 cl p W qQ cl pº. Since p … K.Q/, p … qQ for every q 2 K.Q/. Consequently, cl p \ K.Q/ D ;. On the other hand, for every q 2 cl p, one has qQ \ K.Q/ ¤ ;. Indeed, this is certainly true if q 2 Q. Otherwise p 2 qQ. Then pQ qQ and therefore qQ \ K.Q/ ¤ ;. It follows that for every q 2 cl p, one has qQ ª cl p. Hence, int cl p D ; and so p is nowhere dense. (5) If Q D K.Q/, then by (4) every ultraﬁlter from Q is dense. It follows that T is nodec. If Q ¤ K.Q/ and Q is ﬁnite, then by (4) any ultraﬁlter from Q n K.Q/ is nowhere dense, and so T is not nodec. We now show that every ﬁnite semigroup in G is the ultraﬁlter semigroup of a left invariant topology. Proposition 7.8. Let S be a semigroup with identity, let Q be a ﬁnite semigroup in S , and let F be the ﬁlter on S such that F D Q. Then there is a left invariant topology T on S in which for each s 2 S, ¹sA [ ¹sº W A 2 F º is a neighborhood base at s. If S is left cancellative, then T is a T1 -topology and Ult.T / D Q. Proof. Let N be the ﬁlter on S such that N D Q [ ¹1º. We claim that T (i) N D ¹1º, and (ii) for every U 2 N , ¹x 2 S W x 1 U 2 N º 2 N . Statement (i) is obvious. To show (ii), let U 2 N . For S every Tp; q 2 Q, pick Ap;q 2 p such that Ap;q U and Ap;q q U . Put V D p2Q q2F Ap;q [ ¹1º. Then V 2 N , V U and VQ S U . Now let x 2 V . For every q 2 Q, pick Bx;q 2 q such that xBx;q U . Put Wx D q2Q Bx;q [ ¹1º. Then Wx 2 N and xWx U .

Section 7.2 Regularity

101

It follows from (i)–(ii) and Theorem 4.3 that there is a left invariant topology T on S in which for each a 2 S, aN is a neighborhood base at a. The next example shows that not every closed subsemigroup of G is the ultraﬁlter semigroup of a left invariant topology. Example 7.9. Let .G; T0 / be a regular left topological group and suppose that there is a one-to-one sequence .xn /n

7.2

Regularity

Deﬁnition 7.11. Let G be a group and let Q be a closed subsemigroup of ˇG. We say that Q is left saturated if Q G and for every p 2 ˇG n .Q [ ¹1º/, one has pQ \ Q D ;.

102

Chapter 7 Ultraﬁlter Semigroups

Lemma 7.12. Let .G; T / be a regular left topological group and let Q D Ult.T /. Then Q is left saturated. Proof. Let N be the neighborhood ﬁlter of 1 in T . Since T is regular, N is closed. Then by Lemma 7.3, for every p 2 ˇG n N , pQ \ N D ;. It follows that for every p 2 ˇG n .Q [ ¹1º/, pQ \ Q D ;. Hence, Q is left saturated. Theorem 7.13. Let G be a group and let Q be a closed subsemigroup of G . Suppose that Q is left saturated and has a ﬁnite left ideal. Then there is a regular left invariant topology T on G such that Ult.T / D Q. Proof. Let N be the ﬁlter on G such that N D Q [ ¹1º. We ﬁrst show that T (i) N D ¹1º, and (ii) for every U 2 N , ¹x 2 G W x 1 U 2 N º 2 N . Statement (i) is obvious. To see (ii), let L be a ﬁnite left ideal of Q and let C D G n U . Since Q is left saturated, C Q \TQ D ;. For every q 2 L, choose Wq 2 N such that C q \ Wq D ;, and put W D q2L Wq . Then W 2 N , as L is ﬁnite, and 1 L L. For every q 2 L, .C L/ \ W D ;. Next, since L is a left ideal of Q, QT choose Vq 2 N such that Vq q W , and put V D q2L Vq . Then V 2 N and V L W . We claim that for all x 2 V , x 1 U 2 N . Indeed, otherwise xp 2 C for some x 2 V and p 2 Q. Take any q 2 L. Then, on the one hand, xpq D xp q 2 C L ˇG n W ; and on the other hand, xpq D x pq 2 V L W ; which is a contradiction. It follows from (i)–(ii) and Theorem 4.3 that there is a left invariant T1 -topology T on G for which N is the neighborhood ﬁlter of 1, and so Ult.T / D Q. We now show that T is regular. Assume the contrary. Then there is a neighborhood U of 1 such that for every neighborhood V of 1, .cl V / n U ¤ ;. For every open neighborhood V of 1, choose xV 2 .cl V / n U and pV 2 Q such that V 2 xV pV . Pick any q 2 L. By Lemma 7.1, one has V 2 xV pV q, and pV q 2 L because L is a left ideal. Since L is ﬁnite, it follows that there exist q 2 L and p 2 G n U such that for every neighborhood V of 1, V 2 pq, and so pq 2 Q. We have obtained that Q is not left saturated, which is a contradiction. Deﬁnition 7.14. Given a group G and p 2 G , C.p/ D ¹x 2 G W xp D pº

and

C 1 .p/ D C.p/ [ ¹1º:

Note that C.p/ is a closed subset of G and p 2 C.p/ if and only if p is an idempotent.

103

Section 7.2 Regularity

Lemma 7.15. C.p/ is a closed left saturated subsemigroup of ˇG (if nonempty). Proof. To see that C.p/ is a subsemigroup, let x; y 2 C.p/. Then xy 2 G by Lemma 6.8 and xyp D xp D p, so xy 2 C.p/. To see that C.p/ is left saturated, let xq D r for some x 2 ˇG and q; r 2 C.p/. Then xqp D rp, and so xp D p. If x 2 G, this equality implies that x D 1 by Corollary 6.11. Hence x 2 C.p/ [ ¹1º. From Lemma 7.15 we obtain that Corollary 7.16. C.p/ D ¹x 2 ˇG n ¹1º W xp D pº and C 1 .p/ D ¹x 2 ˇG W xp D pº. Now we deduce from Theorem 7.13 and Lemma 7.15 the following result. Theorem 7.17. Let G be a group and let p 2 G be an idempotent. Then (1) there is a regular left invariant topology T on G such that Ult.T / D C.p/, (2) T is the largest regular left invariant topology on G in which p converges to 1, and (3) T is extremally disconnected. Proof. By Lemma 7.15, C.p/ is a closed left saturated subsemigroup of ˇG. Clearly, ¹pº is a left ideal of C.p/. Then applying by Theorem 7.13 gives us (1). To see (2), let T 0 be any regular left invariant topology on G in which p converges to 1, let Q D Ult.T 0 /, and let x 2 C.p/. We have that xp D p, p 2 Q and x 2 G . Hence by Lemma 7.12, x 2 Q. To see (3), notice that, since ¹pº is a minimal left ideal of C.p/, there is only one minimal right ideal of C.p/. Hence by Proposition 7.7, T is extremally disconnected. Since a regular extremally disconnected space is zero-dimensional, we obtain from Theorem 7.17 that for every idempotent p 2 G , there is a zero-dimensional Hausdorff left invariant topology T on G such that Ult.T / D C.p/. The next theorem tells us that this is true for every p 2 G . Theorem 7.18. For every group G and for every p 2 G , there is a zero-dimensional Hausdorff left invariant topology T on G such that Ult.T / D C.p/. The proof of Theorem 7.18 involves the following notion. Deﬁnition 7.19. Let S be a semigroup, A S, and p 2 ˇS. Then Ap0 D ¹x 2 S W A 2 xpº:

104

Chapter 7 Ultraﬁlter Semigroups

The next lemma contains some simple properties of this notion. Lemma 7.20. Let S be a semigroup, A; B S, and p; q 2 ˇS. Then (1) .A \ B/p0 D Ap0 \ Bp0 , (2) A 2 pq if and only if A0q 2 p, and (3) .Ap0 /0q D A0qp . Proof. (1) Let x 2 G. Then x 2 .A \ B/p0 , A \ B 2 xp , A 2 xp and B 2 xp , x 2 Ap0 and x 2 Bp0 , x 2 Ap0 \ Bp0 : (2) is Lemma 6.6 (ii). (3) Let x 2 G. Then x 2 .Ap0 /0q , Ap0 2 xq , A 2 xqp

by .2/

, x 2 A0qp : Proof of Theorem 7.18. Let B D ¹Ap0 W A 2 pº. Since 1 2 Ap0 for every A 2 p and Ap0 \ Bp0 D .A \ B/p0 , B is a ﬁlter base on G. Now we show that B possesses the following properties: (i) B D C 1 .p/, (ii) for every U 2 B and for every x 2 U , there is V 2 B such that xV U , and (iii) for every U 2 B and for every x 2 GnU , there is V 2 B such that xV \U D ;. To see (i), let q 2 ˇG. Then q 2 B , Ap0 2 q

for every A 2 p

, A 2 qp

for every A 2 p

, qp D p: To see (ii) and (iii), it sufﬁces to show that for every U 2 B and for every q 2 B, D U . To this end, pick A 2 p such that U D Ap0 . Then

Uq0

Uq0 D .Ap0 /0q D A0qp D Ap0 D U: It follows from (i) and (ii) that there is a left invariant T1 -topology T on G such that Ult.T / D C.p/, and by (iii), T is zero-dimensional.

Section 7.3 Homomorphisms

105

We conclude this section by characterizing subsemigroups of a ﬁnite left saturated subsemigroup of ˇG which determine locally regular left invariant topologies. A space is locally regular if every point has a neighborhood which is a regular subspace. Proposition 7.21. Let G be a group, let Q be a ﬁnite left saturated subsemigroup of G , let S be a subsemigroup of Q, and let T be a left invariant topology on G with Ult.T / D S. Then T is locally regular if and only if for every q 2 Q n S, qS \ S ¤ ; implies S q \ S D ;. Proof. Suppose that T is locally regular and let q 2 Q n S be such that qS \ S ¤ ;. We have to show that Sq \ S D ;. Choose a regular open neighborhood X of 1 2 G in T . It sufﬁces to show that Xq \ X D ;, as this implies Xq \ X D ; and then S q \ S D ;. Assume on the contrary that xq 2 X for some x 2 X . Since q … S [ ¹1º, xq does not converge to x, so there is a neighborhood U of x 2 X such that U … xq. Since X is regular, U can be chosen to be closed. We have that X n U 2 xq and X n U is open. Then by Lemma 7.1, xqS X n U . It follows that xqS \ xS D ;, and consequently, qS \ S D ;, a contradiction. Conversely, let F D ¹q 2 Q n S W qS \ S ¤ ;º and suppose that for each q 2 F , S q \ S D ;. Then for each Tq 2 F , there is a neighborhood Xq of 1 in T such that Xq q \ Xq D ;. Put X D q2F Xq . Since F is ﬁnite, X is a neighborhood of 1 in T , and for each q 2 F , one has Xq \ X D ;. We claim that X is regular. Assume the contrary. Then there is x 2 X and a neighborhood U of x such that for every neighborhood V of x, clX .V / n U ¤ ;. Since S is ﬁnite, it follows that there is p 2 S , and for every neighborhood V of x, there is yV 2 X n U such that V 2 yV p. Let r be an ultraﬁlter on G extending the family of subsets YV D ¹yW W W is a neighborhood of x contained in V º; where V runs over neighborhoods of x. Then r 2 X n U and rp 2 xS , so x 1 rp 2 S. Put q D x 1 r. We have that (a) qp 2 S and q ¤ 1, and (b) r D xq. Since Q is left saturated, it follows from (a) that q 2 F . It is clear that xq 2 Xq, and (b) gives us that xq 2 X. Hence Xq \ X ¤ ;, a contradiction.

7.3

Homomorphisms

Constructing homomorphisms of ultraﬁlter semigroups is based on the following lemma. Lemma 7.22. Let S be a semigroup, let F be a ﬁlter on S , and let X 2 F . Let T be a compact Hausdorff right topological semigroup and let f W X ! T . Assume that

106

Chapter 7 Ultraﬁlter Semigroups

(1) for every x 2 X , there is Ux 2 F such that f .xy/ D f .x/f .y/ for all y 2 Ux , and (2) f .X / ƒ.T /. Then for every p 2 X and q 2 F , one has f .pq/ D f .p/f .q/, where f W X ! T is the continuous extension of f . Proof. For every x 2 X , one has f .xq/ D f .

lim

Ux 3y!q

xy/

D lim f .xy/ y!q

D lim f .x/f .y/ y!q

by (1)

D f .x/ lim f .y/ by (2) y!q

D f .x/f .q/: Then f .pq/ D f . lim xq/ X3x!p

D lim f .xq/ x!p

D lim f .x/f .q/ x!p

D . lim f .x//f .q/ x!p

D f .p/f .q/: L Deﬁnition 7.23. Let be an inﬁnite cardinal and let H D Z2 . For every ˛ < , let H˛ D ¹x 2 H W x. / D 0 for each < ˛º, and let T0 denote the group topology on H with a neighborhood base at 0 consisting of subgroups H˛ , where ˛ < . Deﬁne the semigroup H ˇH by H D Ult.T0 /. If D !, we write H instead of H! . The next theorem tells us that the semigroup H admits a continuous homomorphism onto any compact Hausdorff right topological semigroup containing a dense subset of cardinality within the topological center. Theorem 7.24. Let be an inﬁnite cardinal and let T be a compact Hausdorff right topological semigroup. Assume that there is a dense subset A T such that jAj and A ƒ.T /. Then there is a continuous surjective homomorphism W H ! T .

107

Section 7.3 Homomorphisms

Proof. For every ˛ < , let e˛ denote the element of H with supp.e˛ / D ¹˛º. Then every x 2 H n ¹0º can be uniquely written in the form x D e˛1 C C e˛n where n 2 N and ˛1 < < ˛n < . Pick a surjection f0 W ¹e˛ W ˛ < º ! A such that for each a 2 A, one has jf01 .a/j D . Extend f0 to a mapping f W H ! T by f .x/ D f0 .e˛1 / f0 .e˛n / where x D e˛1 C C e˛n and ˛1 < < ˛n . (As f .0/ pick any element of T .) Deﬁne W H ! T by D f jH . Now let x 2 H n ¹0º and let t D max supp.x/ C 1. We claim that for every y 2 H t n ¹0º, one has f .x C y/ D f .x/f .y/. Indeed, let x D e˛1 C C e˛n where ˛1 < < ˛n and let y D eˇ1 C C eˇm where ˇ1 < < ˇm . Then x C y D e˛1 C C e˛n C eˇ1 C C eˇm and ˛1 < < ˛n < ˇ1 < < ˇm . Consequently, f .x C y/ D f .e˛1 / f .e˛n /f .eˇ1 / f .eˇm / D f .x/f .y/: It follows from this and Lemma 7.22 that is a homomorphism. To see that is surjective, it sufﬁces to show that A .H /, because A T is dense and is continuous. Let a 2 A. Pick p 2 U.H / such that f01 .a/ 2 p. Then p 2 H and f .p/ D a. Now we are going to show that for every cancellative semigroup S of cardinality , there is a zero-dimensional Hausdorff left invariant topology T on S 1 such that Ult.T / is topologically and algebraically isomorphic to H . Lemma 7.25. Let S be an inﬁnite cancellative semigroup with identity and let jSj D . Then there are two -sequences .x˛ /1˛< and .y˛ /˛< in S with y0 D 1 such that every element of S is uniquely representable in the form y˛0 x˛1 x˛n where n < ! and ˛0 < ˛1 < < ˛n < . Proof. Enumerate S as ¹s˛ W ˛ < º. Put y0 D 1. Fix 0 < < and suppose that we have constructed .x˛ /1˛< and .y˛ /˛< such that all products y˛0 x˛1 x˛n , where n < ! and ˛0 < ˛1 < < ˛n < , are different. Pick as y the ﬁrst element in the sequence .s˛ /˛< not belonging to the subset S D ¹y˛0 x˛1 x˛n W n < ! and ˛0 < ˛1 < < ˛n < º: S Then pick x 2 S n .S1 S /. (Here, S1 S D x2S x 1 S .) This can be done because jS1 S j jS j2 < . Then whenever n < ! and ˛0 < ˛1 < < ˛n D , one has y˛0 x˛1 x˛n … S . Also if y˛0 x˛1 x˛n and yˇ0 xˇ1 xˇm are different elements of S , the elements y˛0 x˛1 x˛n x and yˇ0 xˇ1 xˇm x are different as well.

108

Chapter 7 Ultraﬁlter Semigroups

Theorem 7.26. Let S be an inﬁnite cancellative semigroup with identity and let jS j D . Then there is a zero-dimensional Hausdorff left invariant topology T on S such that Ult.T / U.S/ and Ult.T / is topologically and algebraically isomorphic to H . Proof. Let .x˛ /1˛< and .y˛ /˛< be sequences guaranteed by Lemma 7.25. For every ˛ 2 Œ1; /, deﬁne B˛ S by B˛ D ¹y0 x˛1 x˛n W n < !; ˛ ˛1 < < ˛n < º: Deﬁne also W S ! by .y˛0 x˛1 x˛n / D ˛n where n < and ˛0 < ˛1 < : : : < ˛n < . It is easy to see that the subsets B˛ possess the following properties: T (i) .B˛ /1˛< is a decreasing sequence of subsets of S with 1˛< B˛ D ¹1º, (ii) for every ˛ 2 Œ1; / and x 2 B˛ , one has xB.x/C1 B˛ , and (iii) for every x 2 S, ˛ 2 Œ.x/ C 1; / and y 2 S n .xB˛ /, one has .yB.y/C1 / \ .xB˛ / D ;. It follows from (i)–(ii) and Corollary 4.4 that there is a left invariant T1 -topology T on S in which for each x 2 S , ¹xB˛ W .x/ C 1 ˛ < º is an open neighborhood base at x. Condition (iii) gives us that T is zero-dimensional. Since T is also a T1 -topology, it is Hausdorff. Obviously, Ult.T / U.S /. To see that Ult.T / is topologically and algebraically isomorphic to H , let X D B1 . For every ˛ < , let e˛ denote the element of H with supp .e˛ / D ¹˛º. Deﬁne f W X ! H by putting f .1/ D 0 and f .y0 x˛1 x˛n / D e˛1 C C e˛n where n 2 N and 0 < ˛1 < : : : < ˛n < . Clearly f is bijective and f .B˛ / D H˛ for every ˛ 2 Œ1; /, so f homeomorphically maps X onto .H; T0 /. Also f satisﬁes conditions of Lemma 7.22, since f .xy/ D f .x/ C f .y/ whenever x 2 X n ¹1º and y 2 B.x/C1 . Let f W clˇS X ! ˇH denote the continuous extension of f and let ' D f jUlt.T / . It then follows that ' W Ult.T / ! H is an isomorphism. We conclude this section by constructing a homomorphism of a semigroup generated by a strongly discrete ultraﬁlter. Lemma 7.27. Let F be a strongly discrete ﬁlter on S and let M W S ! F be a basic mapping. Let x 2 S n ¹1º and let x D x0 xn be an M -decomposition with x0 D 1. Then there is a neighborhood U of 1 in T ŒF such that whenever y 2 U n ¹1º and y D y0 ym is an M -decomposition with y0 D 1, xy D x0 x1 xn y1 ym is an M -decomposition.

Section 7.3 Homomorphisms

109

Proof. Deﬁne N W S ! F by N.y/ D M.xy/ \ M.y/ and put U D ŒN 1 . Let y 2 U n ¹1º. Then there is an N -decomposition y D y0 ym with y0 D 1. Clearly this is also an M -decomposition. We have that xy D x0 x1 xn y1 ym . Since x D x0 xn is an M -decomposition, xiC1 2 M.x0 xi /. Next, we have that y1 2 N.y0 / D N.1/ M.x1/ D M.x0 xn / and, for i > 1, yiC1 2 N.y0 yi / D N.y1 yi / M.xy1 yi / D M.x0 xn y1 yi /: Hence, xy D x0 x1 xn y1 ym is an M -decomposition. Deﬁnition 7.28. Given a semigroup S and p 2 ˇS, let Cp denote the smallest closed subsemigroup of ˇS containing p. Theorem 7.29. Let S be an inﬁnite semigroup with identity and let p be a strongly discrete ultraﬁlter on S . Then there is a continuous homomorphism W Ult.T Œp/ ! ˇN such that .p/ D 1 and .Cp / D ˇN. Proof. Choose a basic mapping M W S ! p such that M.x/ M.1/ for every x 2 S (see the proof of Theorem 4.16). For every x 2 M.1/, put f0 .x/ D 1. Extend f0 W M.1/ ! N to a mapping f W .S; T Œp/ ! N ˇN by f .x/ D f0 .x1 / C C f0 .xn / where x D x0 x1 xn is an M -decomposition with x0 D 1. Deﬁne W Ult.T Œp/ ! ˇN by D f jUlt.T Œp / . Lemma 7.27 gives us that for every x 2 S n ¹1º, there is a neighborhood U of 1 in T Œp such that f .xy/ D f .x/ C f .y/ for all y 2 U n ¹1º. Then by Lemma 7.22, is a homomorphism. Clearly, is continuous and .p/ D 1. It follows that .Cp / is a closed subsemigroup of ˇN containing 1. Hence .Cp / D ˇN.

References Proposition 7.7 and Proposition 7.8 are from [99]. Example 7.9 is from [92] and Lemma 7.10 is from [98]. Theorem 7.13 is a result from [101]. Theorem 7.17 is due to T. Budak (Papazyan) [54]. Proposition 7.21 was proved in [99]. Theorem 7.24 is from [38], where it was also proved that for every inﬁnite cancellative semigroup S of cardinality , S contains copies of H . Theorem 7.26 is from [108]. Theorem 7.29 is a result from [107].

Chapter 8

Finite Groups in ˇG

In this chapter we show that if G is a countable torsion free group, then ˇG contains no nontrivial ﬁnite groups. We also extend in some sense this result to the case where G is an arbitrary countable group. To this end, we develop a special technique based on the concepts of a local left group and a local homomorphism.

8.1

Local Left Groups and Local Homomorphisms

Deﬁnition 8.1. A local left group is a T1 -space X with a partial binary operation and a distinguished point 1 2 X such that (i) x 1 D x for all x 2 X , (ii) .x y/ z D x .y z/ whenever all the products in the equality are deﬁned, and (iii) for every x 2 X n ¹1º, there is a neighborhood U of 1 such that x y is deﬁned for all y 2 U , x U is a neighborhood of x and x W U 3 y 7! x y 2 x U is a homeomorphism. A basic example of a local left group is an open neighborhood of the identity of a T1 left topological group. Let X be a local left group and let Xd be the partial semigroup X reendowed with the discrete topology. As in the case of ˇS, the partial operation of Xd can be naturally extended to ˇXd by pq D lim lim xy; x!p y!q

where x; y 2 X , making ˇXd a right topological partial semigroup. The product pq is deﬁned if and only if ¹x 2 X W ¹y 2 X W xy is deﬁnedº 2 qº 2 p; in particular, if the ultraﬁlter q converges to 1 2 X . Deﬁnition 8.2. Given a local left group X , Ult.X / D ¹p 2 Xd W p converges to 1 2 X º: It is straightforward to check that Ult.X / is a closed subsemigroup in Xd . If X is an open neighborhood of the identity of a left topological group .G; T /, we identify Ult.X / with Ult.T /.

Section 8.1 Local Left Groups and Local Homomorphisms

111

Deﬁnition 8.3. Let X and Y be local left groups. A mapping f W X ! Y is a local homomorphism if f .1X / D 1Y and for every x 2 X n ¹1º, there is a neighborhood U of 1 2 X such that f .xy/ D f .x/f .y/ for all y 2 U . We say that f W X ! Y is a local isomorphism if f is a local homomorphism and homeomorphism. Note that if f W X ! Y is a local isomorphism, so is f 1 W Y ! X . Lemma 8.4. Let f W X ! Y be a continuous local homomorphism. Deﬁne f W Ult.X / ! Ult.Y / by f D f jUlt.X/ , where f W ˇXd ! ˇYd is the continuous extension of f . Then f is a continuous homomorphism. Furthermore, if f is injective, f is a continuous injective homomorphism, and if f is a local isomorphism, f is a topological and algebraic isomorphism. Proof. Apply Lemma 7.22. Deﬁnition 8.5. Let X be a local left group and let S be a semigroup. A mapping f W X ! S is a local homomorphism if for every x 2 X n¹1º, there is a neighborhood U of 1 such that f .xy/ D f .x/f .y/ for all y 2 U n ¹1º. Lemma 8.6. Let T be a compact right topological semigroup and let f W X ! T be a local homomorphism such that f .X / ƒ.T /. Deﬁne f W Ult.X / ! T by f D f jUlt.X/ , where f W ˇXd ! T is the continuous extension of f . Then f is a continuous homomorphism. Furthermore, if for every neighborhood U of 1 2 X , f .U n ¹1º/ is dense in T , then f is surjective. Proof. That f is a homomorphism follows from Lemma 7.22. To check the second statement, let t 2 T . We have that for every neighborhood U of 1 2 X and for every neighborhood V of t 2 T , there exists x 2 U n ¹1º such that f .x/ 2 V . It follows from this that there exists p 2 Ult.T / such that f .p/ D t . L Deﬁnition 8.7. We say that an element a 2 ! Z2 is basic if supp.a/ is a nonempty L interval in !. Each nonzero a 2 ! Z2 can be uniquely written in the form a D a 0 C C ak where (a) for each i k, ai is basic, and (b) for each i k 1, max supp.ai / C 2 min supp.aiC1 /. We call such L a decomposition canonical . Endow ! Z2 with the group topology by taking as a neighborhood base at 0 the subgroups ° ± M Z2 W x.i / D 0 for all i < n Hn D x 2 !

where n < !.

112

Chapter 8 Finite Groups in ˇG

L Lemma 8.8. Let S be a semigroup and let f W ! Z2 ! S. The ﬁrst two of the following statements are equivalent and imply the third: (1) f .a/ D f .a0 / f .ak / whenever a D a0 C C ak is a canonical decomposition, (2) f .a C b/ D f .a/f .b/ whenever max supp.a/ C 2 min supp.b/, (3) f is a local homomorphism. Proof. .1/ ) .2/ Let max supp.a/ C 2 min supp.b/ and let a D a0 C C ak and b D b0 C C bl be the canonical decompositions. Since max supp.a/ D max supp.ak / and min supp.b/ D min supp.b0 /, one has max supp.ak / C 2 min supp.b0 /, so aCb D a0 C Cak Cb0 C Cbl is the canonical decomposition. Hence f .a C b/ D f .a0 C C ak C b0 C C bl / D f .a0 / f .ak /f .b0 / f .bl / D f .a/f .b/: .2/ ) .1/ If a D a0 C C ak is a canonical decomposition, then f .a/ D f .a0 C C ak1 /f .ak / D D f .a0 / f .ak /: L .2/ ) .3/ Let 0 ¤ a 2 ! Z2 and let n D max supp.a/ C 2. Then for every b 2 Hn n ¹0º, max supp.a/ C 2 min supp.b/, and so f .a C b/ D f .a/f .b/. We now come to the main result of this section. Theorem 8.9. Let X be a countable regular local left group and let ¹Ux W x 2 X n¹1ºº be L a family of neighborhoods of 1 2 X . Then there is a continuous bijection h W X ! ! Z2 with h.1/ D 0 such that (1) h1 .Hn / Ux whenever max supp.h.x// C 2 n, and (2) h.xy/ D h.x/ C h.y/ whenever max supp.h.x// C 2 min supp.h.y//. Notice that condition (2) in Theorem 8.9 may be rewritten as (20 ) h1 .ab/ D h1 .a/ C h1 .b/ whenever max supp.a/ C 2 min supp.b/, L so h1 W ! Z2 ! X is a local homomorphism, by Lemma 8.8. Since h is continuous, it follows that h also is a local homomorphism. Finally, if ¹Ux W x 2 X n ¹1ºº is a neighborhood base at 1 2 X , then h is open, and so h is a local isomorphism. To see that h is open, let x 2 X . Put n D max supp.h.x// C 2 (if x D 1, put n D 0). Since h is continuous, there is a neighborhood V of 1 such that h.V / Hn . Now let U be any neighborhood of 1 contained in V . Then h.xU / D h.x/ C h.U / by (2). Pick y 2 X n ¹1º such that Uy U and put m D max supp.h.y// C 2. Then h.Uy / Hm by (1). Hence, h.xU / h.x/ C Hm .

Section 8.1 Local Left Groups and Local Homomorphisms

113

Deﬁnition 8.10. Given m 2, let W D W .Zm / denote the set of all words on the alphabet Zm including the empty word ;. A nonempty word w 2 W is basic if all nonzero letters in w form a ﬁnal subword. In particular, every nonempty zero word (= all the letters are zeros) is basic. For every v; w 2 W such that jvj C 2 jwj and the ﬁrst jvj C 1 letters in w are zeros, deﬁne v C w 2 W to be the result of substituting v for the initial subword of length jvj in w. Each nonempty w 2 W can be uniquely written in the form w D w0 C C wk where (a) for each i k, wi is basic, and (b) for each i < k, wi is nonzero. We call such a decomposition canonical. Proof of Theorem 8.9. Enumerate X without repetitions as ¹1; x1 ; x2 ; : : :º and let W D W .Z2 /. We shall assign to each w 2 W a point x.w/ 2 X and a clopen neighborhood X.w/ of x.w/ such that (i) x.0n / D 1, X.;/ D X , and X.0n / Ux.v/ for all v 2 W with jvj n 2, (ii) X.w _ 0/ \ X.w _ 1/ D ; and X.w _ 0/ [ X.w _ 1/ D X.w/, (iii) x.w/ D x.w0 / x.wk / and X.w/ D x.w0 / x.wk1 /X.wk / where w D w0 C C wk is the canonical decomposition, and (iv) xn 2 ¹x.v/ W v 2 W and jvj D nº. Choose as X.0/ a clopen neighborhood of 1 such that x1 … X.0/. Put X.1/ D X n X.0/, x.0/ D 1 and x.1/ D x1 . Fix n 2 and suppose that X.w/ and x.w/ have been constructed for all w with jwj < n such that conditions (i)–(iv) hold. Notice that the subsets X.w/, jwj D n 1, form a partition of X . So, one of them, say X.u/, contains xn . Let u D u0 C C ur be the canonical decomposition. Then X.u/ D x.u0 / x.ur1 /.X.ur // and xn D x.u0 / x.ur1 /yn for some yn 2 X.ur /. If yn D x.ur /, choose as X.0n / a clopen neighborhood of 1 such that (a) X.0n / Ux.w/ for all w 2 W with jwj n 2, and (b) for each basic w with jwj D n 1, X.w/ n x.w/X.0n / ¤ ;. For each basic w with jwj D n 1, put X.w _ 1/ D X.w/ n x.w/X.0n / and pick as x.w _ 1/ any element of X.w _ 1/. Also put x.0n / D 1. If yn ¤ x.ur /, choose X.0n / in addition so that (c) yn … x.ur /X.0n / and put x.u_ r 1/ D yn . For all nonbasic w with jwj D n, deﬁne X.w/ and x.w/ by condition (iii). Then x.w/ D x.w0 / x.wk / 2 x.w0 / x.wk1 /X.wk / D X.w/

114

Chapter 8 Finite Groups in ˇG

and if xn … ¹x.w/ W jwj D n 1º, _ xn D x.u0 / x.ur1 /x.u_ r 1/ D x.u 1/ 2 ¹x.w/ W jwj D nº:

To check (ii), let jwj D n 1. If w is basic, X.w _ 0/ D x.w/X.0n / X.w/ and X.w _ 1/ D X.w/ n x.w/X.0n /. Suppose that w is nonbasic and let w D w0 C C wk be the canonical decomposition. If wk is zero, then w _ 0 D w0 C Cwk1 C0n is the canonical decomposition, and consequently, X.w _ 0/ D x.w0 / x.wk1 /X.0n / D x.w0 / x.wk1 /X.wk_ 0/: Otherwise w _ 0 D w0 C C wk1 C wk C 0n is the canonical decomposition, and then X.w _ 0/ D x.w0 / x.wk1 /x.wk /X.0n / D x.w0 / x.wk1 /X.wk_ 0/: In any case, X.w _ 0/ D x.w0 / x.wk1 /X.wk_ 0/: Next, since w _ 1 D w0 C C wk1 C wk_ 1 is the canonical decomposition, X.w _ 1/ D x.w0 / x.wk1 /X.wk_ 1/: It follows that X.w _ 0/ [ X.w _ 1/ D x.w0 / x.wk1 /ŒX.wk_ 0/ [ X.wk_ 1/ D x.w0 / x.wk1 /X.wk / D X.w/ and X.w _ 0/ \ X.w _ 1/ D ;. Now for every x 2 X , there is w 2 W with nonzero last letter such that x D x.w/, so ¹v 2 L W W x D x.v/º D ¹w _ 0n W n < !º. Consequently, we can deﬁne h W X ! ! Z2 by putting for every w D 0 n 2 W , h.x.w// D w D . 0 ; : : : ; n ; 0; 0; : : :/: Obviously, h.1/ D 0. It is clear also that h is bijective. Since for every z D . i /i

Section 8.1 Local Left Groups and Local Homomorphisms

115

To check (2), let max supp.h.x// C 2 min supp.h.y//. Pick w; v 2 W with nonzero last letters such that x D x.w/ and y D x.v/. Let w D w0 C C wk and v D v0 C C v t be the canonical decompositions. Then y 2 X.0jwjC1 /, consequently w C v D w0 C C wk C v0 C C v t is the canonical decomposition, and so xy D x.w0 / x.wk /x.v0 / x.v t / D x.w C v/: Hence, h.xy/ D h.x.w C v// DwCv DwCv D h.x.w// C h.x.v// D h.x/ C h.y/: Corollary 8.11. Let X be a countable nondiscrete regular L local left group. Then there is a continuous bijective local homomorphism f W X ! ! Z2 , and consequently, Ult.X / is topologically and algebraically isomorphic to a subsemigroup of H. If X is ﬁrst countable, f can be chosen to be a local isomorphism, and consequently, Ult.X / is topologically and algebraically isomorphic to H. Proof. It is immediate from Theorem 8.9 and Lemma 8.4. Corollary 8.12. Let X and Y be countable nondiscrete regular local left groups. Then there is a bijection f W X ! Y such that both f and f 1 are local homomorphisms. L L Proof. Let h W X ! ! Z2 and g W Y ! ! Z2 be bijections guaranteed by Theorem 8.9. Put f D g 1 ı h. We say that a homomorphisms of an ultraﬁlter semigroup is proper if it can be induced by a local homomorphism as in Lemmas 8.4 and 8.6. Corollary 8.13. Let X be a countable nondiscrete regular local left group and let Q and T be ﬁnite semigroups. Then for every local homomorphism f W X ! Q and for every surjective homomorphism g W T ! Q, there is a local homomorphism h W X ! T such that f D g ı h. Consequently, for every proper homomorphism ˛ W Ult.X / ! Q and for every surjective homomorphism ˇ W T ! Q, there is a proper homomorphism W Ult.X / ! T such that ˛ D ˇ ı . Proof. For every x 2 X n ¹1º, pick a neighborhood UL x of 1 2 X such that f .xy/ D f .x/f .y/ for all y 2 Ux n ¹1º, and let ' W X ! ! Z2 be a bijection guaranteed by by Theorem 8.9. It then follows that f ' 1 .a C b/ D f ' 1 .a/f ' 1 .b/

116

Chapter 8 Finite Groups in ˇG

L whenever max supp.a/ C 2 min supp.b/. For every basic a 2 L ! Z2 , pick .a/ 2 g 1 f ' 1 .a/ T , so g .a/ D f ' 1 .a/. Deﬁne W ! Z2 ! T by .a/ D .a0 / .ak / where a D a0 C C ak is a canonical decomposition. Let h D ı '. It is clear that h is a local homomorphism. To see that g ı h D f , let x 2 X n ¹1º and let '.x/ D a0 C C ak be the canonical decomposition. Then gh.x/ D g '.x/ D g .a0 C C ak / D g. .a0 / .ak // D g .a0 / g .ak / D f ' 1 .a0 / f ' 1 .ak / D f ' 1 .a0 C C ak / D f .x/: Finally, it follows from f D g ı h that f D g ı h . Let X be a local left group and let S be a ﬁnite semigroup. We say that a local homomorphism f W X ! S is surjective if for every neighborhood U of 1 2 X , f .U n ¹1º/ D S. Corollary 8.14. Let X be a countable nondiscrete regular local left group and let Q be a ﬁnite semigroup. Then for every local L homomorphism f W X ! Q and for every surjective local homomorphism g W ! Z2 ! Q, there is a continuous local L homomorphism h W X ! ! Z2 such that f D gıh. Consequently, for every proper homomorphism ˛ W Ult.X / ! Q and for every surjective proper homomorphism ˇ W H ! Q, there is a proper homomorphism W Ult.X / ! H such that ˛ D ˇ ı . Proof. For every x 2 X n ¹1º, pick a neighborhoodL Ux of 1 2 X such that f .xy/ D f .x/f .y/ for all y 2 Ux n ¹1º, and let ' W X ! ! Z2 be a bijection guaranteed by Theorem 8.9. For every n < !, pick ng > n such that g.a C b/ D g.a/g.b/ L whenever max supp.a/ n and min supp.b/ ng . Then for every basic a 2 ! Z2 , L pick a nonzero .a/ 2 g 1 f ' 1 .a/ ! Z2 such that the following condition is satisﬁed: If m < max supp.a/ and n D max¹max supp. .b// W b is basic and max supp.b/ mº; then min supp. .a// ng .

117

Section 8.2 Triviality of Finite Groups in ˇZ

Deﬁne

W

L !

Z2 !

L !

Z2 by

.a/ D

.a0 / C C

.ak /;

where a D a0 C C ak is a canonical decomposition, and let h D from the condition that g. .a0 / C C

ı '. It follows

.ak // D g .a0 / g .ak /;

and so f D g ı h. The condition also implies that min supp. .a// max supp.a/ for every basic a, which gives us that is continuous. To see this, suppose that max supp.a/ D m C 1 and the statement holds for all basic b with max supp.b/ D m. Pick any such b. Then by the inductive assumption, min supp. .b// m. It follows that n m. Now, applying the condition, we obtain that min supp. .a// ng n C 1 m C 1 D max supp.a/:

8.2

Triviality of Finite Groups in ˇZ

Lemma 8.15. Let G be a group and let Q be a group in ˇG. Then Q is contained either in G or in G . Proof. It is immediate from the fact that G is an ideal of ˇG (Lemma 6.8). Deﬁnition 8.16. Given a group G and a group Q in G , G.Q/ D ¹x 2 G W xQ D Qº: If x; y 2 G.Q/, then xy 1 Q D xy 1 yQ D xQ D Q, and so xy 1 2 G.Q/. Hence, G.Q/ is a subgroup of G. Also note that G.Q/ D ¹x 2 G W xu 2 Qº where u is the identity of Q. Indeed, if xu 2 Q, then xQ D x.uQ/ D .xu/Q D Q. Lemma 8.17. G.Q/ 3 x 7! xu 2 Q is an injective homomorphism. Proof. That this mapping is injective follows from Lemma 6.9. To see that this is a homomorphism, let x; y 2 G.Q/. Then .xy/u D x.yu/ D x.u.yu// D .xu/.yu/.

118

Chapter 8 Finite Groups in ˇG

Theorem 8.18. Let G be a countable group and let Q be a ﬁnite group in G . If G.Q/ is trivial, so is Q. Proof. Assume on the contrary that Q is nontrivial while G.Q/ D ¹1º. Without loss of generality one may suppose that Q is a cyclic group of order n > 1. Let u be the identity of Q. Deﬁne C G by C D ¹x 2 G W xu 2 Qº Equivalently, C D ¹x 2 G W xQ D Qº: It is clear that C is a closed subsemigroup of G and Q is a minimal left ideal of C . Furthermore, C is left saturated. Indeed, let xy D z for some x 2 ˇG and y; z 2 C . Then xyQ D zQ, and so xQ D Q. Consequently, x 2 C [ G.Q/. Since G.Q/ D ¹1º, x 2 C [ ¹1º. By Theorem 7.13, there is a regular left invariant topology T on G such that Ult.T / D C . Since Q is a minimal left ideal of C , it follows that C has only one minimal right ideal. Consequently, T is extremally disconnected, by Proposition 7.7. Being regular extremally disconnected, T is zero-dimensional. (Note that we showed zero-dimensionality of T not using the fact that G is countable.) Next for every p 2 Q, let Cp D ¹x 2 C W xu D pº: It is easy to see that ¹Cp W p 2 Qº is a partition of C into closed subsets and p 2 Cp for each p 2 Q. Let Fp be the ﬁlter on G such that Fp D Cp . For every p 2 Q, choose Vp 2 Fp such that Vp \ Vq D ; if p ¤ q. We now show that for each p 2 Q, there is Wp 2 Fp such that Wp Cq Vpq for all q 2 Q. Indeed, let D u jˇGn¹1º . Then Cpq D 1 .pq/. It follows that there exists Apq 2 pq such that 1 .Apq / Vpq ; or equivalently, ¹x 2 ˇG n ¹1º W xu 2 Apq º Vpq : Since Cp q D Cp uq D pq and Q is ﬁnite, there is Wp 2 Fp such that Wp q Apq for all q 2 Q. Then Wp Cq u D Wp q Apq ; and consequently, Wp Cq Vpq .

119

Section 8.2 Triviality of Finite Groups in ˇZ

Choose the subsets Wp in addition so that Wp Vp and [ Wp [ ¹1º XD p2Q

is open in T . Then deﬁne f W X ! Q by f .x/ D p

if x 2 Wp :

The value f .1/ does not matter. We claim that f is a local homomorphism. To see this, let x 2 X n ¹1º. Then x 2 Wp for some p 2 Q. For each q 2 Q, choose Ux;q 2 Fq such that Ux;q Wq

and

xUx;q Vpq : This can be done because Wp Cq Vpq . Then choose a neighborhood Ux of 1 2 X such that [ Ux Ux;q [ ¹1º and xUx X: q2Q

Now let y 2 Ux n ¹1º. Then y 2 Ux;q for some q 2 Q. Since xUx;q Vpq , one has xy 2 Vpq . But then, since xUx X , xy 2 Wpq . Hence f .xy/ D pq D f .x/f .y/: Having checked that f is a local homomorphism, let ˛ D f . Then ˛ W Ult.X / ! Q is a proper homomorphism with the property that ˛jQ D idQ . Now let T be a cyclic group of order n2 and let ˇ W T ! Q be a surjective homomorphism. By Corollary 8.13, there is a proper homomorphism W Ult.X / ! T such that ˛ D ˇı . It follows that .Q/ T is a subgroup of order n. But T has only one subgroup of order n and this is the kernel of ˇ, so ˇ. .Q// D ¹0º, a contradiction. Corollary 8.19. Let G be a countable torsion free group. Then ˇG contains no nontrivial ﬁnite groups. Proof. By Lemma 8.15, every group in ˇG is contained either in G or in G . Let Q be a ﬁnite group in G . By Lemma 8.17, Q contains an isomorphic copy of G.Q/. Consequently, G.Q/ is ﬁnite. Since G is torsion free, it follows that G.Q/ is trivial. Then by Theorem 8.18, Q is trivial as well. As an immediate consequence of Corollary 8.19 we obtain that Corollary 8.20. ˇZ contains no nontrivial ﬁnite groups.

120

Chapter 8 Finite Groups in ˇG

Corollary 8.20 and Corollary 8.11 give us the following. Corollary 8.21. Let X be a countable regular local left group. Then Ult.X / contains no nontrivial ﬁnite groups. Proof. Pick a nondiscrete ﬁrst countable group topology T on Z. By Corollary 8.11, Ult.T / is isomorphic to H and Ult.X / is isomorphic to a subgroup of H, and by Corollary 8.20, Ult.T / contains no nontrivial ﬁnite groups.

8.3

Local Automorphisms of Finite Order

Let X be a set and let f W X ! X . A subset Y X is invariant (with respect to f ) if f .Y / Y . We say that a family F of subsets of X is invariant if for every Y 2 F , f .Y / 2 F . For every x 2 X , let O.x/ D ¹f n .x/ W n < !º. Lemma 8.22. Let X be a space, let f W X ! X be a homeomorphism, and let x 2 X with jO.x/j D s 2 N. Let U be a neighborhood of x such that the family ¹f j .U / W j < sº is disjoint and suppose that there is n 2 N such that f n jU D idU . Then there is an open neighborhood V of x contained in U such that the family ¹f j .V / W j < sº is invariant. If X is zero-dimensional, then V can be chosen to be clopen. Proof. Clearly, n D sl for some l 2 N. Choose an open neighbourhood W of x such that f j Cis .W / f j .U / for all j < s and i < l, in particular, f is .W / U for all i < l. This can be done because f s .x/ D x. Now let V D

[

f is .W /:

i

Then V is an open neighborhood of x contained in U and f s .V / D

[

f .iC1/s .W /:

i

Since f ls D f 0 , f s .V / D

[

f is .W / D V:

i

It follows that ¹f

j .V /

W j < sº is invariant.

A bijection f W X ! X has ﬁnite order if there is n 2 N such that f n D idX , and the smallest such n is the order of f .

Section 8.3 Local Automorphisms of Finite Order

121

Corollary 8.23. Let X be a space with a distinguished point 1 2 X and let f W X ! X be a homeomorphism with f .1/ D 1. Then exactly one of the following two possibilities holds: (1) for every neighborhood U of 1 and n 2 N, there is x 2 U with jO.x/j > n, (2) there is an open invariant neighborhood U of 1 such that f jU has ﬁnite order. Deﬁnition 8.24. Let X be a space with a distinguished point 1 2 X and let f W X ! X be a homeomorphism of ﬁnite order with f .1/ D 1. Deﬁne the spectrum of f by spec.f / D ¹jO.x/j W x 2 X n ¹1ºº; and more generally, for any subset Y X , spec.f; Y / D ¹jO.x/j W x 2 Y n ¹1ºº: We say that f is spectrally irreducible if for every neighborhood U of 1, spec.f; U / D spec.f /: Also, a neighborhood U of a point x 2 X is spectrally minimal if for every neighborhood V of x contained in U , spec.f; V / D spec.f; U /: Corollary 8.25. Let X be a space with a distinguished point 1 2 X and let f W X ! X be a homeomorphism of ﬁnite order with f .1/ D 1. Then there is an open invariant neighborhood U of 1 such that f jU is spectrally irreducible. Corollary 8.25 tells us that we can restrict ourselves in the study of homeomorphisms of ﬁnite order in a neighborhood of a ﬁxed point to considering spectrally irreducible ones. Deﬁnition 8.26. A local automorphism of a local left group X is a local isomorphism of X onto itself. In other words, a mapping f W X ! X is a local automorphism if f is both a homeomorphism with f .1/ D 1 and a local homomorphism, that is, for every x 2 X n ¹1º, there is a neighborhood U of 1 such that f .xy/ D f .x/f .y/ for all y 2 U. The next lemma says that the spectrum of a spectrally irreducible local automorphism of ﬁnite order is a ﬁnite subset of N closed under taking the least common multiple lcm, that is, lcm.s; t / 2 spec.f / for all s; t 2 spec.f /.

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Chapter 8 Finite Groups in ˇG

Lemma 8.27. Let X be a Hausdorff local left group and let f W X ! X be a spectrally irreducible local automorphism of ﬁnite order. Let x0 2 X n ¹1º with jO.x0 /j D s and let U be a spectrally minimal neighbourhood of x0 . Then spec.f; U / D ¹lcm.s; t / W t 2 ¹1º [ spec.f /º: Proof. For each x 2 O.x0 /, let Vx be a neighborhood of 1 such that Vx 3 y 7! xy 2 xVx is a homeomorphism and f .xy/ D f .x/f .y/ for all y 2 Vx . Choose a neighborhood V of 1 such that \ V Vx ; x2O.x0 /

x0 V U , and the subsets xV , where x 2 O.x0 /, are pairwise disjoint. Let n be the order of f . Choose a neighborhood W of 1 such that f i .W / V for all i < n. Then clearly this inclusion holds for all i < !, and furthermore, for every y 2 W , f i .x0 y/ D f i .x0 /f i .y/. Indeed, it is trivial for i D 0, and further, by induction, we obtain that f i .x0 y/ D ff i1 .x0 y/ D f .f i1 .x0 /f i1 .y// D f i .x0 /f i .y/: Now let y 2 W , jO.y/j D t and k D lcm.s; t /. We claim that jO.x0 y/j D k. Indeed, f k .x0 y/ D f k .x0 /f k .y/ D x0 y: On the other hand, suppose that f i .x0 y/ D x0 y for some i . Then f i .x0 /f i .y/ D x0 y. Since the subsets xV , x 2 O.x0 /, are pairwise disjoint, it follows from this that f i .x0 / D x0 , so sji . But then also f i .y/ D y, and so t ji . Hence kji . Now, given any ﬁnite subset of N closed under lcm, we produce a spectrally irreducible local automorphism of the corresponding spectrum. Example 8.28. Let S be a ﬁnite subset P L of N closed under lcm and let m D 1 C s. Consider the direct sum ! Zm of ! copies of the group Zm . Endow Ls2S Z with the group topology by taking as a neighborhood base at 0 the subgroups ! m ° ± M Hn D x 2 Zm W x.i / D 0 for all i < n !

where n < !. Write the elements of S as s1 < < s t . Deﬁne the permutation 0 on Zm by the product of disjoint cycles 0 D .1; : : : ; s1 /.s1 C 1; : : : ; s1 C s2 / .s1 C C s t1 C 1; : : : ; s1 C C s t /: L Let be the coordinatewise permutation on ! Zm induced by 0 , that is, .x/.n/ D 0 .x.n//. Then is a homeomorphism with .0/ D 0, spec.; Hn / D S for each

Section 8.3 Local Automorphisms of Finite Order

123

n < !, and .x C y/ D .x/ C .y/ whenever max supp.x/ < min supp.y/. Hence, is a spectrally irreducible local automorphism of spectrum S. We call the standard permutation of spectrum S. We now come to the main result of this section. Theorem 8.29. Let X be a countable nondiscrete regular local left group, let f W X ! X be a spectrally irreducible local automorphism of ﬁnite order, let P LS D spec.f /, and let m D 1 C s2S s. Let be the standard permutation on ! Zm L of spectrum S. Then there is a continuous bijection h W X ! ! Zm with h.1/ D 0 such that (1) f D h1 h, and (2) h.xy/ D h.x/ C h.y/ whenever max supp.h.x// C 2 min supp.h.y//. If X is ﬁrst countable, then h can be chosen to be a homeomorphism. Proof. The proof is similar to that of Theorem 8.9. W .Zm /. The permutation Enumerate X as ¹xn W n < !º with x0 D 1 and let W D L 0 on Zm , which induces the standard permutation on ! Zm , also induces the permutation 1 on W . If w D 0 n , then 1 .w/ D 0 . 0 / 0 . n /. We will write instead of 0 and 1 . We shall assign to each w 2 W a point x.w/ 2 X and a clopen spectrally minimal neighborhood X.w/ of x.w/ such that (i) x.0n / D 1 and X.;/ D X , (ii) ¹X.w _ / W 2 Zm º is a partition of X.w/, (iii) x.w/ D x.w0 / x.wk1 /x.wk / and X.w/ D x.w0 / x.wk1 /X.wk / where w D w0 C C wk is the canonical decomposition, (iv) f .x.w// D x..w// and f .X.w// D X..w//, and (v) xn 2 ¹x.v/ W v 2 W and jvj D nº. For this, we need the following Lemma 8.30. Let X be a countable nondiscrete regular space with a distinguished point 1 2 X and let f W X ! X be a homeomorphism of ﬁnite order with f .1/ D 1. Let U be a clopen invariant subset of X , let K be a ﬁnite invariant subset of U , and let P be a clopen invariant partition of U such that for each C 2 P , spec.f; K \ C / D spec.f; C /. Then there is a clopen invariant partition ¹U.x/ W x 2 Kº of U inscribed into P such that for each x 2 K, U.x/ is a spectrally minimal neighborhood of x. Proof. Enumerate U as ¹xn W n < !º with x0 2 K. For each x 2 K, we shall construct an increasing sequence .Un .x//n

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Chapter 8 Finite Groups in ˇG

S inscribedSinto P and invariant, and xn 2 Un D x2K Un .x/. Then the subsets U.x/ D n 0 and suppose that we have constructed required Un1 .x/, x 2 K. Without loss of generality one may suppose also that xn … Un1 . Let jO.xn /j D s and let xn 2 Cn 2 P . Using Lemma 8.22, choose a clopen neighborhood Vn of xn such that for each j < s, f j .Vn / is a spectrally minimal neighborhood of f j .xn /, and the family ¹f j .Vn / W j < sº [ ¹Un1 .x/ W x 2 Kº is disjoint, inscribed into P and invariant. Pick zn 2 K \ Cn with jO.zn /j D s. For each j < s, put Un .f j .zn // D Un1 .f j .zn // [ f j .Vn /: For each x 2 K n O.zn /, put Un .x/ D Un1 .x/. Now write the elements of S as s1 < < s t . For each i D 1; : : : ; t , pick a representative i of the orbit in Zm n ¹0º of lengths si . Choose a clopen invariant neighborhood U1 of 1 2 X such that x1 … U1 and spec.f; X n U1 / D spec.f /. Put x.0/ D 1 and X.0/ D U1 . Then pick points S ai 2 X n U1 , i D 1; : : : ; t , with pairwise disjoint orbits of lengths si such that x1 2 tiD1 O.ai /. For each i D 1; : : : ; t and j < si , put x. j .i // D f j .ai /: By Lemma 8.30, there is an invariant partition ¹X. / W 2 Zm n ¹0ºº of X n U1 such that X. / is a clopen spectrally minimal neighborhood of x. /. Fix n > 1 and suppose that X.w/ and x.w/ have been constructed for all w 2 W with jwj < n so that conditions (i)–(v) are satisﬁed. Notice that the subsets X.w/, jwj D n 1, form a partition of X . So one of them, say X.u/, contains xn . Let u D u0 C C uq be the canonical decomposition. Then X.u/ D x.u0 / x.uq1 /X.uq / and xn D x.u0 / x.uq1 /yn for some yn 2 X.uq /. Choose a clopen invariant neighborhood Un of 1 such that for all basic w with jwj D n 1, (a) x.w/Un X.w/, (b) f .xy/ D f .x/f .y/ for all y 2 Un , and (c) spec.f; X.w/ n x.w/Un / D spec.X.w//.

Section 8.3 Local Automorphisms of Finite Order

125

If yn ¤ x.uq /, choose Un in addition so that (d) yn … x.uq /Un . Put x.0n / D 1 and X.0n / D Un . Let w 2 W be an arbitrary nonzero basic word with jwj D n 1 and let O.w/ D ¹wj W j < sº, where wj C1 D .wj / for j < s 1 and .ws1 / D w0 . Put Yj D X.wj / n x.wj /Un . Using Lemma 8.27, choose points bi 2 Y0 , i D 1; : : : ; t , with pairwise disjoint orbits of lengths lcm.si ; s/. If uq 2 O.w/, choose bi in addition so that t [ yn 2 O.bi /: iD1

For each i D 1; : : : ; t and j < si , put x. j .w _ i // D f j .bi /: Then, using Lemma 8.30, inscribe an invariant partition ¹X.v _ / W v 2 O.w/; 2 Zm n ¹0ºº into the partition ¹Yj W j < sº such that X.v _ / is a clopen spectrally minimal neighborhood of x.v _ /. For nonbasic w 2 W with jwj D n, deﬁne x.w/ and X.w/ by condition (iii). To check (ii) and (iv), let jwj D n 1 and let w D w0 C C wk be the canonical decomposition. Then X.w _ 0/ D x.w0 / x.wk /X.0n / D x.w0 / x.wk1 /x.wk /X.0n / and X.w _ / D x.w0 / x.wk1 /X.wk_ /; so (ii) is satisﬁed. Next, f .x.w// D f .x.w0 / x.wk1 /x.wk // D f .x.w0 //f .x.w1 / x.wk1 /x.wk // :: : D f .x.w0 // f .x.wk1 //f .x.wk // D x..w0 // x..wk1 //x..wk // D x..w0 / .wk1 /.wk // D x..w//; so (iv) is satisﬁed as well.

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Chapter 8 Finite Groups in ˇG

To check (v), suppose that xn … ¹x.w/ W jwj D n 1º. Then xn D x.u0 / x.uq1 /yn D x.u0 / x.uq1 /x.u_ q / D x.u_ /: Now, for every x 2 X , there is w 2 W with nonzero last letter such that x D Lx.w/, so ¹v 2 W W x D x.v/º D ¹w _ 0n W n < !º. Hence, we can deﬁne h W X ! ! Zm by putting for every w D 0 n 2 W , h.x.w// D w D . 0 ; : : : ; n ; 0; 0; : : :/: Obviously, h.1/ D 0. It is clear also that h is bijective. Since for every z D . i /i

127

Section 8.3 Local Automorphisms of Finite Order

Proof. Consider an arbitrary relation p1 pk D q1 qs in Ult.X /, where pi ; qj 2 ¹p; f .p/º, pi ¤ piC1 and qj ¤ qj C1 . We prove that p1 D q1 and k D s. Without loss of generality one may suppose that f is spectrally irreducible. Let M hWX ! Zm !

be a bijection guaranteed by Theorem 8.29. Denote C the set of all nonﬁxed points in Zm (with respect to 0 ) and let Y D ¹x 2 X W there is a coordinate of h.x/ belonging to C º: (Equivalently, Y consists of all nonﬁxed points in X .) Note that Y \ Ult.X / is a subsemigroup containing p and f .p/. For every x 2 Y , consider the sequence of coordinates of h.x/ belonging to C and denote ˛.x/ and .x/ the ﬁrst and the last elements in this sequence. Then for every u; v 2 Y \ Ult.X /, ˛.uv/ D ˛.u/. Indeed, let ˛.u/ D c 2 C and let A D ¹x 2 Y W ˛.x/ D cº. Then A 2 u. For every x 2 A, put n.x/ D max supp.h.x// C 2 and Ux D h1 .Hn.x/ /. We have that S x2A xUx 2 uv and for every y 2 Ux , ˛.xy/ D ˛.x/ D c, so ˛.uv/ D c. Similarly, .uv/ D .v/, and if f .u/ ¤ u, then ˛.u/ ¤ ˛.f .u// and .u/ ¤

.f .u//. Applying ˛ and to the relation gives us that ˛.p1 / D ˛.q1 / and .pk / D

.qs /, so p1 D q1 and pk D qs . We now show that k D s. Deﬁne the subset F C 2 by F D ¹. .q/; ˛.q// W q 2 ¹p; f .p/ºº and let n max¹k; sº. For every x 2 X , consider the sequence of coordinates of h.x/ belonging to C and deﬁne .x/ 2 Z.n/ to be the number modulo n of pairs of neighbouring elements in this sequence other than pairs from F . Then for every u; v 2 Ult.X /, ´ .u/ C .v/ if . .u/; ˛.v// 2 F .uv/ D .u/ C .v/ C 1 otherwise. It follows from this that .p1 pk / D .p1 / C C .pk / C k 1 and .q1 qs / D .q1 / C C .qs / C s 1: Also we have that for every q 2 ¹p; f .p/º, .qq/ D 2.q/. Consequently, since q is an idempotent, .q/ D .qq/ D 2.q/. Hence, .q/ D 0. Finally, we obtain that .p1 pk / D k 1 and so k D s.

.q1 qs / D s 1;

128

8.4

Chapter 8 Finite Groups in ˇG

Finite Groups in G

Finite groups in G can be constructed in the following trivial way. Example 8.32. Let G be a group, let F be a ﬁnite subgroup of G, and let u be an idempotent in G which commutes with each element of F . Then F u is a ﬁnite subgroup of G isomorphic to F . The isomorphism is given by F 3 x 7! xu 2 F u: Indeed, that this mapping is injective follows from Lemma 6.9. To see that this is a homomorphism, let x; y 2 F . Then xyu D xyuu D xuyu. In this section we show that if G is a countable group, then all ﬁnite groups in G have such a trivial structure. Theorem 8.33. Let G be a countable group, let Q be a ﬁnite group in G with identity u, and let F D G.Q/. Then u commutes with each element of F and Q D F u. Proof. We ﬁrst show that u commutes with each element of F . Let a 2 F and assume on the contrary that a1 ua ¤ u. Note that both idempotents u and a1 ua belong to the semigroup C.u/ D ¹x 2 G W xu D uº. Indeed, since a 2 G.Q/, one has au 2 Q, so uau D au and then a1 uau D a1 au D u. By Theorem 7.17, there is a regular left invariant topology T on G with Ult.T / D C.u/. Consider the conjugation f W x 7! a1 xa on .G; T /. Clearly, f .u/ D a1 ua. We claim that f is a homeomorphism. To see this, let p 2 C.u/. Then a1 pau D a1 puau D a1 uau D a1 au D u; so a1 pa 2 C.u/. Consequently, f is continuous. Similarly, f 1 is continuous. It follows that f is a local automorphism of ﬁnite order. Since f .u/ ¤ u, idempotents u and f .u/ D a1 ua generate a free product by Theorem 8.31. But this contradicts the equality a1 uau D u. We now show that Q D F u. Without loss of generality one may suppose that Q is a ﬁnite cyclic group with a generator q. Then for every x 2 F , xq D qx. Indeed, since xu D ux and Q is Abelian, xq D xuq D qxu D qux D qx: Choose A 2 q such that xy D yx for all x 2 F and y 2 A. Let H be the subgroup of G generated by the subset A [ F . We have that Q H and F is central in H . Let L D H=F and let g W ˇH ! ˇL be the continuous extension of the canonical

Section 8.4 Finite Groups in G

129

homomorphism H ! L. Then g is a homomorphism and the elements of ˇL are the subsets of the form Fp where p 2 ˇH . Consider the group R D g.Q/ D Q=.F u/ in L . We claim that the subgroup L.R/ D ¹x 2 L W xR D Rº in L is trivial. To see this, let x 2 L.R/. Pick y 2 H with g.y/ D x. Then g.yQ/ D g.Q/. Since yQ and Q are complete preimages with respect to g, it follows that yQ D Q. Consequently y 2 F , and so x D 1 2 L. Since L.R/ is trivial, R is trivial as well by Theorem 8.18, and then Q D F u.

References The results of Sections 8.1 and 8.2 are from [85] (announced in [86]). Theorem 8.33 is due to I. Protasov [58]. Its proof is based on Theorem 8.18 and Theorem 8.31. The latter and Theorem 8.29 were proved in [87]. The exposition of this chapter is based on the treatment in [92].

Chapter 9

Ideal Structure of ˇG

In this chapter we show that for every inﬁnite group G of cardinality , the ideal U.G/ of ˇG consisting of uniform ultraﬁlters can be decomposed (D partitioned) into 22 closed left ideals of ˇG. We also prove that if G is Abelian, then ˇG contains 22 minimal right ideals and the structure group of K.ˇG/ contains a free group on 22 generators. We conclude by showing that if is not Ulam-measurable, then K.ˇG/ is not closed.

9.1

Left Ideals

Let G be an arbitrary inﬁnite group of cardinality . For every A G, let U.A/ denote the set of uniform ultraﬁlters from A. It is easy to see that the set U.G/ of all uniform ultraﬁlters on G is a closed two-sided ideal of ˇG. Deﬁnition 9.1. Let I.G/ denote the ﬁnest decomposition of U.G/ into closed left ideals of ˇG with the property that the corresponding quotient space of U.G/ is Hausdorff. Deﬁnition 9.1 can be justiﬁed by noting that the family of all such decompositions is nonempty (it contains the trivial decomposition ¹U.G/º) and considering the diagonal of the corresponding quotient mappings. Deﬁnition 9.2. For every p 2 U.G/, deﬁne Ip ˇG by \ Ip D cl.GU.A//: A2p

The next theorem is the main result of this section. Theorem 9.3. If is a regular cardinal, then I.G/ D ¹Ip W p 2 U.G/º. Before proving Theorem 9.3 we establish several auxiliary statements. Lemma 9.4. For every p 2 U.G/, Ip is a closed left ideal of ˇG contained in U.G/. Proof. Clearly, Ip is a closed subset of U.G/. In order to show that Ip is a left ideal of ˇG, it sufﬁces to show that for every x 2 G, xIp Ip . Since \ xcl.GU.A// and xcl.GU.A// D cl.xGU.A// D cl.GU.A//; xIp D A2p

it follows that xIp D Ip .

131

Section 9.1 Left Ideals

A decomposition D of aSspace X into closed subsets is called upper semicontinuous if for every open U X , ¹A 2 D W A U º is open in X . Equivalently, D is upper semicontinuous if for every A 2 D and for every neighborhood U of A X , there is a neighborhood V of A X such that if B 2 D and B \ V ¤ ;, then B U . Lemma 9.5. Let D be a decomposition of a compact Hausdorff space X into closed subsets and let Y be the corresponding quotient space of X . Then Y is Hausdorff if and only if D is upper semicontinuous. Proof. Let f W X ! Y denote the natural quotient mapping. Suppose that Y is Hausdorff and let U X be open. It then follows that f .X n U / Y is closed, so f 1 .Y n f .X n U // X is open. It remains to notice that [ f 1 .Y n f .X n U // D ¹A 2 D W A U º: Conversely, suppose that D is upper semicontinuous and let A1 ; A2 be distinct members of D.SPick disjoint neighborhoods U1 ; U2 of A1 ; A2 in X . For each i D 1; 2, let Vi D ¹B 2 D W B Ui º. Then Vi is a neighborhood of Ai X and Vi D f 1 .f .Vi //. It follows that f .V1 /; f .V2 / are disjoint neighborhoods of f .A1 /; f .A2 / 2 Y . Lemma 9.6. Let J be a decomposition of U.G/ into closed left ideals such that the corresponding quotient space of U.G/ is Hausdorff. Then for every J 2 J and p 2 J , Ip J . Proof. It sufﬁces to show that for every neighborhood V of J U.G/, Ip cl.V /. By Lemma 9.5, J is upper semicontinuous. Therefore, one may suppose that for every I 2 J, if I \ V ¤ ;, then I V . It follows from this that GV V . Since V is a neighborhood of p 2 U.G/, there is A 2 p such that U.A/ V . Consequently, GU.A/ V , so cl.GU.A// cl.V /. Hence, Ip cl.V /. Recall that given a set X and a cardinal , ŒX D ¹A X W jAj D º and

ŒX < D ¹A X W jAj < º:

Deﬁnition 9.7. For every p 2 U.G/, let Fp denote the ﬁlter on G with a base consisting of subsets of the form [ x.A n Fx / x2G

where A 2 p and Fx 2

ŒG<

for each x 2 G.

Lemma 9.8. For every p 2 U.G/, Ip D

\ C 2Fp

C:

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Chapter 9 Ideal Structure of ˇG

T Proof. To see that Ip C 2Fp C , let A 2 ŒG and Fx 2 ŒG< for each x 2 G and S let C D x2G x.AnFx /. For every x 2 G, xU.A/ x.A n Fx / D x.A n Fx / C . Consequently, cl.GU.A// C . To see the converse inclusion, let B G and Ip B. It then follows that there is A 2 p such that cl.GU.A// B. (Indeed, G n B D G n B is compact and for every y 2 G n B, there is Ay 2 p such that y … cl.GU.Ay //.) For every x 2 G, one has xU.A/ B, consequently, there is Fx 2 ŒG< such that x.A n Fx / B. Let S C D x2G x.A n Fx /. Then C 2 Fp and C B. Lemma 9.9. Suppose that isSa regular cardinal. Let A 2 ŒG and Fx 2 ŒG< for every x 2 G and let B D G n x2G x.A n Fx /. Then there are Hx ; Kx 2 ŒG< for every x 2 G such that [ [ x.A n Hx / \ x.B n Kx / D ;: x2G

x2G

Proof. Enumerate G as ¹x˛ W ˛ < º. For every ˛ < , deﬁne Hx˛ ; Kx˛ 2 ŒG< by [ [ Hx˛ D Fx 1 x˛ and Kx˛ D x˛1 xˇ Fx˛1 xˇ : ˇ ˛

ˇ

ˇ ˛

Then for every ˛ < and ˇ ˛, xˇ1 x˛ .A n Hx˛ / \ B D ;

and

x˛1 xˇ A \ .B n Kx˛ / D ;;

x˛ .A n Hx˛ / \ xˇ B D ;

and

xˇ A \ x˛ .B n Kx˛ / D ;:

and so Consequently, for every ˛; ˇ < , x˛ .A n Hx˛ / \ xˇ .B n Kxˇ / D ;: Now we are in a position to prove Theorem 9.3. Proof of Theorem 9.3. Let I D ¹Ip W p 2 U.G/º. By Lemma 9.4, all members of I are closed left ideals of ˇG contained in U.G/. To show that I is an upper < semicontinuous decomposition S of U.G/, let p 2 U.G/, A 2 p, and Fx 2 ŒG for every x 2 G, and let B D x2G x.A n Fx /. By Lemma 9.9, there are Hx ; Kx 2 ŒG< for every x 2 G such that Q \ R D ; where [ [ QD x.A n Hx / and R D x.B n Kx /: x2G

x2G

By Lemma 9.8, Ip Q and for every r 2 U.B/, Ir R, consequently, Ir G n Q. This shows that I is a decomposition. It follows from this also that for every

133

Section 9.1 Left Ideals

q 2 U.Q/, Iq G n B, which shows that I is upper semicontinuous. Thus, I is a decomposition of U.G/ into closed left ideals such that the corresponding quotient space of U.G/ is Hausdorff. That I is the ﬁnest decomposition of this kind follows from Lemma 9.6. Now we consider decompositions of U.G/ with an additional property that for every member I of the decomposition, IG I . Deﬁnition 9.10. Let I 0 .G/ denote the ﬁnest decomposition of U.G/ into closed left ideals of ˇG with the property that the corresponding quotient space of U.G/ is Hausdorff and for every member I of the decomposition, IG I . Deﬁnition 9.11. For every p 2 U.G/, deﬁne Ip0 ˇG by \ Ip0 D cl.GU.A/G/: A2p

As in the proof of Lemma 9.8, one shows that \ Ip0 D C C 2Fp0

where Fp0 denotes the ﬁlter on G with a base consisting of subsets of the form [ x.A n Fx;y /y; x;y2G

where A 2 p and Fx;y 2 ŒG< for every x; y 2 G. The next lemma is the corresponding version of Lemma 9.9. Lemma 9.12. Suppose that is a regular cardinal. Let A 2 ŒG and Fx;y 2 ŒG< S for every x; y 2 G and let B D G n x;y2G x.A n Fx;y /y. Then there are Hx;y ; Kx;y 2 ŒG< for every x; y 2 G such that [ [ x.A n Hx;y /y \ x.B n Kx;y /y D ;: x;y2G

x;y2G

Proof. Enumerate G G as ¹.x˛ ; y˛ / W ˛ < º and for every ˛ < , deﬁne Hx˛ ;y˛ ; Kx˛ ;y˛ 2 ŒG< by [ [ Hx˛ ;y˛ D Fx 1 x˛ ;y˛ y 1 and Kx˛ ;y˛ D x˛1 xˇ Fx˛1 xˇ ;yˇ y˛1 yˇ y˛1 : ˇ ˛

ˇ

ˇ

ˇ ˛

Then for every ˛ < and ˇ ˛, xˇ1 x˛ .A n Hx˛ ;y˛ /y˛ yˇ1 \ B D ;

and

x˛1 xˇ Ayˇ y˛1 \ .B n Kx˛ ;y˛ / D ;;

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Chapter 9 Ideal Structure of ˇG

so x˛ .A n Hx˛ ;y˛ /y˛ \ xˇ Byˇ D ;

and

xˇ Ayˇ \ x˛ .B n Kx˛ ;y˛ /y˛ D ;;

and consequently, for every ˛; ˇ < , x˛ .A n Hx˛ ;y˛ /y˛ \ xˇ .B n Kxˇ ;yˇ /yˇ D ;: It is easy to see that the corresponding versions of Lemma 9.4 and Lemma 9.6 also hold. Hence, we obtain the following analogue of Theorem 9.3. Theorem 9.13. If is a regular cardinal, then I 0 .G/ D ¹Ip0 W p 2 U.G/º. The next lemma will allow us to compute the cardinality of I 0 .G/. Lemma 9.14. Let A 2 ŒG . Then there are B 2 ŒA and Fx;y 2 ŒG< for every x; y 2 G such that whenever B0 ; B1 2 ŒB and B0 \ B1 D ;, one has [ [ x.B0 n Fx;y /y \ x.B1 n Fx;y /y D ;: x;y2G

x;y2G

Proof. Enumerate G G as ¹.x˛ ; y˛ / W ˛ < º. Construct inductively a -sequence .a /< in A such that for every < and ˛ , x˛ a y˛ … ¹xˇ aı yˇ W ˇ ı < º: Deﬁne B 2 ŒA and Fx˛ ;y˛ 2 ŒG< for every ˛ < by B D ¹a W < º

and

Fx˛ ;y˛ D ¹aˇ W ˇ < ˛º:

We claim that these are as required. Indeed, assume the contrary. Then x˛ a y˛ D xˇ aı yˇ for some ˛; ˇ < and some distinct ; ı < such that ˛ and ˇ ı. But this is a contradiction.

Corollary 9.15. If is a regular cardinal, then jI 0 .G/j D 22 , and for every I 2 I 0 .G/, I is nowhere dense in U.G/. Proof. Let A 2 ŒG and let B be a subset of A guaranteed by Lemma 9.14. Then jU.B/j D 22 and for any distinct p; q 2 U.B/, Ip \ Iq D ;. To see that I is nowhere dense in U.G/, suppose that U.A/\I ¤ ;. If U.B/\I D ;, we are done. Otherwise I D Ip for some p 2 U.B/. Pick C 2 ŒB such that C … p. Then U.C / \ I D ;.

135

Section 9.1 Left Ideals

The next theorem covers in some sense the case where is a singular cardinal. Theorem 9.16. If > !, then there is a decomposition J of U.G/ into closed left ideals of ˇG such that (1) the corresponding quotient space of U.G/ is homeomorphic to U. /, (2) for every J 2 J, J G J , and (3) for every J 2 J, J is nowhere dense in U.G/. The proof of Theorem 9.16 is based on the following lemma. Lemma 9.17. Let > !. Then there is a surjective function f W G ! such that (a) for every ˛ < , jf 1 .˛/j < , and (b) whenever x; y 2 G and f .x/ < f .y/, one has f .xy/ D f .yx/ D f .y/. Proof. Construct inductively a -sequence .G˛ /˛< of subgroups of G such that (i) for every ˛ < , jG˛ j < , (ii) for every ˛ < , G˛ G˛C1 , (iii) for every limit ordinal ˛ < , G˛ D S (iv) ˛< G˛ D G.

S ˇ <˛

Gˇ , and

Note that G is a disjoint union of nonempty sets G˛C1 n G˛ , where ˛ < , and G0 . Deﬁne f W G ! by ´ ˛ f .x/ D 0

if x 2 G˛C1 n G˛ if x 2 G0 :

Clearly, f is surjective and satisﬁes (a). To check (b), let x; y 2 G and f .x/ < f .y/. Then x 2 Gˇ and y 2 G˛C1 n G˛ for some ˇ ˛ < . It follows that both xy and yx also belong to G˛C1 n G˛ . Hence, f .xy/ D f .yx/ D f .y/. Proof of Theorem 9.16. Let f W G ! be a function guaranteed by Lemma 9.17 and let f W ˇG ! ˇ be the continuous extension of f . Then (i) f .U.G// D U. / and f

1

.U. // D U.G/,

(ii) f .qp/ D f .p/ for all p 2 U.G/ and q 2 ˇG, (iii) f .px/ D f .p/ for all p 2 U.G/ and x 2 G, and (iv) for every u 2 U. /, f

1

.u/ is nowhere dense in U.G/.

136

Chapter 9 Ideal Structure of ˇG

Indeed, (i) follows from surjectivity of f and condition (a). To see (ii), let A 2 p. For every x 2 G, let Ax D A n ¹y 2 G W f .y/ f .x/º. Then Ax 2 p and by condition (b), f .xy/ D f .y/ 2 f .A/ for all y 2 Ax . Consequently, B D S x2G xAx 2 qp and f .B/ f .A/. Hence, f .qp/ D f .p/. The check of (iii) is 1

similar. Finally, to see (iv), let A 2 ŒG and suppose that U.A/ \ f .u/ ¤ ;. Then E D f .A/ 2 u. Pick D 2 ŒE such that D … u and let B D f 1 .D/ \ A. Then 1 1 B A, U.B/ ¤ ;, but f .B/ … u, and so U.B/ \ f .u/ D ;. Hence, f .u/ is nowhere dense in U.G/. 1 Now let J D ¹f .u/ W u 2 U. /º. It then follows from (i)-(iv) that J is as required. Applying Theorem 9.16 in the case > ! and Corollary 9.15 in the case D !, we obtain the following result.

Theorem 9.18. jI 0 .G/j D 22 , and for every I 2 I 0 .G/, I is nowhere dense in U.G/. We conclude this section by the following consequence of Theorem 9.18. Corollary 9.19. ƒ.ˇG/ D G. Proof. We have to show that for every p 2 G , p W ˇG 3 x 7! px 2 ˇG is not continuous. Without loss of generality one may suppose that p 2 U.G/. By Theorem 9.18, there is a nontrivial decomposition I of U.G/ into closed left ideals of ˇG such that for every I 2 I, IG I . Let J be the member of I containing p. Pick any K 2 I different from J and any q 2 K. Then pq 2 K and cl.pG/ J . Consequently, p is discontinuous at q.

9.2

Right Ideals

In this section we prove the following result.

Theorem 9.20. For every inﬁnite Abelian group G of cardinality , ˇG contains 22 minimal right ideals.

The proof of Theorem 9.20 involves some additional concepts. The Bohr compactiﬁcation of a topological group G is a compact group bG together with a continuous homomorphism e W G ! bG such that e.G/ is dense in bG and the following universal property holds: For every continuous homomorphism h W G ! K from G into a compact group K there is a continuous homomorphism hb W bG ! K such that h D hb ı e. In the case where G is a discrete Abelian group, the Bohr compactiﬁcation can be naturally deﬁned in terms of the Pontryagin duality

Section 9.2 Right Ideals

137

as follows. Let GO be the dual group of G and let GO d be the group GO reendowed with the discrete topology. Then bG is the dual group of GO d . The mapping e W G ! bG is given by e.x/./ D .x/, where x 2 G and 2 GO d . It is injective. (See [34, 26.11 and 26.12].) Recall that ﬁlters F and G on a set X are incompatible if A \ B D ; for some A 2 F and B 2 G . A ﬁlter F on a space X is open if F has a base of open subsets of X . In order to prove Theorem 9.20, we show the following.

Theorem 9.21. For every inﬁnite Abelian group G of cardinality , there are 22 pairwise incompatible open ﬁlters on bG converging to zero. Before proving Theorem 9.21, let us show how it implies Theorem 9.20.

Proof of Theorem 9.20. Let T denote the Bohr topology on G, that is, the topology induced by the mapping e W G ! bG, and let S D Ult.T /. By Theorem 9.21, there are 22 pairwise incompatible open ﬁlters on bG converging to zero. Considering the restriction of the ﬁlters to e.G/, we conclude that there are pairwise incompatible open ﬁlters F˛ , ˛ < 22 , on .G; T / converging to zero. For each ˛ < 22 , let J˛ D F˛ n¹0º. Then by Lemma 7.3, each J˛ is a closed right ideal of S, and since the ﬁlters F˛ are pairwise incompatible, the ideals J˛ are pairwise disjoint. Furthermore, since .G; T / is a subgroup of a compact group, by Lemma 7.10, S contains all the idempotents of G , in particular, the idempotents of K.ˇG/. Consequently, S \ K.ˇG/ ¤ ;. But then, by Proposition 6.24, K.S / D K.ˇG/ \ S. It follows from this that every minimal right ideal R of S is contained in a minimal right ideal R0 of ˇG, and the correspondence R 7! R0 is injective. Consequently, the number of minimal right ideals of ˇG is greater than or equal to that of S, and so it is 22 . To prove Theorem 9.21, we need three lemmas. The ﬁrst of them is an elementary fact on inﬁnite Abelian groups. Lemma 9.22. Let G be an inﬁnite Abelian group of cardinality . Then G admits a homomorphism onto one of the following groups: L L (1) Z, ! Zp , Zp1 and p2Q Zp if D !, L L (2) Zp and Zp 1 if > ! and cf. / > !, L L L L L L (3) Zp , Zp 1 , p2Q p Zp and p2Q p Zp 1 if > ! and cf. / D !. Here, p is a prime number and Q is an inﬁnite subset of the primes. Zp1 denotes the quasicyclic p-group. If > ! and cf. / D !, . p /p2Q is an inﬁnite increasing sequence of uncountable cardinals coﬁnal in , that is, supp2Q p D .

138

Chapter 9 Ideal Structure of ˇG

Proof. If G is ﬁnitely generated, then D ! and G admits a homomorphism onto Z. Therefore, one may assume that G is not ﬁnitely generated. We ﬁrst prove that G admits a homomorphism onto a periodic group of cardinality . Let ¹ai W i 2 I º be a maximal independent subset of G L and let A D hai W i 2 I i be the subgroup generated by ¹ai W i 2 I º. Then A D i2I hai i, and for every nonzero g 2 G, one has hgi \ A ¤ ¹0º, so G=A is periodic. If jG=Aj D , we are done. Suppose that jG=Aj < . Then jAj D and jI j D , because G is not ﬁnitely generated. We show that Lthere is a subgroup H of G and a subset I1 I with jI1 j D such that G D H ˚ i2I1 hai i. To this end, choose a complete set S for representatives of the cosets of A in G, and let H0 D hSi \ A. Deﬁne I0 I by I0 D ¹i 2 I W x.i / ¤ 0 for some x 2 H0 º and put I1 D I n I0 . If G=A is ﬁnite, I0 is ﬁnite as well. If G=A is inﬁnite, jI0 j jG=Aj, because jhSij D jG=Aj and then jH0 j jG=Aj. In any case, jI0 j < , and consequently jI1 j D . Let A0 D hai W i 2 I0 i; A1 D hai W i 2 I1 i and H D hS [ A0 i: We claim that G D H ˚A1 . Indeed, since G D hS [A0 [A1 i, one has H CA1 D G. To see that H \A1 D ¹0º, let g 2 H \A1 . Then g D d Cc0 D c1 for some d 2 hSi, c0 2 A0 and c1 2 A1 . Consequently, d D c0 C c1 2 A. But then d 2 H0 A0 . Hence, c1 D 0, and g D 0. L Having established that G D H ˚ i2I1 hai i, we obtain that G admits a homoL morphism onto i2I1 hai i, and so onto a periodic group of cardinality . Now let G be a p-group. Then there is a so-called basic subgroup B of L G (see [28, Theorem 32.3]). We have that B is a direct sum of cyclic groups, say B D j 2J hbj i, L 1 and G=B is divisible, that is, isomorphic to Zp , where 0 . Suppose that jG=Bj D . Then > 0, and D if L> !. It follows that G admits a homomorphism onto Zp1 if D !, and onto Zp1 if > !. Now suppose that jG=Bj < . Then jBj D , and consequently jJ j D . It follows that G D L C ˚ j 2J1 hbj i for some subgroup C of G and a subset J1 J with L jJ1 j D (see the ﬁrst L part of the proof). Hence, G admits a homomorphism onto j 2J1 hbj i, and so onto Zp . L Finally, let G be periodic. Then G D p2M Gp , where M is the set of all primes p such that the p-primary component Gp of G is nontrivial. If jGp j D for some p 2 M , we are done, because then G admits a homomorphism onto Gp , a p-group of cardinality . Suppose that jGp j < for each p 2 M . Then M is inﬁnite and cf. / L D !. If D !, all Gp are ﬁnite, and so G admits a homomorphism onto p2M Zp . Suppose that > !. For each p 2 M , put p D jGp j. Clearly supp2M p D . Choose an inﬁnite subset N M such that . p /p2N is an increasing sequence of uncountable cardinals coﬁnal in . By the previous paragraph, for

Section 9.2 Right Ideals

139

each p 2 N , Gp L admits a homomorphism onto a group Kp of cardinality p which L is isomorphic to p Zp or p Zp1 . It L follows that there is an inﬁnite subset Q NL such that either Kp is isomorphic to p Zp forL all p 2 Q or Kp is isomor1 phic to Z for all p 2 Q. Then the group K D p2Q Kp is isomorphic to L p p L L L 1 p2Q p Zp or p2Q p Zp , jKj D , and G admits a homomorphism onto K. Now, using Lemma 9.22 and the Pontrjagin duality, we prove the following statement on bG. Lemma 9.23. For every inﬁnite discrete Q Abelian Q group G of cardinality , bG admits a continuous homomorphism onto 2 T or 2 Zp . Q Q Here, both products 2 T and 2 Zp are endowed with the product topology. Proof. The dual groups of continuous homomorphic images of bG are the subgroups of GO d and the dual groups of homomorphic images of G are the Q closed subgroups Q of GOL(see [34, Theorems 23.25 and 24.8]). The dual groups of 2 T and 2 Zp L are 2 Z and 2 Zp , respectively. Consequently, in order to prove the lemma, it sufﬁces to show that G admits onto a group whose dual group La homomorphism L contains an isomorphic copy of 2 Z or 2 Zp . Consider two cases. Case 1: L D !. Then G L admits a homomorphism onto one of Q the following 1 1 groups: p . Their dual groups are T , ! Zp , Z.p / Q Z, ! Zp , Zp and p2Q Z1 and p2Q Zp , respectively. Here, Z.p / denotes the group of p-adic integers. The L second group is algebraically isomorphic to 2! Zp . The others contain L torsion-free subgroups of cardinality 2! , and so contain an isomorphic copy of 2! Z. a homomorphism LCase 2:L > !. Then L L onto one of the following groups: L G admits L 1, 1 (the two latter groups apZ , Z Z and p p p p2Q p p2QQ p ZpQ Q Q 1 pear Q if cf. / D !). Their dual groups are Z , p Z.p /, p2Q p Zp Q 1 /, respectively. The ﬁrst group is algebraically isomorphic to and Z.p p2Q p L 2 Zp . The others contain L torsion-free subgroups of cardinality 2 , and so contain an isomorphic copy of 2 Z. The third lemma deals with products of topological spaces. Lemma 9.24. Let be an inﬁnite cardinal. For each ˛ < Q , let X˛ be a space having at least two disjoint nonempty open sets, and let X D ˛< X˛ . Then there are at least 2 many pairwise incompatible open ﬁlters on X converging to the same point. Q Before proving Lemma 9.24 note that if each factor in an inﬁnite product X D n

140

Chapter 9 Ideal Structure of ˇG

subsets of Xn , let xQ n 2 Vn , and let x D .xn /n m: S S It follows that U D m

Q jLj D , and for each ˛ 2 L, n

9.3

The Structure Group of K.ˇG /

In this section we prove the following result. Theorem 9.25. Let G be an inﬁnite group of cardinality embeddable into a direct sum of countable groups. Then the structure group of K.ˇG/ contains a free group on 22 generators.

Section 9.3 The Structure Group of K.ˇG/

141

Since every Abelian group can be isomorphically embedded into a direct sum of groups isomorphic to Q or Zp1 , we obtain from Theorem 9.25 as a consequence that Corollary 9.26. For every inﬁnite Abelian group G of cardinality , the structure group of K.ˇG/ contains a free group on 22 generators. Recall that if a semigroup S has a smallest ideal which is a completely simple semigroup, then for every p 2 E.K.S//, pSp K.S / is a maximal group in S with identity p. The ﬁrst step in the proof of Theorem 9.25 is the following result. L Theorem 9.27. Let A D ¹x 2 ! Z2 W jsupp.x/j D 1º and let p 2 E.K.H//. Then the elements p C q C p 2 K.H/, where q 2 A , generate a free group. Proof. Let q1 ; : : : ; qn be distinct elements of A , let G be the subgroup of p C H C p generated by p C qi C p, i D 1; : : : ; n, and let F be a free group on n generators, say x1 ; : : : ; xn . It sufﬁces to show that there exists a homomorphism of G into F sending p C qi C p to xi for each i D 1; : : : ; n. By Corollary 1.24, F can be algebraically embedded into a compact group K. Without loss of generality one may suppose that F K. Partition A into subsets Ai , i D 1; : : : ; n, such that Ai 2 qi . Deﬁne f0 W A ! K by f0 .a/ D xi if a 2 Ai : For every n < !, let an denote L the element of A with supp.an / D ¹nº. Extend f0 to a local homomorphism f W ! Z2 ! K by f .an1 C C ank / D f0 .an1 / f0 .ank / where 1 k < ! and n1 < < nk < !. Then f W H ! K is a homomorphism such that f .qi / D xi . Since f .p/ is the identity, it follows that f .p C qi C p/ D f .p/f .qi /f .p/ D f .qi / D xi : Lemma 9.28. Let G be a countable group, let T0 be a regular left invariant topology on G, and let .Un /1n

142

Chapter 9 Ideal Structure of ˇG

(3) Vn Vn1 , and (4) yk;m Vn Vk n VkC1 for all k; m < ! with k C m D n 2, where ¹yk;m W m < !º is an enumeration of Vk n VkC1 ﬁxed immediately after VkC1 has been chosen. T Then n

i D 1; : : : ; n;

generate a free group. Hence, the elements pqi p, i D 1; : : : ; n, also generate a free group. Given a set D and A D, let Q.A/ denote the set of countably incomplete ultraﬁlters from A ˇD.

Lemma 9.30. If jAj D !, then jQ.A/j D 22 .

Proof. We show that there are 22 uniform countably incomplete ultraﬁlters on . Let ¹An W n < !º be a partition of such that jAn j D for everySn < ! and let U be the set of uniform ultraﬁlters u on such that for every m < !, mn

Section 9.3 The Structure Group of K.ˇG/

143

Theorem 9.31. Let G be a group of cardinality > ! embeddable into a direct sum of countable groups. Then there is A G with jAj D such that for every p 2 E.K.ˇG//, the elements pqp 2 K.ˇG/, where q 2 Q.A/, generate a free group. L Proof. Let H D ˛< H˛ be a direct sum of countable groups and let G be a subgroup of H . Note that for every ˛ < , there is x 2 G n¹1º with min supp.x/ ˛. L Indeed, since j ˇ <˛ Hˇ j < , there are distinct y; z 2 G such that y.ˇ/ D z.ˇ/ for all ˇ < ˛. Put x D yz 1 . Then x ¤ 1 and for each ˇ < ˛, x.ˇ/ D 1ˇ . Choose inductively a -sequence .x /< in G n ¹1º such that max supp.xˇ / < min supp.x / L whenever ˇ < < . For every < , ˇ 2supp.x / Hˇ is countable. Therefore, without loss of generality one may suppose that jsupp.x /j D 1. Let A D ¹x W < º: We have to show that for any p 2 E.K.ˇG// and for any distinct q1 ; : : : ; qn 2 Q.A/, the elements pq1 p; : : : ; pqn p generate a free group. For every ˛ < , deﬁne the idempotent p˛ 2 ˇH˛ by p˛ D ˛ .p/, where ˛ W H ! H˛ is the projection and ˛ W ˇH ! ˇH˛ is the continuous extension of ˛ . Endow H˛ with a regular left invariant topology in which p˛ converges to 1˛ (Theorem 7.17). Now endow H with the topology induced by the product topology on Q H and let T denote the topology on G induced from H . Note that p 2 Ult.T / ˛ ˛< and A Ult.T /. Partition A into inﬁnite subsets Ai , i D 1; : : : ; n, such that Ai 2 qi , and for each i D 1; : : : ; n, partition Ai into inﬁnite subsets Aij , j < !, such that Aij … qi . Also partition the set ± ° M Z2 W jsupp.x/j D 1 BD x2 !

into inﬁnite subsets Bi , i D 1; : : : ; n, and enumerate Bi as ¹bij W j < !º without repetitions. L For every ˛ < , deﬁne a local homomorphism h˛ W H˛ ! ! Z2 as follows. If there is a 2 A with supp.a/ D ¹˛º, then pick a clopen neighborhood U˛ of 1˛ such that ˛ .a/ … U˛ , pick i; j such that a 2 Aij , and deﬁne h˛ by ´ 0 if x 2 U˛ h˛ .x/ D bij otherwise: If there is no a 2 A with supp.a/ D ¹˛º, deﬁne h ˛ by h˛ .x/ D 0. L Now deﬁne a local homomorphism h W H ! ! Z2 by h.x1 C C x t / D h˛1 .x1 / C C h˛ t .x t /;

144

Chapter 9 Ideal Structure of ˇG

where t 2 N, ˛1 < < ˛ t < , L and xs 2 H˛s for each s D 1; : : : ; t , and let f D hjG . Then f W .G; T / ! ! Z2 is a local homomorphism such that f .Aij / D ¹bij º for every i D 1; : : : ; n and j < !. It follows that M f W Ult.T / ! ˇ Z2 !

: : ; f .qn / are distinct elements from is a surjective homomorphism and f .q1 /; : L B . Since f is surjective, f .p/ 2 E.K.ˇ. ! Z2 /// (Lemma 6.25). Note that M E K ˇ Z2 D E.K.H//: !

Consequently, by Theorem 9.27, the elements f .p/ C f .qi / C f .p/ D f .pqi p/;

i D 1; : : : ; n;

generate a free group. Hence, the elements pqi p, i D 1; : : : ; n, also generate a free group.

9.4

K.ˇG / is not Closed

In this section we prove the following result. Theorem 9.32. Let G be an inﬁnite group of cardinality and assume that is not Ulam-measurable. Then both K.ˇG/ and E.cl K.ˇG// are not closed. Pick p 2 E.K.ˇG// and let T D C.p/ D ¹x 2 G W xp D pº. By Theorem 7.17, there is a regular extremally disconnected left invariant topology T on G such that Ult.T / D T . Note that K.T / is a right zero semigroup consisting of the idempotents from the minimal right ideal p.ˇG/ of ˇG, that is, K.T / D E.p.ˇG//. It follows that in order to prove Theorem 9.32, it sufﬁces to show that Theorem 9.33. There are elements in cl K.T / which are not in .cl K.T //T . Indeed, let q 2 .cl K.T // n ..cl K.T //T /. Then clearly, q 2 cl K.ˇG/. To see that q … K.ˇG/, assume the contrary. Then q 2 K.ˇG/ \ T D K.T /, and so q is an idempotent. Consequently, q D qq 2 .cl K.T //T , a contradiction. To see that q … E.cl K.ˇG//, assume the contrary. Then q D qq 2 .cl K.T //T , a contradiction. To prove Theorem 9.33, we need the following lemma. Lemma 9.34. Let X be a nondiscrete regular extremally disconnected space and assume that jX j is not T Ulam-measurable. Then there is a sequence .Un /n

145

Section 9.4 K.ˇG/ is not Closed

T Proof. Choose a family B of clopen sets in X such that Y D B is not open and D jBj is as small as possible. Since X is extremally disconnected and Y is closed, it follows that cl int Y D int Y , that is, int Y is clopen.TLet C D ¹U n int Y W U 2 Bº. Then jC j D , all members of C are clopen, and C D Z is a nonempty closed nowhere dense set. Enumerate C as ¹U˛ W ˛ < º. Deﬁne a decreasing T -sequence .W˛ /˛< of clopen T subsets of X by putting W0 D X and W D ˛ ˇ <˛ Uˇ for ˛ > 0. Then T W˛ D ˇ <˛ Wˇ if ˛ is a limit ordinal and ˛< W˛ D Z. Deﬁne f W X n Z ! by f .x/ D ˛ if x 2 W˛ n W˛C1 : Pick x 2 Z and let F denote the ﬁlter on with a base consisting T of subsets of the form f .U n Z/ where U is a neighborhood of x 2 X . Clearly, F D ;. We claim that F is a -complete ultraﬁlter, and so D !. To see that F isS an ultraﬁlter, let A . Then X nZ is a disjoint union of open sets VA D f 1 .A/ D ˛2A W˛ n W˛C1 and V nA D f 1 . n A/. Since X is extremally disconnected, there is a neighborhood U of x such that either U \ V nA D ; or U \ VA D ;, so either U n Z VA or U n Z V nA . It follows that either A 2 F or n A 2 F . To see that F is -complete, let < , and for every ˛ < , let A˛ 2 F . Then 1 .A / [ Z is a neighborhood of x. Consequently, by for every ˛ < , V˛ D fT ˛ the minimality of , V D T˛< V˛ is also a neighborhood of x. Deﬁne A 2 F by A D f .V n Z/. Then A ˛< A˛ . Proof of Theorem 9.33. By Theorem 9.18, there is an inﬁnite decomposition I of U.G/ into closed left ideals of ˇG. Pick a sequence .In /n

S Then by Corollary 2.24, either qm 2 cl n

146

Chapter 9 Ideal Structure of ˇG

Remark 9.35. In the case D ! Theorem 9.33 can be strengthened as follows: There are elements in cl K.T / which are not in T 2 . Indeed, inTthis case the sequence .Wn /n

References

It has long been known that U.G/ can be decomposed into 22 left ideals [14]. That it can be decomposed into 22 closed left ideals was established by I. Protasov [61] in the case where is a regular cardinal and by M. Filali and P. Salmi [24] for all . Theorems 9.3, 9.13, and 9.16 are from [109]. Corollary 9.19 is a partial case of the result of A. Lau and J. Pym [44]. Our proof of Corollary 9.19 is from [24]. Theorem 9.20 is from [104]. An introduction to the Pontryagin duality can be found in [55] and [34]. See also [50] and [17]. Theorem 9.25 is from [111]. Corollary 9.26 was proved also independently by S. Ferri, N. Hindman, and D. Strauss [23]. Theorem 9.27 is due to N. Hindman and J. Pym [36]. Theorem 9.32 complements the result from [106] which says that, for every inﬁnite semigroup S embeddable algebraically into a compact group, both K.ˇS / and E.cl K.ˇS // are not closed.

Chapter 10

Almost Maximal Topological Groups

In this chapter almost maximal topological groups and their ultraﬁlter semigroups are studied. A topological (or left topological) group is said to be almost maximal if the underlying space is almost maximal. We show that the ultraﬁlter semigroup of any countable regular almost maximal left topological group is a projective in the category F of ﬁnite semigroups. L Assuming MA, for every projective S in F, we construct a group topology T on ! Z2 such that Ult.T / is isomorphic to S. We show that every countable almost maximal topological group contains an open Boolean subgroup and its existence cannot be established in ZFC. We then describe projectives in F. These are certain chains of rectangular bands. We conclude by showing that the ultraﬁlter semigroup of a countable regular almost maximal left topological group is its topological invariant.

10.1

Construction

Deﬁnition 10.1. An object S in some category is an absolute coretract if for every surjective morphism f W T ! S there exists a morphism g W S ! T such that f ı g D idS . Let C denote the category of compact Hausdorff right topological semigroups. The next lemma gives us some simple examples of absolute coretracts in C. Lemma 10.2. Finite left zero semigroups, right zero semigroups and chains of idempotents are absolute coretracts in C. Proof. Let S be a ﬁnite left zero semigroup, let T be a compact Hausdorff right topological semigroup, and let f W T ! S be a continuous surjective homomorphism. Pick a minimal left ideal L of T . For each e 2 S, f 1 .e/ is a right ideal of T , so pick a minimal right ideal Re f 1 .e/ of T , and let g.e/ be the identity of the group Re \ L. Then ¹g.e/ W e 2 S º is a left zero semigroup and f ı g D idS . The proof for ﬁnite right zero semigroups is similar. Finally, suppose that S is a ﬁnite chain of idempotents, say e1 > > en . Construct inductively a chain of idempotents g.e1 / > > g.en / in T such that f .g.ei // D ei for each i D 1; : : : ; n. As g.e1 / pick any idempotent in T1 D f 1 .e1 /. Now ﬁx k < n and assume that we have constructed idempotents g.e1 / > > g.ek / in T such that f .g.ei // D ei . Let TkC1 D f 1 .¹e1 ; : : : ; ekC1 º/. Pick

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Chapter 10 Almost Maximal Topological Groups

a minimal right ideal RkC1 g.ek /TkC1 of TkC1 . Note that RkC1 f 1 .ekC1 / (Lemma 6.25), so g.ek / … RkC1 . And pick a minimal left ideal LkC1 TkC1 g.ek / of TkC1 . Deﬁne g.ekC1 / to be the identity of the group RkC1 \ LkC1 . Deﬁnition 10.3. Let S be a ﬁnite semigroup. We say that S is an absolute Hcoretract if for every surjective proper homomorphism ˛ W H ! S there exists a homomorphism ˇ W S ! H such that ˛ ı ˇ D idS . Clearly, every ﬁnite absolute coretract in C is an absolute H-coretract. Theorem 10.4. Assume p DL c. Let S be a ﬁnite absolute H-coretract. Then there exists a group topology T on ! Z2 such that Ult.T / is isomorphic to S. The proof of Theorem 10.4 is based on the following lemma. Lemma 10.5. Let G be an inﬁnite group, let S D ¹pi W i < mº be a ﬁnite semigroup in G , and let F be the ﬁlter on G such that F D S. For every i < m, let Ai 2 pi , and for every x 2 G, let Bx 2 F . Then there is a sequence .xn /n

\

¹Bx W x 2 FP..xn /n

Choose C0 2 p0 such that C0 A0 and for each p 2 S, one has C0 p Ai where i is deﬁned by p0 p D pi . Pick x0 2 C0 . Now ﬁx l > 0 and assume that we have constructed a sequence .xn /n

149

Section 10.1 Construction

(v) whenever k < l, n0 < < nk < l and p 2 S, one has xn0 xnk Cl p Ai where i is deﬁned by pi D pin0 pink pj p, T (vi) whenever p 2 S, Cl p ¹Bx W x 2 FP..xn /n

\

¹Bx W x 2 FP..xn /n

Then pick xl 2 Cl . Proof of Theorem 10.4. Let G D basis of G, that is,

L !

yn .m/ D

Z2 and let Y D ¹yn W n < !º be the standard ´

1 0

if m D n otherwise.

Let T0 denote the group topology on G with a neighborhood base at 0 consisting of subgroups FS..yn /mn

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Chapter 10 Almost Maximal Topological Groups

As .x0;n /n

10.2

Properties

Deﬁnition 10.7. An object S in some category is a projective if for every morphism f W S ! Q and for every surjective morphism g W T ! Q there exists a morphism h W S ! T such that g ı h D f . Note that every projective is an absolute coretract. In many categories these notions coincide but not in all.

151

Section 10.2 Properties

Deﬁnition 10.8. Let S be a ﬁnite semigroup. We say that S is an H-projective if for every homomorphism ˛ W S ! Q of S into a ﬁnite semigroup Q and for every surjective proper homomorphism ˇ W H ! Q there exists a homomorphism W S ! H such that ˛ D ˇ ı . Clearly, every H-projective is an absolute H-coretract. Let F denote the category of ﬁnite semigroups. Lemma 10.9. Every H-projective (absolute H-coretract) is a projective (absolute coretract) in F. Proof. Let S be an H-projective, let Q and T be ﬁnite semigroups, let ˛ W S ! Q be a homomorphism, and let ˇ W T ! Q be a surjective homomorphism. Pick a surjective proper homomorphism ı W H ! T . Then ˇ ı ı W H ! Q is a surjective proper homomorphism. Consequently, since S is an H-projective, there is a homomorphism " W S ! H such that ˛ D ˇ ı ı ı ". Deﬁne the homomorphism W S ! T by

D ı ı ". Then ˛ D ˇ ı . The proof for an absolute H-coretract is similar. Theorem 10.10. The ultraﬁlter semigroup of a countable regular almost maximal left topological group is an H-projective. Before proving Theorem 10.10 we establish the following fact. Lemma 10.11. Let .G; T / be an almost maximal T1 left topological group and let S D Ult.T /. Then for every homomorphism ˛ W S ! Q, there are an open neighborhood X of the identity of .G; T / and a local homomorphism f W X ! Q such that f D ˛. Proof. For each p 2 S, choose Ap 2 p such that Ap \ Aq D ; if p ¤ q. Then, for each p 2 S, choose Bp 2 p such that Bp q Apq for all q 2 S. This can be done because the mapping ˇG 3 x 7! xq 2 ˇG is continuous and S is ﬁnite. Choose the subsets Bp in addition so that Bp Ap and XD

[

Bp [ ¹1º

p2S

is open in .G; T /. Deﬁne f W X ! Q by putting for every p 2 S and x 2 Bp , f .x/ D ˛.p/. The value f .1/ does not matter. We claim that f is a local homomorphism and f D ˛. It sufﬁces to check the ﬁrst statement. Let x 2 X n ¹1º. Then x 2 Bp for some p 2 S . For each q 2 S, choose Cq 2 q such S that Cq Aq and xCq Apq . Choose a neighborhood U of 1 2 X such that U q2S Cq [ ¹1º and xU X . Now let

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Chapter 10 Almost Maximal Topological Groups

y 2 U n ¹1º. Then y 2 Cq for some q 2 S, so y 2 Aq and then y 2 Bq . Hence f .x/f .y/ D ˛.p/˛.q/. On the other hand, xy 2 Apq , then xy 2 Bpq , and so f .xy/ D ˛.pq/ D ˛.p/˛.q/: Hence f .xy/ D f .x/f .y/. Proof of Theorem 10.10. Let .G; T / be a countable regular almost maximal left topological group and let S D Ult.T /. Let Q be a ﬁnite semigroup, let ˛ W S ! Q be a homomorphism, and let ˇ W H ! Q be a surjective proper homomorphism. By Lemma 10.11, there are an open neighborhood X of the identity of .G; T / and a local homomorphism f W X ! Q such that f D ˛, so ˛ is proper. Now by Corollary 8.14, there is a proper homomorphism W Ult.T / ! H such that ˛ D ˇ ı . Hence, S is an H-projective. Recall that a band is a semigroup of idempotents. Theorem 10.12. The ultraﬁlter semigroup of a countable regular almost maximal left topological group is a band. Proof. Let .G; T / be a countable regular almost maximal left topological group and let S D Ult.T /. Assume on the contrary that S is not a band. By Corollary 8.21, S contains no nontrivial ﬁnite groups. Consequently, S contains a 2-element null subsemigroup, that is, there are distinct p; q 2 S such that q 2 D qp D pq D p 2 D p: It follows from q 2 D p 2 that for any A 2 p and B 2 q, Bq \ Ap ¤ ;, and hence by Corollary 2.23, either Bq \ Ap ¤ ; or Bq \ Ap ¤ ;. Consider two cases. Case 1: Bq \ Ap ¤ ; for some A 2 p and B 2 q. Then yq D p 0 p for some y 2 B and p 0 2 A, so q D y 1 p 0 p D y 1 p 0 pp D qp D p, a contradiction. Case 2: Bq \ Ap ¤ ; for all A 2 p and B 2 q. Then qA;B q D xA;B p for some 1 q 1 qA;B 2 B and xA;B 2 A, so p D xA;B A;B q D rA;B q where rA;B D xA;B qA;B . Since T is regular, it follows from p D rA;B q that rA;B 2 S. And since S is ﬁnite, it then follows from qA;B D xA;B rA;B that qB D prB for some rB 2 S and qB 2 B. Consequently, q D pr for some r 2 S. But then q D pr D ppr D pq D p, again a contradiction. Theorem 10.12, Theorem 10.10 and Theorem 10.4 raise the question of characterizing ﬁnite bands which are H-projectives. We address this question in Section 10.4. Now we consider the structure of a countable almost maximal topological group. Let X be a space with a distinguished point 1 2 X and let f W X ! X be a homeomorphism with f .1/ D 1. We say that f is trivial if the set of ﬁxed points of f is a neighborhood of f .

Section 10.2 Properties

153

Lemma 10.13. Let X be a countable regular local left group and suppose that K.Ult.X // is ﬁnite. Then every local automorphism on X is trivial. Proof. Assume on the contrary that there is a nontrivial local automorphism f W X ! X . For every n 2 N, let Xn D ¹x 2 X W jO.x/j > nº. Note that Xn is open, f n .x/ ¤ x for all x 2 Xn , and XnC1 Xn . By Corollary 8.23, it sufﬁces to consider the following two cases. Case 1: for every n 2 N, 1 2 cl Xn . Let F be the ﬁlter on X with a base consisting of subsets U \ Xn , where U runs over neighborhoods of 1 2 X and n 2 N, and let R D F . Then for every p 2 R, all elements p; f .p/; .f /2 .p/; : : : are distinct. Indeed, otherwise .f /n .p/ D p for some n 2 N, and consequently by Corollary 2.18, ¹x 2 X W f n .x/ D xº 2 p, which contradicts Xn 2 p. Next, by Corollary 7.3, R is a right ideal of Ult.X /, so there is p 2 R \ K.Ult.X //. Since f is an automorphism on Ult.X /, it follows that .f /n .p/ 2 K.Ult.X // for every n. Hence, K.Ult.X // is inﬁnite, a contradiction. Case 2: f has ﬁnite order. Since f is nontrivial, 1 2 cl X1 . Let R1 D X1 \Ult.X /. Then R1 is a right ideal of Ult.X /, so there is an idempotent p 2 R1 \ K.Ult.X //. We have also that f .p/ ¤ p and clearly f .p/ 2 K.Ult.X //. But then, applying Theorem 8.31, we obtain that the structure group of K.Ult.X // is inﬁnite, again a contradiction. Lemma 10.14. Let G be a countable group endowed with an invariant topology. Suppose that for every a 2 G, the conjugation G 3 x 7! axa1 2 G is a trivial local automorphism. Then the inversion W G 3 x 7! x 1 2 G is a local automorphism. Proof. To see that is a local homomorphism, let a 2 G n ¹1º. Since G 3 x 7! axa1 2 G is a trivial, U D ¹x 2 G W axa1 D xº is a neighborhood of 1. Pick a neighborhood V of 1 such that V D V 1 U . Then for every x 2 V , .ax/ D .ax/1 D x 1 a1 D a1 ax 1 a1 D a1 x 1 D .a/.x/: Theorem 10.15. Every countable almost maximal topological group contains an open Boolean subgroup. Proof. Let G be a countable almost maximal topological group. By Lemma 10.13, the conjugations of G are trivial. Then by Lemma 10.14, the inversion W G 3 x 7! x 1 2 G is a local automorphism. Consequently by Lemma 10.13, is trivial, so ¹x 2 G W .x/ D xº is a neighborhood of 1. Since ¹x 2 G W .x/ D xº D B.G/, G contains an open Boolean subgroup by Lemma 5.3. We conclude this section by showing that the existence of a countable almost maximal topological group cannot be established in ZFC.

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Chapter 10 Almost Maximal Topological Groups

Theorem 10.16. The existence of a countable almost maximal topological group implies the existence of a P -point in ! . Proof. Let .G; T / be a countable almost L maximal topological group. By Theorem 10.15, one may suppose that G D ! Z2 . Let F be the neighborhood ﬁlter of 0 in T and let T denote the topology on G induced by the product topology on 0 Q Z . By Theorem 10.12 and Lemma 7.10, T0 T . Then by Theorem 5.19, each ! 2 point of .F / is a P -point. Combining Theorem 10.16 and Theorem 2.38 gives us that Corollary 10.17. It is consistent with ZFC that there is no countable almost maximal topological group. Remark 10.18. It is easy to see that Theorem 10.15 and Theorem 10.16 remain to be true if to replace ‘countable almost maximal topological group’ by ‘countable topological group .G; T / with ﬁnite K.Ult.T //’.

10.3

Semilattice Decompositions and Burnside Semigroups

This is a preliminary section for the next one. A partially ordered set is a semilattice if every 2-element subset ¹a; bº has a greatest lower bound a^b. Obviously, every semilattice is a commutative band with respect to the operation ^. Conversely, every commutative band is a semilattice with respect to the standard ordering on idempotents. That is, a b if and only if ab D ba D a, and then a ^ b D ab. We shall identify semilattices with commutative bands. A semilattice decomposition of a semigroup S is a homomorphism f W S ! of S onto a semilattice . Equivalently, a semilattice decomposition of S is a partition ¹S˛ W ˛ 2 º of S into subsemigroups with the property that for every ˛; ˇ 2 there is 2 such that S˛ Sˇ S and Sˇ S˛ S . A semigroup which is a union of groups is called completely regular. Theorem 10.19. Every completely regular semigroup decomposes into a semilattice of completely simple semigroups. Proof. Let S be a completely regular semigroup. For every a 2 S, let J.a/ D SaS . We ﬁrst note that if H is a subgroup of S containing a, then H J.a/ and for every h 2 H , J.h/ D J.a/. For h D haa1 2 J.a/ and a D ahh1 2 J.h/. It follows from J.a2 / D J.a/ that J.ab/ D J.ba/ for all a; b 2 S. For ab ab D a ba b 2 J.ba/, so ab 2 J.ba/, and similarly ba 2 J.ab/. Now we show that J.ab/ D J.a/ \ J.b/. That J.ab/ J.a/ \ J.b/ is obvious. Conversely, let c 2 J.a/ \ J.b/. Write c D uav D xby for some u; v; x; y 2 S.

Section 10.3 Semilattice Decompositions and Burnside Semigroups

155

Then c 2 D xbyuav 2 J.byua/ D J.abyu/. Consequently, c 2 J.abyu/ J.ab/, and so J.a/ \ J.b/ J.ab/. Hence J.ab/ D J.a/ \ J.b/. It follows that D ¹J.a/ W a 2 Sº is the semilattice of principal ideals of S under intersection and S 3 a 7! J.a/ 2 is a surjective homomorphism. For every a 2 S, the preimage of the element J.a/ 2 is the set Ja D ¹b 2 S W J.b/ D J.a/º. Being the preimage of an idempotent, Ja is a subsemigroup of S . To see that Ja is simple, let b 2 Ja . Write a D ubv for some u; v 2 S. Then a D eae D eubve where e is the identity of a group containing a. It follows that J.a/ J.eu/ J.e/ D J.a/, so eu 2 Ja . Similarly ve 2 Ja . Hence a 2 Ja bJa . Finally, to show that Ja is completely simple, it sufﬁces to show that Ja e is a minimal left ideal of Ja . Let b 2 Ja e. Write b D ue for some u 2 Ja and, since Ja is simple, e D vbw for some v; w 2 Ja . Then e D vuew D xew where x D vu 2 Ja . Let f be the identity of a group containing x and x 1 the inverse of x with respect to f . Then f e D f xew D xew D e and e D f e D x 1 xe D x 1 vue D x 1 vb 2 Ja b. A band is a completely simple semigroup if and only if it is rectangular, that is, isomorphic to the direct product of a left zero semigroup and a right zero semigroup. Corollary 10.20. Every band decomposes into a semilattice of rectangular bands. The next lemma tells us that a semilattice decomposition of a completely regular semigroup is in fact unique. Lemma 10.21. Let S be a completely regular semigroup and let ¹S˛ W ˛ 2 Y º be a decomposition of S into a semilattice of completely simple semigroups. Then the semigroups S˛ are precisely the maximal completely simple subsemigroups of S. Proof. Let f W S ! Y be a homomorphism of S onto a commutative band Y and let T be a simple subsemigroup of S. Then f .T / is a simple subsemigroup of Y . But simple commutative bands are trivial. Hence, T f 1 .˛/ for some ˛ 2 Y . Maximal completely simple subsemigroups of a completely regular semigroup S are called completely simple components of S. If S is a band, we say rectangular components instead of completely simple components. Recall that Green’s relations R; L; J on any semigroup S are deﬁned by aRb , aS 1 D bS 1 ; aLb , S 1 a D S 1 b; aJb , S 1 aS 1 D S 1 bS 1 : The proof of Theorem 10.19 shows that J-classes of a completely regular semigroup are its completely simple components, R-classes and L-classes are minimal right and left ideals of the components. Deﬁnition 10.22. Let 1 k < ! and 0 m < n < !. The Burnside semigroup B.k; m; n/ is the largest semigroup on k generators satisfying the identity x m D x n .

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Chapter 10 Almost Maximal Topological Groups

Note that B.k; 1; 2/ is the free band on k generators and B.k; 0; n/ is the Burnside group B.k; n/ (see Example 1.25). Theorem 10.23. For every ﬁxed n 2, the following statements are equivalent: (1) the semigroups B.k; 1; n/ are ﬁnite for all k 1, and (2) the groups B.k; 0; n 1/ are ﬁnite for all k 1. Proof. The implication .1/ ) .2/ is obvious. We need to prove .2/ ) .1/. Let B D B.k; 1; n/, let F be the free semigroup on a k-element alphabet A, and let h W F ! B be the canonical homomorphism. Note that for any u; v 2 F , h.u/ D h.v/ if and only if v can be obtained from u by a succession of elementary operations in each of which a subword w of a word is replaced by w n , or vice versa. For every w 2 F , let ct.w/ denote the set of letters of A appearing in w. Lemma 10.24. Let u; v 2 F . Then h.u/Jh.v/ if and only if ct.u/ D ct.v/. Proof. If h.u/Jh.v/, then h.u/ D h.v1 vv2 / and h.v/ D h.u1 uu2 /. Since the function ct is invariant under elementary operations, ct.v1 vv2 / D ct.u/, so ct.v/ ct.u/. Similarly, ct.u/ ct.v/. Hence, ct.u/ D ct.v/. Conversely, suppose that ct.u/ D ct.v/ D ¹x1 ; : : : ; xn º. Since B is a semilattice of its J-classes, it then follows that h.u/Jh.x1 / h.xn / and h.v/Jh.x1 / h.xn /. Hence, h.u/Jh.v/. By Lemma 10.24, the mapping B 3 h.w/ 7! ct.w/ A gives us the semilattice decomposition of B into completely simple components (D J-classes). The semilattice involved is the set of nonempty subsets of A under union. We need to show that J-classes of B are ﬁnite. For every nonempty C A, let JC denote the J-class of B corresponding to C . Clearly, for every a 2 A, J¹aº is just the group ¹a; a2 ; : : : ; an1 º (with identity an1 ). For every w 2 F , let .w/ denote the letter of A which has the latest ﬁrst appearance in w, 0 .w/ the subword of w that precedes the ﬁrst appearance of .w/, and .w/ the subword 0 .w/ .w/. Let .w/ denote the letter of A which has the earliest last appearance in w, 0 .w/ the subword of w that follows the last appearance of .w/, and .w/ the subword .w/0 .w/. Lemma 10.25. Let C A with jC j > 1 and let u; v 2 F with ct.u/ D ct.v/ D C . Then (1) h.u/Rh.v/ if and only if .u/ D .v/ and h.0 .u// D h.0 .v//, (2) h.u/Lh.v/ if and only if .u/ D .v/ and h.0 .u// D h.0 .v//.

Section 10.3 Semilattice Decompositions and Burnside Semigroups

157

Proof. (1) If h.u/Rh.v/, then h.u/ D h.vw/ for some w 2 F with ct.w/ D C . Clearly, .vw/ D .v/. It is easy to see that the functions and h0 are invariant under elementary operations. Consequently, .u/ D .v/ and h.0 .u// D h.0 .v//. Conversely, suppose that .u/ D .v/ and h.0 .u// D h.0 .v//. Then h..u// D h..v//. Consequently, in order to ﬁnish the proof it sufﬁces to show that for every w 2 F , h.w/Rh..w//. We proceed by the induction on the length jwj of w. The statement is obviously true if jwj D 1. Let jwj > 1 and suppose that the statement holds for all words of length < jwj. If w D .w/, there is nothing to prove. Let w ¤ .w/. Then w D w1 aw2 a for some words w1 ; w2 , possibly empty. Consequently, .w/ D .w1 aw2 / and by the inductive assumption, h..w1 aw2 //Rh.w1 aw2 /, so h..w//Rh.w1 aw2 /. But h.w1 aw2 / D h.w1 .aw2 /n / D h.w1 aw2 a/h.w2 .aw2 /n2 / D h.w/h.w2 .aw2 /n2 / and h.w/ D h.w1 aw2 /h.a/, so h.w1 aw2 /Rh.w/. Hence h..w//Rh.w/. The proof of (2) is similar. Lemma 10.25 gives us a one-to-one correspondence between R-classes (L-classes) of the J-class JC and the pairs .x; a/ where a 2 C and x 2 JC n¹aº . Lemma 10.26. Let S be a ﬁnitely generated completely regular semigroup and let C be a be a completely simple component of S. Suppose that C contains a ﬁnite number of minimal left (right) ideals. Then the structure group of C is ﬁnitely generated. Proof. Let T be the subsemigroup of S consisting of all completely simple components of S over C , let X be a ﬁnite generating subset of S, and let Y D X \ T . Then Y is a generating subset of T and C D K.T /. Suppose that C contains a ﬁnite number of minimal left ideals. Let H be a maximal subgroup of C , let R be the minimal right ideal of C containing H , and let E D E.R/. Then T e, where e 2 E, are the minimal left ideals of C and R \ .T e/, where e 2 E, the maximal subgroups of R. Let e1 2 E be the identity of H . We claim that ¹exe1 W x 2 Y [ Y 1 ; e 2 Eº is a generating subset of H . To see this, let h 2 H . Write h D x1 xn for some x1 ; : : : ; xn 2 Y [ Y 1 . Then h D e1 he1 D e1 x1 xn e1 . Deﬁne inductively e2 ; : : : ; en 2 E by ei xi 2 T eiC1 . Then ei xi D ei xi eiC1 D ei xi e1 eiC1 , and so h D .e1 x1 e1 /.e2 x2 e1 / .en xn e1 /. Now for each i D 1; : : : ; k, pick Ci A with jCi j D i and let ci D jJCi j. Clearly, c1 D n 1. Fix i > 1 and assume that ci1 is ﬁnite. Then by Lemma 10.25, the number of minimal right (left) ideals of JCi is i ci1 , and by Lemma 10.26, the cardinality gi of the structure group of JCi is ﬁnite, so ci D .i ci1 /2 gi is also ﬁnite.

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Chapter 10 Almost Maximal Topological Groups

From Theorem 10.23 we obtain the following. Corollary 10.27. The semigroups B.k; 1; 2/ and B.k; 1; 3/ are ﬁnite for all k 1. Proof. Since B.k; 0; 2/ is the Boolean group on k generators, jB.k; 0; 2/j D 2k . Hence, the result follows from Theorem 10.23.

10.4

Projectives

In this section we describe ﬁnite bands which are H-projectives. Let V denote the set of words of the form i1 i2 ip p p1 1 where p 2 N and iq ; q 2 ! for each q D 1; : : : ; p. Deﬁne the operation on V by 8 ˆ if p D q q ˆ : if p < q: i1 ip jpC1 jq q 1 It is easy to see that V is a band being decomposed into a decreasing chain of its rectangular components Vp whose elements are words of length 2p. Now let P denote the family of ﬁnite subsemigroups of V satisfying the following conditions for every p 2 N: (1) if i1 ip p 1 2 S, then both ip ¤ 0 and p ¤ 0, (2) if i1 ip p 1 2 S and iq ¤ 0 for some q 2 Œ1; p 1, then i1 iq 1 1 2 S, and dually, if i1 ip p 1 2 S and q ¤ 0 for some q 2 Œ1; p 1, then 1 1q 1 2 S (here, i1 iq 1 1 and 1 1q 1 denote the elements from Vq ), and (3) either ip D 1 for all i1 iq q 1 2 S with q p or p D 1 for all i1 iq q 1 2 S with q p. Theorem 10.28. Every semigroup from P is a projective in C. Proof. Let S 2 P. Let f W S ! Q be a homomorphism of S into a ﬁnite semigroup Q and let g W T ! Q be a continuous homomorphism of a compact Hausdorff right topological semigroup T onto Q. We adjoin the identities ;; 1Q ; 1T to S; Q; T respectively and extend f; g in the obvious way. We shall inductively construct a homomorphism h W S ! T such that g ı h D f . Let l D max¹p 2 N W S \ Vp ¤ ;º. For each p 2 Œ1; l, let Sp D S \ Vp p and Sp deﬁne ep 2 Sp by ep D 1 11 1. Also put S0 D ¹;º, e0 D ;, and S0 D qD0 Sq . If x D i1 iq 0 0ip p 0 0r 1 2 Sp with iq ; r ¤ 0, we put ? x D i1 iq 1 1 2 Sq and x ?? D 1 1r 1 2 Sr (if q D 0, x ? D ;, and if r D 0, x ?? D ;).

159

Section 10.4 Projectives

Lemma 10.29.

(a) For every x 2 S, x ? x D xx ?? D x.

(b) If x 2 Sp and y 2 Sq , where q < p, then .xy/?? D x ?? y and .yx/? D yx ? . (c) If x; y 2 Sp , then x ?? y ? D ep1 . Proof. (a) is obvious. (b) Let x D i1 ip p 0 0s 1 and y D j1 jq q 1 , where s ¤ 0. Then x ?? D 1 1s 1 and xy D i1 ip p qC1 q 1 . If s D q, then x ?? y D 1 1q 1 and .xy/?? D 1 1q 1 . If s > q, then x ?? y D 1 1s qC1 q 1 and .xy/?? D 1 1s qC1 q 1 . If s < q, then jsC1 D D jq D 1, so x ?? y D 1 1q 1 , and .xy/?? D 1 1q 1 . The proof that .yx/? D yx ? is similar. (c) Let x D i1 ip p 0 0s 1 and y D j1 j t 0 0jp p 1 , where s ; j t ¤ 0. Then x ?? D 1 1s 1 , y ? D j1 j t 1 1, so 8 ˆ ˆ <1 11 1 ?? ? x y D 1 1s tC1 1 1 ˆ ˆ :1 1j j 11 sC1

t

if s D t if s > t if s < t:

Now, if s > t , then s D D tC1 D 1, and if s < t , then jsC1 D D j t D 1. Furthermore, s D p 1 or t D p 1. If x D i1 ip p 1 2 Vp , we put x 0 D i1 ip and x 00 D p 1 , and for each q D 1; : : : ; p, put xq0 D iq and xq00 D q . If x 2 Sp , we put R.x/ D ¹y 2 Sp W y 0 D x 0 º and L.x/ D ¹y 2 Sp W y 00 D x 00 º. Note that these are respectively minimal right and minimal left ideals of Sp containing x. p1 Deﬁne h on S0 by h.;/ D 1T . Suppose that h has been deﬁned on S0 . We shall show that h can be extended to Sp . Let Ip D ¹xp0 W x 2 Sp º. For each i 2 Ip , choose zi 2 Sp such that .zi /p0 D i and min¹q 2 Œ0; p 1 W .zi /0t D 0 for all t 2 Œq C 1; p 1º is as small as possible. Then choose a minimal right ideal Rp .i / in g 1 .f .Sp // with g.Rp .i // f .R..zi ///. Note that for any x 2 Sp with xp0 D i , one has x ? R..zi // R.x/, so g.h.x ? /Rp .i // f .x ? /f .R..zi /// f .R.x//. Consequently, for any x 2 Sp , one has g.h.x ? /Rp .xp0 // f .R.x//. We deﬁne minimal left ideals Lp ./ in g 1 .f .Sp // in the dual way. Now for every x 2 Sp , we deﬁne h.x/ to be the idempotent of the group h.x ? /Rp .xp0 /Lp .xp00 /h.x ?? /. Since gh.x/ 2 g.h.x ? /Rp .xp0 //g.Lp .xp00 /h.x ?? // f .R.x//f .L.x/ D f .¹xº/; we have that gh.x/ D f .x/.

160

Chapter 10 Almost Maximal Topological Groups p

Now we shall show that h.x/h.y/ D h.xy/ for every x 2 Sp and y 2 S0 . The proof that h.y/h.x/ D h.yx/ is similar. p1 First let y 2 S0 . We have that h.x/h.y/ 2 h.x ? /Rp .xp0 /Lp .xp00 /h.x ?? /h.y/. ? ? But x D .xy/ , xp0 D .xy/p0 , xp00 D .xy/p00 , and x ?? y D .xy/?? by Lemma 10.29, so h.x ?? /h.y/ D h.x ?? y/ D h..xy/?? /. It follows that h.x/h.y/ and h.xy/ belong to the same group in g 1 .f .Sp //. Therefore, it sufﬁces to show that h.x/h.y/ is an idempotent. We show this by proving that h.x/h.y/h.x/ D h.x/. Write h.x/ D h.x ? /zh.x ?? / for some z 2 Rp .xp0 /Lp .xp00 /. Then h.x/h.y/h.x/ D h.x ? /zh.x ?? /h.y/h.x ? /zh.x ?? / D h.x ? /zh.x ?? yx ? /zh.x ?? /: Since x ?? yx ? D .xy/?? x ? D ep1 D x ?? x ? by Lemma 10.29, h.x/h.y/h.x/ D h.x ? /zh.x ?? x ? /zh.x ?? / D h.x ? /zh.x ?? /h.x ? /zh.x ?? / D h.x/h.x/ D h.x/: Now let y 2 Sp . Again, we have that h.x/h.y/ 2 h.x ? /Rp .xp0 /Lp .yp00 /h.y ?? / and also x ? D .xy/? , xp0 D .xy/p0 , yp00 D .xy/p00 , and y ?? D .xy/?? . So h.x/h.y/ and h.xy/ belong to the same group. We again show that h.x/h.y/ is an idempotent by proving that h.x/h.y/h.x/ D h.x/. We know that either zp00 D 1 for all z 2 Sp or zp0 D 1 for all z 2 Sp . Suppose that the ﬁrst possibility holds (considering the second is similar). Then h.y/ 2 Lp .1/h.y ?? / and h.x/ 2 Lp .1/h.x ?? /. Consequently, h.y/h.x ? / and h.x/h.y ? / belong to the same minimal left ideal Lp .1/h.ep1 / in g 1 .f .Sp //. We have seen that these elements are idempotents, so h.x/h.y ? /h.y/h.x ? / D h.x/h.y ? /. Hence, h.x/h.y/h.x/ D h.x/h.y ? /h.y/h.x ? /h.x/ D h.x/h.y ? /h.x/ D h.x/h.x ?? /h.y ? /h.x/ D h.x/h.ep1 /h.x/: This statement holds with y replaced by x, and so h.x/ D h.x/h.ep1 /h.x/ D h.x/h.y/h.x/: Theorem 10.30. Let S be a ﬁnite band. If S is an absolute coretract in F, then S is isomorphic to some semigroup from P. Proof. Let k D jSj and B D B.k; 1; 3/. By Corollary 10.27, B is ﬁnite. We can deﬁne a surjective homomorphism f W B ! S. Then, since S is an absolute coretract in F, there exists a homomorphism g W S ! B such that f ı g D idS . Identifying S and g.S /, we may suppose that S is a subsemigroup of B and f jS D idS . Let F be the free semigroup on a k-element alphabet A and let h W F ! B be the canonical homomorphism. Note that h.u/ D h.v/ if and only if v can be obtained from u by

161

Section 10.4 Projectives

a succession of elementary operations in each of which a subword w of a word is replaced by w 3 , or vice versa. By Lemma 10.24, h.u/ and h.v/ belong to the same completely simple component of B if and only if ct.u/ D ct.v/. Recall that ct.v/ denotes the set of letters from A appearing in v. For any w 2 F and C A, let wjC denote the word obtained from w by removing all letters from A n C and let ˛.w; C / and ˇ.w; C / denote the ﬁrst and the last letters in wjC , respectively. It is easy to see that if h.u/ D h.v/, then ˛.u; C / D ˛.v; C / and ˇ.u; C / D ˇ.v; C /. For any w 2 F , C A and C 2 , let .w; C; / denote the number of pairs of neighboring letters in wjC belonging to . Lemma 10.31. If h.u/ D h.v/, then .u; C; / .v; C; / .mod 2/. Proof. It sufﬁces to consider the case where u D w1 ww2 , v D w1 w 3 w2 . Put .t / D .t; C; /. Then ´ .u/ C 2.w/ C 2 if wjC ¤ ; and .ˇ.w; C /; ˛.w; C // 2 .v/ D .u/ C 2.w/ otherwise. Lemma 10.32. S is a chain of its rectangular components. Proof. Assume the contrary. Then there exist u; v 2 h1 .S / with a 2 ct.u/ n ct.v/ and b 2 ct.v/ n ct.u/. Put .w/ D .w; ¹a; bº; ¹.a; b/º/. Then .uv/ D 1 and .uvuv/ D 2, although h.uvuv/ D h.uv/, a contradiction. Let S1 > S2 > > Sl be the rectangular components of S and for each p 2 Œ1; l, let Ap D ¹a 2 A W f h.a/ 2 Sp º: Observe that for any u 2 h1 .S /, h.u/ 2 Sp if and only if p D max¹q l W ct.u/\Aq ¤ ;º. Indeed, if u D a1 an , then h.u/ D f h.u/ D f h.a1 / f h.an /. Also if 1 p q l, let Apq D

q [

Ar

and

Spq D

rDp

q [

Sr ;

rDp

and let Mp D ¹˛.u; Apl / W u 2 h1 .Spl /º

and

Np D ¹ˇ.u; Apl / W u 2 h1 .Spl /º:

Observe that Mp \ Ap ¤ ; and Np \ Ap ¤ ;. Lemma 10.33. For every p 2 Œ1; l, at least one of the sets Mp , Np is a singleton.

162

Chapter 10 Almost Maximal Topological Groups

Proof. Choose u 2 h1 .Sl /. Let a D ˛.u; Apl / and b D ˇ.u; Apl /. Put .w/ D .w; Apl ; ¹.b; a/º/. Since .uu/ D 2.u/ C 1 .u/ .mod 2/, .u/ is odd. Suppose that there exist v1 ; v2 2 h1 .Spl / with ˛.v1 ; Apl / ¤ a and ˇ.v2 ; Apl / ¤ b. Put v D v1 v2 . Since .vv/ D 2.v/ .v/ .mod 2/, .v/ is even. Then .uvu/ D 2.u/ C .v/ is also even. On the other hand, in S, as in any chain of rectangular bands, the following statement holds: if x; z 2 Sq ; y 2 Sr , and r q, then xyz D xz. Therefore h.uvu/ D h.uu/ D h.u/, and so .uvu/ .u/ .mod 2/, a contradiction. Lemma 10.34. If x 2 Sp ; y 2 Sq ; z 2 Sr , and q p; r, then xyz D xz. Proof. Adjoin identities ;, 1B D 1S to F; B; S and to extend h; f in the obvious way. Also put S0 D ¹1S º. Then the lemma is obviously true if q D 0. Fix q > 0 and assume that the lemma holds for all smaller values of q. Pick u 2 h1 .x/, v 2 h1 .y/ and w 2 h1 .z/. By Lemma 10.33, one of the sets Mq , Nq is a singleton. Suppose that Nq D ¹aº. Then we can write u D u1 au2 and v D v1 av2 , where q1 ct.u2 /; ct.v2 / A1 . Since x D f h.u/ and y D f h.v/, it follows from this that x D x1 sx2 and y D y1 sy2 , where s D f h.a/ 2 Sq , x2 D f h.u2 /; y2 D f h.v2 / 2 q1 q S0 and y1 D f h.v1 / 2 S0 . So xyz D x1 sx2 y1 sy2 z and xz D x1 sx2 z. It is clear that sx2 y1 s D s. By our inductive assumption, sy2 z D sz and sx2 z D sz. Hence xyz D x1 sz and xz D x1 sz. The case jMq j D 1 is similar. Enumerate sets Mp \ Ap and Np \ Ap without repetitions as ¹api W 1 i mp º and ¹bp W 1 np º so that ap1 D ˛.u; Ap / and bp1 D ˇ.v; Ap / for some u; v 2 h1 .Sp /. Deﬁne functions 'p and p on Spl as follows. Let x 2 Spl . Pick u 2 h1 .x/ and put ´ ´ 0 if ˛.u; Apl / … Ap 0 if ˇ.u; Apl / … Ap and .x/ D 'p .x/ D p i if ˛.u; Apl / D api if ˇ.u; Apl / D bp : We now deﬁne the mapping W S ! V by putting for every x 2 Sp , .x/ D '1 .x/'2 .x/ 'p .x/

p .x/ p1 .x/

1 .x/:

It is clear that both 'p .x/ ¤ 0 and p .x/ ¤ 0. By Lemma 10.33, either 'p .y/ D 1 for all y 2 Spl or p .y/ D 1 for all y 2 Spl . Lemma 10.35. is injective. Proof. Let x 2 Sp and pick u 2 h1 .x/. Let p1 < p2 < < ps D p be all r 2 Œ1; p with 'r .x/ ¤ 0, q1 < q2 < < q t D p all r 2 Œ1; p with r .x/ ¤ 0, 'pj .x/ D ij and qk .x/ D k . Then u D ap1 i1 u1 ap2 i2 u2 us1 aps is wbq t t v t v2 bq2 2 v1 bq1 1 ;

163

Section 10.4 Projectives p

q

where ct.uj / A1j and ct.vk / A1k . But then, by Lemma 10.34, x D f h.ap1 i1 ap2 i2 aps is bq t t b 2 q2 bq1 1 /; and consequently, x is uniquely determined by .x/. That is a homomorphism follows from the next lemma. Lemma 10.36. Let x 2 Sp and y 2 Sq . Then (a) 'r .xy/ D 'r .x/ if r p, (b) 'r .xy/ D 'r .y/ if p < r q, (c)

r .xy/

D

r .y/

if r q, and

(d)

r .xy/

D

r .x/

if q < r p.

Proof. Let u 2 h1 .x/, v 2 h1 .y/, and w D uv. If r p, then ˛.w; Alr / occurs in u, because ct.u/ \ Ap ¤ ;, so 'r .xy/ D 'r .x/. If p < r q, then ˛.w; Alr / occurs in v, because ct.v/ \ Alr D ;, so 'r .xy/ D 'r .y/. The check of (c) and (d) is similar. It remains to verify that the semigroup .S/ satisﬁes condition (2) in the deﬁnition of the family P. Let x 2 Sp and let 'q .x/ D a ¤ 0 for some q 2 Œ1; p (the case .x/ ¤ 0 is similar). Pick u 2 h1 .x/ and write it in the form u D u1 au2 , q1 where ct.u1 / A1 . Deﬁne y 2 Sq by y D f h.u1 a/. Since x D f h.u1 au2 /, yx D x. By Lemma 10.36 (a), 'r .x/ D 'r .y/ for each r q. By our choice of br1 , there exists vr 2 h1 .Sr / such that ˇ.vr ; Ar / D br1 . Let v D vq vq1 v1 and deﬁne z 2 Sq by z D h.v/. Then yz 2 Sq and by Lemma 10.36, .yz/ D '1 .x/ 'q .x/1 1. From Theorem 10.28 and Theorem 10.30 we obtain the following result. Theorem 10.37. Let S be a ﬁnite band. Then the following statements are equivalent: (1) S is isomorphic to some semigroup from P, (2) S is an H-projective, (3) S is an absolute H-coretract, (4) S is a projective in C, (5) S is an absolute coretract in C, (6) S is a projective in F, (7) S is an absolute coretract in F.

164

Chapter 10 Almost Maximal Topological Groups

Proof. We proceed by the circuits .1/ ) .4/ ) .2/ ) .6/ ) .7/ ) .1/ and .1/ ) .4/ ) .5/ ) .3/ ) .7/ ) .1/. The implications .1/ ) .4/ and .7/ ) .1/ are Theorem 10.28 and Theorem 10.28, .2/ ) .6/ and .3/ ) .7/ is Lemma 10.9, and the remaining implications are obvious. Using Theorem 10.37, we can summarize Theorem 10.10, Theorem 10.12 and Theorem 10.4 as follows: Theorem 10.38. The ultraﬁlter semigroup of any countable regular almost maximal left topological group is isomorphic to some semigroup from L P. Assuming p D c, for every semigroup S 2 P, there is a group topology T on ! Z2 such that Ult.T / is isomorphic to S. From Theorem 10.38 and Proposition 7.7 we obtain the following. Corollary 10.39. Assuming p D c, for each combination of the properties of being extremally disconnected, irresolvable, and nodec, except for the combination .; ; C/, there is a corresponding almost maximal topological group. There is no countable regular almost maximal left topological group corresponding to the combination .; ; C/. Proof. A maximal topological group corresponds to the combination .C; C; C/. For the combinations .; C; C/, .C; ; C/ and .C; C; /, pick topological groups whose ultraﬁlter semigroups are the 2-element left zero semigroup, right zero semigroup and chain of idempotents, respectively. For the combinations .C; ; / and .; C; /, pick the semigroups ¹11; 1111; 1121º and ¹11; 1111; 1211º in P. These are 3-element semigroups consisting of 2 components with the second components being the 2-element right zero semigroup and left zero semigroup, respectively. For the combination .; ; /, pick the semigroup ¹11; 1111; 1110; 1211; 1210º in P. This is a 5-element semigroup consisting of 2 components with the second component being the 2 2 rectangular band. Finally, every rectangular band in P is either a right zero semigroup or a left zero semigroup. Consequently, if a countable regular almost maximal left topological group is nodec, it is either extremally disconnected or irresolvable. As a consequence we also obtain from Theorem 10.37 the following result. Theorem 10.40. Let G be any inﬁnite group, let Q 2 P, and let S be a subsemigroup of Q. Then there is in ZFC a Hausdorff left invariant topology T on G such that Ult.T / U.G/ and Ult.T / is isomorphic to S. Proof. By Theorem 7.26, there is a zero-dimensional Hausdorff left invariant topology T0 on G such that T D Ult.T0 / is topologically and algebraically isomorphic to

Section 10.5 Topological Invariantness of Ult.T /

165

H and T U.G/. Pick a surjective continuous homomorphism g W T ! Q (Theorem 7.24). By Theorem 10.37, Q is an absolute coretract in C. Consequently, there is an injective homomorphism h W Q ! T (such that g ı h D idQ ). By Proposition 7.8, there is a left invariant topology T on G such that Ult.T / D h.S /. Since T0 T , T is Hausdorff. It turns out that Theorem 10.37 remains to be true with ‘ﬁnite band’ replaced by ‘ﬁnite semigroup’. Let FR denote the category of ﬁnite regular semigroups. Theorem 10.41. Let S be a ﬁnite semigroup. Then the following statements are equivalent: (1) S is isomorphic to some semigroup from P, (2) S is an H-projective, (3) S is an absolute H-coretract, (4) S is a projective in C, (5) S is an absolute coretract in C, (6) S is a projective in F, (7) S is an absolute coretract in F, (8) S is a projective in FR. Proof. See [95].

10.5

Topological Invariantness of Ult.T /

In this section we show that the ultraﬁlter semigroup of a countable regular almost maximal left topological group is its topological invariant. We start by pointing out a complete system of nonisomorphic representatives of P. l Let M denote the set of all matrices M D .mp;q /p;qD0 without the main diagonal l .mp;p /pD0 , where l 2 N and mp;q 2 !, satisfying the following conditions for every p 2 Œ1; l: (a) m0;p m1;p mp1;p 2 N and mp;0 mp;1 mp;p1 2 N, (b) either mp1;p D 1 and mp1;pC1 D D mp1;l D 0 or mp;p1 D 1 and mpC1;p1 D D ml;p1 D 0.

166

Chapter 10 Almost Maximal Topological Groups

These are precisely matrices of the form 0

BC B B B B :: B: B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @

1 C :: :

0 1 :: :

:: :

0 0 0 :: :

1 C C 1 0 1 :: :: 0 : :

0

1

0 0 0 :: :

0

C :: :

:: :

0

:: :

:: :

:: :

0 0 0 C 0 0 0 1 C :: : 0

1 C :: :

0 0 1 :: :

:: :

0 0 0 :: :

C 0

0

C C C C C C C C C C C C C C C C 0 C C C C C 0 C C 0 C C 0 C C :: : 0 C C C 1 C C C C 1 C C C 0 1 C A :: :: :: : : : : : : 0

0

and their transposes, where C is a positive integer, is a nonnegative integer, and all rows and columns are nondecreasing up to the main diagonal. l 2 M, let V .M / denote the subsemigroup of V Now, for every M D .mp;q /p;qD0 consisting of all words i1 i2 ip p p1 1 , where p 2 Œ1; l, such that (i) both ip ¤ 0 and p ¤ 0, (ii) for every q < r p, if i t D 0 for all t 2 Œq C 1; r 1, then ir mq;r , and dually, if t D 0 for all t 2 Œq C 1; r 1, then r mr;q . It is obvious that for every M 2 M, V .M / 2 P. Proposition 10.42. For every S 2 P, there is a unique M 2 M such that S is isomorphic to V .M /. Proof. Let l D max¹p 2 N W S \ Vp ¤ ;º and, for each p 2 Œ1; l, let Sp D S \ Vp . For every q < p l, let Iq;p D ¹ip W i1 iq 0 0ip p 1 2 Sp º ƒq;p D ¹p W i1 ip p 0 0q 1 2 Sp º

and

167

Section 10.5 Topological Invariantness of Ult.T /

and let mq;p D jIq;p j and mp;q D jƒp;q j. For every p 2 Œ1; l, choose bijections fp W Ip1;p ! Œ1; mp1;p

and

gp W ƒp;p1 ! Œ1; mp;p1

fp .Iq;p / D Œ1; mq;p

and

gp .ƒp;q / D Œ1; mp;q

such that for each q p 1. Also put fp .0/ D 0 and gp .0/ D 0. An easy check shows that l M D .mp;q /p;qD0 2 M and S 3 i1 ip p 1 7! f1 .i1 / fp .ip /gp .p / g1 .1 / 2 V .M / is an isomorphism. Next, adjoin an identity ; to S and put S0 D ¹;º. For every p 2 Œ0; l, let rp denote the number of minimal right ideals of Sp , and for every different p; q 2 Œ0; l, let ´ Sq1 Sp Sq n kDpC1 Sk Sq if p < q Sp;q D Sp1 Sp Sq n kDqC1 Sp Sk if p > q: Then the uniqueness of M follows from the next lemma. Lemma 10.43. For every different p; q 2 Œ0; l, one has mp;q D

jSp;q j rq : rp jSq j

Proof. To compute jSp;q j, one may suppose that S D V .M /. Then ´ if p < q ¹i1 ip 0 0iq q 1 2 Sq W ip ¤ 0º Sp;q D ¹i1 ip p 0 0q 1 2 Sp W q ¤ 0º if p > q: Since j¹i1 ip W i1 ip p 1 2 Sp ºj D rp j¹q 1 W i1 iq q 1 2 Sq ºj D

and jSq j ; rq

it follows that jSp;q j D rp mp;q

jSq j : rq

Note that it also follows from Lemma 10.43 that the matrix M is uniquely determined by the numbers l and rp and the sets Sp and Sp Sq , where p; q 2 Œ1; l and p ¤ q.

168

Chapter 10 Almost Maximal Topological Groups

Theorem 10.44. If countable regular almost maximal left topological groups are homeomorphic, then their ultraﬁlter semigroups are isomorphic. Proof. Let .G; T / be a countable regular almost maximal left topological group and let S D Ult.T /. By Theorem 10.38, S is isomorphic to some semigroup from P. Let S1 > > Sl be the rectangular components of S. For every p 2 Œ1; l, let Tp be the left invariant topology on G with Ult.Tp / D Sp . For every different p; q 2 Œ1; l, let Tp Tq be the left invariant topology on G with Ult.Tp Tq / D Sp Sq . By Lemma 7.3, the number rp of minimal right ideals of Sp is equal to the number of maximal open ﬁlters on .G; Tp / converging to the identity. Then by Lemma 10.43, in order to show that Ult.T / is a topological invariant of .G; T /, it sufﬁces to show that topologies Tp and Tp Tq are determined purely topologically. O O SpFor every p D Œ1; l, let Tp be the left invariant topology on G with Ult.Tp / D S . Then T D T and by Proposition 7.7, for p < l, a nonprincipal ultraﬁlter l kD1 k U on G converges to a point x 2 G in TOp if and only if U converges to x in TOpC1 and U is nowhere dense in TOpC1 . Consequently, topologies TOp , p D l; l 1; : : : ; 1, are determined purely topologically. But then this holds for topologies Tp as well, since a nonprincipal ultraﬁlter U on G converges to a point x 2 G in Tp if and only if U converges to x in TOp and U is dense in TOp . Finally, a neighborhood base at a point x 2 G in the topology Tp Tq consists of subsets of the form [ ¹xº [ Vy n ¹yº y2U n¹xº

where U is a neighborhood of x in Tp and Vy is a neighborhood of y in Tq . Hence, topologies Tp Tq are also determined purely topologically. In Section 12.1 we will see that every countable homogeneous regular space admits a structure of a left topological group (Theorem 12.5). Deﬁnition 10.45. For every countable homogeneous regular space X , pick a group operation on X with continuous left translations and let Ult.X / denote the ultraﬁlter semigroup of the left topological group .X; /. By Theorem 10.44, Ult.X / does not depend, up to isomorphism, on the choice of the operation , so Ult.X / is a topological invariant of X .

References Theorem 10.4 is a result from [99] and Corollary 10.6 from [84]. Theorem 10.10 was proved in [99], and Theorem 10.12 in [90]. Theorem 10.15 and Theorem 10.16 are from [87].

Section 10.5 Topological Invariantness of Ult.T /

169

Theorem 10.19 is due to A. Clifford [9] and Corollary 10.20 to D. McLean [49]. Theorem 10.23 is a result of J. Green and D. Rees [31]. The deﬁnition of the family P, Theorem 10.28 and Theorem 10.30 are from [93]. Theorem 10.41 is a result from [95]. Its proof is based on Theorem 10.28, Theorem 10.30 and the fact that every projective in FR is a band. The latter is a result of P. Trotter [74, 75] who also characterized projectives in FR. Theorem 10.41 tells us among other things that the semigroups from P are the same that those characterized by Trotter. Theorem 10.44 was proved in [99]. The exposition of this chapter is based on the treatment in [92].

Chapter 11

Almost Maximal Spaces

In this chapter we show that for every inﬁnite group G and for every n 2 N, there is in ZFC a zero-dimensional Hausdorff left invariant topology T on G such that Ult.T / is a chain of n idempotents and Ult.T / U.G/. As a consequence we obtain that for every inﬁnite cardinal and for every n 2 N, there is a homogeneous zero-dimensional Hausdorff space of cardinality with exactly n nonprincipal ultraﬁlters converging to the same point, all of them being uniform. In particular, for every inﬁnite cardinal , there is a homogeneous regular maximal space of dispersion character .

11.1

Right Maximal Idempotents in H

Recall that given anL inﬁnite cardinal , H D Ult.T0 /, where T0 denotes the group topology on H D Z2 with a neighborhood base at 0 consisting of subgroups H˛ D ¹x 2 H W x. / D 0 for each < ˛º, ˛ < . When working with H , the following two functions are also usuful. Deﬁnition 11.1. Deﬁne functions ; W H n ¹0º ! by .x/ D min supp.x/ and

.x/ D max supp.x/

and let ; W ˇH n ¹0º ! ˇ denote their continuous extensions. The main properties of these functions are that for every x 2 ˇH n ¹0º and y 2 H , .x C y/ D .x/ and

.x C y/ D .y/:

In this section we show that for every !, there is a right maximal idempotent p 2 H such that C.p/ D ¹x 2 ˇH n¹0º W xCp D pº is a ﬁnite right zero semigroup, and if is not Ulam-measurable, every right maximal idempotent p 2 H enjoys this property. Note that H is left saturated in ˇH , so for every p 2 H , one has C.p/ H . The proof of the result about right maximal idempotents in H involves right cancelable ultraﬁlters in H . An element p of a semigroup S is called right cancelable if whenever q; r 2 S and qp D rp, one has q D r. Equivalently, p is right cancelable if the right translation by p is injective.

Section 11.1 Right Maximal Idempotents in H

171

Theorem 11.2. For every ultraﬁlter p 2 H , the following statements are equivalent: (1) p is right cancelable in ˇH , (2) p is right cancelable in H , (3) there is no idempotent q 2 H for which p D q C p, (4) there is no q 2 H for which p D q C p. (5) H C p ˇH is discrete, (6) H C p ˇH is strongly discrete, (7) p is strongly discrete. Proof. .1/ ) .2/ is obvious. .2/ ) .3/ Assume on the contrary that there is an idempotent q 2 H for which p D q C p. Clearly q 2 H . For every ˛ < , deﬁne e˛ 2 H by supp.e˛ / D ¹˛º, and let E˛ D ¹eˇ W ˛ ˇ < º. Pick any ultraﬁlter r on H extending the family of subsets E˛ , where ˛ < . Then r; r C q 2 H and r ¤ r C q. Indeed, Y D

[

.e˛ C H˛C1 n ¹0º/ 2 r C q

˛<

and jsupp.y/j > 1 for all y 2 Y , but jsupp.x/j D 1 for all x 2 E0 . On the other hand, it follows from p D q C p that r C p D r C q C p and, since p is right cancelable in H , we obtain that r D r C q, a contradiction. .3/ ) .4/ Assume on the contrary that there is q 2 H for which p D q C p. Then C.p/ ¤ ;. Since C.p/ is a closed subsemigroup of H , it has an idempotent, a contradiction. .4/ ) .5/ Assume on the contrary that H C p ˇH is not discrete. Then there is a 2 H such that a C p 2 cl..H n ¹aº/ C p/. Since cl..H n ¹aº/ C p/ D .ˇH n ¹aº/ C p; we obtain that there is r 2 ˇH n ¹aº such that a C p D r C p, so a C r C p D p. Let q D a C r. Then q C p D p, and since r ¤ a, q 2 C.p/ H H , a contradiction. .5/ ) .6/ Since H C p ˇH is discrete, for every x 2 H , there is Bx 2 p such that y C p … x C Bx for all y 2 H n ¹xº, that is, x C Bx … y C p for all y 2 H n ¹xº. For every x 2 H n ¹0º, let Fx D ¹0º [ ¹y 2 H n ¹0º W .y/ < .x/ and

supp.y/ supp.x/º:

Put A0 D B0 and inductively for every ˛ < and for every x 2 H with .x/ D ˛, choose Ax 2 p such that

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Chapter 11 Almost Maximal Spaces

(i) Ax Bx \ H.x/C1 , and (ii) .x C Ax / \ .y C Ay / D ; for all y 2 Fx . This can be done because Fx is ﬁnite and y C Ay y C By … x C p for all y 2 Fx . We now claim that .x C Ax / \ .y C Ay / D ; for all different x; y 2 H . Indeed, without loss of generality one may suppose that x ¤ 0 and .y/ .x/ or y D 0. If y 2 Fx , the statement holds by (ii). Otherwise supp.y/ n supp.x/ ¤ ; or .y/ D .x/, in any case .y C H.y/C1 / \ .x C H.x/C1 / D ;; so the statement holds by (i). For every x 2 H , x C Ax is a neighborhood of x C p 2 ˇH , and all these neighborhoods are pairwise disjoint. Hence H C p ˇH is strongly discrete. .6/ ) .7/ Since H C p ˇH is strongly discrete, for every x 2 H , there is Ax 2 p such that the subsets x C Ax ˇH , where x 2 H , are pairwise disjoint. Then the subsets x C Ax H , where x 2 H , are pairwise disjoint. It follows that p is strongly discrete. .7/ ) .1/ Since p is strongly discrete, for every x 2 H , there is Ax 2 p such that the subsets x C Ax are pairwise disjoint. Let q; r 2 ˇH and q ¤ r. Choose disjoint Q 2 q and R 2 r and put [ [ x C Ax and B D x C Ax : AD x2Q

x2R

Then A 2 q C p, B 2 r C p and A \ B D ;, so q C p ¤ r C p. Hence p is right cancelable. Recall that given a group G and p 2 ˇG, T Œp is the largest left invariant topology on G in which p converges to 1, and Cp is the smallest closed subsemigroup of ˇG containing p. Corollary 11.3. Let p be a right cancelable ultraﬁlter in H . Then (1) the topology T Œp is zero-dimensional, and (2) there is a continuous homomorphism W Ult.T Œp/ ! ˇN such that .p/ D 1 and .Cp / D ˇN. Proof. By Theorem 11.2, p is a strongly discrete ultraﬁlter on H . Then apply Theorem 4.18 and Theorem 7.29. We now turn to the right maximal idempotents in H . Proposition 11.4. For every right maximal idempotent p 2 H , C.p/ is a right zero semigroup.

Section 11.1 Right Maximal Idempotents in H

173

Proof. Let C D C.p/ and let q 2 C . Suppose that q is not right cancelable in H . Then by Theorem 11.2, there is an idempotent r 2 H such that r Cq D q. It follows that r C q C p D q C p, and so r C p D p. Thus, p R r, and since p is right maximal, r R p, that is, p C r D r. From this we obtain that pCq DpCr Cq Dr Cq Dq and q C q D q C p C q D p C q D q; so q is an idempotent. Hence, p R q and q R p. It then follows that the elements of C which are not right cancelable in H form a right zero semigroup. Now we claim that no element of C is right cancelable in H . Indeed, assume on the contrary that some q 2 C is right cancelable in H . Then by Corollary 11.3, Cq admits a continuous homomorphism onto ˇN. Taking any nontrivial ﬁnite left zero semigroup in ˇN, we obtain, by the Lemma 10.2, that there is a nontrivial left zero semigroup in Cq C , a contradiction. Proposition 11.5. Let C be a compact right zero semigroup in H and let .C / D ¹uº. If u is countably incomplete, then C is ﬁnite. Note that for every right zero semigroup C H , .C / is a singleton. Indeed, if x; y 2 C , then y D x C y, and so .y/ D .x C y/ D .x/: Proof of Proposition 11.5. Assume on the contrary that C is inﬁnite. Pick any countably inﬁnite subset X C and pick p 2 .cl X / n X . Put Y D .H n ¹0º/ C p. Since cl Y D .ˇH n¹0º/Cp and p D p Cp, p 2 cl Y . Consequently, .cl X /\.cl Y / ¤ ;. Also we have that for every x 2 X , x … cl Y . Indeed, otherwise x D y C p for some y 2 ˇH and then x C p D y C p C p D y C p D x: But x C p D p ¤ x, since x 2 X C , p 2 .cl X / n X C and C is a right zero semigroup. Hence, in order to derive a contradiction, it sufﬁces, by Corollary 2.24, to construct a partition ¹An W n < !º of H n¹0º such that .cl X /\.cl Yn / D ; where Yn D An Cp. Since u is countably incomplete, there is a partition ¹Bn W n < !º of such that Bn … u for all n < !, equivalently u … Bn . Put An D 1 .Bn /. Then for every x 2 cl X , .x/ D u, and for every y 2 Yn , .y/ 2 Bn , so for every y 2 cl Yn , .y/ 2 Bn . Hence, .cl X / \ .cl Yn / D ;. Combining Proposition 11.4 and Proposition 11.5, we obtain the following result. Theorem 11.6. Let p be a right maximal idempotent in H . Then C.p/ is a compact right zero semigroup, and if .p/ is countably incomplete, C.p/ is ﬁnite.

174

11.2

Chapter 11 Almost Maximal Spaces

Projectivity of Ult.T /

Theorem 11.7. Let T be a translation invariant topology on H such that T0 T and let X be an open zero-dimensional neighborhood of 0 in T . Then for every homomorphism g W T ! Q of a semigroup T onto a semigroup Q and for every local homomorphism f W X ! Q, there is a local homomorphism h W X ! T such that f D g ı h. The proof of Theorem 11.7 is based on the following notion. Deﬁnition 11.8. A basis in X is a subset A X n ¹0º together with a partition ¹X.a/ W a 2 Aº of X n ¹0º such that for every a 2 A, X.a/ is a clopen neighborhood of a 2 X n ¹0º and X.a/ a X \ H.a/C1 . Lemma 11.9. Whenever ¹Ux W x 2 X n ¹0ºº is a family of neighborhoods of 0 2 X , there is a basis A in X such that for every a 2 A, X.a/ a Ua . Proof. Without loss of generality one may suppose that for every x 2 X n ¹0º, Ux is a clopen neighborhood of 0 2 X and x C Ux X n ¹0º. For every x 2 X n ¹0º, let Fx D ¹y 2 X n ¹0º W .y/ < .x/ and

supp.y/ supp.x/º:

Note that Fx is ﬁnite. For every ˛ < , let X˛ D ¹x 2 X n ¹0º W .x/ D ˛º: Now put A1 D ; and inductively, for every ˛ < , deﬁne a subset A˛ X˛ and for every a 2 A˛ , a clopen neighborhood X.a/ of a 2 X n ¹0º by S S (i) A˛ D X˛ n b2B˛ X.b/, where B˛ D ˇ <˛ Aˇ , and S (ii) X.a/ D .a C Ua \ H.a/C1 / n b2Fa X.b/. S We put A D ˛< A˛ . S It follows from (i) that for every a 2 A˛ , a … b2B˛ X.b/. Then, since Fa is ﬁnite, we obtain from (ii) that X.a/ is indeed a clopen neighborhood of a 2 X n ¹0º. It is clear also that X.a/ a Ua \ H.a/C1 and that the subsets X.a/, a 2 A, cover X n ¹0º. To see that they are disjoint, let a 2 A˛ , b 2 A˛ [ B˛ and a ¤ b. If b 2 Fa , then X.a/ \ X.b/ D ; by (ii). Otherwise supp.b/ n supp.a/ ¤ ; or b 2 A˛ , in any case .a C H.a/C1 / \ .b C H.b/C1 / D ;, so again X.a/ \ X.b/ D ;. Lemma 11.10. Let A be a basis in X . Then (1) every x 2 X n ¹0º can be uniquely written in the form x D a1 C C an , where a1 ; : : : ; an 2 A and ai C C an 2 X.ai / for each i D 1; : : : ; n 1, and (2) every mapping h0 W A ! S of A into a semigroup S extends to a local homomorphism h W X ! S by h.a1 C C an / D h0 .a1 / h0 .an /, where a1 ; : : : ; an 2 A and ai C C an 2 X.ai / for each i D 1; : : : ; n 1.

Section 11.2 Projectivity of Ult.T /

175

Proof. (1) Let x 2 X n ¹0º. Then x 2 X.a1 / for some a1 2 A. If x D a1 , we are done. Otherwise x D a1 C x1 , where x1 D x a1 2 X n ¹0º, and .a1 / < .x1 /. Suppose that we have written x as x D a1 C C ai C x i where a1 ; : : : ; ai 2 A, xi 2 X n ¹0º, and (i) .aj / < .aj C1 / for each j D 1; : : : ; i 1 and .ai / < .xi /, and (ii) aj C C ai C xi 2 X.aj / for each j D 1; : : : ; i . Then xi 2 X.aiC1 / for some aiC1 2 A. Since .aiC1 / D .xi /, .ai / < .aiC1 /. If xi D aiC1 , we are done: x D a1 C C aiC1 and aj C C aiC1 2 X.aj / for each j D 1; : : : ; i . Otherwise xi D aiC1 C xiC1 where xiC1 D xi aiC1 , then x D a1 C C aiC1 C xiC1 and (i) and (ii) are satisﬁed with i replaced by i C 1. After jsupp.x/j steps we obtain the required decomposition. Now suppose that x has two such decompositions, say x D a 1 C C an D b 1 C C b m : We show that n D m and ai D bi for each i D 1; : : : ; n. We proceed by induction on min¹n; mº. Let min¹n; mº D 1, say n D 1. We have that a1 2 X.a1 / and a1 D b1 C C bm 2 X.b1 /: Since ¹X.a/ W a 2 Aº is disjoint, it follows that a1 D b1 . But then also m D 1. Indeed, otherwise b2 C C bm D 0 which contradicts b2 C C bm 2 X.b2 /. Now let min¹n; mº > 1. Again we have that a1 C C an 2 X.a1 / and a1 C C an D b1 C C bm 2 X.b1 /; and so a1 D b1 . But then a2 C C an D b2 C C bm and we can apply the inductive assumption. (2) Let x 2 X n ¹0º. Write x D a1 C C an , where a1 ; : : : ; an 2 A and ai C C an 2 X.ai / for each i D 1; : : : ; n. Deﬁne the neighborhood U of 0 2 X by n \ U D .X.ai / .ai C C an // \ X: iD1

Now let y 2 U n ¹0º. Then ai C C an C y 2 X.ai / for each i D 1; : : : ; n. Write y D anC1 C C anCm , where anC1 ; : : : ; anCm 2 A and anCj C C anCm 2 X.anCj / for each j D 1; : : : ; m 1. We obtain that ai C C anCm 2 X.ai / for each i D 1; : : : ; n C m 1 and h.x C y/ D h.a1 C C an C anC1 C anCm / D h0 .a1 / h0 .an /h0 .anC1 / h0 .anCm / D h.x/h.y/:

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Chapter 11 Almost Maximal Spaces

Now we are in a position to prove Theorem 11.7. Proof of Theorem 11.7. For every x 2 X n ¹0º, pick a neighborhood Ux of 0 2 X such that f .xy/ D f .x/f .y/ for all y 2 Ux n ¹0º. By Lemma 11.9, there is a basis A in X such that for every a 2 A, X.a/ a Ua . By Lemma 11.10, every x 2 X n ¹0º can be uniquely written as x D a1 C C an where a1 ; : : : ; an 2 A and ai C C an 2 X.ai / for each i D 1; : : : ; n 1. Then f .x/ D f .a1 / f .an /. Indeed, a1 C C an 2 X.a1 / and X.a1 / a1 C Ua1 , consequently a2 C C an 2 Ua1 . It follows that f .a1 C a2 C C an / D f .a1 /f .a2 C C an / and then by induction f .a1 C C an / D f .a1 / f .an /. Now for every a 2 A, pick ta 2 T such that g.ta / D f .a/. Deﬁne h W X ! T by h.a1 C C an / D ta1 tan where a1 ; : : : ; an 2 A and ai C C an 2 X.ai / for each i D 1; : : : ; n 1. By Lemma 11.10, h is a local homomorphism, and since g.ta1 tan / D g.ta1 / g.tan / D f .a1 / f .an /; it follows that g ı h D f . As a consequence we obtain from Theorem 11.7 the following result. Corollary 11.11. Let T be a locally zero-dimensional almost maximal translation invariant topology on H such that T0 T . Then Ult.T / is a projective in F. Proof. Let T and Q be ﬁnite semigroups, let ˛ W Ult.T / ! Q be a homomorphism, and let ˇ W T ! Q be a surjective homomorphism. By Lemma 10.11, there are an open zero-dimensional neighborhood X of zero of .H; T / and a local homomorphism f W X ! Q such that f D ˛, so ˛ is proper. Now by Theorem 11.7, there is a local homomorphism h W X ! T such that f D ˇ ı h. Deﬁne W Ult.T / ! T by

D h . Then ˛ D ˇ ı . We will need also the following proposition. Proposition 11.12. Every projective S in F is a chain of rectangular bands satisfying the following conditions: (i) whenever x; y; z 2 S and yRz, xy D xz implies y D z, and dually (ii) whenever x; y; z 2 S and yLz, yx D zx implies y D z. Proof. By Theorem 10.41, S is isomorphic to some semigroup from P. It is easy to see that every subsemigroup of V possesses these properties. We conclude this section by noting that Theorem 11.7 can be used also to prove the following result.

177

Section 11.3 The Semigroup C.p/

Theorem 11.13. For every inﬁnite cardinal , the semigroup H contains no nontrivial ﬁnite groups. Proof. Similar to the proof of Theorem 8.18 with Corollary 8.13 replaced by Theorem 11.7.

11.3

The Semigroup C.p/

Lemma 11.14. Let p 2 H and let C.p/ be ﬁnite. Then for every q; r 2 ˇG, the equality q C p D r C p implies that q 2 r C C 1 or r 2 q C C 1 , where C 1 D C 1 .p/. Proof. Assume the contrary. Then, since C 1 is ﬁnite, there exist A 2 q and B 2 r such that A \ .B C C 1 / D ; and B \ .A C C 1 / D ;: By Theorem 7.18, there is a zero-dimensional translation invariant topology T on H with Ult.T / D C.p/. It follows that for every x 2 A [ B, there exists a clopen neighborhood U of 0 2 H in T such that A \ .x C U / D ; if x 2 B;

and

B \ .x C U / D ; if x 2 A:

Enumerate A [ B as ¹x˛ W ˛ < º so that the sequence ..x˛ //˛< is nondecreasing. For each ˛ < , choose inductively a clopen neighborhood U˛ of 0 in T so that the following conditions are satisﬁed: (i) U˛ H.x˛ /C1 , (ii) A \ .x˛ C U˛ / D ; if x˛ 2 B, and B \ .x˛ C U˛ / D ; if x˛ 2 A, and (iii) .x˛ C U˛ / \ .x C U / D ; for all < ˛ such that supp.x / supp.x˛ / and elements x˛ ; x belong to different sets A; B. To this end, ﬁx ˛ < and suppose that we have already chosen U for all < ˛ satisfying (i)–(iii). Without loss of generality one may suppose also that x˛ 2 A. Let F D ¹ < ˛ W supp.x / supp.x˛ / and x 2 Bº: It follows from (ii) that x˛ …

[

.x C U /:

2F

Since F is ﬁnite and each U is closed, there is a clopen neighborhood U˛ of 0 such that [ .x˛ C U˛ / \ .x C U / D ;; 2F

which means that (iii) is satisﬁed. Obviously, one can choose U˛ to satisfy also (i) and (ii).

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Chapter 11 Almost Maximal Spaces

We now claim that .x˛ C U˛ / \ .x C U / D ; whenever < ˛ < and elements x˛ ; x belong to different sets A; B. Indeed, if supp.x / supp.x˛ /, then .x˛ C U˛ / \ .x C U / D ; by (iii). If supp.x / n supp.x˛ / ¤ ;, then .x˛ C H.x˛ /C1 / \ .x C H.x /C1 / D ;, and consequently, .x˛ C U˛ / \ .x C U / D ; by (i). Thus, we have that [ [ .x˛ C U˛ / \ .x C U / D ;; x˛ 2A

x 2B

so q C p ¤ r C p, which is a contradiction. Theorem 11.15. Let p 2 H and let C.p/ be ﬁnite. Then (1) C.p/ is a projective in F, and (2) C.p/ is a chain of right zero semigroups. Proof. (1) By Theorem 7.18, there is a zero-dimensional translation invariant topology T on H such that Ult.T / D C.p/. Then apply Corollary 11.11. (2) Let C D C.p/. By (1) and Proposition 11.12, C is a chain of rectangular bands. We have to show that for every x; y 2 C , xLy implies x D y. Let K D K.C /. Pick any z 2 K. Then x C z; y C z 2 K and .x C z/L.y C z/. We have also that x C z C p D y C z C p. It follows from this and Lemma 11.14 that either x C z 2 y C z C C 1 or y C z 2 x C z C C 1 , where C 1 D C 1 .p/. Both y C z C C 1 and x C z C C 1 are R-classes of K. Therefore in any case, .x C z/R.y C z/. Since also .x Cz/L.y Cz/, we obtain that x Cz D y Cz and then, by Proposition 11.12 (ii), x D y. In the rest of this section, we show that for every n 2 N, there is an idempotent p 2 H such that C.p/ is a chain of n ﬁnite right zero semigroups. We ﬁrst prove several auxiliary statements. Lemma 11.16. Let p 2 H and let C.p/ be ﬁnite. Then for every q 2 H , j¹x 2 ˇH W x C p D qºj jC 1 .p/j: Proof. Let X D ¹x 2 ˇH W x C p D qº and let C 1 D C 1 .p/. Choose y 2 X with maximally possible jy C C 1 j. For every z 2 C 1 , one has y C z C p D y C p D q, so y C C 1 X . We claim that X D y C C 1 . To see this, let x 2 X . We have that x C p D y C p. Then by Lemma 11.14, either x 2 y C C 1 or y 2 x C C 1 . The ﬁrst possibility is what we wish to show. The second implies that y C C 1 x C C 1 . Since jy C C 1 j is maximally possible, we obtain that y C C 1 D x C C 1 , so again x 2 y C C 1 . It follows from X D y C C 1 that jX j jC 1 j.

179

Section 11.3 The Semigroup C.p/

Lemma 11.17. Let p; q 2 H and let C.p/; C.q/ be ﬁnite. Then jC 1 .p C q/j jC 1 .p/j jC 1 .q/j: Proof. We have that C 1 .p C q/ D ¹x 2 ˇH W x C p C q D p C qº D ¹x 2 ˇH W x C p 2 ¹y 2 ˇH W y C q D p C qºº: Let Y D ¹y 2 ˇH W y C q D p C qº and for each y 2 Y , let X.y/ D ¹x 2 ˇH W x C p D yº. Then [ C 1 .p C q/ D X.y/ y2Y

and, by Lemma 11.16, jY j

jC 1 .q/j

and jX.y/j C 1 .p/.

Lemma 11.18. Let G be a group, let p 2 G , and let Q D Ult.T Œp/. Then Q D .Qp/ [ ¹pº. Proof. Clearly .Qp/ [ ¹pº Q. We have to show that for every q 2 G n ..Qp/ [ ¹pº/, one has q … Q. Pick A 2 q such that 1 … A and A \ ..Qp/ [ ¹pº/ D ;. It sufﬁces to construct a neighborhood U of 1 in T Œp such that U \ A D ;. To this end, we use Theorem 4.8. Since A \ ..Qp/ [ ¹pº/ D ;, there is an open neighborhood V of 1 in T Œp such that A \ .V p/ D ;. For every x 2 V , pick M.x/ 2Sp such that xM.x/ V and .xM.x// \ A D ;. Put U D ŒM 1 . Then U ¹1º [ x2V .xM.x//. It follows that U \ A D ;. Lemma 11.19. For every p 2 H and q 2 Ult.T Œp/, one has .q/ D .p/. Proof. Let A 2 p. Choose M W H ! p such that M.0/ A and for every x 2 H n ¹0º and y 2 M.x/, .x/ < .y/. Then whenever 0 ¤ z 2 ŒM 0 , .z/ 2 .A/. Proposition 11.20. For every idempotent p 2 H with ﬁnite C.p/, there is a right maximal idempotent q 2 H with ﬁnite C.q/ such that for each x 2 C.q/, x

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Chapter 11 Almost Maximal Spaces

We claim that r C p is right cancelable and p … .ˇH / C r C p. To see that r C p is right cancelable, suppose that u C r C p D v C r C p for some u; v 2 ˇH . Then by Lemma 11.14, either u C r 2 v C r C C 1 or v C r 2 u C r C C 1 , where C 1 D C 1 .p/. But .u C r/ D .v C r/ D .r/ and for every x 2 C.p/, .u C r C x/ D .v C r C x/ D .x/ 2 .C.p//: It follows that u C r D v C r and, since r is right cancelable, u D v. To see that p … .ˇH / C r C p, assume on the contrary that p D u C r C p for some u 2 ˇH . Then u C r 2 C.p/, which is a contradiction, since .u C r/ D .r/ and .r/ … .C.p//. Now let Q D Ult.T Œr C p/. Pick any right maximal idempotent q 2 Q. By Corollary 11.3 and Lemma 7.12, Q is left saturated. Consequently, C.q/ Q and q is right maximal in H . By Lemma 11.19, .q/ D .r C p/ D .r/, so .q/ is countably incomplete. Then by Theorem 11.6, C.q/ is ﬁnite. By Lemma 11.18, Q .ˇH / C r C p. For every x 2 .ˇH / C r C p, one has x C p D x. Indeed, x D u C r C p for some u 2 ˇH and then x C p D u C r C p C p D u C r C p D x. It follows that for every x 2 C.q/, x C p D x, so x L p. But p C x ¤ p, since p C x 2 .ˇH / C r C p and p … .ˇH / C r C p. Hence x

Section 11.4 Local Monomorphisms

181

We now come to the main result of this section. Theorem 11.22. For every right maximal idempotent q 2 H with ﬁnite C.q/ and for every n 2 N, there is an idempotent p 2 H such that q 2 C.p/ and C.p/ is a chain of n ﬁnite right zero semigroups. Proof. If n D 1, put p D q. By Theorem 11.6, C.p/ is a ﬁnite right zero semigroup. Now let n > 1 and suppose that we have found an idempotent p 0 2 H such that q 2 C.p 0 / and C.p 0 / is a chain of n 1 ﬁnite right zero semigroups, say C1 > > Cn1 . By Proposition 11.20, there is a right maximal idempotent q 0 2 H with ﬁnite C.q 0 / such that for each x 2 C.q 0 /, x > Cn . Now, having proved Theorem 11.22, we can show that Theorem 11.23. For every n 2 N, there is a locally zero-dimensional translation invariant topology T on H such that T0 T and Ult.T / is a chain of n idempotents. Proof. By Theorem 11.22, there is an idempotent p 2 H such that C.p/ is a chain of n ﬁnite right zero semigroups, say C1 > > Cn . Inductively for each i D 1; : : : ; n, pick qi 2 Ci and deﬁne pi 2 Ci by p1 D q1 and, for i > 1, pi D pi1 C qi C pi1 : Then p1 > > pn . Let C D C.p/ and S D ¹p1 ; : : : ; pn º. We claim that the subsemigroup S C possesses the following property: For every q 2 C n S, .S C q/ \ S D ;. Indeed, let q 2 Ci . Then qRpi and q ¤ pi . It follows that for every r 2 C , one has .r C q/R.r C pi / and, by Theorem 11.15 (1) and Proposition 11.12 (i), r C q ¤ r C pi . If r 2 S, then r C pi 2 S , so r C q … S , since no different elements of S are R-related. Hence .S C q/ \ S D ;. Now let T be the translation invariant topology on H such that Ult.T / D S. It follows from the property above and Proposition 7.21 that T is locally regular. Being a chain of idempotents, S has only one minimal right ideal. Hence by Proposition 7.7, T is extremally disconnected. Let X be an open regular neighborhood of 0 2 H in T . Since extremal disconnectedness is preserved by open subsets and a regular extremally disconnected space is zero-dimensional, we obtain that X is zerodimensional.

11.4

Local Monomorphisms

Given a local left group X and a semigroup S with identity, a local monomorphism is an injective local homomorphism f W X ! S with f .1X / D 1S .

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Chapter 11 Almost Maximal Spaces

Lemma 11.24. Let X be a local left group, let S be a left cancellative semigroup with identity, and let f W X ! S be a local monomorphism. Then there is a left invariant T1 -topology T f on S with a neighborhood base at s 2 S consisting of subsets sf .U /, where U runs over neighborhoods of 1X . Furthermore, let Y D f .X / .S; T f / and let f W ˇXd ! ˇS be the continuous extension of f . Then f homeomorphically maps X onto Y and f isomorphically maps Ult.X / onto Ult.T f /. Proof. Let B be an open neighborhood base at 1X . For every U 2 B and x 2 U , there is V 2 B such that xV U and f .xy/ D f .x/f .y/ for all y 2 T V . Then f .x/f .V / D f .xV / f .U /. Since f is injective, we have also that f .B/ D ¹1º. Consequently by Corollary 4.4, there is a left invariant T1 -topology T f on S in which for each s 2 S , sf .B/ is an open neighborhood base at s. To see that f homeomorphically maps X onto Y , let x 2 X . Choose a neighborhood U of 1X such that xU X and f .xy/ D f .x/f .y/ for all y 2 U . Then whenever V is a neighborhood of 1X and V U , one has f .xV / D f .x/f .V /. Finally, by Lemma 8.4, f isomorphically maps Ult.X / onto Ult.T f /. Deﬁnition 11.25. Let T be a translation invariant topology on H such that T0 T and let X be an open neighborhood of 0 in T . Denote by P .X / the set of all x 2 X n ¹0º which cannot be decomposed into a sum x D y C z where y; z 2 X n ¹0º and .y/ < .z/. Note that jP .X /j D . We say that X satisﬁes the P -condition if there is a neighborhood W of 0 2 X such that jP .X / n W j D . It follows from the next lemma that P .X / is a strongly discrete subset of X with at most one limit point 0. Lemma 11.26. Let x 2 P .X / and y 2 X n ¹0º. If x 2 y C H.y/C1 \ X , then x D y. Proof. Otherwise x D y Cz for some z 2 H.y/C1 \.X n¹0º/ and then .y/ < .z/, which contradicts x 2 P .X /. Suppose that X has the property that, whenever D is a strongly discrete subset of X with exactly one limit point 0, there is A D such that jAj D and 0 is not a limit point of A. Then, obviously, X satisﬁes the P -condition. In particular, X satisﬁes the P -condition if T is almost maximal. In fact, the P -condition is always satisﬁed in the following sense. Lemma 11.27. Let x 2 X n ¹0º, let Fx D ¹y 2 X n ¹0º W supp.y/ supp.x/º, and let Y D X n Fx . Then Y satisﬁes the P -condition. Proof. Choose a subset A Y with jAj D such that whenever y; z 2 A and y ¤ z, one has supp.y/ \ supp.z/ D supp.x/. For every y 2 A, there is zy 2 P .Y / such that supp.zy / supp.z/ and supp.zy / \ supp.x/ ¤ ;. Since Fx \ Y D ;, we have

Section 11.4 Local Monomorphisms

183

that supp.zy / n supp.x/ ¤ ; for all y 2 A. Put B D ¹zy W y 2 Aº. Then B P .Y /, B \ H.x/C1 D ; and jBj D . We now come to the main result about local monomorphisms. Theorem 11.28. Let T be a translation invariant topology on H such that T0 T , let X be an open neighborhood of 0 in T , and let G be a group of cardinality . Suppose that X is zero-dimensional and satisﬁes the P -condition. Then there is a local monomorphism f W X ! G such that the topology T f is zero-dimensional. If G D H , then f can be chosen to be continuous with respect to T0 . Proof. Using the P -condition, choose a clopen neighborhood W of 0 2 X such that jP .X / n W j D . By Lemma 11.9, there is a basis A in X such that (1) for each a 2 A, X.a/ a W , and (2) for each a 2 A n W , X.a/ \ W D ;. Let F be the free semigroup on the alphabet A including the empty word ;. Deﬁne h W X ! F by putting h.0/ D ; and h.a1 C C an / D a1 an where a1 ; : : : ; an 2 A and ai C Can 2 X.ai / for each i D 1; : : : ; n1. By Lemma 11.10, h is a local monomorphism. By Lemma 11.24, h induces a left invariant T1 topology T h on F . We have that Y D h.X / is an open neighborhood of the identity of .F; T h / and h homeomorphically maps X onto Y , so Y is zero-dimensional. Lemma 11.29. T

h

is zero-dimensional.

Proof. It sufﬁces to show that Y is closed in T h . Let a1 an 2 F n Y . Then ai C C an … X.ai / for some i D 1; : : : ; n 1, so aiC1 C C an … X.ai / ai . Taking the biggest such i we obtain that aiC1 C Can 2 X.aiC1 / X . It follows that there is a neighborhood U of 0 2 X such that .aiC1 C C an C U / \ .X.ai / ai / D ;, so .ai C C an C U / \ X.ai / D ;: We claim that .a1 an h.U // \ Y D ;. Indeed, assume on the contrary that a1 an h.y/ 2 Y for some y 2 U n ¹0º. Let h.y/ D anC1 anCm 2 Y . Since a1 an h.y/ D a1 an anC1 anCm 2 Y , we obtain that ai C C an C y D ai C C an C anC1 C C anCm 2 X.ai /; which is a contradiction.

184

Chapter 11 Almost Maximal Spaces

Denote I D A n W . Then jI j D and for every a1 an 2 Y , a 2 ¹a1 ; : : : ; an º \ I

implies a D a1 :

Indeed, by the construction of A and Lemma 11.26, P .X / A, and by the choice of W , jP .X / n W j D , so the ﬁrst statement holds. And since ai C C an 2 X.ai1 / ai1 W for each i D 2; : : : ; n, the second one holds as well. Now let Z denote the subset of F consisting of all words a1 an such that .ai / < .aiC1 / for each i D 1; : : : ; n 1 and a 2 ¹a1 ; : : : ; an º \ I

implies a D a1 :

Clearly Z is a neighborhood of the identity of .F; T h / containing Y . Furthermore, for every ˛ < , Z˛ D ¹b1 bm 2 h.W / W .b1 / ˛º [ ¹;º is a neighborhood of the identity, and for every a1 an 2 Z, a1 an Z.an /C1 Z; so Z is open. In addition, and as distinguished from Y , Z has the property that, whenever a1 an 2 Z and i D 1; : : : ; n 1, one has a1 ai 2 Z. Lemma 11.30. There is a bijective local monomorphism g W Z ! G. If G D H , then g can be chosen to be continuous with respect to T0 . Proof. We shall construct a bijection g W Z ! G such that g.;/ D 1 and g.a1 an / D g.a1 / g.an / for every a1 an 2 Z. That such g is a local homomorphism follows from the last but one sentence preceding the lemma. It sufﬁces to deﬁne g on A so that (i) whenever a1 an and b1 bm are different elements of Z, g.a1 / g.an / and g.b1 / g.bm / are different elements of G, and (ii) for each s 2 G n ¹1º, there is a1 an 2 Z such that g.a1 / g.an / D s. To this end, enumerate A without repetitions as ¹c˛ W ˛ < º so that if a; b 2 A, .a/ < .b/, a D c˛ and b D cˇ , then ˛ < ˇ. This deﬁnes W A ! by .c˛ / D ˛. Note that whenever a1 an 2 Z, one has .a1 / < < .an /. Also enumerate G n ¹1º as ¹s˛ W ˛ < º.

Section 11.4 Local Monomorphisms

185

Fix ˛ < and suppose that values g.cˇ / have already been deﬁned for all ˇ < ˛ so that, whenever a1 an and b1 bm are different elements of Z with .an /; .bm / < ˛, g.a1 / g.an / and g.b1 / g.bm / are different elements of G. Let G˛ D ¹g.a1 / g.an / 2 G W a1 an 2 Z and .an / < ˛º [ ¹1º: Consider two cases. Case 1: c˛ … I . Pick as g.c˛ / any element of G n .G˛1 G˛ /. This can be done because jG˛1 G˛ j jG˛ j2 < . Then whenever a1 an 2 Z and an D c˛ , one has g.a1 / g.an / … G˛ . Indeed, otherwise g.c˛ / D g.an / 2 .g.a1 / g.an1 //1 G˛ G˛1 G˛ : Also if a1 an and b1 bm are different element of Z with an D bm D c˛ , then g.a1 / g.an / ¤ g.b1 / g.bm /, by the inductive hypothesis. Case 2: c˛ 2 I . Then whenever a1 an 2 Z and an D c˛ , one has n D 1. Put g.c˛ / to be the ﬁrst element in the sequence ¹sˇ W ˇ < º n G˛ . It is clear that the mapping g W A ! G so constructed satisﬁes (i), and since jI j D , (ii) is satisﬁed as well. If G D H , the construction remains the same with the only correction in Case 1: we pick g.c˛ / so that .g.cˇ // < .g.c˛ // for all ˇ < ˛. To see that such g is continuous, let ˛ < be given. Choose ˇ < such that .g.cˇ // ˛ and put U˛ D ¹a1 an 2 h.W / W .a1 / > ˇº [ ¹;º: Then U˛ is a neighborhood of identity of Z and g.U˛ / H˛C1 . The bijective local monomorphism g W Z ! G induces a left invariant topology T g on G. Since g homeomorphically maps Z onto .G; T g /, T g is zero-dimensional and Hausdorff. Let f D g ı h. Then f W X ! G is a local monomorphism and T f D T g . If G D H , choose g to be continuous with respect to T0 , then so is f . We will use Theorem 11.28 in a very special situation when the topology T is almost maximal. But it is also interesting in the general case. Corollary 11.31. Let T1 be a locally zero-dimensional translation invariant topology on H such that T0 T1 and let G be a group of cardinality . Then there is a zero-dimensional left invariant topology T on G such that (1) .G; T / and .H; T1 / are locally homeomorphic, (2) Ult.T / is topologically and algebraically isomorphic to Ult.T1 /, and (3) Ult.T / is left saturated in ˇG. If G D H , then T can be chosen to be stronger then T0 .

186

Chapter 11 Almost Maximal Spaces

Proof. Let X be an open zero-dimensional neighborhood of zero of .H; T1 /. By Lemma 11.27, one may suppose that X satisﬁes the P -condition. Then by Theorem 11.28, there is a local monomorphism f W X ! G such that the topology T f on G is zero-dimensional. Put T D T f . By Lemma 11.24, conditions (1) and (2) are satisﬁed, and by Lemma 7.12, (3) is satisﬁed as well. If G D H , choose f to be continuous with respect to T0 . Now, using Theorem 11.23 and Theorem 11.28, we prove the main result of this chapter. Theorem 11.32. For every inﬁnite group G and for every n 2 N, there is a zerodimensional Hausdorff left invariant topology T on G such that Ult.T / is a chain of n idempotents and Ult.T / U.G/. Proof. Let G be a group of cardinality D jH j and let n 2 N. By Theorem 11.23, there is a locally zero-dimensional translation invariant topology T1 on H such that T0 T1 and Ult.T1 / is a chain of n idempotents. Pick an open zero-dimensional neighborhood X of 0 2 H . By Theorem 11.28, there is a local monomorphism f W X ! G such that the topology T f on G is zero-dimensional. Put T D T f . The dispersion character of a space is the minimum cardinality of a nonempty open set. Corollary 11.33. For every inﬁnite cardinal and for every n 2 N, there is a homogeneous zero-dimensional Hausdorff space of cardinality with exactly n nonprincipal ultraﬁlters converging to the same point, all of them being uniform. In particular, for every inﬁnite cardinal , there is a homogeneous regular maximal space of dispersion character . Remark 11.34. If G D H , the topology T in Theorem 11.32 can be chosen to be stronger than T0 , and if G D R, stronger than the natural topology of the real line. To see the second, apply Theorem 11.32 to the circle group T . This gives us a left saturated chain of n uniform idempotents in ˇTd . Since T is a compact group, every idempotent converges to 1 2 T (Lemma 7.10). Then identifying T with the subset Œ 12 ; 12 / R, we obtain a left saturated chain of n uniform idempotents in ˇRd converging to 0 2 R. One can show also that if G D R, the topology T in Theorem 11.32 can be chosen to be stronger than the Sorgenfrey topology.

References The question of whether there exists a regular maximal space was raised by M. Katˇetov [42]. A countable example of such a space was constructed by E. van Douwen

Section 11.4 Local Monomorphisms

187

[78, 80] and that of arbitrary dispersion character by A. El’kin [19]. The ﬁrst consistent example of a homogeneous regular maximal space was produced by V. Malykhin [46]. Right cancelable and right maximal idempotent ultraﬁlters on a countable group G have been studied by N. Hindman and D. Strauss [37, Sections 8.2, 8.5, and 9.1]. In particular, they showed that for every right cancelable ultraﬁlter p on G, the semigroup Cp admits a continuous homomorphism onto ˇN [37, Theorem 8.51], and for every right maximal idempotent p in G , C.p/ is a ﬁnite right zero semigroup [37, Theorem 9.4]. That every countably inﬁnite group admits in ZFC a regular maximal left invariant topology was proved by I. Protasov [60]. Theorem 11.2 and Theorem 11.6 are from [107]. Theorem 11.7 and Theorem 11.13 were proved in [101]. The results of Section 11.3 are from [108] and Theorem 11.28 is from [107]. Theorem 11.32 was proved in [107] for n D 1 and in [108] for any n.

Chapter 12

Resolvability

In this chapter we prove a structure theorem for a broad class of homeomorphisms of ﬁnite order on countable regular spaces. Using this, we show that every countable nondiscrete topological group not containing an open Boolean subgroup is !resolvable. We also show that every inﬁnite Abelian group not containing an inﬁnite Boolean subgroup is absolutely !-resolvable, and consequently, can be partitioned into ! subsets such that every coset modulo inﬁnite subgroup meets each subset of the partition.

12.1

Regular Homeomorphisms of Finite Order

Deﬁnition 12.1. Let X be a space with a distinguished point 1 2 X and let f W X ! X be a homeomorphism with f .1/ D 1. We say that f is regular if for every x 2 X n ¹1º, there is a homeomorphism gx of a neighborhood of 1 onto a neighborhood of x such that fgx jU D gf .x/ f jU for some neighborhood U of 1. Note that if a space X admits a regular homeomorphism, then for any two points x; y 2 X , there is a homeomorphism g of a neighborhood of x onto a neighborhood of y with g.x/ D y, and if in addition X is zero-dimensional and Hausdorff, then g can be chosen to be a homeomorphism of X onto itself. Hence, a zero-dimensional Hausdorff space admitting a regular homeomorphism is homogeneous. The notion of a regular homeomorphism generalizes that of a local automorphism on a local left group. To see this, let X be a local left group and let f W X ! X be a local automorphism. For every x 2 X n ¹1º, choose a neighborhood Ux of 1 such that xy is deﬁned for all y 2 Ux , xUx is a neighborhood of x and x W Ux 3 y 7! xy 2 xUx is a homeomorphism, and put gx D x . Clearly gx .1/ D x. Choose a neighborhood Vx of 1 such that Vx Ux , f .Vx / Uf .x/ and f .xy/ D f .x/f .y/ for all y 2 Vx . Then for every y 2 Vx , fgx .y/ D f .xy/ D f .x/f .y/ D gf .x/ f .y/. We now show that, likewise in the case of a local automorphism, the spectrum of a spectrally irreducible regular homeomorphism of ﬁnite order is a ﬁnite subset of N closed under taking the least common multiple. Lemma 12.2. Let X be a Hausdorff space with a distinguished point 1 and let f W X ! X be a spectrally irreducible regular homeomorphism of ﬁnite order. Let x0 2 X n ¹1º with jO.x0 /j D s and let U be a spectrally minimal neighbourhood of x0 .

Section 12.1 Regular Homeomorphisms of Finite Order

189

Then spec.f; U / D ¹lcm.s; t / W t 2 ¹1º [ spec.f /º: Proof. The proof is similar to that of Lemma 8.27. For each x 2 O.x0 /, let gx be a homeomorphism of a neighborhood Ux of 1 onto a neighborhood of x such that fgx jVx D gf .x/ f jVx for some neighborhood Vx Ux of 1. Choose a neighborhood V of 1 such that \ V Vx ; x2O.x0 /

gx0 .V / U , and the subsets gx .V /, where x 2 O.x0 /, are pairwise disjoint. Let n be the order of f . Choose a neighborhood W of 1 such that f i .W / V for all i < n. Then clearly this inclusion holds for all i < !, and furthermore, for every y 2 W , f i gx0 .y/ D gf i .x0 / f i .y/. Indeed, it is trivial for i D 0, and further, by induction, we obtain that f i gx0 .y/ D ff i1 gx0 .y/ D fgf i1 .x0 / f i1 .y/ D gf i .x0 / f i .y/: Now let y 2 W , jO.y/j D t and k D lcm.s; t /. We claim that jO.gx0 .y//j D k. Indeed, f k .gx0 .y// D gf k .x0 / .f k .y// D gx0 .y/: On the other hand, suppose that f i .gx0 .y// D gx0 .y/ for some i . Then gf i .x0 / .f i .y// D gx0 .y/: Since the subsets gx .V /, x 2 O.x0 /, are pairwise disjoint, it follows from this that f i .x0 / D x0 , so sji . But then also f i .y/ D y, as gx0 is injective, and so t ji . Hence kji . The next theorem is the main result of this section. Theorem 12.3. Let X be a countable nondiscrete regular space with a distinguished point 1 2 X , let f W X ! X be a spectrally irreducible P regular homeomorphism of ﬁnite order, let S D be the stanL spec.f /, and let m D 1 C s2S s. LetL dard L permutation L on Z of spectrum S, and for every a 2 m ! ! Zm , deﬁne a W Z ! Z by .x/ D a C x. Then there is a continuous bijection m m a !L ! h W X ! ! Zm with h.1/ D 0 such that (1) f D h1 h, and (2) for every x 2 X , x D h1 h.x/ h is a homeomorphism of X onto itself. Furthermore, if X is a local left group and f is a local automorphism, then h can be chosen so that (3) x .y/ D xy, whenever max supp.h.x// C 1 < min supp.h.y//.

190

Chapter 12 Resolvability

L Recall that the topology of ! Zm is generated by taking as a neighborhood base at 0 the subgroups ° ± M Hn D x 2 Zm W x.i / D 0 for all i < n !

where n < !. The conclusion of Theorem 12.3 can be rephrased as follows: L One can deﬁne the operation of the group Z ! m on X in such a way that 0 D 1, L the topology of ! Zm is weaker than that of X and (1) D f , and (2) for every x 2 X , x W X 3 y 7! x C y 2 X is a homeomorphism. Furthermore, if X is a local left group and f is a local automorphism, then the operation can be deﬁned so that (3) x C y D xy, whenever max supp.x/ C 1 < min supp.y/. Actually, Theorem 12.3 characterizes spectrally irreducible regular homeomorphisms of ﬁnite order on countable regular spaces. If f W X ! X is a spectrally irreducible homeomorphism and for some m there is a continuous bijection h W X ! L Z with h.1/ D 0 such that ! m L (1) hf h1 is a coordinatewise permutation on ! Zm , and (2) for every x 2 X , h1 h.x/ h is a homeomorphism of X onto itself, then f is regular. To see this, let D hf h1 . For every x 2 X n¹1º, let n.x/ D max supp.h.x//C1, Ux D h1 .Hn.x/ / and gx D h1 h.x/ hjUx . Then for every y 2 Ux , f .y/ 2 Ux D Uf .x/ and fgx .y/ D h1 hh1 h.x/ h.y/ D h1 h.x/ h.y/ D h1 .h.x/ C h.y// D h1 ..h.x// C .h.y/// D h1 .h.f .x// C h.f .y/// D h1 h.f .x// h.f .y// D gf .x/ f .y/: Proof of Theorem 12.3. Let W D W .Z Lm /. The permutation 0 on Zm , which induces the standard permutation on ! Zm , also induces the permutation 1 on W . If w D 0 n , then 1 .w/ D 0 . 0 / 0 . n /. We will write instead of 0 and 1 . For each x 2 X n ¹1º, choose a homeomorphism gx of a neighborhood

Section 12.1 Regular Homeomorphisms of Finite Order

191

of 1 onto a neighborhood of x with gx .1/ D x such that fgx D gf .x/ f jU for some neighborhood U of 1. Also put g1 D idX . If X is a local left group and f is a local automorphism, choose gx so that gx .y/ D xy. Enumerate X as ¹xn W n < !º with x0 D 1. We shall assign to each w 2 W a point x.w/ 2 X and a clopen spectrally minimal neighborhood X.w/ of x.w/ such that (i) x.0n / D 1 and X.;/ D X , (ii) ¹X.w _ / W 2 Z.m/º is a partition of X.w/, (iii) x.w/ D gx.w0 / gx.wk1 / .x.wk // and X.w/ D gx.w0 / gx.wk1 / .X.wk //, where w D w0 C C wk is the canonical decomposition, (iv) f .x.w// D x..w// and f .X.w// D X..w//, (v) xn 2 ¹x.v/ W v 2 W and jvj D nº. Enumerate S as s1 < < s t and for each i D 1; : : : ; t , pick a representative i of the orbit in Zm n ¹0º of length si . Choose a clopen invariant neighborhood U1 of 1 such that x1 … U1 and spec.f; X n U1 / D spec.f /. Put x.0/ D 1 and X.0/ D U1 . Then choose points Sai 2 X n U1 , i D 1; : : : ; t , with pairwise disjoint orbits of lengths si such that x1 2 tiD1 O.ai /. For each i D 1; : : : ; t and j < si , put x. j .i // D f j .ai /: By Lemma 8.30, there is an invariant partition ¹X. / W 2 Zm n ¹0ºº of X n U1 such that X. / is a clopen spectrally minimal neighborhood of x. /. Fix n > 1 and suppose that X.w/ and x.w/ have been constructed for all w 2 W with jwj < n so that conditions (i)–(v) are satisﬁed. Note that the subsets X.w/, jwj D n 1, form a partition of X . So one of them, say X.u/, contains xn . Let u D u0 C C uq be the canonical decomposition. Then X.u/ D gx.u0 / gx.uq1 / .X.uq // and xn D gx.u0 / gx.uq1 / .yn / for some yn 2 X.uq /. Choose a clopen invariant neighborhood Un of 1 such that for all basic w with jwj D n 1, (a) gx.w/ .Un / X.w/, (b) fgx.w/ jUn D gf .x.w// f jUn , and (c) spec.f; X.w/ n gx.w/ .Un // D spec.X.w//. If yn ¤ x.uq /, choose Un in addition so that (d) yn … gx.uq / .Un /. Put x.0n / D 1 and X.0n / D Un . Let w 2 W be an arbitrary nonzero basic word with jwj D n 1 and let O.w/ D ¹wj W j < sº, where wj C1 D .wj / for j < s 1 and .ws1 / D w0 . Put Yj D X.wj / n gx.wj / .Un /. Using Lemma 12.2, choose points bi 2 Y0 , i D 1; : : : ; t ,

192

Chapter 12 Resolvability

with pairwise disjoint orbits of lengths lcm.si ; s/. If uq 2 O.w/, choose bi in addition so that t [ yn 2 O.bi /: iD1

For each i D 1; : : : ; t and j < si , put x. j .w _ i // D f j .bi /: Then, using Lemma 8.30, inscribe an invariant partition ¹X.v _ / W v 2 O.w/; 2 Zm n ¹0ºº into the partition ¹Yj W j < sº such that X.v _ / is a clopen spectrally minimal neighborhood of x.v _ /. For nonbasic w 2 W with jwj D n, deﬁne x.w/ and X.w/ by condition (iii). To check (ii) and (iv), let jwj D n 1 and let w D w0 C C wk be the canonical decomposition. Then X.w _ 0/ D gx.w0 / gx.wk / .X.0n // D gx.w0 / gx.wk1 / .gx.wk / .X.0n /// and X.w _ / D gx.w0 / gx.wk1 / .X.wk_ //; so (ii) is satisﬁed. Next, f .x.w// D fgx.w0 / gx.wk1 / .x.wk // D gf .x.w0 // fgx.w1 / gx.wk1 / .x.wk // :: : D gf .x.w0 // gf .x.wk1 // f .x.wk // D gx..w0 // gx..wk1 // .x..wk /// D x..w0 / .wk1 /.wk // D x..w//; so (iv) is satisﬁed as well. To check (v), suppose that xn … ¹x.w/ W jwj D n 1º. Then xn D gx.u0 / gx.uq1 / .yn / D gx.u0 / gx.uq1 / .u_ q / D x.u_ /:

193

Section 12.1 Regular Homeomorphisms of Finite Order

Now, for every x 2 X , there is w 2 W with nonzero last letter such that x D Lx.w/, so ¹v 2 W W x D x.v/º D ¹w _ 0n W n < !º. Hence, we can deﬁne h W X ! ! Zm by putting for every w D 0 n 2 W , h.x.w// D w D . 0 ; : : : ; n ; 0; 0; : : :/: It is clear that h is bijective and h.1/ D 0. Since for every z D . i /i

L !

Zm ,

h1 .z C Hn / D X. 0 n /; h is continuous. To see (1), let x D x.w/. Then h.f .x.w/// D h.x..w/// D .w/ D .w/ D .h.x.w///: To see (2), let x D x.w/, w D w0 C C wk and n D max supp.h.x// C 1. We ﬁrst show that x jh1 .Hn / D gx.w0 / gx.wk / jh1 .Hn / : Let y 2 h1 .Hn /, y D x.v/ and v D v0 C C vl . Then hgx.w0 / gx.wk / .y/ D hgx.w0 / gx.wk / gx.v0 / gx.vl1 / .x.vl // D h.x.w C v// DwCv DwCv D h.x.w// C h.x.v// D h.x/ h.y/: It follows from (iii) that gx.w0 / gx.wk / homeomorphically maps X.0n /, a neighborhood of 1, onto X.w _ 0/, a neighborhood of x, and so x does. Now, to see that x homeomorphically maps a neighborhood of an arbitrary point y 2 X onto a neighborhood of z D x .y/, it sufﬁces to check that x D z .y /1 . Indeed, z D h1 h.x/ h.y/ D h1 .h.x/ C h.y// and then z .y /1 D h1 h.x/Ch.y/ h.h1 h.y/ h/1 D h1 h.x/Ch.y/ hh1 .h.y/ /1 h D h1 h.x/Ch.y/ h.y/ h D h1 h.x/ h D x :

194

Chapter 12 Resolvability

To see (3), let x D x.w/ and w D w0 C Cwk . If k D 0, then x .y/ D gx .y/ D xy. Continuing, by induction on k, we obtain that x .y/ D gx.w0 / gx.wk / .y/ D gx.w0 / gx.wk1 / .x.wk / y/ D x.w0 C C wk1 / .x.wk / y/ D .x.w0 C C wk1 / x.wk // y D .gx.w0 / gx.wk1 / .x.wk /// y D x.w/ y: The second part of Theorem 12.3, the case where X is a local left group and f is a local automorphism, is Theorem 8.29. The ﬁrst part with f D idX tells us that Corollary 12.4. Every countably inﬁnite homogeneous regular space admits a Boolean group operation with continuous translations. We conclude this section by deducing from Corollary 12.4 and Corollary 8.12 the following result. Theorem 12.5. Let X be a countably inﬁnite homogeneous regular space and let G be a countably inﬁnite group. Then there is a group operation on X such that .X; / is a left topological group algebraically isomorphic to G. Proof. One may suppose that X is nondiscrete. By Corollary 12.4, there is a Boolean group operation C on X with continuous translations. Endowing G with any nondiscrete regular left invariant topology and applying Corollary 8.12, we obtain that there is a bijective local homomorphism f W .X; C/ ! G. For any x; y 2 X , deﬁne x y D f 1 .f .x/f .y//. Obviously, .X; / is a group isomorphic to G. Now given any x 2 X , we can choose a neighborhood U of 0 such that f .x C z/ D f .x/f .z/ for all z 2 U , and then x z D f 1 .f .x/f .z// D f 1 .f .x C z// D x C z. It follows from this that the left translations of .X; / are continuous and open at the identity. Consequently, the left translations of .X; / are continuous.

12.2

Resolvability of Topological Groups

Theorem 12.6. If a countable regular space admits a nontrivial regular homeomorphism of ﬁnite order, then it is !-resolvable. Proof. Let X be a countable regular space with a distinguished point 1 2 X and let f W X ! X be a nontrivial regular homeomorphism of ﬁnite order. By Corollary 8.25 and Corollary 3.27, one may suppose that f is spectrally irreducible. Let h W X !

Section 12.2 Resolvability of Topological Groups

195

L

! Zm be a bijection guaranteed by Theorem 12.3. Pick an orbit C in Zm (with respect to 0 ) of a smallest possible length s > 1. For every x 2 X , consider the sequence of coordinates of h.x/ belonging to C and deﬁne .x/ to be the number of pairs of distinct neighbouring elements in this sequence. Denote also by ˛.x/ and

.x/ the ﬁrst and the last elements in the sequence (if nonempty). Then whenever x; y 2 X and max supp.h.x// C 1 < min supp.h.y//, ´ .x/ C .y/ if .x/ D ˛.y/ .x .y// D .x/ C .y/ C 1 otherwise:

We deﬁne a disjoint family ¹Xn W n < !º of subsets of X by Xn D ¹x 2 X W .x/ 2n .mod 2nC1 /º: To see that every Xn is dense in X , let x 2 X and let U be an open neighbourhood of 1. We have to show that x .U / \ Xn ¤ ;. Put k D 2nC1 and choose inductively x1 ; : : : ; xk 2 U such that (i) jO.xj /j D s, (ii)

max supp.h.xj // C 1 < min supp.h.xj C1 //, and if x ¤ 0, then max supp.h.x// C 1 < min supp.h.x1 //,

(iii) y1 yk .1/ 2 U whenever yj 2 O.xj /. Without loss of generality one may suppose that .xj / D ˛.xj C1 /, and that if x 2 X , then .x/ D ˛.x1 /. For every l D 0; 1; : : : ; k 1, deﬁne zl 2 U by zl D x1 f .x2 / f l .xlC1 / f l .xlC2 / f l .xk / .1/ (in particular, z0 D x1 x2 xk .1/). Then h.x .zl // D h.x/Ch.x1 /Ch.x2 /C C l h.xlC1 /C l h.xlC2 /C C l h.xk /: It follows that .x .z0 // D .x/ C .x1 / C C .xk / and .x .zl // D .z0 / C l. Hence, for some l, .x .zl // 2n .mod 2nC1 /, so x .zl / 2 Xn . The next proposition says that every nondiscrete topological group not containing an open Boolean subgroup admits a nontrivial regular homeomorphism of order 2. Proposition 12.7. Let G be a nondiscrete topological group not containing an open Boolean subgroup. Suppose that for every element x 2 G of order 2, the conjugation G 3 y 7! xyx 1 2 G is a trivial local automorphism. Then the inversion G 3 y 7! y 1 2 G is a nontrivial regular homeomorphism.

196

Chapter 12 Resolvability

In order to prove Proposition 12.7, we need the following lemma. Lemma 12.8. Let X be a homogeneous space with a distinguished point 1 2 X and let f W X ! X be a homeomorphism of ﬁnite order n with f .1/ D 1. Suppose that for every x 2 X n ¹1º with jO.x/j D s < n, there is a homeomorphism gx of a neighborhood U of 1 onto a neighborhood of x with gx .1/ D x such that f s gx .y/ D gx f s .y/ for all y 2 U . Then f is regular. In particular, if for every x 2 X n ¹1º, jO.x/j D n, then f is regular. Proof. Consider an arbitrary orbit in X distinct from ¹1º and enumerate it as ¹xi W i < sº, where xiC1 D f .xi / for i D 0; : : : ; s 2 and f .xs1 / D x0 . If s D n, choose as gx0 any homeomorphism of a neighborhood U of 1 onto a neighborhood of x0 with gx0 .e/ D x0 . If s < n, choose gx0 in addition such that f s gx0 .y/ D gx0 f s .y/ for all y 2 U . For every i D 1; : : : ; s 1, put gxi D f i gx0 f i jU . Then for every i D 0; : : : ; s 1 and y 2 U , fgxi .y/ D ff i gx0 f i .y/ D f iC1 gx0 f .iC1/ f .y/: If i < s 1, then f iC1 gx0 f .iC1/ f .y/ D gxiC1 f .y/, so fgxi .y/ D gxiC1 f .y/. Hence, it remains only to check that fgxs1 .y/ D gx0 f .y/. If s D n, then fgxs1 .y/ D f s gx0 f s f .y/ D idX gx0 idX f .y/ D gx0 f .y/: If s < n, then fgxs1 .y/ D f s gx0 f s f .y/ D gx0 f s f s f .y/ D gx0 f .y/: Proof of Proposition 12.7. Let f denote the inversion and let B D B.G/. We have that f is a homeomorphism of order 2 and B is the set of ﬁxed points of f , in particular, f .1/ D 1. By Lemma 5.3, B is not a neighborhood of 1, so f is nontrivial. To see that f is regular, let x 2 G n ¹1º and jO.x/j < 2. Then x 2 B. But then there is a neighborhood U of 1 such that xyx 1 D y for all y 2 U , that is, xy D yx. Deﬁne gx W U ! xU by gx .y/ D xy. We have that fgx .y/ D .xy/1 D .yx/1 D x 1 y 1 D xy 1 D gx f .y/: Hence, by Lemma 12.8, f is regular. Combining Theorem 12.6 and Proposition 12.7, we obtain that Theorem 12.9. Every countable nondiscrete topological group not containing an open Boolean subgroup is !-resolvable. Note that in the Abelian case Theorem 12.9 can be proved easier. If a topological group is Abelian, then the inversion is a local automorphism. Therefore, it sufﬁces to use Theorem 8.29 instead of Theorem 12.3. Also in the Abelian case the restriction ‘countable’ is redundant.

Section 12.2 Resolvability of Topological Groups

197

Theorem 12.10. Every nondiscrete Abelian topological group not containing a countable open Boolean subgroup is !-resolvable. Since every Abelian group can be isomorphically embedded into a direct sum of countable groups, Theorem 12.10 is immediate from the Abelian case of Theorem 12.9 and the following result. Theorem 12.11. Let > !. ForLevery ˛ < , let G˛ be a countable group and let G be an uncountable subgroup of ˛< G˛ . Then G can be partitioned into ! subsets dense in every group topology of uncountable dispersion character. The proof of Theorem 12.11 involves the functions 2 ; 2 W N ! ! (see Deﬁnition 6.35). Lemma 12.12. Let m 2 N and n < !. Let I denote the integer interval Œ2 .m/ C 1; 2 .m/ C 2nC1 : Then for every k 2 N, there is i 2 I such that 2 .2 .m C k 2i // D n. Proof. For every i 2 .m/ C 1, one has 2 .k 2i / 2 .m/ C 1. It follows that 2 .m C k 2i / D 2 .k 2i / D 2 .k/ C i: Consequently, J D ¹2 .m C k 2i / W i 2 I º is an integer interval of length 2nC1 , and so there is j 2 J such that j 2n .mod 2nC1 /. Proof of Theorem 12.11. Deﬁne a disjoint family ¹Yn W n < !º of subsets of G by Yn D ¹x 2 G W 2 .2 .jsupp.x/j// D nº: Let G be endowed with any group topology of uncountable dispersion character, let U be a neighborhood of 1 2 G, and let x 2 G. We show that .xU / \ Yn ¤ ;. Without loss of generality one may suppose that x ¤ 1. Let m D jsupp.x/j. Put nC1 s D 22 .m/C2 and pick a neighborhood V of 1 such that V s U . Since G has uncountable dispersion character, it follows that for every countable A , there is y 2 V n ¹1º such that supp.y/ \ A D ;. To L see this, pick a neighborhood W of 1 such that W W 1 V . Since jW j > ! and j ˛2A G˛ j !, there exist distinct v; w 2 W such that v.˛/ D w.˛/ for each ˛ 2 A. Put y D vw 1 . Then 1 ¤ y 2 W W 1 V and supp.y/ \ A D ;. Now construct an !1 -sequence .y˛ /˛

198

Chapter 12 Resolvability

Passing to a coﬁnal subsequence, one may suppose that there is k 2 N such that jsupp.y˛ /j D k for all ˛ < . Let i be a natural number guaranteed by Lemma 12.12 i and let y D y1 y2i . Then y 2 V 2 U , jsupp.xy/j D m C k 2i , and 2 .2 .jsupp.xy/j// D n. We conclude this section by showing that the existence of a countable nondiscrete !-irresolvable topological group cannot be established in ZFC. Theorem 12.13. The existence of a countable nondiscrete !-irresolvable topological group implies the existence of a P -point in ! . Proof. Let .G; T / be a countable L !-irresolvable topological group. By Theorem 12.9, one may suppose that Q G D ! Z2 . Let T0 denote the topology on G induced by the product topology on ! Z2 . Then T _ T0 is a nondiscrete !-irresolvable group topology. Therefore, one may suppose that T0 T . By Theorem 3.33, every discrete subset of .G; T / is closed. Let F be the neighborhood ﬁlter of 0 in T . It is easy to 1 see that ¹ .u/ W u 2 .F /º is a partition of Ult.T / into right ideals. It then follows from Theorem 3.35 and Proposition 7.7 that .F / is ﬁnite. Hence by Theorem 5.19, each point of .F / is a P -point. Combining Theorem 12.13 and Theorem 2.38 gives us that Corollary 12.14. It is consistent with ZFC that there is no countable nondiscrete !-irresolvable topological group.

12.3

Absolute Resolvability

Deﬁnition 12.15. Let G be a group. A subset A G is absolutely dense if A is dense in every nondiscrete group topology on G. Given a cardinal 2, G is absolutely -resolvable (absolutely resolvable if D 2) if G can be partitioned into absolutely dense subsets. Lemma 12.16. Let G be an Abelian group and let A G. If A is absolutely dense, then G n A contains no coset modulo inﬁnite subgroup. Proof. Suppose that there exist an inﬁnite subgroup H of G and g 2 G such that g C H G n A. Pick any nondiscrete group topology TH on H and extend it to the group topology T on G by declaring H to be an open subgroup. Then g C H is an open subset of .G; T / disjoint from A, so A is not dense. Hence, A is not absolutely dense. The next proposition is a consequence of Hindman’s Theorem.

Section 12.3 Absolute Resolvability

199

Proposition 12.17. Whenever an inﬁnite Boolean group is partitioned into ﬁnitely many subsets, there is a coset modulo inﬁnite subgroup contained in one subset of the partition. Proof. Let B be an inﬁnite Boolean group and let ¹Ai W i < rº be a ﬁnite partition of B. By Hindman’s Theorem, there exist i < r and a one-to-one sequence .xn /n

In this section we prove the following result. Theorem 12.20. Let G be an Abelian group and let A D G n B.G/ be inﬁnite. Then there is a disjoint family ¹Am W m < !º of subsets of A such that whenever .xn /n

200

Chapter 12 Resolvability

Corollary 12.21 and Lemma 12.16 give us in turn that Corollary 12.22. Every inﬁnite Abelian group not containing an inﬁnite Boolean subgroup can be partitioned into ! subsets such that every coset modulo inﬁnite subgroup meets each subset of the partition. Note that the cardinal number ! in Theorem 12.20, Corollary 12.21 and Corollary 12.22 is maximally possible. We ﬁrst prove Theorem 12.20 in the case where G is a direct sum of ﬁnite groups, not necessarily Abelian. Theorem 12.23. LetL !. For each ˛ < , let G˛ be a ﬁnite group written additively, let G D ˛< G˛ , and let A D G n B.G/ be inﬁnite. Then there is a disjoint family ¹Am W m < !º of subsets of A such that whenever .xn /n d . Proof. Without loss of generality one may suppose that cnC1 cn 5 for all n < !. For each n < !, pick an integer dn 2 such that dn2 C dn 1 cnC1 cn and dn ! 1. To deﬁne ', let k < ! and let a ckC1 . Choose the largest integer l 0 for which ckC1 C ldk2 a, then choose the largest integer i 0 for which ckC1 C ldk2 C idk a, and then put j D a ckC1 ldk2 idk . Thereby we write a in the form a D ckC1 C ldk2 C idk C j;

201

Section 12.3 Absolute Resolvability

where l is a nonnegative integer and i; j 2 ¹0; : : : ; dk 1º. Put ak D ck C jdk C i: Since ak 2 Œck ; ck C dk2 /, the function ' so deﬁned satisﬁes the condition (1). To check (2), let d 2 N be given. Choose n0 < ! such that dn d C 2 for all n n0 and put c D cn0 . Now let a; b 2 N, 0 < ja bj d , u 2 '.a/, v 2 '.b/ and u; v c. Since ckC1 ak D ckC1 ck jdk i dk2 C dk 1 jdk i dk ; one may suppose that u D ak and v D bk for some k n0 . Let a D ckC1 C ldk2 C idk C j; b D ckC1 C l 0 dk2 C i 0 dk C j 0 : Then ak D jdk C i; bk D j 0 dk C i 0 : Thus, we have that a b D Œ.l l 0 /dk C .i i 0 /dk C .j j 0 /; ak bk D .j j 0 /dk C .i i 0 /: Notice that ji i 0 j < dk and jj j 0 j < dk . Then it follows from the second equality that if jj j 0 j > 1, jak bk j > dk . Therefore one may suppose that jj j 0 j 1. But then it follows from the ﬁrst equality that a b differs from a multiple of dk by 0, 1 or 1. Since ja bj d dk 2, this multiple of dk must be zero, so .l l 0 /dk C .i i 0 / D 0: This gives us l D l 0 and i D i 0 . Consequently, jj j 0 j D 1, and we obtain that jak bk j D dk . Let ' W N ! ŒN

202

Chapter 12 Resolvability

and deﬁne .x/ to be the number of pairs of distinct neighbouring elements in the sequence sgn.x.˛l0 //; : : : ; sgn.x.˛ln1 //: (If l 2 Œc0 ; c1 /, then '.l/ D ;, and then supp' .x/ D ; and .x/ D 0.) We deﬁne the disjoint family ¹Am W m < !º of subsets of A by Am D ¹x 2 A W .x/ 2m .mod 2mC1 /º: Now let .xn /n

\

FS..xn /mn

m

(Proposition 6.39) and pick an idempotent p 2 T (Theorem 6.12). We then have that FS..xn /mn

\ ik

The sequence .yn /n

Uyi :

203

Section 12.3 Absolute Resolvability

By Lemma 12.25, one may suppose that supp.g/ \ supp.xi / D ;

and

supp.xi / \ supp.xj / D ;

for all i < j < !. Also one may suppose that the sequence .min supp0 .xn //n

D sup¹min supp0 .xn / W n < !º: Every x 2 G can be uniquely written in the form x D x 0 C x 00 , where x 0 ; x 00 2 G, supp.x 0 / D supp.x/ \

and

supp.x 00 / D supp.x/ n supp.x 0 /:

Put k D 2mC1 1. Choose inductively a sum subsystem .yn /n

(b) each of the intervals .jsupp0 .g 0 /j; jsupp0 .g 0 C y00 /j and 0 /j; .jsupp0 .g 0 C y00 C C yi0 /j; jsupp0 .g 0 C y00 C C yi0 C yiC1

where i < k 1, contains some interval Œcn ; cnC1 /. P Let h D g C y0 C C yk1 , yk1 D n2H xn and n0 D max H C 1. We claim that there is yk 2 FS..xn /n0 n

and

for all z 2 FS..zn /n

Put d D jsupp0 .h C z0 /j and let c be a constant guaranteed by Lemma 12.16. Pick z 2 FS..zn /1n

and

204

Chapter 12 Resolvability

Let a D jsupp0 .h C z0 C z/j and b D jsupp0 .h C z/j. Then 0 < a b D jsupp0 .z0 /j jsupp0 .h C z0 /j D d: Let u D jsupp0 .h C z0 C z/ \ .ı C 1/j, v D jsupp0 .h C z/ \ .ı C 1/j, w D jsupp0 .h C z0 / \ .ı C 1/j and t D jsupp0 .h/ \ .ı C 1/j. Then u D v C w t , and so u v D w t; which is a contradiction, since u 2 '.a/, v 2 '.b/, u; v c and w t w d . Now consider the element g0 D g C y0 C y1 C C yk : By the construction, supp' .g0 / \ supp0 .yk / ¤ ;; and for each i k 1, ; ¤ supp' .g0 / \ supp0 .yi / supp0 .yi0 /: Therefore we have that ˇ0 < ˛1 ˇ1 < ˛2 ˇ2 < < ˛k1 ˇk1 < ˛k ; where ˛i D min.supp' .g0 / \ supp0 .yi //; ˇi D max.supp' .g0 / \ supp0 .yi //: Put "0 D 1 and, by induction on i D 1; : : : ; k, choose "i 2 ¹1; 1º so that "i1 yi1 .ˇi1 / D "i yi .˛i /: Without loss of generality one may suppose that all "i D 1. For each i k, deﬁne gi 2 g C FSI..yn /nk / by gi D g C y0 y1 C C .1/i yi C .1/i yiC1 C C .1/i yk : Then .gi / D .g0 / C i: Hence, there is j k D 2mC1 1 such that .gj / 2m and so gj 2 Am .

.mod 2mC1 /;

Section 12.3 Absolute Resolvability

205

In order to to prove Theorem 12.20 in the general case, we need in addition the following result. Theorem 12.26. Let G be an inﬁnite Abelian group endowed with the largest totally bounded L group topology and let jGj D . Then there is a continuous injection f W G ! Z4 such that (1) f .x/ D f .x/ for all x 2 G, (2) f .x C y/ D fL .x/ C f .y/ for all x; y 2 G with S.f .x// \ S.f .y// D ;, where for each a 2 Z4 , S.a/ D supp.a/ [ .supp.a/ C 1/. L Q Here, Z4 is endowed with the topology induced by the product topology on Z4 . Theorem 12.26 Lallows us to identify an arbitrary Abelian group G of cardinality with a subset of Z4 so that L (a) for every x 2 G, x 2 Z4 is the inverse of x in G, L (b) for every x; y 2 G with S.x/ \ S.y/ D ;, x C y 2 Z4 is the sum of x and y in G, and (c) the largest L totally bounded group topology on G is stronger than that induced from Z4 . And that is all what we need to extend the proof of Theorem 12.23 to Theorem 12.20. We deduce Theorem 12.26 from the following fact. Proposition 12.27. Let G be a countable Abelian topological group. Suppose that B.G/ is neither L discrete nor open. Then there is a continuous bijection f W G ! ! Z4 such that (i) f .x/ D f .x/ for all x 2 G, (ii) f .x C y/ D f .x/ C f .y/ for all x; y 2 G n ¹0º with max supp.f .x// C 2 min supp.f .y//. Proposition 12.27 is an immediate consequence of Theorem 8.29. Before proving Theorem 12.26 we show that condition (ii) in Proposition 12.27 can be replaced by (2) from Theorem 12.26. L Indeed, every nonzero element a 2 ! Z4 has a unique canonical decomposition a D a1 C C a n ; where for each i D 1; : : : ; n, supp.ai / is a nonempty interval in !, and for each i D 1; : : : ; n 1, max supp.ai / C 2 min supp.aiC1 /. And it follows from (ii) that if xi D f 1 .ai / for each i D 1; : : : ; n, then f .x1 C C xn / D a1 C C an :

206

Chapter 12 Resolvability

Let a D f .x/, b D f .y/ and S.a/ \ S.b/ D ;. Let a D a1 C C an and b D b1 C C bm be the canonical decompositions. Since S.a/ \ S.b/ D ;, there is a permutation c1 ; : : : ; cnCm of a1 ; : : : ; an ; b1 ; : : : ; bm such that a C b D c1 C C cnCm is the canonical decomposition. Let xi D f 1 .ai /, yj D f 1 .bj / and zk D f 1 .ck /. Then x D x1 C C xn , y D y1 C C ym and, since G is Abelian, x C y D z1 C C znCm . Finally, we obtain that f .x C y/ D f .z1 C C znCm / D c1 C C cnCm D .a1 C C an / C .b1 C C bm / D f .x/ C f .y/: Proof of Theorem 12.26. Without loss of generality one may suppose that M GD G˛ ; ˛<

where for each ˛ < , jG˛ j D jB.G˛ /j D jG˛ W B.G˛ /j D !: For each ˛ < , endow G˛ with a totally bounded group topology and let M f˛ W G˛ ! Z4 !

be a continuous bijection guaranteed by Proposition 12.27. We deﬁne M M f WG! Z4 ˛< Œ˛;˛C!/

by f D

M

f˛ :

˛<

References The study of resolvability for topological groups was initiated by W. Comfort and J. van Mill [13]. They showed that if G is an inﬁnite Abelian group not containing an inﬁnite Boolean subgroup, then every nondiscrete group topology on G is resolvable. Theorem 12.3 and Theorem 12.6 are results from [105]. Theorem 12.5 was proved in [96]. Theorem 12.9 is a result from [94] and Theorem 12.10 from [89]. Theorem 12.11 is due to I. Protasov [57]. He also proved Theorem 12.13 in the Abelian case [59].

Section 12.3 Absolute Resolvability

207

The question of characterizing absolutely resolvable groups was raised in [13]. Corollary 12.18 is due to I. Protasov [57]. Theorem 12.20 was proved in [102]. That every inﬁnite Abelian group not containing an inﬁnite Boolean subgroup is absolutely resolvable was proved in [91]. Corollary 12.22 and Proposition 12.17 complement the Graham-Rothschild Theorem [29] (see also [30, Section 2.4] ) which can be stated as follows. If an inﬁnite Abelian group of ﬁnite exponent is partitioned into ﬁnitely many subsets, then there are arbitrarily large ﬁnite cosets contained in one subset of the partition.

Chapter 13

Open Problems

In this chapter we list several open questions related to the topic of this book. (1) (Old question) Is there any element of ﬁnite order in ˇN other than idempotents? L (2) Let > ! and Q let T denote the topology on Z2 induced by the product topology on Z2 . Are ﬁnite groups in Ult.T / trivial? (3) Is the structure group of K.H/ a free group? (4) (Old question) Does every point of Z lie in a maximal proper principal left ideal of ˇZ? (5) (Old question) Is there an inﬁnite increasing sequence of principal left ideals of ˇZ? (6) (N. Hindman and D. Strauss) Is there an inﬁnite increasing (L -increasing) sequence of idempotents in ˇZ? (7) (N. Hindman and D. Strauss) Is there in ZFC a left maximal idempotent in Z ? (8) Can every compact Hausdorff right topological semigroup be topologically and algebraically embedded into a compact Hausdorff right topological semigroup with a dense topological center? (9) Let bZ denote the Bohr compactiﬁcation of the discrete group Z and let T be the topology on N induced by bZ. Is the largest semigroup compactiﬁcation of .N; T / different from bZ? (10) (A. Arhangel’ski˘ı, 1967) Is there in ZFC a (countable) nondiscrete extremally disconnected topological group? (11) (W. Comfort and J. van Mill) Is there in ZFC a nondiscrete irresolvable (!irresolvable, almost maximal) topological group? (12) (V. Malykhin) Is there an irresolvable (!-irresolvable, almost maximal) topological group of uncountable dispersion character? (13) Is there a (countable) nondiscrete !-irresolvable topological group different from almost maximal? (14) (I. Protasov and Y. Zelenyuk) Does every maximal (almost maximal) topological group have a neighborhood base at zero consisting of subgroups? (15) (A. Arhangel’ski˘ı and P. Collins) Is there in ZFC a countable nondiscrete nodec topological group?

209 (16) (I. Protasov) Is there in ZFC a countable nondiscrete topological group in which every discrete subset is closed? (17) Is it true that for every projective S in F, there exists in ZFC a countable homogeneous regular almost maximal space X with Ult.X / isomorphic to S? (18) Let .G1 ; T1 / and .G2 ; T2 / be countable regular left topological groups and suppose that they are homeomorphic. Are Ult.T1 / and Ult.T2 / topologically and algebraically isomorphic?

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Index

Symbols Ap0 , 103 A , 29 B.G/, 70 B.k; m; n/, 155 C.p/, 102 C 1 .p/, 102 Cp , 109 E.S /, 88 G.Q/, 117 Ip0 , 133 Ip , 130 K.S /, 87 S 1 , 86 Sn , 8 U.A/, 130 U.D/, 33 V .M /, 166 W .Zm /, 113 Xd , 110 Y 0 , 60 ŒM a , 55 ŒX , 131 ŒX < , 131 ƒ.T /, 83 ˇD, 29 ˇS, 84 FP..xn /1 nD1 /, 23 FSI..xn /n

M, 165 P, 158 I 0 .G/, 133 I.G/, 130 M.GI I; ƒI P /, 89 P .X /, 4 Pf .X /, 23 T .F /, 7 T ŒF , 54 T f , 182 C, 147 F, 151 p, 14 A, 29 A, 34 f , 31 , 170 2 .x/, 93 , 170 2 .x/, 93 a1 U , 52 bG, 136 f , 111 A absolute H-coretract, 148 absolute coretract, 147 B band, 152 rectangular, 155 basic mapping, 56 Bohr compactiﬁcation, 136 Burnside semigroup, 155 C canonical decomposition, 111 character, 14 D dispersion character, 186

218 F F -syndetic, 94 ﬁlter, 4 closed, 41 dense, 41 Fréchet, 57 image, 28 locally Fréchet, 61 nowhere dense, 41 open, 41 strongly discrete, 58 trace, 28 ﬁlter base, 6 ﬁnite intersection property, 26 G Green’s relations, 155 group absolutely -resolvable, 198 absolutely resolvable, 198 Boolean, 70 topologizable, 11 H H-projective, 151 homeomorphism of ﬁnite order, 120 regular, 188 spectrally irreducible, 121 trivial, 152 I ideal, 85 left (right), 85 minimal left (right), 87 smallest, 87 idempotent, 86 minimal, 91 right (left) maximal, 91 invariant subset (family), 120 L left saturated subsemigroup, 101 left topological group, 52 left topological semigroup, 52 local automorphism, 121 local homomorphism, 111 surjective, 116

Index local isomorphism, 111 local left group, 110 local monomorphism, 181 P preorderings on the idempotents, 90 projective, 150 proper homomorphism, 115 pseudo-intersection number, 14 R right cancelable, 170 right topological semigroup, 83 S semigroup cancellative, 54 completely simple, 88 left (right) zero, 88 semigroup compactiﬁcation, 84 semilattice, 154 semilattice decomposition, 154 space -irresolvable, 46 hereditarily -irresolvable, 46 open-hereditarily -irresolvable, 46 -resolvable, 46 -compact, 33 almost maximal, 46 extremally disconnected, 32 ﬁrst countable, 3 homogeneous, 2 locally regular, 105 maximal, 45 nodec, 42 strongly extremally disconnected, 43 submaximal, 45 zero-dimensional, 2 spectrum, 121 standard permutation, 123 ˇ Stone–Cech compactiﬁcation of a discrete semigroup, 82 of a discrete space, 29 strong ﬁnite intersection property, 14 structure group, 90 subset of a space locally maximal discrete, 59 perfect, 44

Index strongly discrete, 48 syndetic, 94 T topological center, 83 topological group, 1 almost maximal, 147 maximal, 71 maximally nondiscrete, 79 metrizable, 3 totally bounded, 12 topology group, 1 invariant, 63 left invariant, 52 U Ulam-measurable cardinal, 39 ultraﬁlter, 26 P -point, 38 countably complete, 39 nonprincipal, 27 principal, 27 Ramsey, 36 uniform, 28 ultraﬁlter semigroup, 98 upper semicontinuous decomposition, 131

219

Yevhen G. Zelenyuk

Ultrafilters and Topologies on Groups

De Gruyter

Mathematical Subject Classification 2010: 22A05, 22A15, 54D80, 54G05.

ISBN 978-3-11-020422-3 e-ISBN 978-3-11-021322-5 ISSN 0938-6572 Library of Congress Cataloging-in-Publication Data Zelenyuk, Yevhen G. Ultrafilters and topologies on groups / by Yevhen G. Zelenyuk. p. cm. ⫺ (De Gruyter expositions in mathematics ; 50) Includes bibliographical references and index. ISBN 978-3-11-020422-3 (alk. paper) 1. Topological group theory. 2. Ultrafilters (Mathematics) I. Title. QA166.195.Z45 2011 5121.55⫺dc22 2010050782

Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de. ” 2011 Walter de Gruyter GmbH & Co. KG, Berlin/New York Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ⬁ Printed on acid-free paper Printed in Germany www.degruyter.com

Preface

This book presents the relationship between ultraﬁlters and topologies on groups. It shows how ultraﬁlters are used in constructing topologies on groups with extremal properties and how topologies on groups serve in deriving algebraic results about ultraﬁlters. The contents of the book fall naturally into three parts. The ﬁrst, comprising Chapters 1 through 5, introduces to topological groups and ultraﬁlters insofar as the semigroup operation on ultraﬁlters is not required. Constructions of some important topological groups are given. In particular, that of an extremally disconnected topological group based on a Ramsey ultraﬁlter. Also one shows that every inﬁnite group admits a nondiscrete zero-dimensional topology in which all translations and the inversion are continuous. ˇ In the second part, Chapters 6 through 9, the Stone–Cech compactiﬁcation ˇG of a discrete group G is studied. For this, a special technique based on the concepts of a local left group and a local homomorphism is developed. One proves that if G is a countable torsion free group, then ˇG contains no nontrivial ﬁnite groups. Also the ideal structure of ˇG is investigated. In particular, one shows that for every inﬁnite jGj Abelian group G, ˇG contains 22 minimal right ideals. In the third part, using the semigroup ˇG, almost maximal topological and left topological groups are constructed and their ultraﬁlter semigroups are examined. Projectives in the category of ﬁnite semigroups are characterized. Also one shows that every inﬁnite Abelian group with ﬁnitely many elements of order 2 is absolutely !resolvable, and consequently, can be partitioned into ! subsets such that every coset modulo inﬁnite subgroup meets each subset of the partition. The book concludes with a list of open problems in the ﬁeld. Some familiarity with set theory, algebra and topology is presupposed. But in general, the book is almost self-contained. It is aimed at graduate students and researchers working in topological algebra and adjacent areas. Johannesburg, November 2010

Yevhen Zelenyuk

Contents

Preface 1

v

Topological Groups 1.1 The Notion of a Topological Group . . . . . . 1.2 The Neighborhood Filter of the Identity . . . 1.3 The Topology T .F / . . . . . . . . . . . . . 1.4 Topologizing a Group . . . . . . . . . . . . . 1.5 Metrizable Reﬁnements . . . . . . . . . . . . 1.6 Topologizability of a Countably Inﬁnite Ring

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Ultraﬁlters 2.1 The Notion of an Ultraﬁlter . . . 2.2 The Space ˇD . . . . . . . . . . 2.3 Martin’s Axiom . . . . . . . . . 2.4 Ramsey Ultraﬁlters and P -points 2.5 Measurable Cardinals . . . . . .

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26 26 29 34 36 39

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Topological Spaces with Extremal Properties 3.1 Filters and Ultraﬁlters on Topological Spaces . . . . . . . . . . . . . 3.2 Spaces with Extremal Properties . . . . . . . . . . . . . . . . . . . . 3.3 Irresolvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 42 46

4

Left Invariant Topologies and Strongly Discrete Filters 4.1 Left Topological Semigroups . . . . . . . . . . . . . 4.2 The Topology T ŒF . . . . . . . . . . . . . . . . . . 4.3 Strongly Discrete Filters . . . . . . . . . . . . . . . 4.4 Invariant Topologies . . . . . . . . . . . . . . . . .

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52 52 54 56 63

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68 68 71 72 76

The Semigroup ˇS 6.1 Extending the Operation to ˇS . . . . . . . . . . . . . . . . . . . . . 6.2 Compact Right Topological Semigroups . . . . . . . . . . . . . . . .

82 82 86

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Topological Groups with Extremal Properties 5.1 Extremally Disconnected Topological Groups 5.2 Maximal Topological Groups . . . . . . . . . 5.3 Nodec Topological Groups . . . . . . . . . . 5.4 P -point Theorems . . . . . . . . . . . . . .

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viii

Contents

6.3 6.4

Hindman’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . Ultraﬁlters from K.ˇS/ . . . . . . . . . . . . . . . . . . . . . . . . .

91 94

7

Ultraﬁlter Semigroups 97 7.1 The Semigroup Ult.T / . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.2 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.3 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

8

Finite Groups in ˇG 8.1 Local Left Groups and Local Homomorphisms 8.2 Triviality of Finite Groups in ˇZ . . . . . . . . 8.3 Local Automorphisms of Finite Order . . . . . 8.4 Finite Groups in G . . . . . . . . . . . . . . .

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110 110 117 120 128

Ideal Structure of ˇG 9.1 Left Ideals . . . . . . . . . . . 9.2 Right Ideals . . . . . . . . . . 9.3 The Structure Group of K.ˇG/ 9.4 K.ˇG/ is not Closed . . . . .

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130 130 136 140 144

10 Almost Maximal Topological Groups 10.1 Construction . . . . . . . . . . . . . . . . . . . . . . . 10.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Semilattice Decompositions and Burnside Semigroups 10.4 Projectives . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Topological Invariantness of Ult.T / . . . . . . . . . .

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147 147 150 154 158 165

11 Almost Maximal Spaces 11.1 Right Maximal Idempotents in H 11.2 Projectivity of Ult.T / . . . . . . . 11.3 The Semigroup C.p/ . . . . . . . 11.4 Local Monomorphisms . . . . . .

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170 170 174 177 181

12 Resolvability 12.1 Regular Homeomorphisms of Finite Order . . . . . . . . . . . . . . . 12.2 Resolvability of Topological Groups . . . . . . . . . . . . . . . . . . 12.3 Absolute Resolvability . . . . . . . . . . . . . . . . . . . . . . . . .

188 188 194 198

13 Open Problems

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Bibliography

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Index

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Chapter 1

Topological Groups

In this chapter some basic concepts and results about topological groups are presented. The largest group topology in which a given ﬁlter converges to the identity is described. As an application Markov’s Criterion of topologizability of a countable group is derived. Another application is computing the minimum character of a nondiscrete group topology on a countable group which cannot be reﬁned to a nondiscrete metrizable group topology. We conclude by proving Arnautov’s Theorem on topologizability of a countably inﬁnite ring.

1.1

The Notion of a Topological Group

Deﬁnition 1.1. A group G endowed with a topology is a topological group if the multiplication W G G 3 .x; y/ 7! xy 2 G and the inversion

W G 3 x 7! x 1 2 G

are continuous mappings. A topology which makes a group into a topological group is called a group topology. The continuity of the multiplication and the inversion is equivalent to the continuity of the function 0 W G G 3 .x; y/ 7! xy 1 2 G: Indeed, 0 .x; y/ D .x; .y//, .x/ D 0 .1; x/ and .x; y/ D 0 .x; .y//. The continuity of 0 means that whenever a; b 2 G and U is a neighborhood of ab, there are neighborhoods V and W of a and b, respectively, such that V W 1 U: It follows that whenever a1 ; : : : ; an 2 G, k1 ; : : : ; kn 2 Z and U is a neighborhood of a1k1 : : : ankn 2 G, there are neighborhoods V1 ; : : : ; Vn of a1 ; : : : ; an , respectively, such that V1k1 Vnkn U: Another immediate property of a topological group G is that the translations and the inversion of G are homeomorphisms. Indeed, for each a 2 G, the left translation a W G 3 x 7! ax 2 G

2

Chapter 1 Topological Groups

and the right translation a W G 3 x 7! xa 2 G are continuous mappings, being restrictions of the multiplication. The inversion is continuous by the deﬁnition. Since we have also that .a /1 D a1 , .a /1 D a1 and 1 D , all of them are homeomorphisms. A topological space X is called homogeneous if for every a; b 2 X , there is a homeomorphism f W X ! X such that f .a/ D b. If G is a topological group and a; b 2 G, then ba1 W G ! G is a homeomorphism and ba1 .a/ D ba1 a D b. Thus, we have that Lemma 1.2. The space of a topological group is homogeneous. Now we establish some separation properties of topological groups. Lemma 1.3. Every topological group satisfying the T0 separation axiom is regular and hence Hausdorff. In this book, by a regular space one means a T3 -space. Proof. Let G be a T0 topological group. We ﬁrst show that G is a T1 -space. Since G is homogeneous, it sufﬁces to show that for every x 2 G n¹1º, there is a neighborhood U of 1 not containing x. By T0 , there is an open set U containing exactly one of two points 1; x. If 1 2 U , we are done. Otherwise xU 1 is a neighborhood of 1 not containing x. Now we show that for every neighborhood U of 1, there is a closed neighborhood of 1 contained in U . Choose a neighborhood V of 1 such that V V 1 U . Then for every x 2 G n U , one has xV \ V D ;. Indeed, otherwise xa D b for some a; b 2 V , which gives us that x D ba1 2 V V 1 U , a contradiction. Hence cl V U . In fact, the following stronger statement holds. Theorem 1.4. Every Hausdorff topological group is completely regular. Proof. See [55, Theorem 10]. Theorem 1.4 is the best possible general separation result. However, for countable topological groups, it can be improved. A space is zero-dimensional if it has a base of clopen (D both closed and open) sets. Note that if a T0 -space is zero-dimensional, then it is completely regular. Proposition 1.5. Every countable regular space is normal and zero-dimensional.

3

Section 1.1 The Notion of a Topological Group

Proof. Let X be a countable regular space and let A and B be disjoint closed subsets of X . Enumerate A and B as A D ¹an W n < !º

and

B D ¹bn W n < !º:

Inductively, for each n < !, choose neighborhoods Un and Vn of an and bn respectively such that (a) cl Un \ B D ; and A \ cl Vn D ;, S S (b) Un \ . i

[ Ui \ Vi D ;:

in

It follows that U D

[ n

in

Un

and

V D

[

Vn

n

are disjoint neighborhoods of A and B, respectively. Now, having established that X is normal, let U be an open neighborhood of a point x 2 X . Without loss of generality one may suppose that U ¤ X . Then by Urysohn’s Lemma, there is a continuous function f W X ! Œ0; 1 such that f .x/ D 0 and f .X n U / D ¹1º. Since X is countable, there is r 2 Œ0; 1 n f .X /. Then f 1 .Œ0; r// D f 1 .Œ0; r/ is a clopen neighborhood of x contained in U . It follows from Lemma 1.3 and Proposition 1.5 that Corollary 1.6. Every countable Hausdorff topological group is normal and zerodimensional. Note that every ﬁrst countable Hausdorff topological group is also normal. (A space is ﬁrst countable if every point has a countable neighborhood base.) This is immediate from the fact that every metric space is normal and the following result. Theorem 1.7. A Hausdorff topological group is metrizable if and only if it is ﬁrst countable. In this case, the metric can be taken to be left invariant. Proof. See [34, Theorem 8.3]. Starting from Chapter 5, all topological groups are assumed to be Hausdorff.

4

Chapter 1 Topological Groups

1.2

The Neighborhood Filter of the Identity

For every set X , P .X / D ¹Y W Y X º: Deﬁnition 1.8. Let X be a nonempty set. A ﬁlter on X is a family F P .X / with the following properties: (1) X 2 F and ; … F , (2) if A; B 2 F , then A \ B 2 F , and (3) if A 2 F and A B X , then B 2 F . In other words, a ﬁlter on X is a nonempty family of nonempty subsets of X closed under ﬁnite intersections and supersets. A classic example of a ﬁlter is the set Nx of all neighborhoods of a point x in a topological space X called the neighborhood ﬁlter of x. By a neighborhood of x one means any set whose interior contains x. The system ¹Nx W x 2 X º of all neighborhood ﬁlters on X is called the neighborhood system of X . Theorem 1.9. Let X be a space and let ¹Nx W x 2 X º be the neighborhood system of X . Then (i) for every x 2 X and U 2 Nx , x 2 U , and (ii) for every x 2 X and U 2 Nx , ¹y 2 X W U 2 Ny º 2 Nx . Conversely, given a set X and a system ¹Nx W x 2 X º of ﬁlters on X satisfying conditions (i)–(ii), there is a unique topology T on X for which ¹Nx W x 2 X º is the neighborhood system. Proof. That the neighborhood system ¹Nx W x 2 X º of a space X satisﬁes (i)–(ii) is obvious. We need to prove the converse. Deﬁne the operator int on the subsets of X by putting for every A X int A D ¹x 2 X W A 2 Nx º: We claim that it satisﬁes the following conditions: (a) int X D X , (b) int A A, (c) int .int A/ D int A, and (d) int .A \ B/ D .int A/ \ .int B/.

Section 1.2 The Neighborhood Filter of the Identity

5

Indeed, for every x 2 X , X 2 Nx , consequently x 2 int X , and so (a) is satisﬁed. For (b), if x 2 int A, then A 2 Nx , and so by (i), x 2 A. To check (c), let x 2 int A. Then A 2 Nx . Applying (ii) we obtain that int A 2 Nx . It follows that x 2 int .int A/. Hence int A int .int A/. The converse inclusion follows from (b). To check (d), let x 2 .int A/\.int B/. Then A 2 Nx and B 2 Nx , so A\B 2 Nx . It follows that x 2 int .A \ B/. Hence .int A/ \ .int B/ int .A \ B/. Conversely, let x 2 int .A\B/. Then A\B 2 Nx , consequently A 2 Nx and B 2 Nx . It follows that x 2 .int A/ \ .int B/. Hence int .A \ B/ .int A/ \ .int B/. It follows from (a)–(d) that there is a unique topology T on X such that int is the interior operator for .X; T /. We have that a subset U X is a neighborhood of a point x 2 X in T if and only if x 2 int U , and so if and only if U 2 Nx . Hence, ¹Nx W x 2 X º is the neighborhood system for .X; T /. In a topological group, the neighborhood system is completely determined by the neighborhood ﬁlter of the identity. Lemma 1.10. Let G be a topological group and let N be the neighborhood ﬁlter of 1. Then for every a 2 G, aN D N a is the neighborhood ﬁlter of a. Here, aN D ¹aB W B 2 N º

and

N a D ¹Ba W B 2 N º:

Proof. Since both a and a are homeomorphisms and a .1/ D a .1/ D a, aN D a .N / D a .N / D N a is the neighborhood ﬁlter of a. The next theorem characterizes the neighborhood ﬁlter of the identity of a topological group. Theorem 1.11. Let .G; T / be a topological group and let N be the neighborhood ﬁlter of 1. Then (1) for every U 2 N , there is V 2 N such that V V U , (2) for every U 2 N , U 1 2 N , and (3) for every U 2 N and x 2 G, xUx 1 2 N . Conversely, given a group G and a ﬁlter N on G satisfying conditions (1)–(3), there is a unique group topology T on G for which N is the neighborhood ﬁlter of 1. The topology T is Hausdorff if and only if T (4) N D ¹1º.

6

Chapter 1 Topological Groups

Note that conditions (2) and (3) in Theorem 1.11 are equivalent, respectively, to (20 ) N 1 D N , and (30 ) for every x 2 G, xN x 1 D N , where N 1 D ¹A1 W A 2 N º and xN x 1 D ¹xAx 1 W A 2 N º. Proof. That the neighborhood ﬁlter of 1 satisﬁes (1)–(3) follows from the continuity of the multiplication .x; y/ at .1; 1/ and the mappings .x/ and x .x 1 .y// at 1. To prove the converse, consider the system ¹xN W x 2 Gº. We claim that it satisﬁes the conditions of Theorem 1.9. To check (i), let x 2 G and U 2 N . It follows from (1)–(2) that there is V 2 N such that V V 1 U . Then x 2 xV V 1 xU . To check (ii), let x 2 G and U 2 N . It follows from (1) that there is V 2 N such that V V U . For every y 2 xV , yV xV V xU , consequently xV ¹y 2 G W xU 2 yN º; and so ¹y 2 G W xU 2 yN º 2 xN : Now by Theorem 1.9, there is a unique topology T on G such that for each x 2 G, xN is the neighborhood ﬁlter of x, that is, the neighborhoods of x are of the form xU , where U is a neighborhood of 1. To see that T is a group topology, let a; b 2 G be given and let U be a neighborhood of 1. Using conditions (1)–(3) choose a neighborhood V of 1 such that bV V 1 b 1 U . Then aV .bV /1 D aV V 1 b 1 D ab 1 bV V 1 b 1 ab 1 U: Since T is a group T topology, it is Hausdorff if and only if it is a T1 -topology, and so if and only if N D ¹1º. The notion of a ﬁlter is closely related to that of a ﬁlter base. Deﬁnition 1.12. Let X be a nonempty set. A ﬁlter base on X is a nonempty family B P .X / with the following properties: (1) ; … B, and (2) for every A; B 2 B there is C 2 B such that C A \ B. Equivalently, B P .X / is a ﬁlter base if F D ¹A X W A B for some B 2 Bº is a ﬁlter, and in this case we say that B is a base for F . Note that if F is a ﬁlter, then B F is a base for F if and only if for every A 2 F there is B 2 B such that B A.

Section 1.3 The Topology T .F /

7

If X is a topological space and x 2 X , then a base for the neighborhood ﬁlter of x is called a neighborhood base at x. As a consequence we obtain from Theorem 1.11 the following. Corollary 1.13. Let B be a ﬁlter base on G satisfying the following conditions: (1) for every U 2 B, there is V 2 B such that V V U , (2) for every U 2 B, U 1 2 B, and (3) for every U 2 B and x 2 G, xUx 1 2 B. Then there is a unique group topology T on G for which B is a neighborhood base at 1. The topology T is Hausdorff if and only if T (4) B D ¹1º.

1.3

The Topology T .F /

Deﬁnition 1.14. For every ﬁlter F on a group G, let T .F / denote the largest group topology on G in which F converges to 1. Deﬁnition 1.14 is justiﬁed by the factW that for every family ¹Ti W i 2 I º of group topologies on G, the least upper bound i2I Ti taken in the lattice of all topologies on G is a group topology. Deﬁnition 1.15. For every ﬁlter F on a group G, let FQ denote the ﬁlter with a base consisting of subsets of the form [ 1 x.Ax [ A1 ; x [ ¹1º/x x2G

where for each x 2 G, Ax 2 F . Lemma 1.16. For every ﬁlter F on a group G, FQ is the largest ﬁlter contained in F such that T (i) 1 2 FQ , (ii) FQ 1 D FQ , and (iii) for every x 2 G, x FQ x 1 D FQ . Proof. That FQ satisﬁes (i) is obvious. To check (ii) and (iii), let Ax 2 F for each x 2 G. Then [ 1 [ 1 1 x.Ax [ A1 D x.Ax [ A1 : x [ ¹1º/x x [ ¹1º/x x2G

x2G

8

Chapter 1 Topological Groups

Consequently, FQ 1 D FQ . Next, for every y 2 G, [ [ 1 1 y x.Ax [ A1 yx.Ax [ A1 y 1 D x [ ¹1º/x x [ ¹1º/.yx/ x2G

x2G

D

[

1 x.Ay 1 x [ Ay1 1 x [ ¹1º/x

x2G

D

[

x.Bx [ Bx1 [ ¹1º/x 1 ;

x2G

where Bx D Ay 1 x for each x 2 G. It follows that y FQ y 1 D FQ . To see that FQ is the largest ﬁlter on G contained in F and satisfying (i)–(iii), let G be any such ﬁlter and let A 2 G . Then 1 2 A and for each x 2 G, there is Ax 2 G 1 A. Since G F , A 2 F for each x 2 G. Deﬁne such that x.Ax [ A1 x x /x Q B 2 F by [ 1 BD x.Ax [ A1 : x [ ¹1º/x x2G

Then B A, and so A 2 FQ . For every n 2 N, let Sn denote the group of all permutations on ¹1; : : : ; nº. The next theorem describes the topology T .F /. Theorem 1.17. For every ﬁlter F on a group G, the neighborhood ﬁlter of 1 in T .F / has a base consisting of subsets of the form 1 [ Y n [

B.i/ ;

nD1 2Sn iD1

Q where .Bn /1 nD1 is a sequence of members of F . Proof. It is clear that these subsets form a ﬁlter base on G. In order to show that this is the neighborhood ﬁlter of 1 in a group topology, it sufﬁces to check conditions Q (1)–(3) of Corollary 1.13. Let .Bn /1 nD1 be any sequence of members of F . 1 Q To check (1), deﬁne the sequence .Cn /nD1 in F by Cn D B2n \ B2n1 . Then for for every n 2 N and ; 2 Sn , n Y iD1

C.i/

n Y

C.i/

iD1

where 2 S2n is deﬁned by .j / D

n Y

B2.i/1

iD1

´

2.j / 1 2.j n/

n Y

B2.i/ D

iD1

if j n if j > n:

2n Y j D1

B.j /

Section 1.3 The Topology T .F /

It follows that 1 [ Y n [

C.i/

nD1 2Sn iD1

9

1 [ Y n [

1 [ Y n [ C.i/ B.i/ ;

nD1 2Sn iD1

nD1 2Sn iD1

1 Q To check (2), deﬁne the sequence .Cn /1 nD1 in F by Cn D Bn (Lemma 1.16). Then for every n 2 N and 2 Sn , n Y

B.i/

1

D

iD1

n Y

1 B.i/

D

iD1

n Y

C.i/

iD1

where 2 Sn is deﬁned by .i / D .n C 1 i /. Consequently, 1 [ Y n [

B.i/

1

nD1 2Sn iD1

D

1 [ Y n [

C.i/ :

nD1 2Sn iD1

1 Q To check (3), let x 2 G. Deﬁne the sequence .Cn /1 nD1 in F by Cn D xBn x (Lemma 1.16). Then for every n 2 N and 2 Sn ,

x

n Y

n n Y Y B.i/ x 1 D xB.i/ x 1 D C.i/ :

iD1

iD1

iD1

Consequently, x

1 [ Y n [

1 [ Y n [ B.i/ x 1 D C.i/ :

nD1 2Sn iD1

nD1 2Sn iD1

Now let G be endowed with any group topology in which F converges to 1 and let U be a neighborhood of 1. Note that every neighborhood of 1 is a member of FQ (Lemma 1.16). Choose inductively a sequence .Vn /1 nD0 of neighborhoods of 1 such that V0 D U and for every n, VnC1 VnC1 VnC1 Vn : Then whenever n1 ; : : : ; nk are distinct numbers in N, one has Vn1 Vnk Vn ; where n D min¹n1 ; : : : ; nk º 1. (To see this, pick i 2 ¹1; : : : ; kº such that ni D min¹n1 ; : : : ; nk º and write Vn1 Vnk as .Vn1 Vni1 /Vni .VniC1 Vnk /.) It follows that 1 [ Y n [ V.i/ U; nD1 2Sn iD1

and so U is a neighborhood of 1 in T .F /.

10

1.4

Chapter 1 Topological Groups

Topologizing a Group

Deﬁnition 1.18. Let G be a countably inﬁnite group. Enumerate G as ¹gn W n < !º without repetitions and with g0 D 1. 1 (i) For every inﬁnite sequence .an /1 nD1 in G, deﬁne U..an /nD1 / G by U..an /1 nD1 / D

1 [ Y n [

B.i/ ;

nD1 2Sn iD1

S ˙1 ; a˙1 1 where Bi D j1D0 gj ¹1; aiCj iCj C1 ; : : :ºgj . (ii) For every ﬁnite sequence a1 ; : : : ; an in G, deﬁne U.a1 ; : : : ; an / G by n [ Y

U.a1 ; : : : ; an / D

n B.i/ ;

2Sn iD1

Sni

˙1 ; a˙1 ˙1 1 where Bin D j D0 gj ¹1; aiCj iCj C1 ; : : : ; an ºgj . That is, U.a1 ; : : : ; an / consists of all elements of the form

gjn cn gj1 ; gj1 c1 gj1 n 1 ˙1 ; : : : ; an˙1 º for each i D 1; : : : ; n, where ji 2 ¹0; : : : ; n .i /º and ci 2 ¹1; a.i/Cj i and 2 Sn . In particular, U.a1 / D ¹1; a1˙1 º. Also put U.;/ D ¹1º. (iii) For every ﬁnite sequence a1 ; : : : ; an1 in G, let T .a1 ; : : : ; an1 ; x/ denote the set of group words f .x/ in the alphabet G [¹xº in which variable x occurs and which have the form f .x/ D gj1 c1 gj1 gjn cn gj1 ; n 1 ˙1 ˙1 ; x ˙1 º for each i D ; : : : ; an1 where ji 2 ¹0; : : : ; n .i /º and ci 2 ¹1; a.i/Cj i 1; : : : ; n, and 2 Sn . In particular, T .x/ consists of two group words x and x 1 .

Of course, in the case where G is Abelian, all these deﬁnitions look simpler. In particular, Bin D ¹0; ˙ai ; : : : ; ˙an º and

U.a1 ; : : : ; an / D

n X

Bin :

iD1

Theorem 1.19. For every sequence .an /1 nD1 in G, the following statements hold: 1 (1) U..an /1 nD1 / is a neighborhood of 1 in T ..an /nD1 /, S 1 (2) U..an /1 nD1 U.a1 ; : : : ; an /, nD1 / D

(3) U.a1 ; : : : ; an / D U.a1 ; : : : ; an1 / [ ¹f .an / W f .x/ 2 T .a1 ; : : : ; an1 ; x/º for every n 2 N, and (4) for every n 2 N and f .x/ 2 T .a1 ; : : : ; an1 ; x/, f .1/ 2 U.a1 ; : : : ; an1 /.

11

Section 1.4 Topologizing a Group

Proof. (1) follows from Theorem 1.17, and (2)–(4) from Deﬁnition 1.18. Theorem 1.19 gives us a method of topologizing a countable group. We illustrate it by proving Markov’s Criterion. We say that a group is topologizable if it admits a nondiscrete Hausdorff group topology. An inequality over a group G is any expression of the form f .x/ ¤ b, where f .x/ is a group word in the alphabet G [ ¹xº and b 2 G. Theorem 1.20 (Markov’s Criterion). A countable group G is topologizable if and only if every ﬁnite system of inequalities over G having a solution has also another solution. Proof. Necessity is obvious. Indeed, let T be a nondiscrete Hausdorff group topology on G. Consider any ﬁnite system of inequalities over G, say fi .x/ ¤ bi , where i D 1; : : : ; n, having a solution, say a 2 G, that is, fi .a/ ¤ bi for each i D 1; : : : ; n. Since T is a Hausdorff group topology, there is a neighborhood U of a in T such that bi … fi .U / for each i D 1; : : : ; n. Then every element of U is a solution of the system, and since T is nondiscrete, U n ¹aº ¤ ;. The proof of sufﬁciency is based on Theorem 1.19. It is enough to construct a sequence .an /1 nD1 in G n ¹1º such that gi … U.ai ; aiC1 ; : : : ; an / for each n 2 N and i D 1; : : : ; n. This implies that gi … U..an /1 nDi / for each i 2 N. Then the topology T ..an /1 nD1 / would be nondiscrete and Hausdorff. At the ﬁrst step pick any a1 2 G n ¹1; g1˙1 º. Then g1 … U.a1 / D ¹1; a1˙1 º. Now ﬁx n 2 N and suppose that elements a1 ; : : : ; an 2 G have already been chosen so that gi … U.ai ; : : : ; an / for each i D 1; : : : ; n. We need to ﬁnd anC1 2 G n ¹1º such that gi … U.ai ; : : : ; anC1 / for each i D 1; : : : ; n C 1. Since U.ai ; : : : ; anC1 / D U.ai ; : : : ; an / [ ¹f .anC1 / W f .x/ 2 T .ai ; : : : ; an ; x/º; it follows that gi … U.ai ; : : : ; anC1 / for each i D 1; : : : ; n C 1 if and only if anC1 is a solution of the system of inequalities f .x/ ¤ gi ; where i D 1; : : : ; n C 1 and f .x/ 2 T .ai ; : : : ; an ; x/. Since f .1/ 2 U.ai ; : : : ; an / for each i D 1; : : : ; n C 1 and f .x/ 2 T .ai ; : : : ; an ; x/, 1 is a solution of this system. Hence, there is a solution anC1 ¤ 1.

12

Chapter 1 Topological Groups

In connection with Theorem 1.20, A. Markov asked whether every inﬁnite group is topologizable. The next two theorems show that for inﬁnite Abelian groups and for free groups the answer is “Yes”. Theorem 1.21. For every Abelian group G and for every a 2 G n ¹0º, there is a homomorphism f W G ! T such that f .a/ ¤ 1. Proof. We ﬁrst deﬁne a homomorphism f0 W hai ! T such that f0 .a/ ¤ 1. If a has ﬁnite order, say n, deﬁne 2 ik f .ka/ D e n for every k D 1; : : : ; n. If a has inﬁnite order, pick any x 2 T n ¹1º and put f .ka/ D x k for every k 2 Z. Now consider the set of all pairs .H; g/, where H is a subgroup of G containing hai and g W H ! T is a homomorphism extending f0 , ordered by .H1 ; g1 / .H2 ; g2 / if and only if

H1 H2 and g2 jH1 D g1 :

By Zorn’s Lemma, there is a maximal pair .H; f /. We claim that H D G. Indeed, assume on the contrary that there is c 2 G n H . To derive a contradiction, let H 0 D H C hci and deﬁne a homomorphism f 0 W H 0 ! T extending f . If there is n 2 N with nc 2 H , choose the smallest such n and then z 2 T such that z n D f .nc/, and if there is no such n, choose arbitrary z 2 T . Deﬁne f 0 .b C kc/ D f .b/ z k for every b 2 H and k 2 Z. A topological group G is called totally bounded if it is Hausdorff and for every nonempty open U G there is a ﬁnite F G such that F U D G. A topological group is totally bounded if and only if it can be topologically and algebraically embedded into a compact group (see [4, Corollary 3.7.17]). By a compact group one means a compact Hausdorff topological group. Corollary 1.22. Every inﬁnite Abelian group admits a totally bounded group topology. Proof. Let G be an inﬁnite Abelian group. By Theorem 1.21, for everyQ a 2 G n ¹0º, there is a homomorphism fa W G ! T with fa .a/ ¤ 1. Let K D a2Gn¹0º Ta where Ta D T . Deﬁne f W G ! K by .f .x//a D fa .x/. Clearly f is an injective homomorphism. Being a subgroup of a compact group, f .G/ is totally bounded. Hence, the topology on G consisting of subsets f 1 .U /, where U ranges over open subsets of f .G/, is as required.

Section 1.4 Topologizing a Group

13

Theorem 1.23. Let X be a set and let F be the free group generated by X . Then for every w 2 F n ¹;º, there exist n 2 N and a homomorphism f W F ! SnC1 such that f .w/ ¤ , where is the identity permutation. Proof. Write w as x1"1 x2"2 : : : xn"n where each "i D ˙1 and "i D "iC1 whenever xi D xiC1 . It sufﬁces to deﬁne a mapping X 3 x 7! x 2 SnC1 such that x"11 ı x"22 ı ı x"nn ¤ : Given x 2 X and " D ˙1, let D" .x/ D ¹i 2 ¹1; : : : ; nº W xi D x and "i D "º: Note that D1 .x/ \ .D1 .x/ C 1/ D ; and .D1 .x/ C 1/ \ D1 .x/ D ;, so .D1 .x/ [ .D1 .x/ C 1// \ .D1 .x/ [ .D1 .x/ C 1// D ;: First deﬁne x on D1 .x/ [ .D1 .x/ C 1/ by ´ i C 1 if i 2 D1 .x/ x .i / D i 1 if i 2 D1 .x/ C 1: Then extend it in any way to a member x 2 SnC1 . Now we claim that for each i D 1; : : : ; n, x"ii .i C 1/ D i . Indeed, if "i D 1, then x"ii .i C 1/ D xi .i C 1/ D i . If "i D 1, then xi .i / D i C 1, and consequently, x"ii .i C 1/ D i . It follows that x"11 ıx"22 ı ıx"nn .nC1/ D 1 and hence x"11 ıx"22 ı ıx"nn ¤ . Corollary 1.24. Every free group admits a totally bounded group topology. Markov’s question had been open for a long time. However, eventually it was solved in the negative. Example 1.25. Let m; n 2 N, m 2, n 665 and n is odd. Consider the Adian group A.m; n/. This is a torsion free m-generated group whose center is an inﬁnite cyclic group hci and the quotient A.m; n/=hci is an inﬁnite group of period n. More precisely, A.m; n/=hci is the Burnside group B.m; n/, the largest group on m generators satisfying the identity x n D 1. Let x 2 A.m; n/ n hci. It is clear that x n 2 hci, because A.m; n/=hci has period n. We claim that x n … hc n i. To see this, assume the contrary. So x n D .c n /k D .c k /n for some k 2 Z. Let z D xc k . Then z … hci and z n D x n c k n D 1, since hci is the center. But this contradicts that A.m; n/ is torsion free.

14

Chapter 1 Topological Groups

Now let G D A.m; n/=hc n i and let D D hci=hc n i. Then G is inﬁnite, D D ¹1; d1 ; : : : ; dn1 º G and for every x 2 G n D, one has x n 2 ¹d1 ; : : : ; dn1 º. It follows that for every T1 -topology on G in which 1 is not an isolated point, the mapping G 3 x 7! x n 2 G is discontinuous at 1. Hence G is nontopologizable.

1.5

Metrizable Reﬁnements

The character of a space is the minimum cardinal such that every point has a neighborhood base of cardinality . Deﬁnition 1.26. For every countable topologizable group G, let pG denote the minimum character of a nondiscrete Hausdorff group topology on G which cannot be reﬁned to a nondiscrete metrizable group topology. Equivalently, pG is the supremum of all cardinals such that every nondiscrete Hausdorff group topology on G of character < has a nondiscrete metrizable reﬁnement. In this section we show that the cardinal pG is equal to a well-known cardinal invariant of the continuum. We say that a family T F P .X / has the strong ﬁnite intersection property if for every ﬁnite H F , H is inﬁnite. A subset A X is a pseudo-intersection of a family F P .X / if A n B is ﬁnite for all B 2 F . Deﬁnition 1.27. The pseudo-intersection number p is the minimum cardinality of a family F P .!/ having the strong ﬁnite intersection property but no inﬁnite pseudo-intersection. We show that Theorem 1.28. For every countable topologizable group G, pG D p. Before proving Theorem 1.28 we establish several auxiliary statements. Lemma 1.29. Let T be a nondiscrete group topology on G, let .Vn /1 nD1 be a sequence of neighborhoods of 1, and let A G be such that 1 2 clT A n A. Then there is a sequence .an /1 nD1 in A such that U..an /1 nDi / Vi for every i 2 N.

15

Section 1.5 Metrizable Reﬁnements

Proof. Construct inductively a sequence .an /1 nD1 in A such that U.ai ; : : : ; an / Vi for every n 2 N and i D 1; : : : ; n. Without loss of generality one may assume that all Vn are open. Pick a1 2 A \ V1 \ V11 . Then U.a1 / V1 . Now ﬁx n > 1 and suppose that elements a1 ; : : : ; an1 2 A have already been chosen so that U.ai ; : : : ; an1 / Vi for every i D 1; : : : ; n 1. Choose a neighborhood Wn of 1 such that for every i D 1; : : : ; n 1 and f .x/ 2 T .ai ; : : : ; an1 ; x/, one has f .Wn / Vi . This can be done because f .1/ 2 U.ai ; : : : ; an1 /. Pick an 2 A \ Vn \ Vn1 \ Wn . Then U.an / Vn and for every i D 1; : : : ; n 1, U.ai ; : : : ; an / D U.ai ; : : : ; an1 / [ ¹f .an / W f .x/ 2 T .ai ; : : : ; an1 ; x/º Vi : Lemma 1.30. Let T be a nondiscrete Hausdorff group topology on G and let A G be such that 1 2 clT A n A. Then there is a sequence .an /1 nD1 in A such that (i) an … U.a1 ; : : : ; an1 / for every n 2 N, and (ii) whenever k 2 N and 1 n1 < < nk < !, U.an1 ; : : : ; ank / n U.an1 ; : : : ; ank1 / U.a1 ; a2 ; : : : ; ank / n U.a1 ; a2 ; : : : ; ank 1 /: Proof. Construct inductively a sequence .an /1 nD1 in A such that for every n 2 N and f .x/ 2 T .a1 ; : : : ; an1 ; x/, either f .an / D f .1/ or f .an / … U.a1 ; : : : ; an1 /. Then (i) is satisﬁed because x 2 T .a1 ; : : : ; an1 ; x/ and 1 … A. To check (ii), let g 2 U.an1 ; : : : ; ank / n U.an1 ; : : : ; ank1 /: Then g D f .ank / for some f .x/ 2 T .an1 ; : : : ; ank1 ; x/. Since T .an1 ; : : : ; ank1 ; x/ T .a1 ; : : : ; ank 1 ; x/; f .x/ 2 T .a1 ; : : : ; ank 1 ; x/, and since g … U.an1 ; : : : ; ank1 /, f .ank / ¤ f .1/. Hence by the construction, g D f .ank / … U.a1 ; : : : ; ank 1 /. Lemma 1.31. Let T be a nondiscrete Hausdorff group topology on G and let U T . Then there is a Hausdorff group topology T 0 on G such that U T 0 T and the character of T 0 does not exceed max¹!; jUjº.

16

Chapter 1 Topological Groups

Proof. It sufﬁces to show that for every U 2 U, there is a HausdorffW metrizable group topology TU on G such that U 2 TU T . Then the topology T 0 D U 2U TU would be as required. Let V D ¹Ux 1 W x 2 U º. Enumerate V and G n ¹1º as ¹Vn W n 2 Nº and ¹xn W n 2 Nº, respectively. Construct inductively a sequence .Wn /1 nD0 of neighborhoods of 1 in T such that W0 D G and for every n 2 N the following conditions are satisﬁed: (a) Wn Vn , (b) xn … Wn , (c) Wn Wn 1 Wn1 , and (d) xk Wn xk 1 Wn1 for all k D 1; : : : ; n. It follows from (b)–(d) that there is a group topology TU on G for which ¹Wn W n 2 Nº is a neighborhood base at 1. Then TU is metrizable, TU T and by (a), U 2 TU . For every group topology T0 on G, let T04 denote the lattice of all group topologies T such that that T0 T . Let F0 denote the Fréchet ﬁlter on N and F04 the lattice of all ﬁlters F such that F0 F . Theorem 1.32. Let T 0 be a nondiscrete metrizable group topology on G. Then there is a metrizable group topology T0 T 0 and a mapping h W G ! N such that T04 3 T 7! h.N .T // 2 F04 is a surjective order preserving mapping. Here, N .T / denotes the neighborhood ﬁlter of 1 in T , and h.N .T // the ﬁlter on N with a base consisting of subsets h.U / where U 2 N .T /. 0 Proof. Pick any sequence .bn /1 nD1 in G n ¹1º converging to 1 in T and let A D ¹bn W n 2 Nº. By Lemma 1.30, there is a sequence .an /1 nD1 in A such that

(i) an … U.a1 ; : : : ; an1 / for every n 2 N, and (ii) whenever k 2 N and 1 n1 < < nk < !, U.an1 ; : : : ; ank / n U.an1 ; : : : ; ank1 / U.a1 ; a2 ; : : : ; ank / n U.a1 ; a2 ; : : : ; ank 1 /: 0 Clearly, .an /1 nD1 converges to 1 in T . By Lemma 1.31, there is a metrizable group topology T0 on G such that

T 0 T0 T ..an /1 nD1 /: Deﬁne a function h W G ! N by the condition that for every n 2 N and x 2 U.a1 ; : : : ; an / n U.a1 ; : : : ; an1 /, one has h.x/ D n.

17

Section 1.5 Metrizable Reﬁnements

It is clear that the mapping T04 3 T 7! h.N .T // 2 F04 is order preserving. We need to show that it is surjective. Let I be an inﬁnite subset of N. Write I as ¹nk W k 2 Nº, where .nk /1 is an kD1 increasing sequence in N. Deﬁne the group topology TI on G by TI D T ..ank /1 kD1 /: Then h.N .TI // is the ﬁlter on N consisting of all subsets J N with ﬁnite I n J . Indeed, for every x 2 U.an1 ; : : : ; ank / n U.an1 ; : : : ; ank1 /, one has h.x/ D nk , and ank 2 U.an1 ; : : : ; ank / n U.an1 ; : : : ; ank1 /. Now let F be any ﬁlter on N containing F0 . Deﬁne the group topology T on G by T D

_

TI :

I 2F

Then h.N .T // D F . Now we are in a position to prove Theorem 1.28. Proof of Theorem 1.28. To prove that pG p, let T 0 be a nondiscrete Hausdorff group topology on G of character < p. We need to ﬁnd a metrizable group topology T0 T 0 . Let ¹V˛ W ˛ < º be a neighborhood base at 1 in T 0 . Since < p, there is an inﬁnite pseudo-intersection of the neighborhood base, and consequently, a oneto-one sequence .an /1 nD1 in G converging to 1. Using Lemma 1.29 and that < p, 1 inductively for each ˛ , construct a subsequence .an˛ /1 nD1 of .an /nD1 such that (i) for each ˛ < , U..an˛ /1 nD1 / V˛ , and

(ii) for each ˛ and < ˛, ¹an˛ W n 2 Nº n ¹an W n 2 Nº is ﬁnite. 1 Then .an /1 nD1 is a subsequence of .an /nD1 with the following property: for each ˛ < , there is i.˛/ 2 N such that U..an /1 / V˛ . By Lemma 1.31, the nDi.˛/ topology T ..an /1 / can be weakened to a metrizable group topology T0 in which for nD1 each i 2 N, U..an /1 / remains a neighborhood of 1. It then follows that T 0 T0 . nDi To prove that p pG , let A be a family of subsets of N having the strong ﬁnite intersection property and with jAj D < pG . We need to ﬁnd an inﬁnite pseudointersection of A. Deﬁne the ﬁlter F on N by

\ ® ¯ F D ANWA B for some ﬁnite B A : Without loss of generality one may suppose that F contains the Fréchet ﬁlter F0 . Since G is topologizable, there is a nondiscrete Hausdorff group topology T 0 on G,

18

Chapter 1 Topological Groups

and by Lemma 1.31, T 0 can be chosen to be metrizable. By Theorem 1.28, there is a metrizable group topology T0 T 0 and a mapping h W G ! N such that T04 3 T 7! h.N .T // 2 F04 is a surjective order preserving mapping. Pick T 2 T04 such that h.N .T // D F . Choosing a family U of open neighborhoods of 1 in T such that jUj D and ¹h.U / W U 2 Uº is a base for F and applying Lemma 1.31, one may suppose that T has character . Since < pG , there is a nondiscrete metrizable group topology T1 T . Then the ﬁlter F1 D h.N .T1 // has a countable base, and consequently, an inﬁnite pseudo-intersection A N. Since F1 F A, A is also a pseudo-intersection of A.

1.6

Topologizability of a Countably Inﬁnite Ring

A ring is topologizable if it admits a nondiscrete Hausdorff ring topology, that is, a topology in which the addition, the additive inversion, and the multiplication are continuous. In this section we show that Theorem 1.33 (Arnautov’s Theorem). Every countably inﬁnite ring is topologizable. Note that the ring in Theorem 1.33 is not assumed to be associative. The proof of Theorem 1.33 is based on two auxiliary results. First is the ring version of Theorem 1.20. Proposition 1.34. A countable ring R is topologizable if and only if every ﬁnite system of inequalities over R having a solution has also another solution. By an inequality over R one means any expression of the form f .x/ ¤ b, where f .x/ is a ring word in the alphabet R [ ¹xº and b 2 R. The proof of Proposition 1.34 is similar to that of Theorem 1.20. First of all we need the following lemma. Lemma 1.35. A ﬁlter N on a ring R is the neighborhood ﬁlter of 0 in a ring topology if and only if the following conditions are satisﬁed: (1) for every U 2 N , there is V 2 N such that V C V U , (2) for every U 2 N , U 2 N , (3) for every U 2 N , there is V 2 N such that V V U , (4) for every U 2 N and x 2 R, there is V 2 N such that xV U and V x U .

Section 1.6 Topologizability of a Countably Inﬁnite Ring

19

Proof. Necessity is obvious. We need to check sufﬁciency. It follows from (1), (2) and Theorem 1.11 that there is a topology T on R in which, for each x 2 R, x C N is the neighborhood ﬁlter of x, and the addition and the additive inversion are continuous mappings. To see that the multiplication is continuous, let x; y 2 R and let Uxy be a neighborhood of xy 2 R. Put U D xy C Uxy . Then U is a neighborhood of 0 and Uxy D xy C U . Choose a neighborhood V of 0 such that V C V C V U . Applying (3) and (4) we obtain that there is a neighborhood W of 0 such that W W V , xW V and W y V . Then Wx D x C W is a neighborhood of x, Wy D y C W is a neighborhood of y and Wx Wy D .x C W /.y C W / D xy C xW C W y C W W xy C V C V C V xy C U D Uxy : Proof of Proposition 1.34. Necessity is obvious. We have to prove sufﬁciency. Assume that every ﬁnite system of inequalities over R having a solution has also another solution. Enumerate R without repetitions as ¹bn W n < !º with b0 D 0. Deﬁne inductively a set P of ring words in the alphabet ¹x1 ; x2 ; : : :º and to each f D f .x1 ; : : : ; xk / 2 P assign a vector rEf D .r1 ; : : : ; rk / 2 N k as follows: (i) x1 2 P and rEx1 D .1/, (ii) if f 2 P and rEf D .r1 ; : : : ; rk /, then g D g.x1 ; : : : ; xk / D f .x1 ; : : : ; xk / 2 P and rEg D .r1 ; : : : ; rk /, (iii) if f; g 2 P , rEf D .r1 ; : : : ; rk / and rEg D .s1 ; : : : ; sl /, then h D h.x1 ; : : : ; xkCl / D f .x1 ; : : : ; xk / C g.xkC1 ; : : : ; xkCl / 2 P; t D t .x1 ; : : : ; xkCl / D f .x1 ; : : : ; xk / g.xkC1 ; : : : ; xkCl / 2 P; and rEh D rEt D .2r1 ; : : : ; 2rk ; 2s1 ; : : : ; 2sl /, (iv) if f 2 P and rEf D .r1 ; : : : ; rk /, then for each m 2 N, h D h.x1 ; : : : ; xk / D bm f .x1 ; : : : ; xk / 2 P; t D t .x1 ; : : : ; xk / D f .x1 ; : : : ; xk / bm 2 P; and rEh D rEt D .r1 C m; : : : ; rk C m/. Lemma 1.36. For every ﬁlter F on R, the subsets [ f .Ar1 ; : : : ; Ark /; f 2P

where .r1 ; : : : ; rk / D rEf and .An /1 nD1 is a sequence of members of F , form a neighborhood base at 0 in the largest ring topology on R in which F converges to 0.

20

Chapter 1 Topological Groups

Proof. Let .An /1 nD1 be any sequence of members of F and let U D

[

f .Ar1 ; : : : ; Ark /:

f 2P

By (ii), U D U . Deﬁne the sequence .Bn /1 nD1 of members of F by Bn D A2n and let [ V D f .Br1 ; : : : ; Brk /: f 2P

Then by (iii), V C V U . Indeed, f .Br1 ; : : : ; Brk / C g.Bs1 ; : : : ; Bsl / D f .A2r1 ; : : : ; A2rk / C g.A2s1 ; : : : ; A2sl / D h.A2r1 ; : : : ; A2rk ; A2s1 ; : : : ; A2sl /: Similarly, V V U . Next, for each m 2 N, deﬁne the sequence .Cn /1 nD1 of members of F by Cn D AnCm and let W D

[

f .Cr1 ; : : : ; Crk /:

f 2P

Then by (iv), bm W U and W bm U . Applying Lemma 1.35, we obtain that there is a ring topology T on R in which the above subsets form a neighborhood base at 0, and so F converges to 0. It remains to check that T is the largest such topology. Let T 0 be any ring topology on R in which F converges to 0 and let U be any neighborhood of 0 in T 0 . We need to show that U is a neighborhood of 0 in T . 0 Choose a sequence .Un /1 nD0 of neighborhoods of 0 in T with U0 D U0 U such that for each n 2 N and m D 1; : : : ; n, Un D Un ; Un C Un Un1 ; Un Un Un1 ; bm Un Un1 and Un bm Un1 : We show that for every f 2 P with rE.f / D .r1 ; : : : ; rk / and for every j D 0; 1; : : : , one has f .Ur1 Cj ; : : : ; Urk Cj / Uj ; in particular, f .Ur1 ; : : : ; Urk / U0 . Suppose that the statement holds for some f; g 2 P with rEf D .r1 ; : : : ; rk / and rEg D .s1 ; : : : ; sl /. Consider the words h D h.x1 ; : : : ; xkCl / D f .x1 ; : : : ; xk / C g.xkC1 ; : : : ; xkCl / 2 P; t D t .x1 ; : : : ; xk / D bm f .x1 ; : : : ; xk / 2 P

Section 1.6 Topologizability of a Countably Inﬁnite Ring

21

with rEh D .2r1 ; : : : ; 2rk ; 2s1 ; : : : ; 2sl / and rEt D .r1 C m; : : : ; rk C m/. We have that h.U2r1 Cj ; : : : ; U2sl Cj / D f .U2r1 Cj ; : : : ; U2rk Cj / C g.U2s1 Cj ; : : : ; U2sl Cj / f .Ur1 Cj C1 ; : : : ; Urk Cj C1 / C g.Us1 Cj C1 ; : : : ; Usl Cj C1 / Uj C1 C Uj C1 Uj and t .Ur1 CmCj ; : : : ; Urk CmCj / D bm f .Ur1 CmCj ; : : : ; Urk CmCj / bm UmCj UmCj 1 Uj : Considering the words f .x1 ; : : : ; xk /g.xkC1 ; : : : ; xkCl / and f .x1 ; : : : ; xk /bm is similar, and the case of the word f .x1 ; : : : ; xk / is trivial. Now for every n 2 N, denote by Pn the subset of P consisting of all f 2 P such that max¹r1 ; : : : ; rk º n, where .r1 ; : : : ; rk / D rEf , in particular, P1 D ¹x1 ; x1 º. Note that Pn is ﬁnite. For every ﬁnite sequence a1 ; : : : ; an in R, deﬁne the subset U.a1 ; : : : ; an / R by [ U.a1 ; : : : ; an / D ¹0º [ f .¹ar1 ; : : : ; an º; : : : ; ¹ark ; : : : ; an º/; f 2Pn

where .r1 ; : : : ; rk / D rEf . We then obtain that for every inﬁnite sequence .an /1 nD1 in R, 1 [ [ U.a1 ; : : : ; an / D f .Ar1 ; : : : ; Ark /; f 2P

nD1

where Ai D ¹ai ; aiC1 ; : : :º for all i 2 N. Consequently by Lemma 1.36, 1 [

U.a1 ; : : : ; an /

nD1

is a neighborhood of 0 in the largest ring topology on R in which .an /1 nD1 converges to 0. Hence, in order to prove that R is topologizable, it sufﬁces to construct a sequence .an /1 nD1 in R such that an ¤ 0 and bi … U.ai ; aiC1 ; : : : ; an / for all n 2 N and i D 1; : : : ; n. To this end, for every ﬁnite sequence a1 ; : : : ; an1 in R, denote by T .a1 ; : : : ; an1 ; x/ the set of ring words in the alphabet ¹a1 ; : : : ; an1 ; xº in which variable x occurs and which are obtained from words f 2 Pn , say f D f .x1 ; : : : ; xk / with rEf D .r1 ; : : : ; rk /, by substituting xi 2 ¹ari ; : : : ; an1 ; xº into f .x1 ; : : : ; xk / for each i D 1; : : : ; k, in particular, T .x/ D ¹x; xº. Then for every ﬁnite sequence a1 ; : : : ; an in R,

22

Chapter 1 Topological Groups

(a) U.a1 ; : : : ; an / D U.a1 ; : : : ; an1 / [ ¹f .an / W f .x/ 2 T .a1 ; : : : ; an1 ; x/º and U.a1 / D ¹0º [ ¹f .a1 / W f .x/ 2 T .x/º, (b) for every f .x/ 2 T .a1 ; : : : ; an1 ; x/, f .0/ 2 U.a1 ; : : : ; an1 /, and for f .x/ 2 T .x/, f .0/ D 0. Statement (a) is obvious and (b) follows from the next lemma. Lemma 1.37. Let f 2 P and let rEf D .r1 ; : : : ; rk /. Then (1) f .0; : : : ; 0/ D 0, and (2) whenever ; ¤ ¹i1 ; : : : ; il º ¹1; : : : ; kº and g.xi1 ; : : : ; xil / is the word obtained from f .x1 ; : : : ; xk / by substituting xi D 0 for each i 2 ¹1; : : : ; kº n ¹i1 ; : : : ; ik º, there exists h 2 P with rEh D .s1 ; : : : ; sl / such that sj rij for all j D 1; : : : ; l and g.c1 ; : : : ; cl / D h.c1 ; : : : ; cl / for all c1 ; : : : ; cl 2 R. Proof. Suppose that the lemma holds for some f1 ; f2 2 P with rEf1 D .r10 ; : : : ; rk0 1 / and rEf2 D .r100 ; : : : ; rk002 /. Let k D k1 C k2 and consider the word f D f .x1 ; : : : ; xk / D f1 .x1 ; : : : ; xk1 / C f2 .xk1 C1 ; : : : ; xk / 2 P with rEf D .2r10 ; : : : ; 2rk0 1 ; 2r100 ; : : : ; 2rk002 /. Clearly f .0; : : : ; 0/ D f1 .0; : : : ; 0/ C f2 .0; : : : ; 0/ D 0 C 0 D 0: Let ; ¤ ¹i1 ; : : : ; il º ¹1; : : : ; kº and let g.xi1 ; : : : ; xil / be the word obtained from f .x1 ; : : : ; xk / by substituting xi D 0 for each i 2 ¹1; : : : ; kº n ¹i1 ; : : : ; ik º. Without loss of generality one may assume that ¹i1 ; : : : ; il º \ ¹1; : : : ; k1 º ¤ ;. Let ¹i1 ; : : : ; il º \ ¹1; : : : ; k1 º D ¹i1 ; : : : ; il1 º and let g1 .xi1 ; : : : ; xil1 / be the word obtained from f1 .x1 ; : : : ; xk1 / by substituting xi D 0 for each i 2 ¹1; : : : ; k1 º n ¹i1 ; : : : ; il1 º. By the hypothesis, there exists h1 2 P with rEh1 D .s10 ; : : : ; sl01 / such that sj0 ri0j for all j D 1; : : : ; l1 and g1 .c1 ; : : : ; cl1 / D h1 .c1 ; : : : ; cl1 / for all c1 ; : : : ; cl1 2 R. Suppose ﬁrst that l1 < l. Let ¹i1 ; : : : ; il º \ ¹k1 C 1; : : : ; kº D ¹k1 C q1 ; : : : ; k1 C ql2 º and let g2 .xq1 ; : : : ; xql2 / be the term obtained from f2 .x1 ; : : : ; xk2 / by substituting xq D 0 for each q 2 ¹1; : : : ; k2 º n ¹q1 ; : : : ; ql2 º. By the hypothesis, there exists h2 2 P with rEh2 D .s100 ; : : : ; sl002 / such that sj00 rq00j for all j D 1; : : : ; l2 and g2 .c1 ; : : : ; cl2 / D h2 .c1 ; : : : ; cl2 / for all c1 ; : : : ; cl2 2 R. Then the word h D h.x1 ; : : : ; xl / D h1 .x1 ; : : : ; xl1 / C h2 .xl1 C1 ; : : : ; xl / 2 P

Section 1.6 Topologizability of a Countably Inﬁnite Ring

23

with rEh D .2s10 ; : : : ; 2sl01 ; 2s100 ; : : : ; 2sl002 / is as required. Indeed, whenever c1 ; : : : ; cl 2 R, we have that g.c1 ; : : : ; cl / D g1 .c1 ; : : : ; cl1 / C g2 .cl1 C1 ; : : : ; cl / D h1 .c1 ; : : : ; cl1 / C h2 .cl1 C1 ; : : : ; cl / D h.c1 ; : : : ; cl / and 2sj0 2ri0j for all j D 1; : : : ; l1 and 2sj00 2rq00j for all j D 1; : : : ; l2 . Suppose now that l1 D l. Then the word h D h.x1 ; : : : ; xl / D h1 .x1 ; : : : ; xl1 / 2 P with rEh D .s10 ; : : : ; 2sl01 / is as required. Indeed, whenever c1 ; : : : ; cl 2 R, we have that g.c1 ; : : : ; cl / D g1 .c1 ; : : : ; cl1 / C f2 .0; : : : ; 0/ D h1 .c1 ; : : : ; cl1 / C 0 D h.c1 ; : : : ; cl / and sj0 ri0j for all j D 1; : : : ; l1 . Considering the words f1 .x1 ; : : : ; xk1 /f2 .xk1 C1 ; : : : ; xk /, bm f1 .x1 ; : : : ; xk1 / and f1 .x1 ; : : : ; xk1 /bm is similar, and the case of the word f .x1 ; : : : ; xk / is trivial. Using (a), (b) and our assumption, the required sequence .an /1 nD1 can be constructed in the same way as in the proof of Theorem 1.20. Another result that we need is Hindman’s Theorem. Deﬁnition 1.38. Given a set X , Pf .X / is the set of ﬁnite nonempty subsets of X . Deﬁnition 1.39. Let S be a semigroup. Given an inﬁnite sequence .xn /1 nD1 in S, the set of ﬁnite products of the sequence is deﬁned by °Y ± FP..xn /1 / D x W F 2 P .N/ : n f nD1 n2F

Given a ﬁnite sequence .xn /m nD1 in S, °Y ± / D x W F 2 P .¹1; : : : ; mº/ : FP..xn /m n f nD1 n2F

If S is an additive semigroup, we write FS instead of FP and say ﬁnite sums instead of ﬁnite products.

24

Chapter 1 Topological Groups

We state Hindman’s Theorem in the form involving product subsystem. Deﬁnition 1.40. Let S be a semigroup and let .xn /1 nD1 be a sequence in S. A se1 1 quence .yn /nD1 is a product subsystem of .xn /nD1Q if there is a sequence .Hn /1 nD1 in Pf .N/ such that max Hn < min HnC1 and yn D i2Hn xi for each n 2 N. If S is an additive semigroup, we say sum subsystem instead of product subsystem. Theorem 1.41. Let S be a semigroup and let .xn /1 nD1 be a sequence in S. Whenever FP..xn /1 / is partitioned into ﬁnitely many subsets, there exists a product subsystem nD1 1 such that FP..y /1 / is contained in one subset of the partition. .yn /1 of .x / n n nD1 nD1 nD1 We shall prove Theorem 1.41 in Chapter 6. Now we need the following. Corollary 1.42. Let S be an inﬁnite cancellative semigroup and let A S with jAj < jS j. Whenever S n A is partitioned into ﬁnitely many subsets, there exists a 1 sequence .yn /1 nD1 such that FP..yn /nD1 / is contained in one subset of the partition. 1 Proof. Construct inductively a sequence .xn /1 nD1 in S such that FP..xn /nD1 / \ A D ;. Then apply Theorem 1.41.

Given a ring R, let RŒx denote the set of all ring words in the alphabet R [ ¹xº. Using ring identities every f .x/ 2 RŒx can be rewritten as a noncommutative polynomial. We denote by deg f .x/ the degree of that polynomial. Lemma 1.43. For every f .x/ 2 RŒx with deg f .x/ > 0, there is g.x/ 2 RŒx such that deg g.x/ < deg f .x/ and g.a/ D f .a C b/ f .a/ f .b/ for all a; b 2 R. Proof. Rewriting f .x C b/ as a polynomial of x we obtain that f .x C b/ D f .x/ C f .b/ C g.x/; where deg g.x/ < deg f .x/. Now we are in a position to prove Theorem 1.33. Proof of Theorem 1.33. Let R be a countably inﬁnite ring and assume on the contrary that R is not topologizable. By Proposition 1.34, there is a ﬁnite sequence f1 .x/; : : : ; fm .x/

Section 1.6 Topologizability of a Countably Inﬁnite Ring

25

in RŒx with the following properties: (i) fi .0/ ¤ 0 for each i D 1; : : : ; m, and (ii) for every a 2 R n ¹0º, there is i D 1; : : : ; m such that fi .a/ D 0. Let k D max¹deg fi .x/ W i D 1; : : : ; mº. By Theorem 1.41, applied to the additive group of R, there exist a sequence .an /kC1 nD1 in R n ¹0º and i D 1; : : : ; m such that fi .a/ D 0 for every a 2 FS..an /kC1 nD1 /: Let f .x/ D fi .x/. Then by Lemma 1.43, there is g.x/ 2 RŒx with deg g.x/ < deg f .x/ such that g.a/ D f .a C b/ f .a/ f .b/ for all a; b 2 R. It follows that g.0/ D f .0 C 0/ f .0/ f .0/ D f .0/ ¤ 0 and for every a 2 FS..an /knD1 /, g.a/ D f .a C akC1 / f .a/ f .akC1 / D 0: After at most k such reductions we obtain a ring word h.x/ with deg h.x/ D 0 such that h.0/ D ˙f .0/ ¤ 0 and h.a1 / D 0, which is a contradiction.

References The standard references for topological groups are [55] and [34]. A great deal of information about topological groups can be found also in [7], [11], and [4]. Theorem 1.17 is from [62], a result of collaboration with I. Protasov. Its Abelian case was proved in [83]. Theorem 1.20 is due to A. Markov [48]. The ﬁrst example of a nontopologizable group was produced by S. Shelah [68]. It was an uncountable group and its construction used the Continuum Hypothesis CH. G. Hesse [32] showed that CH can be dropped in Shelah’s construction. Example 1.25 is due to A. Ol’šanski˘ı [53]. For the Adian group see [1]. That the Burnside group B.m; n/ is inﬁnite for m 2 and for n sufﬁciently large and odd was proved by P. Novikov and S. Adian [51]. Theorem 1.28 is from [110]. For more information about p and other cardinal invariants of the continuum see [79]. Theorem 1.33 is due to V. Arnautov [5]. Theorem 1.41 is due to N. Hindman [35]. Our proof of Theorem 1.33, as well as Theorem 1.20, is based on the treatment in [62].

Chapter 2

Ultraﬁlters

ˇ This chapter contains some basic facts about ultraﬁlters and the Stone–Cech compactiﬁcation of a discrete space. We also discuss Ramsey ultraﬁlters, P -points, and countably complete ultraﬁlters.

2.1

The Notion of an Ultraﬁlter

Let D be a nonempty set. Recall that a ﬁlter on D is a family F P .D/ with the following properties: (1) D 2 F and ; … F , (2) if A; B 2 F , then A \ B 2 F , and (3) if A 2 F and A B D, then B 2 F . Deﬁnition 2.1. An ultraﬁlter on D is a ﬁlter on D which is not properly contained in any other ﬁlter on D. In other words, an ultraﬁlter is a maximal ﬁlter. Deﬁnition 2.2. A family A P .D/ has the ﬁnite intersection property if for every ﬁnite B A.

T

B¤;

It is clear that every ﬁlter has the ﬁnite intersection property. Conversely, every family A P .D/ with the ﬁnite intersection property generates a ﬁlter ﬂt.A/ on D by \ ﬂt.A/ D ¹A D W A B for some ﬁnite B Aº: Proposition 2.3. Let A P .D/. Then the following statements are equivalent: (1) A is an ultraﬁlter, (2) A is a maximal family with the ﬁnite intersection property, (3) A is a ﬁlter and for every A D, either A 2 A or D n A 2 A. Proof. .1/ ) .3/ Consider two cases. Case 1: there is B 2 A such that B \ A D ;. Then B D n A and, since A is closed under supersets, D n A 2 A.

Section 2.1 The Notion of an Ultraﬁlter

27

Case 2: for every B 2 A, B \ A ¤ ;. Then ¹B \ A W B 2 Aº is a family of nonempty sets closed under ﬁnite intersections, because A is closed under ﬁnite intersections. Consequently, F D ¹C D W C B \ A for some B 2 Aº is a ﬁlter. Clearly A F and A 2 F . Since A is an ultraﬁlter, A D F . Hence A 2 A. .3/ ) .2/ Let A D and A … A. Then D n A 2 A. Since A \ .D n A/ D ;, A [ ¹Aº has no ﬁnite intersection property. .2/ ) .1/ Since A has the ﬁnite intersection property, so does ﬂt.A/, and since A is maximal, ﬂt.A/ D A. It follows that A is a ﬁlter, and consequently, an ultraﬁlter. It is obvious that for every a 2 D, ¹A D W a 2 Aº is an ultraﬁlter. Such ultraﬁlters are called principal. Ultraﬁlters which are not principal are called nonprincipal. Lemma 2.4. Let U be an ultraﬁlter on D. Then the following statements are equivalent: (1) U is a nonprincipal ultraﬁlter, T (2) U D ;, (3) for every A 2 U, jAj !. T Proof. .1/ ) .2/ If a 2 U, then U ¹A D W a 2 Aº, and since U is a maximal ﬁlter, U D ¹A D W a 2 Aº. .2/ ) .3/ Suppose that some A 2 U is ﬁnite. Then, applying Proposition 2.3, we obtain that there is a 2 A such that ¹aº 2 U. It follows that U D ¹A D W a 2 Aº. .3/ ) .1/ is obvious. The existence of nonprincipal ultraﬁlters involves the Axiom of Choice. Proposition 2.5 (Ultraﬁlter Theorem). Every ﬁlter on D can be extended to an ultraﬁlter on D. Proof. Let F be a ﬁlter on D and let P D ¹G P .D/ W F G and G is a ﬁlter on Dº: S S Given any chain C in P , C is a ﬁlter, and so C is an upper bound of C . Hence, by Zorn’s Lemma, P has a maximal element U. Clearly, U is a maximal ﬁlter.

28

Chapter 2 Ultraﬁlters

An ultraﬁlter U on D is uniform if for every A 2 U, jAj D jDj. Corollary 2.6. There are uniform ultraﬁlters on any inﬁnite set. Proof. Let D be any inﬁnite set and let F D ¹A D W jD n Aj < jDjº: By Proposition 2.3, there is an ultraﬁlter U on D containing F . If A D and jAj < jDj, then D n A 2 F U, so A … U. Hence U is uniform. Deﬁnition 2.7. Let F be a ﬁlter on D. If C D and C \ A ¤ ; for every A 2 F , then F jC D ¹C \ A W A 2 F º is a ﬁlter on C called the trace of F on C . If f W D ! E, then f .F / D ¹f .A/ W A 2 F º is a ﬁlter base on E called the image of F with respect to f . Note that if f is surjective, then f .F / is a ﬁlter. It is clear that if F is an ultraﬁlter, so is F jC . Lemma 2.8. If F is an ultraﬁlter, f .F / is an ultraﬁlter base. Proof. Let B E and let A D f 1 .B/. Since F is an ultraﬁlter, either A 2 F or D n A 2 F . Then either B f .A/ 2 f .F / or E n B f .D n A/ 2 f .F /. Recall that a space X is called compact if every open cover of X has a ﬁnite subcover. Equivalently, X is compact if every family of closed subsets of X with the ﬁnite intersection property has a nonempty intersection. A ﬁlter base B on a space X converges to a point x 2 X if for every neighborhood U of x 2 X , there is A 2 B such that A U . Note that in the case where B is a ﬁlter, B converges to x if and only if B contains the neighborhood ﬁlter of x. Proposition 2.9. A space X is compact if and only if every ultraﬁlter on X is convergent. Proof. Let X be a compact space and let U be an ultraﬁlter on X . Assume on the contrary that for every point x 2 X , there is a neighborhood Ux of x such that Ux … U. Clearly one can choose Ux to be open. Then, since U is an ultraﬁlter, X nUx 2 U. The open sets Ux , where x 2 X , cover X . Since X is compact, there is a ﬁnite T subcover ¹Uxi W i < nº of ¹Ux W x 2 X º. But then ; D i

29

Section 2.2 The Space ˇD

Conversely, suppose that every ultraﬁlter on X is convergent and let A be a family of closed T subsets of X with the ﬁnite intersection property. Assume on the contrary that A D ;. By Proposition 2.5, there is an ultraﬁlter U on XTsuch that A U. We claim that U is not convergent. Indeed, let x 2 X . Since A D ;, there is Fx 2 A such that x … Fx . Then Ux D X n Fx is a neighborhood of x and Ux … U. Hence U is not convergent, a contradiction.

2.2

The Space ˇD

Deﬁnition 2.10. Let D be a nonempty set and let ˇD denote the set of all ultraﬁlters on D. For every A D, deﬁne A ˇD by A D ¹p 2 ˇD W A 2 pº: Thus, for every A D and p 2 ˇD, p 2 A if and only if A 2 p. We have that A\B DA\B for all A; B D, so the family ¹A W A 2 P .D/º is closed under ﬁnite intersections. Consequently, this family is a base for a topology on ˇD. For every pair of different p; q 2 ˇD, there are A 2 p and B 2 q such that A \ B D ;, and so A \ B D ;. Hence, this topology is Hausdorff. Note also that the set of principal ultraﬁlters on D is a discrete open dense subset of ˇD with respect to this topology. Deﬁnition 2.11. Given a discrete space D, ˇD is the space of ultraﬁlters on D with the topology generated by taking as a base the subsets A, where A 2 P .D/. The principal ultraﬁlters are being identiﬁed with the points of D. For every A D, A D A n A. We have that D n A D ˇD n A for every A D. Consequently, the subset A ˇD is closed as well as open. Since also A is dense in A, it follows that clˇD A D A: ˇ Deﬁnition 2.12. The Stone–Cech compactiﬁcation of a discrete space D is a compact Hausdorff space Y containing D as a dense subspace such that every mapping f W D ! Z from D into any compact Hausdorff space Z can be extended to a continuous mapping f W Y ! Z. ˇ It is easy to see that the Stone–Cech compactiﬁcation of D is unique up to a homeomorphism leaving D pointwise ﬁxed.

30

Chapter 2 Ultraﬁlters

ˇ Theorem 2.13. ˇD is the Stone–Cech compactiﬁcation of D. Before proving Theorem 2.13, let us consider the following general situation. Suppose that X is a dense subset of a space Y and f W X ! Z is a continuous mapping from X into a Hausdorff space Z. For every p 2 Y , denote by Fp the trace of the neighborhood ﬁlter of p 2 Y on X and let limX3x!p f .x/ denote the limit of the ﬁlter base f .Fp / in Z, if exists. It is clear that if f has a continuous extension f W Y ! Z, then it is unique and f .p/ D

lim f .x/

X3x!p

for every p 2 Y . Lemma 2.14. Suppose that Z is regular and for every p 2 Y, there is limX3x!p f .x/. Deﬁne f W Y ! Z by f .p/ D lim f .x/: X3x!p

Then f is the continuous extension of f . Proof. It follows from the deﬁnition of f that for every A X , f .clY A/ clZ f .A/: To see this, let q 2 clY A and let W be a neighborhood of f .q/ 2 Z. Since f .q/ D limx!q f .x/, there is B 2 Fq such that f .B/ W . We have that B \ A ¤ ; and f .B \ A/ W . Hence, f .q/ 2 clZ f .A/. Now to see that f is continuous, let p 2 Y and let U be a neighborhood of f .p/ 2 Z. Since Z is regular, one may suppose that U is closed. Choose A 2 Fp such that f .A/ U . Put V D clY A. Then V is a neighborhood of p 2 Y and f .V / U . Now we are ready to prove Theorem 2.13. Proof of Theorem 2.13. Since D n A D ˇD n A, the sets A form also a base for the closed sets (that is, every closed set in ˇD is the intersection of some sets A). Consequently, in order to show that ˇD is compact, it sufﬁces to show that every family A of sets of the form A with the ﬁnite intersection property has a nonempty intersection. Let B D ¹A D W A 2 Aº Clearly B has the ﬁnite intersectionT property. Then by Proposition 2.5, there is p 2 ˇD such that B p. But then p 2 A. Now let f W D ! Z be any mapping from D into any compact Hausdorff space Z and let p 2 ˇD. The trace of the neighborhood ﬁlter of p 2 ˇD on D is the ultraﬁlter

31

Section 2.2 The Space ˇD

p. Since Z is compact, the ultraﬁlter base f .p/ converges, so limD3x!p f .x/ exists. Hence by Lemma 2.14, f W ˇD 3 p 7!

lim

D3x!p

f .x/ 2 Z

is the continuous extension of f . Deﬁnition 2.15. For every mapping f W D ! Z of a discrete space D into a compact Hausdorff space Z, let f W ˇD ! Z denote the continuous extension of f . Note that if f W D ! E ˇE and p 2 ˇD, then f .p/ is the ultraﬁlter on E with a base f .p/. Theorem 2.16. Let f W D ! D be a mapping with no ﬁxed points. Then there is a 3-partition ¹Ai W i < 3º of D such that Ai \ f .Ai / D ; for each i < 3. Proof. Consider the set G of all functions g such that (i) dom.g/ D and ran.g/ ¹0; 1; 2º, (ii) f .dom.g// dom.g/, and (iii) g.a/ ¤ g.f .a// for each a 2 dom.g/,

S ordered by . Note that G is nonempty, and if C is a chain in G , then C 2 G . Consequently by Zorn’s Lemma, there is a maximal element g 2 G . We claim that dom.g/ D D. Let Dg D dom.g/ and assume on the contrary that there is b 2 D n Dg . To obtain a contradiction, it sufﬁces to deﬁne an extension h of g such that (a) dom.h/ D Dg [ ¹f n .b/ W n < !º, where f 0 .b/ D b and f nC1 .b/ D f .f n .b//, and (b) g.f n .b// ¤ g.f nC1 .b// for each n < !. We distinguish between three cases. Case 1: there is n < ! such that f k .b/ … Dg for all k n and f nC1 .b/ 2 Dg . Let i D g.f nC1 .b// and pick j < 2 different from i . For each k n, put ´ j if k is even h.f nk .b// D i otherwise: Case 2: all elements f n .b/, where n < !, are different and do not belong to Dg . For each n < !, put ´ 0 if n is even h.f n .b// D 1 otherwise:

32

Chapter 2 Ultraﬁlters

Case 3: there are m < n < ! such that all elements f k .b/, where k n, are different, do not belong to Dg , and f nC1 .b/ D f m .b/. If n m is odd, for each k n, put ´ 0 if k is even k h.f .b// D 1 otherwise: If n m is even, we correct the latter deﬁnition at f n .b/ by putting h.f n .b// D 2. Having established that dom.g/ D D, let Ai D g 1 .i / for each i < 3. Corollary 2.17. If f W D ! D has no ﬁxed points, neither does f W ˇD ! ˇD. Proof. By Theorem 2.16, there is a partition ¹Ai W i < 3º of D such that Ai \f .Ai / D ; for each i < 3. Let p 2 ˇD. Then Aj 2 p for some j < 3. Consequently f .Aj / 2 f .p/. We have that p 2 Aj , f .p/ 2 f .Aj / and Aj \ f .Aj / D ;. Hence f .p/ ¤ p. Corollary 2.18. Let f W D ! D and let p 2 ˇD. Then f .p/ D p if and only if ¹a 2 D W f .a/ D aº 2 p: Proof. Denote F D ¹a 2 D W f .a/ D aº and A D D n F . Let f .p/ D p and assume on the contrary that F … p. Then A 2 p. Pick any b 2 A and deﬁne g W D ! D by ´ f .x/ if x 2 A g.x/ D b if x 2 F: Then g has no ﬁxed points and gjA D f jA , so gjA D f jA . Since p 2 A, we obtain that g.p/ D f .p/ D p. But by Corollary 2.17, g.p/ ¤ p, a contradiction. Conversely, let F 2 p. Then p 2 F and f .x/ D x for all x 2 F . Hence f .p/ D p. Deﬁnition 2.19. A space is extremally disconnected if the closure of an open set is open. Equivalently, a space is extremally disconnected if the closures of disjoint open sets are disjoint. Lemma 2.20. ˇD is extremally disconnected. Proof. Let U be an open subset of ˇD. Put A D U \ D. Then U clˇD A. Hence clˇD U D A. Remark 2.21. Although ˇD is extremally disconnected, D is not (see [22, Example 6.2.31]).

33

Section 2.2 The Space ˇD

A space is called -compact if it can be represented as a countable union of compact subsets. Theorem 2.22. Let X be a regular extremally disconnected space and let A and B be -compact subsets of X . If .cl A/ \ B D A \ .cl B/ D ;, then .cl A/ \ .cl B/ D ;. S S Proof. Write A D n

It follows that U D

[ n

in

Un

and

V D

[

Vn

n

are disjoint open neighborhoods of A and B respectively. Since X is extremally disconnected, .cl U / \ .cl V / D ; and so .cl A/ \ .cl B/ D ;. Corollary 2.23. Let A and B be countable subsets of ˇD. If .cl A/ \ B D A \ .cl B/ D ;, then .cl A/ \ .cl B/ D ;. S S Corollary 2.24. Let X D n

Theorem 2.26. Let D be an inﬁnite set of cardinality . Then jU.D/j D jˇDj D 22 .

Proof. Since every ultraﬁlter is a member of P .P .D//, jˇDj 22 . Let U D U.D/. In order to show that jU j 22 , it sufﬁces to construct a mapping of U onto a set of cardinality 22 .

34

Chapter 2 Ultraﬁlters

Let Z be the product of 2 copies of the discrete space ¹0; 1º. Then jZj D 22 . By the Hewitt–Marczewski–Pondiczery Theorem (see [22, Theorem 2.3.15]), Z has a dense subset E of cardinality . Enumerate E as ¹q˛ W ˛ < º. Now let ¹A˛ W ˛ < º be a partition of D into subsets of cardinality . For each ˛ < , there is p˛ 2 U with A˛ 2 p. Deﬁne f W D ! E by f .A˛ / D ¹q˛ º. Then f .p˛ / D q˛ . Since U ˇD is closed, it follows that f .U / is a compact subset of Z containing E. Hence f .U / D Z. We conclude this section by establishing a one-to-one correspondence between nonempty closed subsets of ˇD and ﬁlters on D. Deﬁnition 2.27. Given a family A P .D/, deﬁne A ˇD by \ AD A: A2A

Lemma 2.28. For every ﬁlter F on D, F is a nonempty closed subset of ˇD consisting of all p 2 ˇD such that F p. Conversely, for every nonempty closed subset X ˇD, the intersection of all ultraﬁlters from X is a ﬁlter F on D such that F D X. Proof. The ﬁrst part of the lemma is obvious, so it sufﬁces to prove the second. It is clear that F is a ﬁlter and X F . To see the reverse inclusion, let p 2 F . Assume on the contrary that p … X . Then there is A 2 p such that A \ X D ;. It follows that for each q 2 X , D n A 2 q. Hence D n A 2 F p, a contradiction.

2.3

Martin’s Axiom

Let P D .P; / be a partially ordered set. A ﬁlter in P is a nonempty subset G P such that (i) for every a; b 2 G, there is c 2 G such that c a and c b, and (ii) for every a 2 G and b 2 P , a b implies b 2 G. Elements a; b 2 P are incompatible if there is no c 2 P such that c a and c b. An antichain in P is a subset of pairwise incompatible elements. P has the countable chain condition if every antichain in P is countable. A subset D P is dense if for every a 2 P there is b 2 D such that b a. Deﬁnition 2.29. MA is the following assertion: whenever .P; / is a partially ordered set with the countable chain condition and D is a family of < 2! dense subsets of P , there is a ﬁlter G in P such that G \ D ¤ ; for all D 2 D. We ﬁrst show that MA follows from the Continuum Hypothesis CH.

35

Section 2.3 Martin’s Axiom

Lemma 2.30. CH implies MA. Proof. Let .P; / be a partially ordered set with the countable chain condition and let D be a family of < 2! dense subsets of P . Then by CH, D is countable. Enumerate D as ¹Dn W n < !º. Construct inductively a sequence .an /n

if and only if

F F 0 ; H H 0 and F 0 n F

\

H:

It is routine to verify that is a partial order. Next notice that if .F; H / and .F 0 ; H 0 / are incompatible elements of P , then F ¤ F 0 . Indeed, otherwise .F; H [ H 0 / .F; H / and .F; H [ H 0 / .F; H 0 /. Since the set of ﬁnite subsets of ! is countable, it follows that P has the countable chain condition. For each B 2 F , let DB D ¹.F; H / 2 P W B 2 H º: Given any .F; H / 2 P , one has .F; H [ ¹Bº/ 2 DB and .F; H [ ¹Bº/ .F; H /. So DB is dense. Also for each n < !, let Dn D ¹.F; H / 2 P W max F nº: T Given any .F; H / 2 P , there is m 2 H such that m n. Then .F [¹mº; H / 2 Dn and .F [ ¹mº; H / .F; H /. So Dn is dense. Now by MA, there is a ﬁlter G in P such that G \ DB ¤ ; for each B 2 F and G \ Dn ¤ ; for each n < !. Let AD

[ .F ;H /2G

F:

36

Chapter 2 Ultraﬁlters

Since G \ Dn ¤ ; for each n < !, it follows that A is inﬁnite. To show that A n B is ﬁnite for each B 2 F , pick .F; H / 2 G \ DB . We claim that A n B F . Indeed, let x 2 A n B. Then there is .F 0 ; H 0 / 2 G such that x 2 F 0 . Since G is a ﬁlter, there 00 ; H 00 / .F; H / and .F 00 ; H 00 / .F 0 ; H 0 /. We have is .F 00 ; H 00 / 2 G such that .F T that x 2 F 0 F 00 , F 00 n F H and B 2 H . Hence x 2 F .

2.4

Ramsey Ultraﬁlters and P-points

Given a set X and a cardinal k, ŒX k D ¹Y X W jY j D kº: Ramsey’s Theorem says that whenever k; r 2 N and Œ!k is r-colored, there exists an inﬁnite A ! such that ŒAk is monochrome (see [30]). Deﬁnition 2.32. An ultraﬁlter p on ! is Ramsey if whenever k; r 2 N and Œ!k is r-colored, there exists A 2 p such that ŒAk is monochrome. Theorem 2.33. Let p be an ultraﬁlter on !. Then the following statements are equivalent: (1) p is Ramsey, (2) for every 2-coloring of Œ!2 , there exists A 2 p such that ŒA2 is monochrome, and (3) for every partition ¹An W n < !º of !, either An 2 p for some n < ! or there exists A 2 p such that jA \ An j 1 for all n < !. Proof. .1/ ) .2/ is obvious. .2/ ) .3/ Deﬁne W Œ!2 ! ¹0; 1º by ´ 0 if ¹i; j º An for some n < ! .i; j / D 1 otherwise: By (2), there is A 2 p be such that ŒA2 is monochrome. Suppose that An … p for all n < !. It then follows that .ŒA2 / D ¹1º, and consequently, jA \ An j 1 for all n < !. .3/ ) .1/ We prove (1) for ﬁxed r by induction on k. For k D 1, it is obvious. Now assume (1) holds for some k and let W Œ!kC1 ! ¹0; 1; : : : ; r 1º be given. For each i < !, deﬁne i W Œ!k ! ¹0; 1; : : : ; r 1º by ´ .¹i º [ x/ if min x > i i .x/ D 0 otherwise:

37

Section 2.4 Ramsey Ultraﬁlters and P -points

By the inductive assumption, for each i < !, there are Bi 2 p and li < r such that i .ŒBi k / D ¹li º. Clearly one may suppose that min Bi > i and the sequence .Bi /i

n

There is s < 2 such that ¹bi W i 2 Is º 2 p. Then whenever i; j 2 Is and i < j , there is n < ! such that i < in < inC1 j , and so j bi . Finally, there is l < r, such that ¹bi W i 2 Is and lbi D lº 2 p: Denote this member of p by A. Then .ŒAkC1 / D ¹lº. Theorem 2.34. Assume p D c. Then there exists a nonprincipal Ramsey ultraﬁlter on !. Proof. Let ¹X˛ W ˛ < 2! º be an enumeration of P .!/ and let ¹˛ W ˛ < 2! º be an enumeration of 2-colorings Œ!2 ! ¹0; 1º. We shall construct inductively a 2! sequence .A˛ /˛<2! of inﬁnite subsets of ! such that for every ˛ < 2! , the following conditions are satisﬁed: (i) for every ˇ < ˛, jA˛ n Aˇ j < !, (ii) ŒA˛ 2 is monochrome with respect to ˛ , and (iii) either A˛ X˛ or A˛ ! n X˛ . By Ramsey’s Theorem, there is an inﬁnite C0 ! such that ŒC0 2 is monochrome with respect to 0 . If jC0 \ X0 j D !, put A0 D C0 \ X0 . Otherwise put A0 D C0 \ .! n X0 /. Now ﬁx < 2! and suppose that we have already constructed a sequence .A˛ /˛< satisfying conditions (i)–(iii) for all ˛ < . Since p D c, there is B ! such that jB n A˛ j < ! for all ˛ < . Choose an inﬁnite C B such that ŒC 2 is monochrome with respect to and put ´ if jC \ X j D ! C \ X A D C \ .! n X / otherwise:

38

Chapter 2 Ultraﬁlters

Having constructed .A˛ /˛<2! , let p D ¹A ! W jA˛ n Aj < ! for some ˛ < 2! º: Then p is a nonprincipal Ramsey ultraﬁlter. A point x in a topological space X is called a P -point if the intersection of countably many neighborhoods of x is again a neighborhood of x. Deﬁnition 2.35. We say that a nonprincipal ultraﬁlter p on ! is a P -point if p is a P -point in ! . Lemma 2.36. Let p be a nonprincipal ultraﬁlter on !. Then p is a P -point if and only if whenever ¹An W n < !º is a partition of ! and An … p for all n < !, there exists A 2 p such that A \ An is ﬁnite for all n < !. Proof. Necessity. Let ¹An W n < !º be a partition of ! and AS n … p for all n < !. Deﬁne the sequence ¹B W n < !º of members of p by B D n n n

Section 2.5 Measurable Cardinals

2.5

39

Measurable Cardinals

Deﬁnition 2.39. Let and be inﬁnite cardinals. A ﬁlter F on is -complete if T G 2 F whenever G F and jG j < . Every ﬁlter is !-complete. An ! C -complete ﬁlter is called countably complete. Lemma 2.40. An ultraﬁlter p on is -complete if and only if whenever < and ¹A˛ W ˛ < º is a partition of , there is ˛ < such that A˛ 2 p. Proof. Necessity. Assume on the contrary T that for every ˛ < , one T has A˛ … p, so nA˛ 2 p. Then by -completeness of p, ˛< nA˛ 2 p. But ˛< nA˛ D ;, a contradiction. Sufﬁciency. Assume on the contrary that p is not -complete. Choose < and T ¹C˛ T W ˛ < º p such that ˛< C˛ … p and T is as small as possible. Taking C˛ n ˛< C˛ instead of C˛ , one may suppose that ˛< T C˛ D ;. Using minimality of , deﬁne ¹B˛ W ˛ < º p by B0 D and B˛ D ˇ <˛ C˛ if ˛ > 0. Then the sets A˛ D B˛ n B˛C1 , ˛ < , form a partition of and A˛ … p for every ˛ < , a contradiction. Corollary 2.41. An ultraﬁlter p on is countably complete if and only if whenever ¹An W ˛ < !º is a partition of , there is n < ! such that An 2 p. Deﬁnition 2.42. A cardinal is measurable (Ulam-measurable) if there is a -complete (countably complete) nonprincipal ultraﬁlter on . Note that ! is a measurable cardinal, but not Ulam-measurable. Theorem 2.43. A cardinal is Ulam-measurable if and only if it is greater than or equal to the ﬁrst uncountable measurable cardinal. Proof. It is clear that every uncountable measurable cardinal is Ulam-measurable, and if a cardinal is Ulam-measurable, then any greater cardinal is also Ulam-measurable. Therefore, it sufﬁces to prove that the ﬁrst Ulam-measurable cardinal is measurable. Let be the ﬁrst Ulam-measurable cardinal and let p be a nonprincipal countably complete ultraﬁlter on . We show that p is -complete. Assume the contrary. Then by Lemma 2.40, there are < and a partition ¹A W < º of such that A … p for every < . Deﬁne f W ! by f .x/ D if x 2 A . Now let q D f .p/. It is easy to see that q is a nonprincipal countably complete ultraﬁlter on , contradicting the choice of . A cardinal is called strongly inaccessible if is regular and 2 < whenever < . Note that ! is a strongly inaccessible cardinal. Theorem 2.44. A measurable cardinal is strongly inaccessible.

40

Chapter 2 Ultraﬁlters

Proof. Let be a measurable cardinal and let p be a nonprincipal -complete ultraﬁlter on . We ﬁrst show that is regular. Let D cf. / and assume on the contrary that < . Pick an increasing coﬁnal T -sequence . /< of cardinals in . If n 2 p for every < , then ; D < n 2 p, since p is -complete. Consequently, 2 p for Tsome < . But then, since p is nonprincipal and -complete, we obtain that ; D ˛< n ¹˛º 2 p, a contradiction. Now let < and assume on the contrary that 2 . Pick an injective function F W ! 2. For every < and i < 2, let Xi D ¹˛ < W F .˛/. / D i º: g./

Note that ¹Xi W i < 2º is a partition of . Deﬁne a function g W ! 2 by X T g./ and let X D < X . Then X 2 p, by -completeness of p, and

2 p,

X D ¹˛ < W F .˛/. / D g. / for all < º D ¹˛ < W F .˛/ D gº D F 1 .g/: Since F is injective, it follows that X is a singleton. But this contradicts the assumption that p is nonprincipal. Theorem 2.45. It is consistent with ZFC that there is no uncountable strongly inaccessible cardinal. Proof. See [43, Corollary IV 6.9].

References Ultraﬁlters were introduced by F. Riesz [64] and S. Ulam [76]. The Ultraﬁlter Theorem is equivalent to the Prime Ideal Theorem, a well-known weaker form of the Axiom of Choice AC (see [41, Section 2.3]). The existence of a nonprincipal ultraﬁlter ˇ on ! cannot be established in ZF [41, Problem 5.24]. The Stone–Cech compactiﬁˇ cation was produced independently by M. Stone [72] and E. Cech [82]. A great deal ˇ of information about ultraﬁlters and the Stone–Cech compactiﬁcation can be found in [12]. Concerning ˇ! see also [81]. Theorem 2.16 goes back to N. de Bruijn and P. Erd˝os [16]. Theorem 2.22 is apparently due to Z. Frolík [26]. Theorem 2.31 is a result of D. Booth [6]. For more information about MA see [43] and [25]. Our proof of Theorem 2.33 is based on the treatment in [10]. P -points were invented by W. Rudin [66] who used them to show that under CH, ! is not homogeneous. Theorem 2.38 is due to S. Shelah [69]. Measurable cardinals were introduced by S. Ulam [77].

Chapter 3

Topological Spaces with Extremal Properties

In this chapter we discuss extremally disconnected, irresolvable, maximal and other spaces with extremal properties.

3.1

Filters and Ultraﬁlters on Topological Spaces

Deﬁnition 3.1. Let F be a ﬁlter on a topological space. (a) F is open (closed) if it has a base of open (closed) sets. (b) F is nowhere dense if there is A 2 F such that int cl A D ;. (a) F is dense if for every A 2 F , int cl A 2 F . Lemma 3.2. Every ultraﬁlter on a space is either dense or nowhere dense. Proof. Let U be an ultraﬁlter on a space X . Suppose that U is not dense. Then there is a closed A 2 F such that int A … U. Let B D A n int A. It follows that B 2 U and int cl B D ;. Hence, U is nowhere dense. Corollary 3.3. An ultraﬁlter U on a space is dense if and only if for every A 2 U, int cl A ¤ ;. Note that a ﬁlter F on a space is nowhere dense if and only if every ultraﬁlter U F is nowhere dense. Lemma 3.4. Let F be a ﬁlter on a space X and suppose that F is not nowhere dense. Deﬁne the ﬁlter G on X by taking as a base the subsets of the form U nY where U 2 F and Y ranges over nowhere dense subsets of X . Then for every ultraﬁlter U F , U is dense if and only if U G . If F is open, G is open as well. Proof. It follows from the deﬁnition of G that the ultraﬁlters containing G are precisely those of ultraﬁlters containing F which are not nowhere dense, and so by Lemma 3.2, these are all dense ultraﬁlters containing F . If F is open, then G has a base consisting of subsets of the form U n Y where U 2 F is open and Y X is closed nowhere dense, so G is open as well. We say that ﬁlters F and G are incompatible if A \ B D ; for some A 2 F and B 2 G.

42

Chapter 3 Topological Spaces with Extremal Properties

Lemma 3.5. Every open ﬁlter is contained in a maximal open ﬁlter. Different maximal open ﬁlters are pairwise incompatible. Proof. The ﬁrst statement is immediate from Zorn’s Lemma. To see the second, let F and G be maximal open ﬁlters on a space X . Suppose that U \ V ¤ ; for all U 2 F and V 2 G . Then the sets U \ V ¤ ;, where U 2 F and V 2 G , form a base for a ﬁlter H . Since H is open and contains F and G , it follows that F D G . Lemma 3.6. Let F be a maximal open ﬁlter on a space X and let A X be such that A \ U ¤ ; for all U 2 F . Then int cl A 2 F . Proof. Since F is open, it sufﬁces to show that cl A 2 F . Assume the contrary. Then U n cl A ¤ ; for all U 2 F . Consequently, the sets U n cl A, where U 2 F , form a base for an open ﬁlter G F . Since F is maximal open, F D G . It follows that .cl A/ \ U D ; for some U 2 F , and so A \ U D ;, a contradiction. Corollary 3.7. Every ultraﬁlter containing a maximal open ﬁlter is dense. Lemma 3.8. Let F be an open ﬁlter on a space and let U be a dense ultraﬁlter containing F . Then there is a maximal open ﬁlter G such that U G F . Proof. For every A 2 U, int cl A 2 U. Consequently, there is a maximal open ﬁlter G such that int cl A 2 G for all A 2 U. Let V 2 G be open and let A 2 U. Then V \ .int cl A/ ¤ ;, and so V \ A ¤ ;. It follows that G U. Summarizing, we obtain the following. Proposition 3.9. Let F be an open ﬁlter on a space X , let G be the ﬁlter with a base consisting of subsets U n Y where U 2 F and Y ranges over nowhere dense subsets of X , and let ¹Gi W i 2 I º be the family of all maximal open ﬁlters containing F . Then (1) for every ultraﬁlter U F , U is dense if and only if U G , T (2) the ﬁlters Gi , where i 2 I , are pairwise incompatible and i2I Gi D G , and (3) for every i 2 I , ultraﬁlter U Gi and A 2 U, one has int cl A 2 Gi .

3.2

Spaces with Extremal Properties

Deﬁnition 3.10. A topological space X is nodec if every nowhere dense subset of X is closed. Equivalently, a space is nodec if every nowhere dense subset is discrete.

43

Section 3.2 Spaces with Extremal Properties

Proposition 3.11. A T1 -space X is nodec if and only if every nonprincipal converging ultraﬁlter on X is dense. Proof. A T1 -space X contains no nonclosed nowhere dense set if and only if there is no nonprincipal converging nowhere dense ultraﬁlter on X , consequently by Lemma 3.2, if and only if every nonprincipal converging ultraﬁlter on X is dense. Recall that a space is extremally disconnected if the closures of disjoint open sets are disjoint. Proposition 3.12. A space X is extremally disconnected if and only if for every x 2 X , there is exactly one maximal open ﬁlter on X converging to x. Proof. Suppose that X is not extremally disconnected. Then there are disjoint open U; V X and x 2 X such that x 2 .cl U / \ .cl V /. Let F be the neighborhood ﬁlter of x and let FU D ﬂt.F [ ¹U º/

and

FV D ﬂt.F [ ¹V º/:

Then FU and FV are incompatible open ﬁlters on X converging to x. Hence, maximal open ﬁlters extending FU and FV (Lemma 3.5) are different. Conversely, suppose that there are at least two different maximal open ﬁlters on X , say G and H , converging to some x 2 X . Since they are incompatible (Lemma 3.5), there are disjoint open U 2 G and V 2 H . Clearly x 2 .cl U / \ .cl V /. Hence, X is not extremally disconnected. An important property of extremally disconnected spaces is contained in the following theorem. Theorem 3.13. Let X be an extremally disconnected Hausdorff space and let f W X ! X be a homeomorphism. Then the set M D ¹x 2 X W h.x/ D xº of all ﬁxed points of f is clopen. Proof. By Zorn’s Lemma, there is a maximal open set U X such that f .U / \ U D ;. Since X is extremally disconnected, U is clopen. Let F D U [ f .U / [ f 1 .U /: Then F is clopen and M \ F D ;. We claim that M D X n F . Indeed, assume on the contrary that f .x/ ¤ x for some x 2 X n F . Choose an open neighborhood V of x such that V \ F D ; and f .V / \ V D ;. It follows that f .V / \ U D ;, since V \ f 1 .U / D ;. But then f .U [ V / \ .U [ V / D ;, which contradicts maximality of U . Deﬁnition 3.14. A space X is strongly extremally disconnected if for every open nonclosed U X , there exists x 2 X n U such that U [ ¹xº is open.

44

Chapter 3 Topological Spaces with Extremal Properties

Proposition 3.15. A space X is strongly extremally disconnected if and only if it is extremally disconnected and every nonempty nowhere dense subset of X has an isolated point. Proof. Necessity. To see that X is extremally disconnected, let U be an open subset of X and let V D int cl U . We claim that V D cl U . Indeed, otherwise V is nonclosed and so there is x 2 cl V n V D cl U n V such that V [ ¹xº is open, which contradicts V D int cl U . Now let Y be a nonempty nowhere dense subset of X . One may suppose that Y is closed. Then U D X n Y is open and nonclosed. Consequently, there is x 2 Y such that U [ ¹xº is open. It follows that x is an isolated point of Y . Sufﬁciency. Let U be an open nonclosed subset of X . Then Y D cl U n U is a nonempty nowhere dense subset. Consequently, there is an isolated point x 2 Y . Also we have that cl U is open. It follows that U [ ¹xº is open. Note that to say that every nonempty nowhere dense subset of a space has an isolated point is the same as saying that every nonempty perfect subset has a nonempty interior. A subset of a space is called perfect if it is closed and dense in itself. Corollary 3.16. A dense in itself space X is strongly extremally disconnected if and only if every perfect subset of X is open. Proof. Necessity. Let F be a perfect subset of X and let U D int F . We have to show that U D F . Assume on the contrary that F n U ¤ ;. Then U is nonclosed. Indeed, otherwise F n U is perfect, so by Proposition 3.15, int .F n U / ¤ ;, which contradicts U D int F . Since U is nonclosed, there is x 2 F n U such that U [ ¹xº is open, a contradiction with U D int F . Sufﬁciency. By Proposition 3.15, we have to check only that X is extremally disconnected. Let U be a nonempty open subset of X . Since X is dense in itself, so is U . Consequently, cl U is perfect, and so it is open. Deﬁnition 3.17. A space is resolvable or irresolvable depending on whether or not it can be partitioned into two dense subsets. A space X is open-hereditarily irresolvable if every nonempty open subset of X is irresolvable. Proposition 3.18. A space X is open-hereditarily irresolvable if and only if every (converging) maximal open ﬁlter on X is an ultraﬁlter. Proof. Let F be a maximal open ﬁlter on X and suppose that F is not an ultraﬁlter. Then there are two different ultraﬁlters U and V on X containing F . Pick pairwise disjoint A 2 U and B 2 V . By Corollary 3.7, both U and V are dense, so int cl A 2 F and int cl B 2 F . Deﬁne an open U 2 F by U D .int cl A/ \ .int cl B/:

Section 3.2 Spaces with Extremal Properties

45

Then both A \ U and B \ U are dense in U , so U is resolvable. Hence, X is not open-hereditarily irresolvable. Conversely, suppose that every converging maximal open ﬁlter on X is an ultraﬁlter. Let U be any nonempty open subset of X and let ¹A0 ; A1 º be a partition of U . Pick any converging maximal open ﬁlter F on X containing U . By the assumption F is an ultraﬁlter, so Ai 2 F for some i < 2, say A0 2 F . Since F is open, int A0 ¤ ;. Consequently, A1 is not dense in U . Hence, X is not open-hereditarily irresolvable.

Deﬁnition 3.19. A space is submaximal if every dense subset is open. Proposition 3.20. A space is submaximal if and only if it is open-hereditarily irresolvable and nodec. Proof. Suppose that X is submaximal. To see that X is open-hereditarily irresolvable, let U be a nonempty open subset of X and let A be a dense subset of U . Then .X nU /[A is dense in X , and consequently, open. It follows that U n A is is not dense in U . Hence, U is irresolvable. To see that X is nodec, let Y be a nowhere dense subset of X . Then X n Y is dense, and consequently, open. So Y is closed. Now suppose that X is open-hereditarily irresolvable and nodec. To show that X is submaximal, let A be a dense subset of X . Since X is open-hereditarily irresolvable, int A is dense in X . Consequently, X n int A is nowhere dense, and so discrete, since X is nodec. It follows from this that int A D A. Deﬁnition 3.21. A space .X; T / is maximal if T is maximal among all dense in itself topologies on X . Equivalently, a space is maximal if it has no isolated point but it does have an isolated point in any stronger topology. Theorem 3.22. For a dense in itself Hausdorff space X , the following statements are equivalent: (1) X is maximal, (2) X is extremally disconnected, open-hereditarily irresolvable and nodec, (3) for each x 2 X , there is exactly one nonprincipal ultraﬁlter on X converging to x. Proof. Let X D .X; T /. .1/ ) .2/ By Proposition 3.20, it sufﬁces to show that that X is extremally disconnected and submaximal.

46

Chapter 3 Topological Spaces with Extremal Properties

Assume that X is not extremally disconnected. Then there is an open subset V X and x 2 cl V n V such that V [ ¹xº is not open. Deﬁne the topology on X by taking as a base T [ ¹U \ .V [ ¹xº/ W U 2 T º: This topology is dense in itself and stronger than T . Hence, T is not maximal, a contradiction. Now assume that X is not submaximal. So there is a dense nonopen subset A X . Deﬁne the topology on X by taking as a base T [ ¹U \ A W U 2 T º: Clearly, this topology is dense in itself and stronger than T , a contradiction. .2/ ) .3/ It follows from extremal disconnectedness of X that there is exactly one maximal open ﬁlter F on X converging to x (Proposition 3.12). Since X is nodec, every nonprincipal ultraﬁlter on X converging to x is dense (Proposition 3.11) and so contains F (Proposition 3.9). Finally, open-hereditary irresolvability of X implies that F is an ultraﬁlter (Proposition 3.18). .3/ ) .1/ Let T1 be any topology on X stronger than T . Then there is x 2 X and a nonprincipal ultraﬁlter U on X converging to x in T but not converging to x in T1 . It follows that x is an isolated point in T1 . Hence, T is maximal. The equivalence .1/ , .3/ in Theorem 3.22 justiﬁes the following deﬁnition. Deﬁnition 3.23. We say that a space X is almost maximal if it is dense in itself and for every x 2 X there are only ﬁnitely many ultraﬁlters on X converging to x.

3.3

Irresolvability

The notion of irresolvability (Deﬁnition 3.17) naturally generalizes as follows. Deﬁnition 3.24. Given a cardinal 2, a space is -resolvable or -irresolvable depending on whether or not it can be partitioned into -many dense subsets. A space X is hereditarily -irresolvable (open-hereditarily -irresolvable) if every nonempty subset of X (every nonempty open subset of X ) is -irresolvable. The next lemma contains the main simple properties of -resolvability. Lemma 3.25. For every 2, the following statements hold: (1) an open subset of a -resolvable space is -resolvable, (2) the closure of a -resolvable subset of a space is -resolvable, and (3) the union of a family of -resolvable subsets of a space is -resolvable.

47

Section 3.3 Irresolvability

Proof. (1) and (2) are obvious, so it sufﬁces to S show (3). Let R be a family of resolvable subsets of a space X and let Y D R. We have to show that Y is -resolvable. Consider the set P of all -sequences A S D ¹A˛ W ˛ < º consisting of pairwise disjoint nonempty subsets A˛ Y dense in A. Deﬁne the order on P by AB

if and only if

A˛ B˛ for all ˛ < :

Every chain .Ai /i2I in P , where Ai D ¹Ai˛ W ˛ < º, has an upper bound ± °[ Ai˛ W ˛ < : i2I

ConsequentlySby Zorn’s Lemma, there is a maximal element A D ¹A˛ W ˛ < º in P . Let Z D ˛< A˛ . Obviously, Z is -resolvable. It is clear also that Z is closed. Indeed, otherwise the element B D ¹B˛ W ˛ < º in P deﬁned by ´ A0 [ cl Z n Z if ˛ D 0 B˛ D A˛ otherwise is greater than A. We claim that Z D Y . To see this, assume the contrary. Then there is R 2 R such that R n Z ¤ ;. Since Z is closed, S R n Z 2 R as well. It follows that there is C D ¹C˛ W ˛ < º in P such that C \ Z D ;. Then the element D D ¹D˛ W ˛ < º in P deﬁned by D˛ D A˛ [ C˛ is greater than A, a contradiction. Lemma 3.25 leads to the following construction. Lemma 3.26. Given a space X and 2, let R .X / denote the union of all resolvable subsets of X . Then (i) R .X / is the largest -resolvable subset of X , (ii) R .X / is closed, (iii) X is -resolvable if and only if R .X / D X , and (iv) if X is -irresolvable, then X n R .X / is hereditarily -irresolvable. Proof. It is immediate from Lemma 3.25. Corollary 3.27. A homogeneous space is -irresolvable if and only if it is hereditarily -irresolvable.

48

Chapter 3 Topological Spaces with Extremal Properties

Proof. Let X be a homogeneous -irresolvable space. Assume on the contrary that X is not hereditarily -irresolvable. Then R .X / is a proper subset of X . Pick x 2 R .X / and y 2 X n R .X /. Let f W X ! X be a homeomorphism with f .x/ D y. Then f .R .X // is a -resolvable subset of X and f .R .X // n R .X / ¤ ;, which is a contradiction. Of special interest are ﬁnitely and !-irresolvable spaces. Recall that for every ﬁlter F on a set X , F denotes the set of all ultraﬁlters on X containing F . Proposition 3.28. A space X is open-hereditarily n-irresolvable if and only if for every (converging) maximal open ﬁlter F on X , jF j < n. Proof. The same as that of Proposition 3.18. Lemma 3.29. Let X be a space and suppose that there is an open ﬁlter F on X such that jF j < n for some n < !. Then X is n-irresolvable. Proof. Assume on the contrary that there is aSpartition of X into n dense sets Ai , where i < n. Then there is j < n such that j ¤i

49

Section 3.3 Irresolvability

Lemma 3.34. Every countable discrete subset of a regular space is strongly discrete. Proof. Let X be a regular space and let D be a countably inﬁnite discrete subset of X . Enumerate D without repetitions as ¹xn W n < !º. For each n < !, pick a closed neighborhood Un of xn 2 X such that Un \ D D ¹xn º. For each n < !, deﬁne a neighborhood Vn of xn 2 X by [ Vn D Un n Ui : i

Then the subsets Vn , where n < !, are pairwise disjoint. Proof of Theorem 3.33. Let D be a countable discrete nonclosed subset of X and let a 2 cl D n D. For each x 2 X , there is a homeomorphism fx W X ! X with fx .a/ D x. Construct inductively a sequence .Dn /1 nD1 of discrete subsets of X such that Dn cl DnC1 n DnC1 : Put D1 D D. Fix k 2 N and suppose that the sequence .Dn /knD1 has already been constructed satisfying that condition. Denote ´S k1 if k > 1 nD1 cl Dn Yk1 D ; otherwise: It follows from the condition that Dk \ Yk1 D ;. By Lemma 3.34, for each x 2 Dk , there is an open neighborhood Ux of x 2 X n Yk1 such that the sets Ux , where x 2 Dk , are pairwise disjoint. Then the set [ DkC1 D fx .D \ fx1 .Ux // x2Dk

is as required. Now deﬁne Z X by ZD

1 [

Dn :

nD1

We claim that Z is !-resolvable. Indeed, let ¹In W n 2 Nº be any partition of N into inﬁnite sets. Deﬁne the partition ¹An W n 2 Nº of Z by [ An D Di : i2In

Then each An is dense in Z. Finally, since X contains a nonempty !-resolvable subset, X itself is !-resolvable by Corollary 3.27.

50

Chapter 3 Topological Spaces with Extremal Properties

We conclude this section by showing that !-irresolvable spaces are in fact ﬁnitely irresolvable. Theorem 3.35. If a space is n-resolvable for every n < !, then it is !-resolvable. To prove Theorem 3.35, we need two lemmas. Lemma 3.36. Given a space X , let W D W .X / denote the union of all open sets in X containing a dense hereditarily irresolvable subset. Then (i) W is the largest open set in X containing a dense open-hereditarily irresolvable subset, and (ii) every dense subset of X n cl W is resolvable, and consequently, !-resolvable. Proof. By Zorn’s Lemma, there is a maximal family U of pairwise S disjoint open sets in X containing a dense hereditarily irresolvable subset. Note that U is dense in W .SFor each U 2 U, pick a dense hereditarily irresolvable DU U . Then D D U 2U DU is a dense open-hereditarily irresolvable subset of W . Now let A be a dense subset of X n cl W . Assume on the contrary that A is irresolvable. It then follows from Lemma 3.26 that there is a nonempty open set V in X n cl W such that V \ A is hereditarily irresolvable. Clearly, V \ A is also dense in V . But this contradicts maximality of U. Lemma 3.37. Let D be an open-hereditarily irresolvable subset of a space X and suppose that X is .n C 1/-resolvable for some n. Then X n D is dense in X and n-resolvable. Proof. By Lemma 3.26, it sufﬁces to show that for every nonempty open set U X , U n D contains a nonempty n-resolvable subset. One may suppose that D \ U ¤ ;. Partition X into dense subsets A1 ; : : : ; AnC1 . Then D \ U is partitioned into subsets A1 \ D \ U; : : : ; AnC1 \ D \ U . Since no space is the union of ﬁnitely many nowhere dense sets, at least one of those sets, say AnC1 \ D \ U , is not nowhere dense in D \ U . So there is an open U0 U such that ; ¤ D \ U0 clD\U .AnC1 \ D \ U /: It follows that AnC1 \ D \ U0 is dense in D \ U0 . Then, since D \ U0 is irresolvable, .D \ U0 / n AnC1 is not dense in D \ U0 . Consequently, there is an open U1 U0 such that ; ¤ D \ U1 AnC1 \ D \ U0 , and so ; ¤ D \ U1 AnC1 \ U1 . It follows that A1 \ U1 ; : : : ; An \ U1 are pairwise disjoint dense subsets of U1 n D. Proof of Theorem 3.35. By Lemma 3.26, it sufﬁces to show that every nonempty open set X0 X contains a nonempty !-resolvable subset. Being open subset of X , X0 is

Section 3.3 Irresolvability

51

n-resolvable for each n. Consider the subset W .X0 / X0 given by Lemma 3.36. If it is not dense, then X0 n cl W .X0 / is !-resolvable and we are done. Otherwise pick a dense open-hereditarily irresolvable subset D1 W .X0 / and put X1 D W .X0 / n D1 . Being an open subset of X0 , W .X0 / is n-resolvable for each n, and then by Lemma 3.37, X1 is n-resolvable for each n. If W .X1 / X1 is not dense, then X1 n cl W .X1 / is !-resolvable. Otherwise pick a dense open-hereditarily irresolvable subset D2 W .X1 / and put X2 D W .X1 / n D2 , and so on. If at some m, W .Xm / Xm is not dense, then Xm n cl W .Xm / is the required nonempty !-resolvable subset of X0 . Otherwise we construct an inﬁnite sequence .Dm /1 mD1 of pairwise disjoint dense subsets of X0 , and then X0 itself is !-resolvable.

References The study of maximal and irresolvable spaces was initiated by E. Hewitt [33]. Theorem 3.13 is due to Z. Frolík [27]. Theorem 3.22 is from [99]. Another characterizations of maximal spaces can be found in [80]. Corollary 3.32 is a result of A. El’kin [18]. Theorem 3.35 is due to A. Illanes [40].

Chapter 4

Left Invariant Topologies and Strongly Discrete Filters

In this chapter left invariant topologies on semigroups are studied. We describe the largest left invariant topology in which a given ﬁlter converges to the identity. A special attention is paid to the case where the ﬁlter is strongly discrete, then the topology possesses certain important properties. We conclude by showing that every inﬁnite group admits a nondiscrete zero-dimensional Hausdorff invariant topology.

4.1

Left Topological Semigroups

Deﬁnition 4.1. A semigroup (group) S endowed with a topology is a left topological semigroup (group) and the topology itself a left invariant topology if for every a 2 S, the left translation a W S 3 x 7! ax 2 S is continuous. A topology T on a semigroup S is left invariant if and only if for every a 2 S and U 2 T , a1 U 2 T where a1 U D 1 a .U / D ¹x 2 S W ax 2 U º: Note that in a left topological group left translations are homeomorphisms. Consequently, if S is a group, a left invariant topology on S is completely determined by the neighborhood ﬁlter N of 1: for every a 2 S, the neighborhood ﬁlter of a is aN D ¹aU W U 2 N º. Topologies with this property on semigroups are characterized by the following lemma. Lemma 4.2. Let S be a semigroup with identity, let T be a topology on S, and let N be the neighborhood ﬁlter of 1 in T . Then the following statements are equivalent: (a) for every a 2 S, aN is a neighborhood base at a, (b) for every a 2 S, the left translation a W S 3 x 7! ax 2 S is continuous and open, and (c) for every a 2 S and U 2 T , both a1 U 2 T and aU 2 T .

Section 4.1 Left Topological Semigroups

53

Proof. (a) ) (b) To see that a is continuous, let b 2 S and let U be a neighborhood of a .b/ D ab. Pick V 2 N such that abV U . Then bV is a neighborhood of b and a .bV / D abV U . To see that a is open, let b 2 S and let U be a neighborhood of b. Pick V 2 N such that bV U . Then abV is a neighborhood of a .b/ and a .U / a .bV / D abV . (b) ) (c) Since a is continuous, a1 U D 1 a .U / 2 T , and since a is open, aU D a .U / 2 T . (c) ) (a) If U is an open neighborhood of 1, then a 2 aU 2 T , so aU is an open neighborhood of a. Conversely, let V be an open neighborhood of a and let U D a1 V . Then U is an open neighborhood of 1 and aU V . The next theorem characterizes the neighborhood ﬁlter of the identity of a left topological semigroup. Theorem 4.3. Let S be a left topological semigroup with identity and let N be the neighborhood ﬁlter of 1. Then (1) for every U 2 N , 1 2 U , and (2) for every U 2 N , ¹x 2 S W x 1 U 2 N º 2 N . Conversely, given a semigroup S with identity and a ﬁlter N on S satisfying conditions (1)–(2), there is a left invariant topology on S in which for each a 2 S, aN is a neighborhood base at a. Note that condition (2) in Theorem 4.3 is equivalent to (20 ) for every U 2 N , there are V 2 N and V 3 x 7! Wx 2 N such that xWx U . Proof. Suppose that S is a left topological semigroup with identity and let U 2 N . Clearly 1 2 U . Putting V D int U , we obtain that V 2 N and for every x 2 V , x 1 U 2 N . Conversely, suppose that N is a ﬁlter on S satisfying conditions (1)–(2). For every x 2 S , let Nx be the ﬁlter with a base xN . We show that, whenever x 2 S and U 2 Nx , one has (i) x 2 U , and (ii) ¹y 2 S W U 2 Ny º 2 Nx . (i) follows from (1) and deﬁnition of Nx . To show (ii), let U0 D x 1 U , V0 D ¹z 2 S W U0 2 Nz º; and V D xV0 . Clearly U0 2 N , so by (2), V0 2 N , and consequently V 2 Nx . We claim that for every y 2 V , U 2 Ny . To see this, write y D xz for some z 2 V0 . Then x 1 U D U0 2 Nz D zN , and so U 2 xzN D Ny . Now by Theorem 1.9, there is a topology T on S for which ¹Nx W x 2 Sº is the neighborhood system. By Lemma 4.2, T is left invariant.

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Chapter 4 Left Invariant Topologies and Strongly Discrete Filters

A semigroup is called cancellative (left cancellative, right cancellative) if all translations (left translations, right translations) are injective. T T Note that whenever f W XT! Y and T B P .X /, one has f . B/ f .B/, and if f is injective, then f . B/ D f .B/. From this and Theorem 4.3 we obtain as a consequence the following. Corollary 4.4. Let S be a semigroup with identity and let B be a ﬁlter base on S satisfying the following conditions: (1) for every U 2 B, 1 2 U , and (2) for every U 2 B and a 2 U , there is V 2 B such that aV U . Then there is a left invariant topology T on S in which for each a 2 S, aB is an open neighborhood base at a. Furthermore, if S is left cancellative, then T is a T1 -topology if and only if T (3) B D ¹1º. Theorem 4.3 gives us also one more characterization of topologies from Lemma 4.2. Corollary 4.5. Let S be a semigroup with identity, let T be a topology on S, and let N be the neighborhood ﬁlter of 1 in T . Then the following statements are equivalent: (a) for every a 2 S , aN is a neighborhood base at a, and (d) T is the largest left invariant topology on S in which N is the neighborhood ﬁlter of 1. Proof. (a) ) (d) That T is left invariant follows from Lemma 4.2. To see that it is the largest one, let T 0 be any left invariant topology on S in which N is the neighborhood ﬁlter of 1. Then whenever a 2 S and U is a neighborhood of a in T 0 , there is V 2 N such that aV U . Hence T 0 T . (d) ) (a) By Theorem 4.3, there is a left invariant topology T 0 on S in which for each a 2 S, aN is a neighborhood base at a. Applying (a))(d) to T 0 , we obtain that T 0 is the largest left invariant topology on S in which N is the neighborhood ﬁlter of 1. Hence T 0 D T .

4.2

The Topology T ŒF

Deﬁnition 4.6. Let S be a semigroup with identity. For every ﬁlter F on S, let T ŒF denote the largest left invariant topology on S in which F converges to 1. Note that by Corollary 4.5, if N is the neighborhood ﬁlter of 1 in T ŒF , then for every a 2 S , aN is a neighborhood base at a, and by Lemma 4.2, the left translations in T ŒF are continuous and open.

Section 4.2 The Topology T ŒF

55

Deﬁnition 4.7. For every mapping M W S ! P .S / and a 2 S, deﬁne ŒM a 2 P .S / by ŒM a D ¹x0 x1 xn W n < !; x0 D a and xiC1 2 M.x0 xi / for each i < nº: The next theorem describes the topology T ŒF . Theorem 4.8. Let S be a semigroup with identity and let F be a ﬁlter on S. Then for every a 2 S, the subsets ŒM a , where M W S ! F , form an open neighborhood base at a in T ŒF . Proof. The subsets ŒM a , where a 2 S and M W S ! F , possess the following properties: (i) for every a 2 S and M; N W S ! F , there is K W S ! F such that ŒKa ŒM a \ ŒN a , (ii) for every a 2 S and M W S ! F , a 2 ŒM a , (iii) for every a 2 S, M W S ! F and x 2 ŒM a , one has ŒM x ŒM a , and (iv) for every a 2 S and M W S ! F , there is N W S ! F such that ŒM a D aŒN 1 . For (i), deﬁne K W S ! F by K.x/ D M.x/ \ N.x/. (ii) and (iii) are obvious. To see (iv), deﬁne N W S ! F by N.x/ D M.ax/. Then ŒM a D ¹x0 x1 xn W n < !; x0 D a and xiC1 2 M.x0 xi / for each i < nº D a¹x0 x1 xn W n < !; x0 D 1 and xiC1 2 M.ax0 xi / for each i < nº D a¹x0 x1 xn W n < !; x0 D 1 and xiC1 2 N.x0 xi / for each i < nº D aŒN 1 : By (i), the subsets ŒM 1 , where M W S ! F , form a ﬁlter base B on S , and by (ii), 1 2 U for all U 2 B. (iii) and (iv) give us that for every U 2 B and a 2 U , there is V 2 B such that aV U . Hence by Corollary 4.4, there is a left invariant topology T on S in which for every a 2 S, aB is an open neighborhood base at a. (iii) and (iv) give us also that for every a 2 S and M W S ! F , ŒM a is an open neighborhood of a in T . On the other hand, let U be an open neighborhood of a 2 S in any left invariant topology on S in which F converges to 1. Deﬁne M W S ! F by ´ x 1 U if x 2 U M.x/ D S otherwise. Then ŒM a U . It follows that the subsets ŒM a , where M W S ! F , form an open neighborhood base at a in T and T D T ŒF .

56

Chapter 4 Left Invariant Topologies and Strongly Discrete Filters

Corollary S 4.9. IfA 2 F and for every x 2 A, Ux is a neighborhood of 1 in T ŒF , then x2A xUx [ ¹1º is a neighborhood of 1 in T ŒF . Proof. Without loss of generality one may assume that for every x 2 A, Ux is open, S so U D x2A xUx is open. Deﬁne M W S ! F by 8 ˆ

if x D 1 if x 2 U n ¹1º otherwise.

Then ŒM 1 U [ ¹1º. Corollary 4.10. If S is left cancellative and

T

F D ;, then T ŒF is a T1 -topology.

Proof. Let 1 ¤ a 2 S . Since S is left cancellative and F such that a … xM.x/ for all x 2 S. Then a … ŒM 1 .

T

F D ;, there is M W S !

Of special importance is the case where F is an ultraﬁlter. Theorem 4.11. For every nonprincipal ultraﬁlter F on S, T ŒF is strongly extremally disconnected. Proof. Let U be an open nonclosed subset of .S; T ŒF / and let C D S n U . Consider two cases. Case 1: there is a 2 C such that U 2 aF . Choose M W S ! F such that aM.a/ U and for every x 2 U , xM.x/ U . Then ŒM a U [ ¹aº. Hence, U [ ¹aº is open. Case 2: for every x 2 C , C 2 xF . Choose M W S ! F such that for every x 2 C , xM.x/ C . Then ŒM b C for every b 2 C . Consequently, C is open, and so U is closed. Hence, this case is impossible.

4.3

Strongly Discrete Filters

Deﬁnition 4.12. Let S be a semigroup with identity. Given a mapping M W S ! P .S /, an M -product is any product of elements of S of the form x0 xn where xiC1 2 M.x0 xi / for each i < n, and an M -decomposition of an element x 2 S is a decomposition x D x0 xn , where x0 xn is an M -product. An M -decomposition x D x0 xn is trivial if n D 0 (and then x0 D x). A mapping M W S ! P .S / is basic if every x 2 S has a unique M -decomposition x D x0 xn with x0 D 1.

Section 4.3 Strongly Discrete Filters

57

Lemma 4.13. Let M W S ! P .S / be a basic mapping. Then (1) there is no nontrivial M -decomposition of 1, (2) the subsets xM.x/, where x 2 S, are pairwise disjoint, and (3) for each x 2 S, the mapping M.x/ 3 y 7! xy 2 xM.x/ is injective. Proof. (1) It follows from the deﬁnition of a basic mapping that there is no nontrivial M -decomposition 1 D x0 xn with x0 D 1. So suppose that 1 D x0 xn is an M decomposition and x0 ¤ 1. Then there is an M -decomposition x0 D y0 ym with y0 D 1 and m > 0. But then 1 D y0 ym x1 xn is a nontrivial M - decomposition with y0 D 1, which is a contradiction. (2) Suppose that xu D yv D z for some x; y; z 2 S, u 2 M.x/ and v 2 M.y/. Let x D x0 xn and y D y0 ym be M -decompositions with x0 D y0 D 1. Then z D x0 xn u and z D y0 ym v are M -decompositions with x0 D y0 D 1. It follows that u D v, n D m and xi D yi for each i n, and so x D y. (3) Suppose that xu D xv for some x 2 S and u; v 2 M.x/. Let x D x0 xn be M -decomposition with x0 D 1. Then x0 xn u and x0 xn v are M -products and x0 xn u D x0 xn v. Hence u D v. Lemma 4.14. Let M W S ! P .S / and suppose that the subsets xM.x/, where x 2 S , are pairwise disjoint and for each x 2 S, M.x/ 3 y 7! xy 2 xM.x/ is injective. Let x0 xn and y0 ym be M -products and let x0 xn D y0 ym . Then either (1) x0 xnm D y0 and xnmCj D yj , 1 j m, if n m, or (2) x0 D y0 ymn and xi D ymnCi , 1 i n, if n < m. Proof. We proceed by induction on min¹n; mº. It is trivial for min¹n; mº D 0. Let min¹n; mº > 0. We claim that x0 xn1 D y0 ym1 and then xn D ym , since M.x0 xn1 / 3 y 7! x0 xn1 y 2 x0 xn1 M.x0 xn1 / is injective. Indeed, if x0 xn1 ¤ y0 ym1 , then x0 xn ¤ y0 ym , since x0 xn1 M.x0 xn1 / \ y0 ym1 M.y0 ym1 / D ;: We then apply the inductive assumption. Recall that a subset D of a space X is called strongly discrete if for every x 2 D, there is a neighborhood Ux of x 2 X such that the subsets Ux , where x 2 D, are pairwise disjoint. The Fréchet ﬁlter on an inﬁnite set X consists of all subsets A X such that jX n Aj < jX j.

58

Chapter 4 Left Invariant Topologies and Strongly Discrete Filters

Deﬁnition 4.15. Let S be a semigroup with identity and let F be a ﬁlter on S. We say that F is strongly discrete if (1) F contains the Fréchet ﬁlter, and (2) there is M W S ! F such that the subsets xM.x/, where x 2 S, are pairwise disjoint, and for every x 2 S, M.x/ 3 y 7! xy 2 xM.x/ is injective. Theorem 4.16. For every strongly discrete ﬁlter F on S, there is a basic mapping M WS !F. Proof. Let D jSj and enumerate S as ¹s˛ W ˛ < º with s0 D 1. Choose M0 W S ! F such that (i) the subsets xM0 .x/, where x 2 S, are pairwise disjoint, and for every x 2 S, M0 .x/ 3 y 7! xy 2 xM0 .x/ is injective, and (ii) s˛ … sˇ M0 .sˇ / for all ˛ ˇ < . It follows from (ii) that 1 … xM0 .x/ for all x 2 S, which in turn implies that there is no nontrivial M0 -decomposition of 1. Construct inductively a -sequence .a /< with maximally possible by putting a0 D 1 and taking a , for > 0, to S be the ﬁrst element in the sequence ¹s˛ W ˛ < º such that a … X where X D ı< ŒM0 aı . We deﬁne M W S ! F by ´ M0 .1/ [ ¹a W 0 < < º M.x/ D M0 .x/

if x D 1 otherwise.

Clearly, for every x 2 S , M.x/ 3 y 7! xy 2 xM.x/ is injective. Let us check that the subsets xM.x/, where x 2 S, are pairwise disjoint. Assume the contrary. Then by the deﬁnition of M and (i), a 2 xM0 .x/ for some 1 ¤ x 2 S and 0 < < . We have also that a D s˛ and x D sˇ for some ˛; ˇ < , so s˛ 2 sˇ M0 .sˇ /. It follows from this and (ii) that ˛ > ˇ. But then by the choice of a D s˛ , one has sˇ 2 X . This implies that also every M0 -product x0 xn with x0 D sˇ belongs to X . Since s˛ 2 sˇ M0 .sˇ /, we obtain that s˛ D a 2 X , which is a contradiction. Now we claim that M is basic. Indeed, for every x 2 S , there is < such that x 2 ŒM0 a , so there is an M0 decomposition x D x0 xn with x0 D a . If D 0, this is M -decomposition with x0 D 1. Otherwise so is x D 1x0 xn . To see that every x has only one M -decomposition x D x0 xn with x0 D 1, it sufﬁces, by Lemma 4.14, to check that if 1 D x0 xn is an M -decomposition with x0 D 1, then n D 0. Assume on the contrary that n > 0. Then 1 D x1 xn is a nontrivial M0 -decomposition of 1, which is a contradiction.

Section 4.3 Strongly Discrete Filters

59

Deﬁnition 4.17. Let D be a discrete subset of a space X and let x be an accumulation point of D. We say that D is locally maximal with respect to x if for every discrete subset E X such that D \ U E \ U for some neighborhood U of x, there is a neighborhood V of x such that D \ V D E \ V . The next theorem gives us the main properties of topologies determined by strongly discrete ﬁlters. Theorem 4.18. Let F be a strongly discrete ﬁlter on S and let S D .S; T ŒF /. Then (1) S is zero-dimensional and Hausdorff, (2) there is D 2 F such that (i) D is a strongly discrete subset of S with exactly one accumulation point, and (ii) D is a locally maximal discrete subset of S with respect to 1. Furthermore, if in addition F is an ultraﬁlter, then (3) S is strongly extremally disconnected. Proof. (1) To see that T ŒF is a T1 -topology, let a; b be distinct elements of S. Pick N W S ! F such that b … xN.x/ for all x 2 S. Then b … ŒN a . Now to show that T ŒF is zero-dimensional, let a 2 S. By Theorem 4.16, there is a basic mapping M W S ! F , and let N W S ! F be any mapping such that N.x/ M.x/ for all x 2 S. We show that ŒN a is closed. To this end, pick K W S ! F such that for every x 2 S, K.x/ M.x/ and a … xK.x/. We claim that for every b 2 S n ŒN a , one has ŒKb \ ŒN a D ;. Indeed, assume the contrary. Then x0 xn D y0 ym for some n; m < ! and xi ; yj 2 S such that x0 D b, xiC1 2 K.x0 xi / for i < n, y0 D a, and yj C1 2 N.y0 yj / for j < m. Since K.x/; N.x/ M.x/ for all x 2 S, we obtain by Lemma 4.14, that either x0 xnm D y0 (if n m) or x0 D y0 ymn (if n < m). The ﬁrst possibilty gives us that a 2 ŒKb , a contradiction with a … xK.x/ for all x 2 S. And the second gives us that b 2 ŒN a , again a contradiction. (2) Pick a basic mapping M W S ! F and let D D M.1/. (i) We claim that the subsets ŒM a , where a 2 D, are pairwise disjoint. Indeed, let x1 ; y1 2 D and let ŒM x1 \ ŒM y1 ¤ ;. Then x1 xn D y1 ym for some n; m 1 and xi ; yj such that xiC1 2 M.x1 xi / and yj C1 2 M.y1 yj /. But then x0 x1 xn D y0 y1 ym where x0 D y0 D 1. Since x1 ; y1 2 M.1/, it follows from this and Lemma 4.14 that n D m and xi D yi for each i n, in particular, x1 D y1 . To see that D [ ¹1º is closed, let 1 ¤ x1 2 ŒM 1 and let ŒM x1 \ D ¤ ;. Then x1 xn D y1 for some y1 2 D, n 1 and xi such that xiC1 2 M.x1 xi /. But then x0 x1 xn D y0 y1 where x0 D y0 D 1. It follows from this that n D 1 and x1 D y1 .

60

Chapter 4 Left Invariant Topologies and Strongly Discrete Filters

(ii) Let E be a discrete subset of S such that D\U E\U for some neighborhood U of 1. One may suppose that U is open. For every x 2 D \ U S, choose a neighborhood Vx of 1 such that xVx U and xVx \E D ¹xº. Put V D x2D\U xVx [¹1º. By Corollary 4.9, V is a neighborhood of 1, and by the construction, D \V D E \V . (3) is immediate from Theorem 4.11. The next proposition shows how naturally strongly discrete ﬁlters arise on left topological groups. Proposition 4.19. Let .S; T / be a regular left topological group such that the intersection of < jSj open sets is open and let D be a discrete subset of .S; T / with exactly one accumulation point 1. Then any ﬁlter on S containing D and converging to 1 is strongly discrete. Proof. Let F be a ﬁlter on S containing D and converging to 1 and let D jSj. Enumerate S as ¹s˛ W ˛ < º. Construct inductively a sequence .M.s˛ //˛< such that M.s˛ / 2 F , M.s˛ / D and [ s˛ M.s˛ / \ sˇ M.sˇ / D ;: ˇ <˛

This can be done because [

0 sˇ M.sˇ / D ¹sˇ W ˇ < ˛º:

ˇ <˛

(For every subset Y of a space X , Y 0 denotes the set of accumulation points of Y X .) We now show that strongly discrete ﬁlters exist in profusion on any inﬁnite cancellative semigroup. Proposition 4.20. Let S be a cancellative semigroup with identity and let jSj D !. Enumerate S without repetitions as ¹s˛ W ˛ < º. Then there is a one-to-one sequence .x˛ /˛< in S such that the subsets s˛ X˛ , where X˛ D ¹xˇ W ˛ ˇ < º, are pairwise disjoint. Proof. Pick as x0 any element of S. Now let 0 < < and assume that we have constructed a one-to-one sequence .x˛ /˛< such that the subsets s˛ X˛; , where X˛; D ¹xˇ W ˛ ˇ < º, are pairwise disjoint. Choose x 2 S satisfying the condition s˛ x ¤ sˇ xı , where ˛ and ˇ ı < (this can be done because S is left cancellative). The condition s˛ x ¤ sˇ x , where ˛; ˇ and ˛ ¤ ˇ, is satisﬁed automatically (because S is right cancellative). Hence, the subsets s˛ X˛;C1 , where ˛ < C 1, are also pairwise disjoint.

61

Section 4.3 Strongly Discrete Filters

Let F and G be ﬁlters on sets X and Y , respectively. We say that F and G are isomorphic if there is a bijection f W X ! Y such that f .F / D G . Deﬁnition 4.21. Let F be a ﬁlter on an inﬁnite set X containing the Fréchet ﬁlter. We say that F is locally Fréchet if there is A 2 F such that F jA is the Fréchet ﬁlter on A. A locally Fréchet ﬁlter is proper if it is not Fréchet. Note that all proper locally Fréchet ﬁlters on X are isomorphic, and if F is not locally Fréchet, then for every A 2 F , F is isomorphic to F jA . To see the latter, pick B 2 F such that B A and jAnBj D jX j and take any bijection g W X nB ! AnB. Deﬁne f W X ! A by ´ x if x 2 B f .x/ D g.x/ otherwise. Then f is a bijection and f .F / D F jA . Now we obtain from Proposition 4.20 the following. Corollary 4.22. Let S be a cancellative semigroup with identity and let jSj D !. For every ﬁlter F on properly containing the Fréchet ﬁlter, there is a strongly discrete ﬁlter G on S isomorphic to F . Proof. Let .x˛ /˛< be a sequence in S guaranteed by Proposition 4.20, let A D ¹x˛ W ˛ < º, and let H be the ﬁlter on S such that A 2 H and H jA is the Fréchet ﬁlter. Then H is proper locally Fréchet and strongly discrete. Now let F be a ﬁlter on properly containing the Fréchet ﬁlter. If F is locally Fréchet, it is isomorphic to H . Otherwise take a ﬁlter G on S such that A 2 G and G jA is isomorphic to F . Then G is isomorphic to F and is strongly discrete. We conclude the section with a topological classiﬁcation of topologies determined by strongly discrete ﬁlters. Deﬁnition 4.23. Given a space X with exactly one accumulation point 1 2 X , deﬁne an increasing sequence .Xn /1 nD1 of extensions of X D X1 as follows. For each n 1, put [ XnC1 D Xn0 [ X.y/; y2Xn nXn0

where X.y/ is a copy of X with y D 1 (we suppose that X.y/ \ Xn D ¹yº and that X.y/ \ X.z/ D ; if y ¤ z), and topologize XnC1 by declaring a subset U XnC1 to be open if and only if U \ Xn is open in Xn and for every y 2 Xn n Xn0 , U \ X.y/ is open in X.y/. Finally, we deﬁne X! D lim Xn ; n!1

62

Chapter 4 Left Invariant Topologies and Strongly Discrete Filters

S that is, X! D 1 nD1 Xn and a subset U X is open if and only if U \ Xn is open in Xn for every n. Theorem 4.24. Let F be a strongly discrete ﬁlter on S and let X be a space with exactly one accumulation point 1 2 X such that the ﬁlter on X with a base consisting of subsets U n¹1º, where U runs over neighborhoods of 1, is isomorphic to F if F is not locally Fréchet and to the Fréchet ﬁlter otherwise. Then .S; T ŒF / is homeomorphic to X! . Proof. Suppose ﬁrst that F is not locally Fréchet. By Theorem 4.16, there is a basic M W S ! F . Deﬁne an increasing sequence .Yn /1 nD1 of subspaces of .S; T ŒF // by putting Y1 D M.1/ and [ .M.y/ [ ¹yº/: YnC1 D Yn0 [ y2Yn nYn0

Then .S; T ŒF / D lim Yn : n!1

For every y 2 S, M.y/ [ ¹yº is homeomorphic to X . It follows that for every n, Yn is homeomorphic to Xn , and consequently, .S; T ŒF / is homeomorphic to X! . Now suppose that F is locally Fréchet, so there is B 2 F such that F jB is the Fréchet ﬁlter. Choose a basic M W S ! F such that M.x/ B for all x 2 S. Then for every N W S ! F such that N.x/ M.x/ and for every a 2 S, ŒN a is homeomorphic to X! , in particular, ŒM 1 is homeomorphic to X! . We shall show that there are N W S ! F and A S with 1 2 A such that N.x/ M.x/, jAj D jS j D , and the subsets ŒN a , where a 2 A, form a partition of ŒM 1 . This implies that both ŒM 1 and .S; T ŒF / are homeomorphic to the sum of copies of X! , and consequently, .S; T ŒF / is homeomorphic to X! . For each x 2 M.1/, pick yx 2 M.x/, and let C D ¹xyx W x 2 M.1/º. Then jC j D , Sthe subsets ŒM a ŒM 1 , where a 2 C , are pairwise disjoint, and U D ŒM 1 n a2C ŒM a is an open neighborhood of 1 (see Theorem 4.18 (2)). Deﬁne N W S ! F by ´ .x 1 U / \ M.x/ if x 2 U N.x/ D M.x/ otherwise. The mapping N , in turn, determines D S such that the subsets ŒN a , where a 2 D, form a partition of S. Put A D D \ ŒM 1 . Remark 4.25. Let be an inﬁnite cardinal, let X be a space with jX j D and exactly one accumulation point 1X 2 X such that the ﬁlter on X with a base consisting of subsets U n ¹1X º, where U runs over neighborhoods of 1X , is isomorphic to the Fréchet ﬁlter on , and let Y be a space with jY j D and exactly one accumulation

Section 4.4 Invariant Topologies

63

point 1Y 2 Y such that the ﬁlter on Y with a base consisting of subsets U n ¹1Y º, where U runs over neighborhoods of 1Y , is isomorphic to the proper locally Fréchet ﬁlter on . It is not hard to see that the space Y! is homeomorphic to the sum of copies of X! , and so just to X! . Hence Theorem 4.24 can be stated also as follows: Let F be a strongly discrete ﬁlter on S and let X be a space with exactly one accumulation point 1 2 X such that the ﬁlter on X with a base consisting of subsets U n ¹1º, where U runs over neighborhoods of 1, is isomorphic to F . Then .S; T ŒF / is homeomorphic to X! . Corollary 4.26. Let F and G be strongly discrete ﬁlters on S. Then the topologies T ŒF and T ŒG are homeomorphic if and only if the ﬁlters F and G are isomorphic. Proof. Let f W .S; T ŒF / ! .S; T ŒG / be a homeomorphism. Without loss of generality one may suppose that f .1/ D 1. We claim that f .F / D G . Let DF 2 F and DG 2 G be sets satisfying Theorem 4.18 (2). It sufﬁces to show that for every A 2 F with A DF , f .A/ 2 G . Assume the contrary. Then B D DG n f .A/ is a discrete subset of .S; T ŒG / with cl B D B [ ¹1º. Consequently, C D A [ f 1 .B/ is a discrete subset of .S; T ŒF / such that A C and for every neighborhood U of 1, .C n A/ \ U ¤ ;. But this contradicts Theorem 4.18 (2). Conversely, let F and G be isomorphic. Then T ŒF and T ŒG are homeomorphic by Theorem 4.24.

4.4

Invariant Topologies

Deﬁnition 4.27. We say that a topology on a group is invariant if all translations and the inversion are continuous. In this section we show that Theorem 4.28. Every inﬁnite group admits a nondiscrete zero-dimensional Hausdorff invariant topology. By Theorem 4.18 (1), in order to prove Theorem 4.28, it sufﬁces to show that for every inﬁnite group G, there is a strongly discrete ﬁlter F on G such that the topology T ŒF is invariant. We ﬁrst derive a simple sufﬁcient condition for the topology T ŒF to be invariant. Lemma 4.29. Let T be a left invariant topology on G and let N be the neighborhood ﬁlter of 1 in T . Then T is invariant if and only if N 1 D N and xN x 1 D N for all x 2 G. Proof. Necessity is obvious. Sufﬁciency follows from the fact that for every U 2 N and x; y 2 G, one has .xU /y D xy.y 1 Uy/ and .xU /1 D x 1 .xU 1 x 1 /.

64

Chapter 4 Left Invariant Topologies and Strongly Discrete Filters

Lemma 4.30. Let F be a ﬁlter on G such that F 1 D F and xF x 1 D F for all x 2 G. Then the topology T ŒF is invariant. Proof. By Lemma 4.29 and Theorem 4.8, it sufﬁces to show that (a) for every M W G ! F and a 2 G, there is N W G ! F such that aŒN 1 a1 ŒM 1 , and (b) for every M W G ! F , there is N W G ! F such that ŒN 1 1 ŒM 1 . To see (a), deﬁne N W G ! F by N.x/ D a1 M.axa1 /a: Let x 2 ŒN 1 and let x D x0 xn be N -decomposition with x0 D 1. Then axa1 D ax0 xn a1 D ax0 a1 axn a1 and for each i < n, one has axiC1 a1 2 aN.x0 xi /a1 D M.ax0 xi a1 / D M.ax0 a1 axi a1 /: To see (b), deﬁne N W G ! F by N.x/ D x 1 .M.x 1 //1 x: Let x 2 ŒN 1 and let x D x0 xn be N -decomposition with x0 D 1. Then x 1 D .x0 xn /1 D x01 x0 x11 x01 .x0 xn1 /xn1 .x0 xn1 /1 and for each i < n, one has 1 .x0 xi /xiC1 .x0 xi /1

2 .x0 xi /.N.x0 xi //1 .x0 xi /1 D M..x0 xi /1 / D M.x01 x0 x11 x01 .x0 xi1 /xi1 .x0 xi1 /1 /: In the case where F is a strongly discrete ﬁlter, the condition in Lemma 4.30 is also necessary for the topology T ŒF to be invariant. Proposition 4.31. Let F be a strongly discrete ﬁlter on G. Then the topology T ŒF is invariant if and only if F 1 D F and xF x 1 D F for all x 2 G.

Section 4.4 Invariant Topologies

65

Proof. Sufﬁciency is immediate from Lemma 4.30. Necessity follows from Theorem 4.18 (2). Thus, by Theorem 4.18 (1) and Lemma 4.30, in order to prove Theorem 4.28, it sufﬁces to show that Theorem 4.32. For every inﬁnite group G, there is a strongly discrete ﬁlter F on G such that F 1 D F and xF x 1 D F for all x 2 G. Proof. We will prove the assertion: For every inﬁnite subgroup H of G and for every subset F G with jF j < jGj, there is a ﬁlter F on G and a mapping M W H ! F such that (1) F 1 D F and xF x 1 D F for all x 2 H , (2) F jM.1/ contains the Fréchet ﬁlter on M.1/, jM.1/j D jH j and M.x/ M.1/ for all x 2 H , (3) the subsets xM.x/, where x 2 H , are pairwise disjoint and disjoint from F . Here, F jM.1/ is the trace of F on M.1/. We proceed by induction on D jH j. Let D !. Enumerate H as ¹xn W n < !º. It sufﬁces to construct a one-to-one sequence .yn /n

where n W ! ! !, and the mapping M W H ! F by M.xn / D Zn (see Lemma 1.16). Pick as y0 any element of G such that x0 y0 x01 … F . Fix 0 < k < ! and assume that we have constructed an one-to-one sequence .yn /n

66

Chapter 4 Left Invariant Topologies and Strongly Discrete Filters

Lemma 4.33. Every ﬁnite system of inequalities of the form ayb ¤ y " , where a; b 2 G, " D ˙1 and ab ¤ 1, has jGj-many solutions in G. Proof. Assume the contrary. Then there is a subset F G with jF j < jGj such that for every x 2 G n F , there exists an inequality ayb ¤ y " of the system with axb D x " . By Corollary 1.42, there is a 3-sequence .xn /n<3 in G and an inequality ayb ¤ y " of the system such that axb D x " for every x 2 FP..xn /n<3 /. Consider two cases. Case 1: " D 1. Then on the one hand ax1 x2 b D x1 x2 and on the other hand ax1 x2 b D ax1 b b 1 a1 ax2 b D x1 .ab/1 x2 ; so

x1 .ab/1 x2 D x1 x2

which contradicts ab ¤ 1. Case 2: " D 1. Then for any x 2 FP..xn /1n<3 /, on the one hand ax0 xb D .x0 x/1 D x 1 x01 and on the other hand ax0 xb D ax0 b b 1 a1 axb D x01 .ab/1 x 1 ; so and consequently Hence

x01 .ab/1 x 1 D x 1 x01 x01 .ab/1 x 1 x0 D x 1 : .x01 /x.abx0 / D x

and since x01 abx0 ¤ 1, we can apply Case 1. Fix > !, assume that the assertion has been proved for all inﬁnite < , and prove it for D . Choose S an increasing sequence .H˛ /˛< of inﬁnite subgroups of H with jH˛ j < and ˛< H˛ D H . For every ˛ < , construct a ﬁlter F˛ on G and a mapping M˛ W H˛ ! F˛ such that (1) F˛1 D F˛ and xF˛ x 1 D F˛ for all x 2 H˛ , (2) F jM˛ .1/ contains the Fréchet ﬁlter on M˛ .1/, jM˛ .1/j D jH˛ j and M˛.x/ M˛ .1/ for all x 2 H˛ , and

67

Section 4.4 Invariant Topologies

(3) the subsets xM˛ .x/, where x 2 H˛ , are pairwise disjoint and disjoint from F and all yM .y/, where < ˛ and y 2 H . Deﬁne the ﬁlter F on G by taking as a base the subsets of the form [ A˛ ; ˛<

where < and A˛ 2 F˛ , and the mapping M W H ! F by [ M.x/ D M˛ .x/; ˛<

where D min¹˛ W x 2 H˛ º. Then F is as required. jGj

Remark 4.34. In fact, the proof of Theorem 4.32 shows more: there are 22 nondiscrete zero-dimensional Hausdorff invariant topologies T˛ on G such that for any disjGj tinct ˛; < 22 , the topology T˛ _ T is discrete. (The latter means that there are neighborhoods U˛ , U of 1 in T˛ , T respectively with U˛ \ U D ¹1º.) To see this, let H D G and for every uniform ultraﬁlter p on , deﬁne the ﬁlter F .p/ on G by taking as a base the subsets of the form [ A˛ ; ˛2P

where P 2 p and A˛ 2 F˛ . It is clear that .F .p//1 D F .p/, xF .p/x 1 D F .p/ for all x 2 G, and for any distinct p and q, there exist A.p/ 2 F .p/ and A.q/ 2 F .q/ with A.p/ \ A.q/ D ;. That T .F .p// _ T .F .q// is discrete follows from Theorem 4.18 (2) and Corollary 4.9.

References The results about the topology T ŒF and strongly discrete ﬁlters in the case where S is a group were proved in [103]. That every inﬁnite group admits a nondiscrete zero-dimensional Hausdorff invariant topology was proved in [100].

Chapter 5

Topological Groups with Extremal Properties

In this chapter, assuming MA, we construct important topological groups with extremal properties. We also show that some of them cannot be constructed in ZFC. Recall that starting from this chapter, all topological groups are assumed to be Hausdorff.

5.1

Extremally Disconnected Topological Groups

Theorem 5.1.LFor every n < !, let Gn be a nontrivial ﬁnite group written additively, and let G D n

Section 5.1 Extremally Disconnected Topological Groups

69

Since p is Ramsey, there exists A 2 p such that A B0 and ŒA2 is monochrome. Observe that .ŒA2 / D ¹1º. We claim that HA ŒM 0 . To see this, let 0 ¤ x 2 HA and let supp.x/ D ¹n1 ; : : : ; nk º, where n1 < < nk . We have that n1 2 B0 and niC1 2 Bni for each i D 1; : : : ; k 1. Write x as x D x1 C C xk , where supp.xi / D ¹ni º. Then x1 2 DB0 M.0/ and xiC1 2 DBni M.x1 C C xi /: Hence x 2 ŒM 0 . .2/ ) .3/ Let E be a discrete subset of .G; T / such that D \ U E \ U for some neighborhood U of 0. One may suppose that U is open. For every x 2 D \ U , choose a neighborhood Vx of 0 such that x C Vx U and .x C Vx / \ E D ¹xº. Let [ V D .x C Vx / [ ¹0º: x2D\U

By Corollary 4.9, V is a neighborhood of 0 and by the construction, D \ V D E \ V . .3/ ) .1/ Let ¹An W n < !º be any partition of ! such that An … p for all n < !. Deﬁne E G by E D ¹x 2 G W supp.x/ An for some n < ! and jsupp.x/j D 2º: Notice that every point from E is isolated in D [ E. Since An … p for all n < !, it follows that also every point from D is isolated in D [ E. Consequently D [ E is discrete. Since D is a locally maximal discrete subset with respect to 0, there is a neighborhood U of 0 such that U \ E D ;. Choose A 2 p such that HA U . Then jA \ An j 1 forL all n < !. Now let G D ! Z2 . Then F is an ultraﬁlter. .2/ ) .4/ and .2/ ) .5/ follow from Theorem 4.11 and Proposition 3.15. To show .4/ ) .3/, let E be a discrete subset of G D .G; T / such that D \ U E \U for some neighborhood U of 0. For every x 2 E, choose an open neighborhood Ux of 0 such that the subsets x C Ux are pairwise disjoint. Put [ [ UD D .x C Ux / and UE nD D .x C Ux /: x2D

x2E nD

Then UD and UE nD are disjoint open subsets and, obviously, 0 2 cl UD . But then, since G is extremally disconnected, 0 … cl UE nD . Hence, V D U n UE nD is a neighborhood of 0 and D \ V D E \ V . To show .5/ ) .1/, suppose that p is not Ramsey. Then there is a partition ¹An W n < !º of ! such that An … p for all n < ! and for every A 2 p, jA \ An j 2 for some n < !. Deﬁne the subset F of .G; T / by F D ¹x 2 G W jsupp.x/ \ An j 1 for all n < !º: We claim that F is perfect and nowhere dense.

70

Chapter 5 Topological Groups with Extremal Properties

Clearly, F is closed. To see that F is dense in itself, let x 2 F . Pick n0 < ! such that max supp.x/ < min An for all n n0 and deﬁne A 2 p by AD

[

An :

n0 n

Then x C DA F and x is an accumulation point of x C DA . Finally, to see that F is nowhere dense, let x 2 G and let U be a neighborhood of 0 in T . Pick A 2 p such that max supp.x/ < min A and HA U . Then jA \ An j 2 for some n. Consequently, there is y 2 HA such that supp.y/ is a 2-element subset of An , and so x C y … F . Since F is closed, this shows that F is nowhere dense. Corollary 5.2. Assume p D c. Then there exists a countable nondiscrete extremally disconnected topological group. A group of exponent 2 is called Boolean. Note that if G is a Boolean group, then 1 1 1 for every x; y 2 G, one has xy D L.xy/ D y x D yx, so G is Abelian. It then follows that G is isomorphic to Z2 for some cardinal . For every group G, let B.G/ D ¹x 2 G W x 2 D 1º: Note that if G is Abelian, then B.G/ is the largest Boolean subgroup of G. Lemma 5.3. Let G be a topological group. If B.G/ is a neighborhood of 1, then G contains an open Boolean subgroup. Proof. Let B D B.G/. Since B is a neighborhood of 1, there is a neighborhood W of 1 such that W 2 B. Then for every x; y 2 W , xy 2 B and consequently xy D .xy/1 D y 1 x 1 D yx. It then follows that for every x1 ; : : : ; xn 2 W , .x1 xn /2 D x12 xn2 D 1 and so x1 xn 2 B. Hence, hW i is an open Boolean subgroup of G. Theorem 5.4. Every extremally disconnected topological group contains an open Boolean subgroup. Proof. Let G be an extremally disconnected topological group and let W G 3 x 7! x 1 2 G. Then by Theorem 3.13, ¹x 2 G W .x/ D xº is clopen. Since ¹x 2 G W .x/ D xº D B.G/, G contains an open Boolean subgroup by Lemma 5.3.

Section 5.2 Maximal Topological Groups

5.2

71

Maximal Topological Groups

We say that a topological group (or a group topology) is maximal if the underlying space is maximal. Theorem 5.5. Assume p D c. Then there exists a maximal topological group. L Proof. Let G D ! Z2 . Enumerate the subsets of G as ¹Z˛ W ˛ < cº with Z0 D G. For every ˛ < c, we shall construct a sequence .x˛;n /n 1Sand suppose that the statement holds for all positive integers less than n. Let X D A and for every x 2 X , let Ax D ¹A 2 A W x 2 Aº. Consider two cases.

72

Chapter 5 Topological Groups with Extremal Properties

Case 1: for every x 2 X , Ax is countable. By transﬁnite recursion on ˛ < !1 , pick so that A˛ \ A D ; for all < ˛. This can be done because S A˛ 2 A S ¹Ax W x 2 <˛ A º is countable. Let B D ¹A˛ W ˛ < !1 º. Then B is a

-system with root ;. Case 2: there is x 2 X with uncountable Ax . Let A0 D ¹A n ¹xº W A 2 Ax º. By the inductive hypothesis, there is an uncountable -system B 0 A0 . Let R denote the root of B 0 . Deﬁne B Ax by B D ¹A [ ¹xº W A 2 B 0 º. Then B is a -system with root R [ ¹xº. Theorem 5.7. Every maximal topological group contains a countable open Boolean subgroup. Proof. Let G be a maximal topological group. Then G is extremally disconnected. L Consequently by Theorem 5.4, one may suppose that G D Z2 for some inﬁnite cardinal . For every a 2 N, let 2 .a/ D max¹n < ! W 2n jaº: Deﬁne the partition ¹Ai W i < 2º of G n ¹0º by Ai D ¹x 2 G n ¹0º W 2 .jsupp.x/j/ i .mod 2/º: Since G is maximal, there exists a neighborhood U of 0 such that either U n ¹0º A0 or U n ¹0º A1 . Choose a neighborhood V of 0 such that V C V C V C V U: We claim that V is countable. Indeed, assume the contrary. Then applying the -system Lemma gives us an uncountable X V and a ﬁnite F such that, whenever x and y are different elements of X , one has supp.x/ \ supp.y/ D F and jsupp.x/j D jsupp.y/j. Pick different x0 ; x1 ; x2 ; x3 2 X and let x D x0 C x1 and y D x0 C x1 C x2 C x3 . Then x; y 2 U and jsupp.y/j D 2 jsupp.x/j, so x; y belong to different sets A0 ; A1 , a contradiction.

5.3

Nodec Topological Groups

Theorem 5.8. Assume p D c. Then every nondiscrete group topology on an Abelian group of character < c can be reﬁned to a nondiscrete nodec group topology. Before proving Theorem 5.8 we establish several auxiliary statements.

73

Section 5.3 Nodec Topological Groups

Lemma 5.9. Let Y be a topological space, let X be a nowhere dense subset of Y , and let F be a ﬁnite set of continuous open mappings f W Y ! Y . Then for every nonempty open set U Y , there is a nonempty open set V U such that f .V / \ X D ; for all f 2 F . Proof. Let f 2 F . Since f is open and X is nowhere dense, there is a nonempty open Wf f .U / such that Wf \ X D ;. Then, since f is continuous, there is a nonempty open Vf U such that f .Vf / Wf , and so f .Vf / \ X D ;. Now let F D ¹f1 ; : : : ; fn º. Construct inductively a decreasing sequence .Vi /niD1 of nonempty open subsets of U such that fi .Vi / \ X D ; for each i D 1; : : : ; n. Then the set V D Vn is as required. Lemma 5.10. Let G be a nondiscrete metrizable Abelian topological group of prime period and let X be a nowhere dense subset of G such that 0 … X . Then there is a nondiscrete subgroup H of G such that H \ X D ;. Proof. Let ¹Un W n 2 Nº be a neighborhood base at 0 2 G. It sufﬁces to construct a sequence .an /1 nD1 in G such that 0 ¤ an 2 Un and ha1 ; : : : ; an i \ X D ; for every n < !. Then the group H D han W n 2 Ni D

1 [

ha1 ; : : : ; an i

nD1

would be as required. Without loss of generality one may suppose that X [ ¹0º is closed. Let p be the period of G. Since p is prime, the mapping x 7! mx is a homeomorphism for each m D 1; : : : ; p 1. Then by Lemma 5.9, there is a1 2 U1 n¹0º such that ha1 i\X D ;. Now ﬁx n < ! and suppose that we have constructed a sequence .ai /niD1 in G such that ha1 ; : : : ; an i \ X D ;. Then there is a neighborhood V of 0 such that .ha1 ; : : : ; an i n ¹0º C V / \ X D ;: Choose a neighborhood W of 0 such that W C W… V: „ C ƒ‚ p1

By Lemma 5.9, there is anC1 2 .W \ UnC1 / n ¹0º such that hanC1 i \ X D ;. Since ha1 ; : : : ; anC1 i n ha1 ; : : : ; an i D .ha1 ; : : : ; an i n ¹0º/ C hanC1 i and hanC1 i V , it follows that ha1 ; : : : ; anC1 i \ X D ;. Lemma 5.11. Let .G; T / be an Abelian topological group with no open subgroup of ﬁnite period. Then T can be reﬁned to a nondiscrete group topology T 0 of the same character that T and such that for every m 2 N, the mapping x 7! mx is open in T 0 .

74

Chapter 5 Topological Groups with Extremal Properties

Proof. Deﬁne the group topology T 0 on G by taking as a neighborhood base at 0 the subsets of the form mU , where U runs over a neighborhood base at 0 in T and m 2 N. Lemma 5.12. Let .G; T / be a nondiscrete metrizable Abelian topological group such that for every m 2 N, the mapping x 7! mx is open, and let X be a nowhere dense subset of .G; T / such that 0 … X . Then the topology T can be reﬁned to a nondiscrete metrizable group topology T 0 such that 0 … clT 0 X . Proof. Let ¹Un W n 2 Nº be a neighborhood base at 0 in T . We shall construct a sequence .an /1 nD1 in G such that 0 ¤ an 2 Un and n X

Bin \ X D ;

iD1

for all n 2 N, where Bin D ¹0; ˙ai ; : : : ; ˙an º. Without loss of generality one may suppose that X [ ¹0º is closed. Clearly, there is a1 2 U1 n ¹0º such that ˙a1 … X . Fix n 2 N and suppose that we have constructed a sequence .ai /niD1 such that n X

Bin \ X D ;:

iD1

Then there is a neighborhood V of 0 such that n X

Bin n ¹0º C V \ X D ;:

iD1

Choose a neighborhood W of 0 such that W C W… V: „ C ƒ‚ nC1

By Lemma 5.9, there is anC1 2 W \ UnC1 n ¹0º such that ¹˙anC1 ; : : : ; ˙.n C 1/anC1 º \ X D ;: Then

nC1 X

BinC1 \ X D ;:

iD1

Now let T1 D T

..an /1 nD1 /

and let U D

1 X n [ nD1 iD1

Bin

Section 5.3 Nodec Topological Groups

75

By Theorem 1.19, U is a neighborhood of 0 in T1 , and by the construction, U \ X D ;. Since an 2 Un , .an /1 nD1 converges to 0 in T , and consequently, T T1 . By Lemma 1.31, T1 can be weakened to a metrizable group topology T 0 in which U remains a neighborhood of 0. Now we are ready to prove Theorem 5.8. Proof of Theorem 5.8. Let .G; T / be a nondiscrete Abelian topological group of character < c. Without loss of generality one may assume that G is countable. By Theorem 1.28, one may assume also that T is metrizable. Enumerate the subsets of G n ¹0º into a sequence .X /

76

Chapter 5 Topological Groups with Extremal Properties

Fix ˛ < c and suppose that for every < ˛, the topology T has already been deﬁned and the mappings x 7! mx, where m 2 N, are open in T . If ˛ is a limit ordinal, deﬁne T˛ by (ii). Then the mappings x 7! mx, where m 2 N, are open in T˛ as well. Let ˛ D ı C 1 for some ı < c. By Theorem 1.28 and Lemma 5.17, there is a nondiscrete metrizable group topology T˛0 on G such that Tı T˛0 and the mappings x 7! mx, where m 2 N, are open in T˛0 . Let X.˛/ be the ﬁrst set in the sequence .X /

from Œı; c/, X is a nowhere dense set in T with 0 2 clT X , then X is the ﬁrst such set and so one canStake C 1 as ˛. /. Now let T 0 D ˛

5.4

P-point Theorems

In this section we prove two theorems. The ﬁrst of them is concerned with extremally disconnected topological groups containing countable nonclosed discrete subsets. Lemma 5.13. Let X be an extremally disconnected space, let D be a strongly discrete subset of X , and let x 2 cl DnD. Then there is exactly one ultraﬁlter on X containing D and converging to x. Proof. Assume on the contrary that there are two different ultraﬁlters on G containing D and converging to x. Then there are disjoint subsets A and B of D with x 2 .cl A/\.cl B/. For every y 2 D, there is a neighborhood S Uy of y such that the S subsets Uy , where y 2 D, are pairwise disjoint. Let UA D y2A Uy and UB D y2B Uy . Then UA and UB are disjoint open subsets of X with x 2 .cl UA / \ .cl UB /, which is a contradiction. Theorem 5.14. Let G be an extremally disconnected topological group and let p be a nonprincipal converging ultraﬁlter on G. Suppose that p contains a countable

77

Section 5.4 P -point Theorems

discrete subset of G. Then there is a mapping f W G ! ! such that f .p/ is a P -point. Proof. Without loss of generality one may assume that G is Boolean and p converges to 0. Let D be a countable discrete subset of G such that D 2 p. Then by Lemma 3.34, D is strongly discrete. Enumerate D without repetitions as ¹xn W n < !º. Construct a decreasing sequence .Un /n

Then P and Q are disjoint open subsets of G, F Q, and 0 2 cl P . It follows that 0 … cl Q and so 0 … cl F . Choose neighborhoods V and W of 0 such that V \ F D ; and W C W V . Let E D D \ W and A D f .E/. We claim that for every m < !, A \ Am is ﬁnite. Indeed, assume the contrary. Pick any xn 2 E \ Dm . Then there exists xk 2 E \ Dm \ Un . But then, on the one hand, xn C xk 2 E C E W C W V and, on the other hand, xn C xk 2 xn C .Dm \ Un / F G n V; a contradiction.

78

Chapter 5 Topological Groups with Extremal Properties

It follows from Theorem 5.14 and Theorem 2.38 that Corollary 5.15. It is consistent with ZFC that there is no extremally disconnected topological group containing a countable discrete nonclosed subset. Our second theorem deals with countable topological Boolean groups containing no nonclosed discrete subset, and even more generally, containing no subset with exactly one accumulation point. A nonempty subset of a topological space is called a P-set if the intersection of any countable family of its neighborhoods is again its neighborhood. Note that every isolated point of a P -set is a P -point. In particular, every point of a ﬁnite P -set is a P -point. Deﬁnition 5.16. We say that a set of nonprincipal ultraﬁlters on ! is a P -set if it is a P -set in ! . Lemma 5.17. Let F be a ﬁlter on ! with are equivalent:

T

F D ;. Then the following statements

(1) F is a P-set, (2) for every decreasing sequence .An /n

and

.x/ D max supp.x/:

79

Section 5.4 P -point Theorems

L Theorem 5.19. Let T be a nondiscrete Q group topology on ! Z2 ﬁner than that induced by the product topology on ! Z2 and let F denote the neighborhood ﬁlter L of 0 in T . Suppose that . ! Z2 ; T / contains no subset with exactly one accumulation point. Then .F / is a P -set. In particular, if .F / is ﬁnite, then each of its points is a P -point. Proof. To show that .F / is a P -set, we use characterization (3) from Lemma 5.17. Let f W ! ! !. Denote S D F n ¹0º and for each p 2 S, pick Ap 2 p such that either f ..x// .x/ for all x 2 Ap or f ..x// < .x/ for all x 2 Ap . Consider two cases. Case 1: there is p 2 S such that f ..x// < .x/ for all x 2 Ap . Pick an accumulation point a ¤ 0 of Ap . Then there is q 2 S and Q 2 q such that a C Q Ap . Choose Q in addition so that for every x 2 Q, one has .a/ < .x/. Then for every x 2 Q, we have that f ..x// D f ..a C x// < .a C x/ D .a/: Hence f ..Q// is ﬁnite. Case 2: for every p 2 S, f ..x// .x/ for all x 2 Ap . Deﬁne the neighborhood V of 0 in T by [ V D Ap [ ¹0º p2S

and choose a neighborhood W of 0 such that W C W V . We claim that for every n < !, f 1 .n/ \ .W / is ﬁnite. Indeed, assume the contrary. Then there exist m < ! and a sequence .xn /n m. Hence f ..x // > m, which is a contradiction. We say that a topological group .G; T / is maximally nondiscrete if T is maximal among nondiscrete group topologies on G. By Zorn’s Lemma, every nondiscrete group topology can be reﬁned to a maximally nondiscrete group topology. Clearly, every maximal topological group is maximally nondiscrete. Lemma 5.20. Let .G; T0 / be a totally bounded topological group and let T1 be a maximally nondiscrete group topology on G. Then T0 T1 . Proof. Pick any nonprincipal ultraﬁlter U on G converging to 1 in T1 . Considering .G; T0 / as a subgroup of a compact group shows that the ﬁlter UU1 (with a base of subsets AA1 , where A 2 U) converges to 1 in T0 . Clearly UU1 converges to 1 in T1 as well. Since T1 is maximally nondiscrete, it follows that T1 D T .UU1 /. Hence T0 T1 .

80

Chapter 5 Topological Groups with Extremal Properties

Now as a consequence we obtain from from Theorem 5.19 that Corollary 5.21. The existence of a maximal topological group implies the existence of a P -point in ! . Proof. Let .G;L T / be a maximal topological group. By Theorem 5.7, one may suppose that G D ! Z2 . Let F be the nonprincipal ultraﬁlter on G converging Q to 0 in T and let T0 be the topology on G induced by the product topology on ! Z2 . By Lemma 5.20, T0 T . Hence by Theorem 5.19, .F / is a P -point. Combining Corollary 5.21 and Theorem 2.38 gives us that Corollary 5.22. It is consistent with ZFC that there is no maximal topological group. As another consequence we obtain from Theorem 5.19 the following result. Corollary 5.23. Let p be a nonprincipal ultraﬁlter on ! not being a P-point and let L T p be the group topology on G D ! Z2 with a neighborhood base at 0 consisting of subgroups HA D ¹x 2 G W supp.x/ Aº where A 2 p. Then for every nondiscrete group topology T on G ﬁner than T p , .G; T / has a discrete subset with exactly one accumulation point. Proof. Let F be the neighborhood ﬁlter of 0 in T . It is clear that .F / D p. Now assume on the contrary that .G; T / has a subset with exactly one accumulation point. Then by Theorem 5.19, .F / D p is a P -point, which is a contradiction. Lemma 5.24. There exists a nonprincipal ultraﬁlter on ! not being a P -point. Proof. Let ¹An W n < !º be a partition of ! into inﬁnite subsets. Deﬁne the ﬁlter F on ! by taking as a base the subsets of the form [

.Ai n Fi /

ni

where n < ! and Fi is a ﬁnite subset of Ai for each i . It then follows that every ultraﬁlter on ! containing F is not a P -point. Combining Corollary 5.23 and Lemma 5.24 gives us that Corollary 5.25. There exists a maximally nondiscrete group topology T on G D L Z such that .G; T / has a discrete subset with exactly one accumulation point. 2 !

Section 5.4 P -point Theorems

81

References The question whether there exists in ZFC a nondiscrete extremally disconnected topological group was raised by A. Arhangel’ski˘ı [3]. The ﬁrst consistent example of such a group L was constructed by S. Sirota [71], the implication .1/ ) .4/ in Theorem 5.1 for ! Z2L . The equivalence .1/ , .2/ and the fact that T ŒF is extremally disconnected for ! Z2 are due to A. Louveau [45]. Theorem 5.1, except for statement (5), is from [103]. Theorem 5.5 is a result of V. Malykhin [46]. Theorem 5.4 and Theorem 5.7 are also due to him [46, 47]. Theorem 5.8 was proved in [88]. Theorems 5.14 and 5.19 are from [97]. Corollary 5.21 is due to I. Protasov [56]. Corollary 5.25 is a partial case of a result from [65].

Chapter 6

The Semigroup ˇS

In this chapter we extend the operation of a discrete semigroup S to ˇS making ˇS a right topological semigroup with S contained in its topological center. We show that every compact Hausdorff right topological semigroup has an idempotent and, as a consequence, a smallest two sided ideal which is a completely simple semigroup. The structure of a completely simple semigroup is given by the Rees-Suschkewitsch Theorem. As a combinatorial application of the semigroup ˇS we prove Hindman’s Theorem. We conclude by characterizing ultraﬁlters from the smallest ideal of ˇS.

6.1

Extending the Operation to ˇS

Theorem 6.1. The operation of a discrete groupoid S extends uniquely to ˇS so that (1) for each a 2 S , the left translation a W ˇS 3 x 7! ax 2 ˇS is continuous, and (2) for each q 2 ˇS, the right translation q W ˇS 3 x 7! xq 2 ˇS is continuous. If S is a semigroup, the extended operation is associative.

Proof. For each a 2 S, the left translation la W S 3 x 7! ax 2 S extends uniquely to a continuous mapping la W ˇS ! ˇS. For each a 2 S and q 2 ˇS, deﬁne a q 2 ˇS by a q D la .q/: Next, for each q 2 ˇS, the mapping rq W S 3 x 7! x q 2 ˇS extends uniquely to a continuous mapping rq W ˇS ! ˇS. For each p 2 S and q 2 ˇS, deﬁne p q 2 ˇS by p q D rq .p/: Under the extended operation a D la and q D rq , consequently, conditions (1) and (2) are satisﬁed. Furthermore, whenever the operation of S extends to ˇS so that conditions (1) and (2) are satisﬁed, one has a D la and q D rq , hence such an extension is unique.

83

Section 6.1 Extending the Operation to ˇS

Finally, suppose that S is a semigroup and let p; q; r 2 ˇS. Then .pq/r D . lim xq/r x!p

because q is continuous

D . lim lim xy/r

because x is continuous

D lim lim .xy/r

because r is continuous

x!p y!q

x!p y!q

D lim lim lim .xy/z because xy is continuous x!p y!q z!r

D lim lim lim xyz x!p y!q z!r

and p.qr/ D lim .x.qr// x!p

D lim .x lim yr/ x!p

y!q

because qr is continuous because r is continuous

D lim .x lim lim yz/ because y is continuous x!p

y!q z!r

D lim lim lim x.yz/ because x is continuous x!p y!q z!r

D lim lim lim xyz; x!p y!q z!r

so .pq/r D p.qr/. Deﬁnition 6.2. A semigroup T endowed with a topology is a right topological semigroup if for each p 2 T , the right translation p W T 3 x 7! xp 2 T is continuous. The topological center of a right topological semigroup T , denoted ƒ.T /, consists of all a 2 T such that the left translation a W T 3 x 7! ax 2 T is continuous. Theorem 6.1 tells us that the operation of a discrete semigroup S extends uniquely to ˇS so that ˇS is a right topological semigroup with S ƒ.ˇS /. Lemma 6.3. Let R and T be Hausdorff right topological semigroups, let S be a dense subsemigroup of R such that S ƒ.R/, and let ' W R ! T be a continuous mapping such that '.S/ ƒ.T /. If 'jS is a homomorphism, so is '.

84

Chapter 6 The Semigroup ˇS

Proof. First let x 2 S and q 2 R. Then '.xq/ D '. lim xy/ S3y!q

because x is continuous

D lim '.xy/

because ' is continuous

D lim '.x/'.y/

because 'jS is a homomorphism

y!q y!q

D '.x/ lim '.y/ because '.x/ is continuous y!q

D '.x/'.q/: Now let p; q 2 R. Then '.pq/ D '. lim xq/ S3x!p

D lim '.xq/ x!p

because q is continuous because ' is continuous

D lim '.x/'.q/ x!p

D . lim '.x//'.q/ because '.q/ is continuous x!p

D '.p/'.q/: Corollary 6.4. Let S be a discrete semigroup and let ' W S ! T be any homomorphism of S into a compact Hausdorff right topological semigroup T such that '.S/ ƒ.T /. Then the continuous extension ' W ˇS ! T of ' is a homomorphism. Deﬁnition 6.5. Given a semigroup S endowed with a topology, a semigroup compactiﬁcation of S is a pair .'; T / where T is a compact right topological semigroup and ' W S ! T is a continuous homomorphism such that '.S / is dense in T and '.S/ ƒ.T /. ˇ Corollary 6.4 tells us that the Stone–Cech compactiﬁcation ˇS of a discrete semigroup S is the largest semigroup compactiﬁcation of S. ˇ From now on, for any semigroup S , ˇS denotes the Stone–Cech compactiﬁcation of the discrete semigroup S . The next lemma describes the operation of ˇS in terms of ultraﬁlters. That is, given ultraﬁlters p; q 2 ˇS, one characterizes the subsets of S which are members of the ultraﬁlter pq. Recall that given a semigroup S, A S and s 2 S, s 1 A D ¹x 2 S W sx 2 Aº D 1 s .A/:

Section 6.1 Extending the Operation to ˇS

85

Lemma 6.6. Let S be a semigroup, A S, s 2 S and p; q 2 ˇS. Then (1) A 2 sq if and only if s 1 A 2 q, and (2) A 2 pq if and only if ¹x 2 S W x 1 A 2 qº 2 p. Proof. (1) Let A 2 sq. Then A is a neighborhood of sq. Since s is continuous, there is Q 2 q such that sQ A. It follows that sQ A and so s 1 A 2 q. Conversely, let s 1 A 2 q. Assume on the contrary that A … sq. Consequently, S n A 2 sq. Then by the already established necessity, s 1 .S n A/ 2 q. But .s 1 A/ \ .s 1 .S n A// D ;; a contradiction. (2) Let A 2 pq. Since q is continuous, there is P 2 p such that P q A. Then for every x 2 P , A 2 xq and so by (1), x 1 A 2 q. Hence, ¹x 2 S W x 1 A 2 qº 2 p. Conversely, let ¹x 2 S W x 1 A 2 qº 2 p. Assume on the contrary that A … pq. Consequently, S n A 2 pq. Then by the already established necessity, ¹x 2 S W x 1 .S n A/ 2 qº 2 p. But .x 1 A/ \ .x 1 .S n A// D ; for each x 2 S. It follows that ¹x 2 S W x 1 A 2 qº \ ¹x 2 S W x 1 .S n A/ 2 qº D ;; a contradiction. Corollary 6.7. Let S be a semigroup, a 2 S and p; q 2 ˇS. Then (1) the ultraﬁlter aq has a base consisting of subsets of the form aQ where Q 2 q, and S (2) the ultraﬁlter pq has a base consisting of subsets of the form x2P xQx where P 2 p and Qx 2 q. We conclude this section by showing that if S is a cancellative semigroup, then S is a two-sided ideal of ˇS and the translations of ˇS are injective on S. A nonempty subset I of a semigroup S is a left ideal (two-sided ideal or just ideal) if SI I (both SI I and IS I ). Lemma 6.8. Let S be a cancellative (left cancellative, right cancellative) semigroup. Then S is an ideal (left ideal, right ideal) of ˇS. Proof. Suppose that S is left cancellative. Let p 2 ˇS and q 2 S . To see that pq 2 S , let A 2 pq. Then there exist x 2 S and B 2 q such that xB A. It follows that A is inﬁnite. Now suppose that S is right cancellative. Let p 2 S and q 2 ˇS. Assume on the contrary that pq D a 2 S. Then there exist A 2 p and, for each x 2 A, Bx 2 q such that xBx D ¹aº. Pick distinct x; y 2 A and any z 2 Bx \ By . Then xz D yz, a contradiction.

86

Chapter 6 The Semigroup ˇS

Lemma 6.9. Let S be a cancellative semigroup, let a and b be distinct elements of S , and let p 2 ˇS. Then ap ¤ bp and pa ¤ pb. Proof. Deﬁne f W S ! S by putting f .ax/ D bx for every x 2 S and f .y/ D a2 for every y 2 S n aS. Then f has no ﬁxed points and f .ap/ D bp. Hence, ap ¤ bp by Corollary 2.17. The proof that pa ¤ pb is similar. As usual, for every semigroup S, we use S 1 to denote the semigroup with identity obtained from S by adjoining one if necessary. Lemma 6.10. If S is a cancellative semigroup, so is S 1 . Proof. We ﬁrst show that if e is an idempotent in S, then e D 1. To see this, let x 2 S . Multiplying the equality e D ee by x from the left gives us xe D xee. Then cancellating the latter equality by e from the right we obtain x D xe. Similarly x D ex. Now assume on the contrary that S 1 is not cancellative. Then some translation in 1 S is not injective. One may suppose that this is a left translation. It follows that there exist a; b 2 S such that ab D a. But then ab 2 D ab, and so b 2 D b. Hence b D 1 2 S , a contradiction. Combining Lemma 6.9 and Lemma 6.10, we obtain the following. Corollary 6.11. Let S be a cancellative semigroup, let a 2 S, and let p 2 ˇS. If ap D p, then a D 1.

6.2

Compact Right Topological Semigroups

Recall that an element p of a semigroup is an idempotent if pp D p. Theorem 6.12. Every compact Hausdorff right topological semigroup has an idempotent. Proof. Let S be a compact Hausdorff right topological semigroup. Consider the set P of all closed subsemigroups of S partially ordered by the inclusion. Since S 2 P , T P ¤ ;. For every chain C in P , C 2 P . Hence by Zorn’s Lemma, C has a minimal element A. Pick x 2 A. We shall show that xx D x. (It will follow that A D ¹xº, but we do not need this.) We start by showing that Ax D A. Let B D Ax. Clearly B is nonempty. Since B D x .A/, B is closed. Also BB AxAx AAAx Ax D B, so B is a subsemigroup. Thus B 2 P . But then B D A, since B D Ax AA A and A is minimal.

Section 6.2 Compact Right Topological Semigroups

87

Now let C D ¹y 2 A W yx D xº. Since x 2 A D Ax, C is nonempty. And since C D x1 .x/, C is closed. Given any y; z 2 C , yz 2 AA A and yzx D yx D x, so C is a subsemigroup. Thus C 2 P . But then C D A, since C A and A is minimal. Hence x 2 C and so xx D x as required. A left ideal L of a semigroup S is minimal if S has no left ideal strictly contained in L. Corollary 6.13. Let S be a compact Hausdorff right topological semigroup. Then S contains a minimal left ideal. Every minimal left ideal of S is closed and has an idempotent. Proof. If L is any left ideal of S and x 2 L, then S x D x .S / is a closed left ideal contained in L. It follows that every minimal left ideal of S is closed and, by Theorem 6.12, has an idempotent. Thus we need only to show that S contains a minimal left ideal. To this end, consider the set P of all closed left ideals of S , partially ordered by the inclusion. Applying Zorn’s Lemma gives us a minimal element L 2 P . Since L is minimal among closed left ideals and every left ideal contains a closed left ideal, it follows that L is a minimal left ideal. A semigroup can have many minimal left (right) ideals. However, it can have at most one minimal two-sided ideal. Indeed, if K is a minimal ideal, then for any ideal I , KI is an ideal and KI K \ I , so K I . Deﬁnition 6.14. For every semigroup S, let K.S / denote the smallest ideal of S, if exists. Lemma 6.15. Let S be a semigroup and assume that there is a minimal left ideal L of S. Then (1) for every a 2 S , La is a minimal left ideal of S , (2) every left ideal of S contains a minimal left ideal, (3) different minimal left ideals of S are disjoint and their union is K.S /. Proof. (1) If M is a left ideal of S contained in La, then L0 D ¹x 2 L W xa 2 M º is a left ideal of S contained in L, so L0 D L and consequently M D La. (2) Let N be a left ideal of S. Pick a 2 N . Then La N and by (1), La is a minimal left ideal. (3) If M and N are minimal left ideals and L D M \ N ¤ ;, then L is a left ideal, so M D L D N . S Now let K D LS D a2S La. Clearly K is an ideal. If I is any ideal of S, then IL I \ L L, so IL D I \ L D L and consequently L I . Hence, K D LS IS I .

88

Chapter 6 The Semigroup ˇS

A semigroup is simple (left simple) if it has no proper ideal (left ideal). It is easy to see that a smallest ideal (a minimal left ideal) is a simple (left simple) semigroup. A semigroup S is completely simple if it is simple and there is a minimal left ideal of S which has an idempotent. Corollary 6.16. Every compact Hausdorff right topological semigroup has a smallest ideal which is a completely simple semigroup. Deﬁnition 6.17. Given a semigroup S, let E.S/ denote the set of idempotents of S. Lemma 6.18. Let S be a left simple semigroup with E.S / ¤ ; and let e 2 E.S /. Then (1) e is a right identity of S, and (2) eS is a group. Proof. (1) Let s 2 S. Since Se D S, there is t 2 S such that t e D s. Then se D t ee D t e D s. (2) Clearly, eS is a semigroup and e is a left identity of eS . By (1), e is also a right identity, so e is identity of eS. Now let x 2 eS . We have to ﬁnd y 2 eS such that yx D xy D e. Since Sx D S, there is s 2 S such that sx D e. Let y D es. Then yx D esx D ee D e. To see that xy D e, note that there is t 2 S such that ty D e, since Sy D S. It then follows that t D t e D tyx D ex D x. A semigroup satisfying the identity xy D x (xy D y) is called a left (right) zero semigroup. By Lemma 6.18 (1), if S is a left simple semigroup and E.S / ¤ ;, then E.S/ is a left zero semigroup. Proposition 6.19. Let S be a left simple semigroup with E.S / ¤ ;. Let e 2 E.S / and deﬁne f W E.S/ eS ! S by f .x; a/ D xa. Then f is an isomorphism. Proof. To see that f is a homomorphism, let .u1 ; a1 /; .u2 ; a2 / 2 E.S / eS. Then f ..u1 ; a1 /.u2 ; a2 // D f .u1 u2 ; a1 a2 / D u1 u2 a1 a2 D u1 a1 a2 D u1 a1 u2 a2 D f .u1 ; a1 /f .u2 ; a2 /: To see that f is surjective, let x 2 S. Since S is left simple, there is y 2 S such that yx D e. Then xyxy D xey D xy, so xy 2 E.S /. It follows that f .xy; ex/ D xyex D xyx D xe D x: To see that f is injective, let .u; a/ 2 E.S/ eS and let x D f .u; a/. We claim that a D ex and u D xy where y is the inverse of ex 2 eS. Indeed, a D ea D eua D ex and u D ue D uexy D uxy D xy, since ux D uua D ua D x.

Section 6.2 Compact Right Topological Semigroups

89

Lemma 6.20. Let S be a semigroup and let e 2 E.S /. If Se is a minimal left ideal of S , then eS is a minimal right ideal. Proof. Let a 2 S . We have to show that e 2 eaS . Since ae 2 Se and Se is a minimal left ideal, there is b 2 S such that aeb D u 2 E.Se/. Then e D eu D eaeb 2 eaS . Lemma 6.21. Let S be a semigroup and let R and L be minimal right and left ideals of S . Then R \ L D RL is a maximal subgroup of S . Proof. Let G D RL. Then for every a 2 G, aG D aRL D RL D G and Ga D RLa D RL D G, so G is a group. Clearly G R\L. To see the converse inclusion, let b 2 R \L and let e denote the identity of G. Since b 2 L and e 2 E.L/, it follows that b D be, and consequently b 2 RL. Finally, if G 0 is any subgroup of S with the identity e, then G 0 D eG 0 R and G 0 D G 0 e L, so G 0 R \ L D G. Deﬁnition 6.22. Let G be a group, let I; ƒ be nonempty sets, and let P D .p i / be a ƒ I matrix with entries in G. The Rees matrix semigroup over the group G with ƒ I sandwich matrix P , denoted M.GI I; ƒI P /, is the set I G ƒ with the operation deﬁned by .i; a; /.j; b; / D .i; ap j b; /: It is straightforward to check that M.GI I; ƒI P / is a completely simple semigroup, the minimal right ideals being the subsets ¹i º G ƒ, where i 2 I , the minimal left ideals the subsets I G ¹º, where 2 ƒ, and the maximal groups the subsets ¹i ºG ¹º, where i 2 I and 2 ƒ. The next theorem tells us that every completely simple semigroup is isomorphic to some Rees matrix semigroup. Theorem 6.23. Let S be a completely simple semigroup. Pick e 2 E.S / such that Se is a minimal left ideal. Let I D E.Se/, ƒ D E.eS /, G D eSe, and p i D i , and deﬁne f W M.GI I; ƒI P / ! S by f .i; a; / D i a. Then f is an isomorphism. Proof. Let .i; a; / and .j; b; / be arbitrary elements of M.GI I; ƒI P /. Then f ..i; a; /.j; b; // D f .i; ajb; / D i ajb D f .i; a; /f .j; b; /; so f is a homomorphism. Since E.Se/eSeE.eS/ D E.Se/eSeeSeE.eS / D SeeS D SeS; f is surjective. To see that f is injective, let i a D jb. Then a D eae D ei ae D ejbe D ebe D b; i D i e D i aa1 D i aea1 D jaea1 D jaa1 D je D j; D e D a1 a D a1 ei a D a1 eja D a1 a D e D :

90

Chapter 6 The Semigroup ˇS

If S is a completely simple semigroup and M.GI I; ƒI P / is a Rees matrix semigroup isomorphic to S , then every maximal subgroup of S is isomorphic to G. We call G the structure group of S. The next proposition and lemma are useful in identifying the smallest ideal of subsemigroups and homomorphic images. Proposition 6.24. Let S be a semigroup and let T be a subsemigroup S. Suppose that both S and T have a smallest ideal which is a completely simple semigroup. If K.S / \ T ¤ ;, then K.T / D K.S/ \ T . Proof. Clearly, K.S/ \ T is an ideal of T , so K.T / K.S / \ T . For the reverse inclusion, let x 2 K.S/ \ T . Then Sx is a minimal left ideal of S, x 2 S x, and T x is a left ideal of T . Since K.T / is completely simple, there is an idempotent e 2 K.T / \ T x. Then Se D Sx, so x 2 Se. It follows that x D xe 2 T e K.T /. Lemma 6.25. Let S and T be semigroups and let f W S ! T be a surjective homomorphism. If S has a smallest ideal, so does T and K.T / D f .K.S //. Proof. By surjectivity, f .K.S// is a two-sided ideal of T . If K 0 is any ideal of T , then f 1 .K 0 / is an ideal of S, so contains K.S / by minimality. Thus f .K.S // K 0 , whence f .K.S// is the smallest ideal of T . We conclude this section with discussing standard preorderings on the idempotents of a semigroup. Deﬁnition 6.26. Let S be a semigroup with E.S / ¤ ;. Deﬁne the relations L , R , and on E.S / by (a) e L f if and only if ef D e, (b) e R f if and only if f e D e, and (a) e f if and only if e L f and e R f , that is, ef D f e D e. Note that e L f if and only if Se Sf . Indeed, if ef D e, then Se D Sef Sf . Conversely, if Se Sf , then e D ee D sf for some s 2 S, and consequently ef D sff D sf D e. Similarly, e R f if and only if eS f S . It is easy to see that the relations L , R , and are reﬂexive and transitive, and , in addition, is antisymmetric. Thus, L and R are preorderings on E.S / and is an ordering. Given a preordering on a set E, an element e 2 E is minimal (maximal) if for every f 2 E, f e implies e f (e f implies f e). Lemma 6.27. Let S be a semigroup and let e 2 E.S /. Then the following statements are equivalent:

Section 6.3 Hindman’s Theorem

91

(a) e is minimal with respect to L , (b) e is minimal with respect to R , and (c) e is minimal with respect to . Proof. It sufﬁces to show the equivalence of (a) and (c). (a) ) (c) Assume that e is minimal with respect to L and let f e. Since e is L -minimal, it follows from f L e that e L f , that is, e D ef . But ef D f , since f R e. Hence, e D f . (c) ) (a) Assume that e is minimal with respect to and let f L e. Denote g D ef . Then gg D ef ef D eff D ef D g 2 E.S /. Also, ge D ef e D ef D g and eg D eef D ef D g, so g e. Consequently g D e. Thus ef D e and so e L f . Hence, e is L -minimal. We say that an idempotent e of a semigroup S is minimal if e is minimal with respect to any of the preorderings L , R , or on E.S /. Thus, to say that a semigroup S contains a minimal left ideal which has an idempotent is the same as saying that S has a minimal idempotent. So a completely simple semigroup may be deﬁned as a simple one having a minimal idempotent. It is clear also that if a semigroup S has a minimal idempotent, then the minimal idempotents of S are precisely the idempotents of K.S/. An idempotent e of a semigroup S is right (left) maximal if e is maximal in E.S / with respect to R (L ). Theorem 6.28. Let S be a compact Hausdorff right topological semigroup. Then for every idempotent e 2 S, there is a right maximal idempotent f 2 S with e R f . Proof. Let P D ¹x 2 E.S/ W e R xº. Clearly P ¤ ;. It sufﬁces to show that any chain C in P has an upper bound. Then by Zorn’s Lemma, C would have a maximal element. For each p 2 C , let Tp D ¹x 2 S W xp D pº. If x; y 2 Tp , then xyp D xp D p, so xy 2 Tp . Also Tp D p1 .p/. Thus, Tp is a closed subsemigroup of S. Furthermore, if p R q and x 2 Tq , then xp T D xqp D qp D p, so x 2 Tp and, consequently, Tp Tq . It follows that T D p2C Tp is a closed T subsemigroup of S. Now by Theorem 6.12, there is an idempotent q 2 T . Since p2C Tp D ¹x 2 S W xp D p for all p 2 C º, q is an upper bound of C .

6.3

Hindman’s Theorem

In this section we prove Hindman’s Theorem using the semigroup ˇS. Lemma 6.29. Let S be a semigroup and let p 2 ˇS. Then p is an idempotent if and only if for every A 2 p, one has ¹x 2 S W x 1 A 2 pº 2 p. Proof. It is immediate from Lemma 6.6.

92

Chapter 6 The Semigroup ˇS

Proposition 6.30. Let S be a semigroup and let p be an idempotent in S . Then there is a left invariant topology T on S 1 in which for each s 2 S 1 , ¹sA [ ¹sº W A 2 pº is a neighborhood base at s. Furthermore, if S is cancellative, then T is Hausdorff. Proof. Let N be the ﬁlter on S 1 with a base consisting of subsets A [ ¹1º where A 2 p. Then for every U 2 N , 1 2 U and and by Lemma 6.29, ¹x 2 S W x 1 U 2 N º 2 N . Hence by Theorem 4.3, there is a left invariant topology T on S 1 in which for each s 2 S 1 , sN is a neighborhood base at s. Now suppose that S is cancellative. Then (i) ap is a nonprincipal ultraﬁlter for every a 2 S 1 , and (ii) ap ¤ bp for all distinct a; b 2 S 1 . It follows that T is Hausdorff. Lemma 6.31. Let S be a left topological semigroup with identity and let U be an open subset of S such that 1 2 cl U . Then there is a sequence .xn /1 nD1 in S such that FP..xn /1 / U . nD1 Proof. Pick x1 2 U . Fix n 2 N and suppose we have chosen a sequence .xi /niD1 such that FP..xi /niD1 / U . Let F D FP..xi /niD1 /. Then there is a neighborhood V of 1 such that F V U . Pick xnC1 2 U \ V . Theorem 6.32. Let S be a semigroup, let p be an idempotent in ˇS, and let A 2 p. 1 Then there is a sequence .xn /1 nD1 in S such that FP..xn /nD1 / A. Proof. The statement is obvious if p 2 S . Let p 2 S . By Proposition 6.30, there is a left invariant topology T on S 1 in which for each s 2 S 1 , ¹sA [ ¹sº W A 2 pº is a neighborhood base at s. Let U D intT A. Note that U D A \ ¹x 2 S W x 1 A 2 pº. We have that U is an open subset of .S 1 ; T /, 1 2 clT U , and U A. Then apply Lemma 6.31. S Corollary 6.33. Let S be a semigroup, let r 2 N, and let S D riD1 Ai . Then there 1 exist i 2 ¹1; 2; : : : ; rº and a sequence .xn /1 nD1 in S such that FS..xn /nD1 / Ai . Proof. Pick by Theorem 6.12 an idempotent p 2 ˇS and pick i 2 ¹1; 2; : : : ; rº such that Ai 2 p. Then apply Theorem 6.32. As a special case of Corollary 6.33 we obtain Hindman’s Theorem. S Corollary 6.34 (Hindman’s Theorem). Let r 2 N and let N D riD1 Ai . Then there 1 exist i 2 ¹1; 2; : : : ; rº and a sequence .xn /1 nD1 in N such that FS..xn /nD1 / Ai . Corollary 6.34 is strong enough to derive all other versions of Hindman’s Theorem including Theorem 1.41.

93

Section 6.3 Hindman’s Theorem

Deﬁnition 6.35. For every x 2 N, deﬁne supp2 .x/ 2 Pf .!/ by X xD 2i i2supp2 .x/

and let 2 .x/ D min supp2 .x/

and

2 .x/ D max supp2 .x/:

In other words, supp2 .x/ is the set of indexes of nonzero digits in the binary expansion of x, and 2 .x/ and 2 .x/ are the indexes of the ﬁrst and the last nonzero digit, respectively. Equivalently, 2 .x/ D max¹i < ! W 2i jxº and

2 .x/ D max¹i < ! W 2i xº:

2 .x/ can be deﬁned also as n < ! such that x 2n

.mod 2nC1 /:

Note that we have already used the function 2 .x/ in the proof of Theorem 5.7. Lemma 6.36. Let .xn /1 nD1 be a sequence in N. Then there is a sum subsystem 1 .yn /1 of .x / such that 2 .yn / < 2 .ynC1 / for every n 2 N. n nD1 nD1 Proof. It sufﬁces to show that for every m; k < !, there is F 2 Pf .N/ such that P min F > m and 2k j n2F xn . We proceed by induction on k. If k D 0, put F D ¹m C 1º. Now assume the statement holds for some k. Then there are F1 ; F2 2 Pf .N/ such that m < min F1 , P P max F1 < min F2 , and 2k j n2Fi xn for each i 2 ¹1; 2º. If 2kC1 j n2Fi xn for P P some j 2 ¹1; 2º, put F D Fj . Otherwise 2kC1 j n2F1 xn C n2F2 xn , so put F D F1 [ F2 . Sr Corollary 6.37. Let r 2 N and let Pf .N/ D iD1 Ai . Then there exist i 2 ¹1; 2; : : : ; rº and a sequence .Gn /1 in P .N/ such that FU..Gn /1 f nD1 nD1 / Ai and for each n 2 N, max Gn < min GnC1 . Proof. Consider the bijection N 3 x 7! supp2 .x/ 2 Pf .!/ and apply Corollary 6.34 and Lemma 6.36. Corollary 6.37 gives us, in turn, the following. 1 Corollary 6.38. Sr Let S be a semigroup, let .xn /nD1 be a sequence in S, let r 2 N, and let S D iD1 Ai . Then there exist i 2 ¹1; 2; : : : ; rº and a product subsystem 1 1 .yn /1 nD1 of .xn /nD1 such that FP..yn /nD1 / Ai .

The next proposition tells us that the relationship between idempotents and ﬁnite products is even more intimate than indicated by Theorem 6.32.

94

Chapter 6 The Semigroup ˇS

Proposition 6.39. Let S be a semigroup and let .xn /1 nD1 be a sequence in S. Then T1 1 / is a closed subsemigroup of ˇS. Consequently, there is an idemFP..x / n nDm mD1 potent p 2 ˇS such that FP..xn /1 nDm / 2 p for every m 2 N. Before proving Proposition 6.39 we establish the following simple general fact. Lemma 6.40. Let S be a semigroup and let F be a ﬁlter on S . Suppose that for every A 2 F , ¹x 2 S W x 1 A 2 F º 2 F . Then F is a closed subsemigroup of ˇS. Proof. Clearly, F is a closed subset of ˇS. To see that it is a subsemigroup, let p; q 2 F and let A 2 F . We have to show that A 2 pq. Let B D ¹x 2 S W x 1 A 2 F º, and for every x 2 B, let Cx D x 1 A. Then B 2 F p, Cx 2 F q, and S x2B xCx A. Hence, A 2 pq. 1 Proof of Proposition 6.39. Let m Q2 N and let x 2 FP..xn /nDm /. Pick F 2 Pf .N/ with min F m such that x D n2F xn . Let k D max F C 1. Then 1 x FP..xn /1 nDk / FP..xn /nDm /:

T 1 Consequently, by Lemma 6.40, 1 mD1 FP..xn /nDm / is a closed subsemigroup of ˇS. For the second part, apply Theorem 6.12.

6.4

Ultraﬁlters from K.ˇS /

In this section we characterize ultraﬁlters from K.ˇS /. Deﬁnition 6.41. Let S be a semigroup. (a) A subset A S is syndetic if there is a ﬁnite F S such that F 1 A D S. (b) Let F and G be ﬁlters on S . A subset A S is .F ; G /-syndetic if for every V 2 F , there is a ﬁnite F V such that F 1 A 2 G . If F D G , we say F -syndetic instead of .F ; G /-syndetic. Note that if F G and A is either F -syndetic or G -syndetic, then A is also .F ; G /-syndetic. Lemma 6.42. Let T be a closed subsemigroup of ˇS, let L be a minimal left ideal of T , let F and G be the ﬁlters on S such that F D T and G D L, and let A S . Then the following statements are equivalent: (1) A \ L ¤ ;, (2) A is G -syndetic, and (3) A is .F ; G /-syndetic.

Section 6.4 Ultraﬁlters from K.ˇS/

95

Proof. .1/ ) .2/ Pick p 2 A \ L. Then for every q 2 L, one has p 2 Lq. Now to show that A is G -syndetic, let V 2 G . For every q 2 L, there is r 2 L such that p D rq, consequently, there is x 2 V such that xq 2 A, and so q 2 x 1 A. Thus, the sets of the form x 1 A, where x 2 V , cover the compact L. Hence, there is a ﬁnite F V such that L F 1 A, and so F 1 A 2 G . .2/ ) .3/ is obvious. .3/ ) .1/ Pick q 2 L. For every V 2 F , there is a ﬁnite F V such that F 1 A 2 G , and so F 1 A 2 q. It follows that for every V 2 F , there is xV 2 V such that A 2 xV q. Pick r 2 T \ c`ˇS ¹xV W V 2 G º. Then A 2 rq and rq 2 L. Hence, A \ L ¤ ;. The next theorem characterizes ultraﬁlters from K.T /. Theorem 6.43. Let T be a closed subsemigroup of ˇS , let F be the ﬁlter on S such that F D T , and let p 2 T . Then p 2 K.T / if and only if for every A 2 p, ¹x 2 S W x 1 A 2 pº is F -syndetic. Proof. Suppose that p 2 K.T /. Let L D Tp and let G be the ﬁlter on S such that G D L. Now let A 2 p, let B D ¹x 2 S W x 1 A 2 pº, and let V 2 F . By Lemma 6.42, A is G -syndetic, so there is F V such that F 1 A 2 G . Since L D Tp, there is W 2 F such that Wp F 1 A. We claim that W F 1 B. Indeed, let y 2 W . Then there is x 2 F such that yp 2 x 1 A. It follows that .xy/1 A 2 p. Hence, xy 2 B, and then y 2 x 1 B. Conversely, suppose that p … K.T /. Pick q 2 K.T /. Then p … T qp, since T qp K.T /. It follows that there is A 2 p such that A \ T qp D ;. Now let B D ¹x 2 S W x 1 A 2 pº. We claim that B is not F -syndetic. To show this, pick a minimal left ideal L of T contained in Tp and let G be the ﬁlter on S such that G D L. Assume on the contrary that B is F -syndetic. Then B is also .F ; G /syndetic. Hence by Lemma 6.42, B \ L ¤ ;. Consequently, B 2 rq for some r 2 T , and so ¹x 2 S W x 1 A 2 pº 2 rq. But then A 2 rqp, which is a contradiction. As a consequence we obtain the following characterization of ultraﬁlters from K.ˇS /. Corollary 6.44. Let p 2 ˇS. Then p 2 K.ˇS / if and only if for every A 2 p, ¹x 2 S W x 1 A 2 pº is syndetic.

References The fact that the operation of a discrete semigroup S can be extended to ˇS was implicitly established by M. Day [15] using a multiplication of the second conjugate of a Banach algebra, in this case l1 .S /, ﬁrst introduced by R. Arens [2] for arbitrary

96

Chapter 6 The Semigroup ˇS

Banach algebras. P. Civin and B. Yood [8] explicitly stated that if S is a discrete group, then the above operation produced an operation on ˇS, viewed as a subspace of that second dual. R. Ellis [21] carried out the extension in ˇS viewed as a space of ultraﬁlters, again assuming that S is a group. Theorem 6.12 was proved by K. Numakura [52] for topological semigroups and by R. Ellis [20] in the general case. Theorem 6.23 is a special case of the ReesSuschkewitsch Theorem proved by A. Suschkewitsch [73] for ﬁnite semigroups and by D. Rees [63] in the general case. Theorem 6.28 is due to W. Ruppert [67]. For more information about compact right topological semigroups, including references, see [67]. An introduction to this topic can be found also in [39]. Corollary 6.34 is Hindman’s Theorem [35] known also as the Finite Sums Theorem. The proof based on the semigroup ˇN is due to F. Galvin and S. Glazer. For other combinatorial applications of ˇS see [37]. The exposition of Section 6.4 is based on the treatment in [70].

Chapter 7

Ultraﬁlter Semigroups

Given a T1 left topological group .G; T /, the set Ult.T / of all nonprincipal ultraﬁlters on G converging to 1 in T is a closed subsemigroup of ˇG called the ultraﬁlter semigroup of T . In this chapter we study the relationship between algebraic properties of Ult.T / and topological properties of .G; T /. Not every closed subsemigroup of G is the ultraﬁlter semigroup of a left invariant topology on G. However, every ﬁnite subsemigroup is. A special attention is paid to the question when a closed subsemigroup of G is the ultraﬁlter semigroup of a regular left invariant topology. We conclude by showing how to construct homomorphisms of Ult.T /.

7.1

The Semigroup Ult.T /

Lemma 7.1. Let .S; T / be a left topological semigroup with identity and let N be the neighborhood ﬁlter of 1. Then (1) N is a closed subsemigroup of ˇS, (2) for every open subset U of .S; T /, U N U , and (3) if T satisﬁes the T1 separation axiom, then N n ¹1º D ¹p 2 S W p converges to 1 in T º is a closed subsemigroup. Proof. (1) follows from Theorem 4.3 and Lemma 6.40. (2) Let p 2 U and q 2S N . Since U is open, for every x 2 U , there is Vx 2 N such that xVx U . Then x2U xVx U . Since U 2 p and Vx 2 q, it follows that U 2 pq, so pq 2 U . T (3) Obviously, N n ¹1º is a closed subset of ˇS. Since T is a T1 -topology, N D ¹1º. It follows that N n¹1º S , and so N n¹1º D ¹p 2 S W p converges to 1 in T º. To see that N n ¹1º is a subsemigroup, let p; q 2 N n ¹1º. We have to show that for every U 2 N , pq 2 U n ¹1º. Clearly, one may suppose that U is open. Then U n ¹1º is also open, because T is a T1 -topology. Since p 2 U n ¹1º, we obtain by (2) that pq 2 U n ¹1º D U n ¹1º. Deﬁnition 7.2. Let .S; T / be a T1 left topological semigroup with identity. Deﬁne Ult.T / ˇS by Ult.T / D ¹p 2 S W p converges to 1 in T º:

98

Chapter 7 Ultraﬁlter Semigroups

By Lemma 7.1, Ult.T / is a closed subsemigroup of ˇS (if nonempty). We refer to Ult.T / as to the ultraﬁlter semigroup of T (or .S; T /). Recall that a ﬁlter on a space is called open (closed) if it has a base consisting of open (closed) sets. Lemma 7.3. Let .G; T / be a T1 left topological group, let F be a ﬁlter on G, and let Q D Ult.T /. (1) If F is open, then F Q F . (2) If F is closed, then pQ \ F D ; for every p 2 ˇG n F . In the case where Q is ﬁnite, the converses of the statements (1)–(2) also hold. Proof. (1) For every open U 2 F , F Q U Q, and by Lemma 7.1, U Q U , so F Q U . It follows that F Q F . Conversely, suppose that F Q F and Q is ﬁnite. To show that F is open, let U 2 F . For every p 2 F and q 2 Q, pick Ap;q 2 p such that Ap;q U and Ap;q q U . Let [ \ Ap;q : V D p2F q2Q

Then V 2 F , V U , and V Q U . It follows that V int U , and so int U 2 F . To see this, for every x 2 V and q 2 Q, pick Bx;q 2 q such that xBx;q U , and let [ Wx D ¹xº [ Bx;q : q2Q

Then Wx is a neighborhood of 1 and xWx U . (2) Since F is closed, for every p 2 ˇS n F , there is an open U 2 p such that U \ Q D ;. By Lemma 7.1, U Q U . It follows that pQ \ Q D ;. Conversely, suppose that pQ \ F D ; for every p 2 ˇG n F and that Q is ﬁnite. To show that F is closed, assume the contrary. Then there is U 2 F such that for every V 2 F , .cl V / n U ¤ ;. Since Q is ﬁnite, it follows that there is q 2 Q and, for every V 2 F , xV 2 G n U such that V 2 xV q. Let p be an ultraﬁlter on G extending the family of subsets XV D ¹xW W V W 2 F º where V 2 F . Then p 2 G n U and V 2 pq for every V 2 F . Consequently, p 2 ˇG n F and pq 2 F , a contradiction. Deﬁnition 7.4. For every ﬁlter F on a space X , denote by int F (respectively cl F ) the largest open (closed) ﬁlter on X , possibly improper, containing (contained in) F . Corollary 7.5. Let .G; T / be a T1 left topological group, let F be a ﬁlter on .G; T /, and let Q D Ult.T /. Suppose that Q is ﬁnite. Then

Section 7.1 The Semigroup Ult.T /

99

(1) int F D ¹p 2 F W pQ F º, and (2) cl F D F [ ¹p 2 ˇG W pQ \ F ¤ ;º. Proof. (1) By Lemma 7.3, int F Q int F F , so int F ¹p 2 F W pQ F º. To see the converse inclusion, note that ¹p 2 F W pQ F º is closed. Let G denote the ﬁlter on G such that G D ¹p 2 F W pQ F º. For every p 2 G and q 2 Q, one has pqQ pQ F , so G Q G . Now, since Q is ﬁnite, applying Lemma 7.3 gives us that G is open. Hence G int F . (2) If pQ \ F ¤ ; for some p 2 ˇG, then pQ \ cl F ¤ ;, and so p 2 cl F by Lemma 7.3. Consequently, F [ ¹p 2 ˇG W pQ \ F ¤ ;º cl F . To see the converse inclusion, note that ¹p 2 ˇG W pQ \ F ¤ ;º is closed, since Q is ﬁnite. Let G denote the ﬁlter on G such that G D F [ ¹p 2 ˇG W pQ \ F ¤ ;º. We claim for every p 2 ˇG n G , pQ \ G ¤ ;. Indeed, otherwise there exists q 2 Q such that pqQ \ F ¤ ;. It then follows that pQ \ F ¤ ;, and so p 2 G , which is a contradiction. Now, since Q is ﬁnite, applying Lemma 7.3 gives us that G is closed. Hence cl F G . Recall that an ultraﬁlter p on a space is dense if for every A 2 p, int cl A ¤ ;. Lemma 7.6. Let .G; T / be a T1 left topological group. Then the set of all dense ultraﬁlters on .G; T / converging to 1 is a closed ideal of Ult.T /. Proof. Let F be the ﬁlter on .G; T / with a base consisting of subsets of the form U n Y where U is a neighborhood of 1 and Y is a nowhere dense subset of .G; T /. Then by Proposition 3.9, F is the set of all dense ultraﬁlters on .G; T / converging to 1. Since F is open, F is a right ideal of Ult.T /, by Lemma 7.3. To see that F is a left ideal, let p 2 Ult.T /, q 2 F , and A 2 pq. Then there is x 2 G and B 2 q such that xB A. Since q is dense, int cl B ¤ ;. But x.int cl B/ int.x.cl B// and int.x.cl B// int cl.xB/. It follows that int cl A ¤ ;. Hence pq 2 F . The next proposition shows how the ultraﬁlter semigroup of a left topological group reﬂects topological properties of the group itself. Proposition 7.7. Let .G; T / be a T1 left topological group and let Q D Ult.T /. (1) If Q has only one minimal right ideal, then T is extremally disconnected. (2) If T is irresolvable, then K.Q/ is a left zero semigroup. (3) If T is n-irresolvable, then a minimal right ideal of Q consists of < n elements. (4) If p 2 K.Q/, then p is dense. (5) If Q D K.Q/, then T is nodec. In the case where Q is ﬁnite, the converses of the statements (1)–(5) also hold.

100

Chapter 7 Ultraﬁlter Semigroups

Proof. (1) Suppose that T is not extremally disconnected. Then there are two disjoint open subsets U and V of .G; T / such that 1 2 .cl U / \ .cl V /. But then by Lemma 7.1, U \ Q and V \ Q are two disjoint right ideals of Q. Conversely, suppose that there are two disjoint right ideals R and J of Q and that Q is ﬁnite. Let F and G denote ﬁlters on G such that F D R and G D J . Then, since Q is ﬁnite, both F and G are open by Lemma 7.3, so there are disjoint open U 2 F and V 2 G . But then 1 2 .cl U / \ .cl V /. Hence T is not extremally disconnected. (2) is a special case of (3). (3) Suppose that T is n-irresolvable. Then by Theorem 3.31, there is an open ﬁlter F on .G; T / converging to 1 with jF j < n. Then by Corollary 7.3, F n ¹1º is a right ideal of Q. Hence, a minimal right ideal of Q consists of jF j elements. Conversely, let R be a minimal right ideal of Q and suppose that jRj < n and Q is ﬁnite. Let F denote the ﬁlter on G such that F D R. Then by Lemma 7.3, F is open. Hence by Theorem 3.31, T is n-irresolvable. (4) By Lemma 7.6, the set of all dense ultraﬁlters from Q is an ideal of Q. It follows that every ultraﬁlter from K.Q/ is dense. Conversely, suppose that p … K.Q/ and Q is ﬁnite. Then by Corollary 7.5, cl p D ¹pº [ ¹q 2 ˇG W p 2 qQº and int cl p D ¹q 2 cl p W qQ cl pº. Since p … K.Q/, p … qQ for every q 2 K.Q/. Consequently, cl p \ K.Q/ D ;. On the other hand, for every q 2 cl p, one has qQ \ K.Q/ ¤ ;. Indeed, this is certainly true if q 2 Q. Otherwise p 2 qQ. Then pQ qQ and therefore qQ \ K.Q/ ¤ ;. It follows that for every q 2 cl p, one has qQ ª cl p. Hence, int cl p D ; and so p is nowhere dense. (5) If Q D K.Q/, then by (4) every ultraﬁlter from Q is dense. It follows that T is nodec. If Q ¤ K.Q/ and Q is ﬁnite, then by (4) any ultraﬁlter from Q n K.Q/ is nowhere dense, and so T is not nodec. We now show that every ﬁnite semigroup in G is the ultraﬁlter semigroup of a left invariant topology. Proposition 7.8. Let S be a semigroup with identity, let Q be a ﬁnite semigroup in S , and let F be the ﬁlter on S such that F D Q. Then there is a left invariant topology T on S in which for each s 2 S, ¹sA [ ¹sº W A 2 F º is a neighborhood base at s. If S is left cancellative, then T is a T1 -topology and Ult.T / D Q. Proof. Let N be the ﬁlter on S such that N D Q [ ¹1º. We claim that T (i) N D ¹1º, and (ii) for every U 2 N , ¹x 2 S W x 1 U 2 N º 2 N . Statement (i) is obvious. To show (ii), let U 2 N . For S every Tp; q 2 Q, pick Ap;q 2 p such that Ap;q U and Ap;q q U . Put V D p2Q q2F Ap;q [ ¹1º. Then V 2 N , V U and VQ S U . Now let x 2 V . For every q 2 Q, pick Bx;q 2 q such that xBx;q U . Put Wx D q2Q Bx;q [ ¹1º. Then Wx 2 N and xWx U .

Section 7.2 Regularity

101

It follows from (i)–(ii) and Theorem 4.3 that there is a left invariant topology T on S in which for each a 2 S, aN is a neighborhood base at a. The next example shows that not every closed subsemigroup of G is the ultraﬁlter semigroup of a left invariant topology. Example 7.9. Let .G; T0 / be a regular left topological group and suppose that there is a one-to-one sequence .xn /n

7.2

Regularity

Deﬁnition 7.11. Let G be a group and let Q be a closed subsemigroup of ˇG. We say that Q is left saturated if Q G and for every p 2 ˇG n .Q [ ¹1º/, one has pQ \ Q D ;.

102

Chapter 7 Ultraﬁlter Semigroups

Lemma 7.12. Let .G; T / be a regular left topological group and let Q D Ult.T /. Then Q is left saturated. Proof. Let N be the neighborhood ﬁlter of 1 in T . Since T is regular, N is closed. Then by Lemma 7.3, for every p 2 ˇG n N , pQ \ N D ;. It follows that for every p 2 ˇG n .Q [ ¹1º/, pQ \ Q D ;. Hence, Q is left saturated. Theorem 7.13. Let G be a group and let Q be a closed subsemigroup of G . Suppose that Q is left saturated and has a ﬁnite left ideal. Then there is a regular left invariant topology T on G such that Ult.T / D Q. Proof. Let N be the ﬁlter on G such that N D Q [ ¹1º. We ﬁrst show that T (i) N D ¹1º, and (ii) for every U 2 N , ¹x 2 G W x 1 U 2 N º 2 N . Statement (i) is obvious. To see (ii), let L be a ﬁnite left ideal of Q and let C D G n U . Since Q is left saturated, C Q \TQ D ;. For every q 2 L, choose Wq 2 N such that C q \ Wq D ;, and put W D q2L Wq . Then W 2 N , as L is ﬁnite, and 1 L L. For every q 2 L, .C L/ \ W D ;. Next, since L is a left ideal of Q, QT choose Vq 2 N such that Vq q W , and put V D q2L Vq . Then V 2 N and V L W . We claim that for all x 2 V , x 1 U 2 N . Indeed, otherwise xp 2 C for some x 2 V and p 2 Q. Take any q 2 L. Then, on the one hand, xpq D xp q 2 C L ˇG n W ; and on the other hand, xpq D x pq 2 V L W ; which is a contradiction. It follows from (i)–(ii) and Theorem 4.3 that there is a left invariant T1 -topology T on G for which N is the neighborhood ﬁlter of 1, and so Ult.T / D Q. We now show that T is regular. Assume the contrary. Then there is a neighborhood U of 1 such that for every neighborhood V of 1, .cl V / n U ¤ ;. For every open neighborhood V of 1, choose xV 2 .cl V / n U and pV 2 Q such that V 2 xV pV . Pick any q 2 L. By Lemma 7.1, one has V 2 xV pV q, and pV q 2 L because L is a left ideal. Since L is ﬁnite, it follows that there exist q 2 L and p 2 G n U such that for every neighborhood V of 1, V 2 pq, and so pq 2 Q. We have obtained that Q is not left saturated, which is a contradiction. Deﬁnition 7.14. Given a group G and p 2 G , C.p/ D ¹x 2 G W xp D pº

and

C 1 .p/ D C.p/ [ ¹1º:

Note that C.p/ is a closed subset of G and p 2 C.p/ if and only if p is an idempotent.

103

Section 7.2 Regularity

Lemma 7.15. C.p/ is a closed left saturated subsemigroup of ˇG (if nonempty). Proof. To see that C.p/ is a subsemigroup, let x; y 2 C.p/. Then xy 2 G by Lemma 6.8 and xyp D xp D p, so xy 2 C.p/. To see that C.p/ is left saturated, let xq D r for some x 2 ˇG and q; r 2 C.p/. Then xqp D rp, and so xp D p. If x 2 G, this equality implies that x D 1 by Corollary 6.11. Hence x 2 C.p/ [ ¹1º. From Lemma 7.15 we obtain that Corollary 7.16. C.p/ D ¹x 2 ˇG n ¹1º W xp D pº and C 1 .p/ D ¹x 2 ˇG W xp D pº. Now we deduce from Theorem 7.13 and Lemma 7.15 the following result. Theorem 7.17. Let G be a group and let p 2 G be an idempotent. Then (1) there is a regular left invariant topology T on G such that Ult.T / D C.p/, (2) T is the largest regular left invariant topology on G in which p converges to 1, and (3) T is extremally disconnected. Proof. By Lemma 7.15, C.p/ is a closed left saturated subsemigroup of ˇG. Clearly, ¹pº is a left ideal of C.p/. Then applying by Theorem 7.13 gives us (1). To see (2), let T 0 be any regular left invariant topology on G in which p converges to 1, let Q D Ult.T 0 /, and let x 2 C.p/. We have that xp D p, p 2 Q and x 2 G . Hence by Lemma 7.12, x 2 Q. To see (3), notice that, since ¹pº is a minimal left ideal of C.p/, there is only one minimal right ideal of C.p/. Hence by Proposition 7.7, T is extremally disconnected. Since a regular extremally disconnected space is zero-dimensional, we obtain from Theorem 7.17 that for every idempotent p 2 G , there is a zero-dimensional Hausdorff left invariant topology T on G such that Ult.T / D C.p/. The next theorem tells us that this is true for every p 2 G . Theorem 7.18. For every group G and for every p 2 G , there is a zero-dimensional Hausdorff left invariant topology T on G such that Ult.T / D C.p/. The proof of Theorem 7.18 involves the following notion. Deﬁnition 7.19. Let S be a semigroup, A S, and p 2 ˇS. Then Ap0 D ¹x 2 S W A 2 xpº:

104

Chapter 7 Ultraﬁlter Semigroups

The next lemma contains some simple properties of this notion. Lemma 7.20. Let S be a semigroup, A; B S, and p; q 2 ˇS. Then (1) .A \ B/p0 D Ap0 \ Bp0 , (2) A 2 pq if and only if A0q 2 p, and (3) .Ap0 /0q D A0qp . Proof. (1) Let x 2 G. Then x 2 .A \ B/p0 , A \ B 2 xp , A 2 xp and B 2 xp , x 2 Ap0 and x 2 Bp0 , x 2 Ap0 \ Bp0 : (2) is Lemma 6.6 (ii). (3) Let x 2 G. Then x 2 .Ap0 /0q , Ap0 2 xq , A 2 xqp

by .2/

, x 2 A0qp : Proof of Theorem 7.18. Let B D ¹Ap0 W A 2 pº. Since 1 2 Ap0 for every A 2 p and Ap0 \ Bp0 D .A \ B/p0 , B is a ﬁlter base on G. Now we show that B possesses the following properties: (i) B D C 1 .p/, (ii) for every U 2 B and for every x 2 U , there is V 2 B such that xV U , and (iii) for every U 2 B and for every x 2 GnU , there is V 2 B such that xV \U D ;. To see (i), let q 2 ˇG. Then q 2 B , Ap0 2 q

for every A 2 p

, A 2 qp

for every A 2 p

, qp D p: To see (ii) and (iii), it sufﬁces to show that for every U 2 B and for every q 2 B, D U . To this end, pick A 2 p such that U D Ap0 . Then

Uq0

Uq0 D .Ap0 /0q D A0qp D Ap0 D U: It follows from (i) and (ii) that there is a left invariant T1 -topology T on G such that Ult.T / D C.p/, and by (iii), T is zero-dimensional.

Section 7.3 Homomorphisms

105

We conclude this section by characterizing subsemigroups of a ﬁnite left saturated subsemigroup of ˇG which determine locally regular left invariant topologies. A space is locally regular if every point has a neighborhood which is a regular subspace. Proposition 7.21. Let G be a group, let Q be a ﬁnite left saturated subsemigroup of G , let S be a subsemigroup of Q, and let T be a left invariant topology on G with Ult.T / D S. Then T is locally regular if and only if for every q 2 Q n S, qS \ S ¤ ; implies S q \ S D ;. Proof. Suppose that T is locally regular and let q 2 Q n S be such that qS \ S ¤ ;. We have to show that Sq \ S D ;. Choose a regular open neighborhood X of 1 2 G in T . It sufﬁces to show that Xq \ X D ;, as this implies Xq \ X D ; and then S q \ S D ;. Assume on the contrary that xq 2 X for some x 2 X . Since q … S [ ¹1º, xq does not converge to x, so there is a neighborhood U of x 2 X such that U … xq. Since X is regular, U can be chosen to be closed. We have that X n U 2 xq and X n U is open. Then by Lemma 7.1, xqS X n U . It follows that xqS \ xS D ;, and consequently, qS \ S D ;, a contradiction. Conversely, let F D ¹q 2 Q n S W qS \ S ¤ ;º and suppose that for each q 2 F , S q \ S D ;. Then for each Tq 2 F , there is a neighborhood Xq of 1 in T such that Xq q \ Xq D ;. Put X D q2F Xq . Since F is ﬁnite, X is a neighborhood of 1 in T , and for each q 2 F , one has Xq \ X D ;. We claim that X is regular. Assume the contrary. Then there is x 2 X and a neighborhood U of x such that for every neighborhood V of x, clX .V / n U ¤ ;. Since S is ﬁnite, it follows that there is p 2 S , and for every neighborhood V of x, there is yV 2 X n U such that V 2 yV p. Let r be an ultraﬁlter on G extending the family of subsets YV D ¹yW W W is a neighborhood of x contained in V º; where V runs over neighborhoods of x. Then r 2 X n U and rp 2 xS , so x 1 rp 2 S. Put q D x 1 r. We have that (a) qp 2 S and q ¤ 1, and (b) r D xq. Since Q is left saturated, it follows from (a) that q 2 F . It is clear that xq 2 Xq, and (b) gives us that xq 2 X. Hence Xq \ X ¤ ;, a contradiction.

7.3

Homomorphisms

Constructing homomorphisms of ultraﬁlter semigroups is based on the following lemma. Lemma 7.22. Let S be a semigroup, let F be a ﬁlter on S , and let X 2 F . Let T be a compact Hausdorff right topological semigroup and let f W X ! T . Assume that

106

Chapter 7 Ultraﬁlter Semigroups

(1) for every x 2 X , there is Ux 2 F such that f .xy/ D f .x/f .y/ for all y 2 Ux , and (2) f .X / ƒ.T /. Then for every p 2 X and q 2 F , one has f .pq/ D f .p/f .q/, where f W X ! T is the continuous extension of f . Proof. For every x 2 X , one has f .xq/ D f .

lim

Ux 3y!q

xy/

D lim f .xy/ y!q

D lim f .x/f .y/ y!q

by (1)

D f .x/ lim f .y/ by (2) y!q

D f .x/f .q/: Then f .pq/ D f . lim xq/ X3x!p

D lim f .xq/ x!p

D lim f .x/f .q/ x!p

D . lim f .x//f .q/ x!p

D f .p/f .q/: L Deﬁnition 7.23. Let be an inﬁnite cardinal and let H D Z2 . For every ˛ < , let H˛ D ¹x 2 H W x. / D 0 for each < ˛º, and let T0 denote the group topology on H with a neighborhood base at 0 consisting of subgroups H˛ , where ˛ < . Deﬁne the semigroup H ˇH by H D Ult.T0 /. If D !, we write H instead of H! . The next theorem tells us that the semigroup H admits a continuous homomorphism onto any compact Hausdorff right topological semigroup containing a dense subset of cardinality within the topological center. Theorem 7.24. Let be an inﬁnite cardinal and let T be a compact Hausdorff right topological semigroup. Assume that there is a dense subset A T such that jAj and A ƒ.T /. Then there is a continuous surjective homomorphism W H ! T .

107

Section 7.3 Homomorphisms

Proof. For every ˛ < , let e˛ denote the element of H with supp.e˛ / D ¹˛º. Then every x 2 H n ¹0º can be uniquely written in the form x D e˛1 C C e˛n where n 2 N and ˛1 < < ˛n < . Pick a surjection f0 W ¹e˛ W ˛ < º ! A such that for each a 2 A, one has jf01 .a/j D . Extend f0 to a mapping f W H ! T by f .x/ D f0 .e˛1 / f0 .e˛n / where x D e˛1 C C e˛n and ˛1 < < ˛n . (As f .0/ pick any element of T .) Deﬁne W H ! T by D f jH . Now let x 2 H n ¹0º and let t D max supp.x/ C 1. We claim that for every y 2 H t n ¹0º, one has f .x C y/ D f .x/f .y/. Indeed, let x D e˛1 C C e˛n where ˛1 < < ˛n and let y D eˇ1 C C eˇm where ˇ1 < < ˇm . Then x C y D e˛1 C C e˛n C eˇ1 C C eˇm and ˛1 < < ˛n < ˇ1 < < ˇm . Consequently, f .x C y/ D f .e˛1 / f .e˛n /f .eˇ1 / f .eˇm / D f .x/f .y/: It follows from this and Lemma 7.22 that is a homomorphism. To see that is surjective, it sufﬁces to show that A .H /, because A T is dense and is continuous. Let a 2 A. Pick p 2 U.H / such that f01 .a/ 2 p. Then p 2 H and f .p/ D a. Now we are going to show that for every cancellative semigroup S of cardinality , there is a zero-dimensional Hausdorff left invariant topology T on S 1 such that Ult.T / is topologically and algebraically isomorphic to H . Lemma 7.25. Let S be an inﬁnite cancellative semigroup with identity and let jSj D . Then there are two -sequences .x˛ /1˛< and .y˛ /˛< in S with y0 D 1 such that every element of S is uniquely representable in the form y˛0 x˛1 x˛n where n < ! and ˛0 < ˛1 < < ˛n < . Proof. Enumerate S as ¹s˛ W ˛ < º. Put y0 D 1. Fix 0 < < and suppose that we have constructed .x˛ /1˛< and .y˛ /˛< such that all products y˛0 x˛1 x˛n , where n < ! and ˛0 < ˛1 < < ˛n < , are different. Pick as y the ﬁrst element in the sequence .s˛ /˛< not belonging to the subset S D ¹y˛0 x˛1 x˛n W n < ! and ˛0 < ˛1 < < ˛n < º: S Then pick x 2 S n .S1 S /. (Here, S1 S D x2S x 1 S .) This can be done because jS1 S j jS j2 < . Then whenever n < ! and ˛0 < ˛1 < < ˛n D , one has y˛0 x˛1 x˛n … S . Also if y˛0 x˛1 x˛n and yˇ0 xˇ1 xˇm are different elements of S , the elements y˛0 x˛1 x˛n x and yˇ0 xˇ1 xˇm x are different as well.

108

Chapter 7 Ultraﬁlter Semigroups

Theorem 7.26. Let S be an inﬁnite cancellative semigroup with identity and let jS j D . Then there is a zero-dimensional Hausdorff left invariant topology T on S such that Ult.T / U.S/ and Ult.T / is topologically and algebraically isomorphic to H . Proof. Let .x˛ /1˛< and .y˛ /˛< be sequences guaranteed by Lemma 7.25. For every ˛ 2 Œ1; /, deﬁne B˛ S by B˛ D ¹y0 x˛1 x˛n W n < !; ˛ ˛1 < < ˛n < º: Deﬁne also W S ! by .y˛0 x˛1 x˛n / D ˛n where n < and ˛0 < ˛1 < : : : < ˛n < . It is easy to see that the subsets B˛ possess the following properties: T (i) .B˛ /1˛< is a decreasing sequence of subsets of S with 1˛< B˛ D ¹1º, (ii) for every ˛ 2 Œ1; / and x 2 B˛ , one has xB.x/C1 B˛ , and (iii) for every x 2 S, ˛ 2 Œ.x/ C 1; / and y 2 S n .xB˛ /, one has .yB.y/C1 / \ .xB˛ / D ;. It follows from (i)–(ii) and Corollary 4.4 that there is a left invariant T1 -topology T on S in which for each x 2 S , ¹xB˛ W .x/ C 1 ˛ < º is an open neighborhood base at x. Condition (iii) gives us that T is zero-dimensional. Since T is also a T1 -topology, it is Hausdorff. Obviously, Ult.T / U.S /. To see that Ult.T / is topologically and algebraically isomorphic to H , let X D B1 . For every ˛ < , let e˛ denote the element of H with supp .e˛ / D ¹˛º. Deﬁne f W X ! H by putting f .1/ D 0 and f .y0 x˛1 x˛n / D e˛1 C C e˛n where n 2 N and 0 < ˛1 < : : : < ˛n < . Clearly f is bijective and f .B˛ / D H˛ for every ˛ 2 Œ1; /, so f homeomorphically maps X onto .H; T0 /. Also f satisﬁes conditions of Lemma 7.22, since f .xy/ D f .x/ C f .y/ whenever x 2 X n ¹1º and y 2 B.x/C1 . Let f W clˇS X ! ˇH denote the continuous extension of f and let ' D f jUlt.T / . It then follows that ' W Ult.T / ! H is an isomorphism. We conclude this section by constructing a homomorphism of a semigroup generated by a strongly discrete ultraﬁlter. Lemma 7.27. Let F be a strongly discrete ﬁlter on S and let M W S ! F be a basic mapping. Let x 2 S n ¹1º and let x D x0 xn be an M -decomposition with x0 D 1. Then there is a neighborhood U of 1 in T ŒF such that whenever y 2 U n ¹1º and y D y0 ym is an M -decomposition with y0 D 1, xy D x0 x1 xn y1 ym is an M -decomposition.

Section 7.3 Homomorphisms

109

Proof. Deﬁne N W S ! F by N.y/ D M.xy/ \ M.y/ and put U D ŒN 1 . Let y 2 U n ¹1º. Then there is an N -decomposition y D y0 ym with y0 D 1. Clearly this is also an M -decomposition. We have that xy D x0 x1 xn y1 ym . Since x D x0 xn is an M -decomposition, xiC1 2 M.x0 xi /. Next, we have that y1 2 N.y0 / D N.1/ M.x1/ D M.x0 xn / and, for i > 1, yiC1 2 N.y0 yi / D N.y1 yi / M.xy1 yi / D M.x0 xn y1 yi /: Hence, xy D x0 x1 xn y1 ym is an M -decomposition. Deﬁnition 7.28. Given a semigroup S and p 2 ˇS, let Cp denote the smallest closed subsemigroup of ˇS containing p. Theorem 7.29. Let S be an inﬁnite semigroup with identity and let p be a strongly discrete ultraﬁlter on S . Then there is a continuous homomorphism W Ult.T Œp/ ! ˇN such that .p/ D 1 and .Cp / D ˇN. Proof. Choose a basic mapping M W S ! p such that M.x/ M.1/ for every x 2 S (see the proof of Theorem 4.16). For every x 2 M.1/, put f0 .x/ D 1. Extend f0 W M.1/ ! N to a mapping f W .S; T Œp/ ! N ˇN by f .x/ D f0 .x1 / C C f0 .xn / where x D x0 x1 xn is an M -decomposition with x0 D 1. Deﬁne W Ult.T Œp/ ! ˇN by D f jUlt.T Œp / . Lemma 7.27 gives us that for every x 2 S n ¹1º, there is a neighborhood U of 1 in T Œp such that f .xy/ D f .x/ C f .y/ for all y 2 U n ¹1º. Then by Lemma 7.22, is a homomorphism. Clearly, is continuous and .p/ D 1. It follows that .Cp / is a closed subsemigroup of ˇN containing 1. Hence .Cp / D ˇN.

References Proposition 7.7 and Proposition 7.8 are from [99]. Example 7.9 is from [92] and Lemma 7.10 is from [98]. Theorem 7.13 is a result from [101]. Theorem 7.17 is due to T. Budak (Papazyan) [54]. Proposition 7.21 was proved in [99]. Theorem 7.24 is from [38], where it was also proved that for every inﬁnite cancellative semigroup S of cardinality , S contains copies of H . Theorem 7.26 is from [108]. Theorem 7.29 is a result from [107].

Chapter 8

Finite Groups in ˇG

In this chapter we show that if G is a countable torsion free group, then ˇG contains no nontrivial ﬁnite groups. We also extend in some sense this result to the case where G is an arbitrary countable group. To this end, we develop a special technique based on the concepts of a local left group and a local homomorphism.

8.1

Local Left Groups and Local Homomorphisms

Deﬁnition 8.1. A local left group is a T1 -space X with a partial binary operation and a distinguished point 1 2 X such that (i) x 1 D x for all x 2 X , (ii) .x y/ z D x .y z/ whenever all the products in the equality are deﬁned, and (iii) for every x 2 X n ¹1º, there is a neighborhood U of 1 such that x y is deﬁned for all y 2 U , x U is a neighborhood of x and x W U 3 y 7! x y 2 x U is a homeomorphism. A basic example of a local left group is an open neighborhood of the identity of a T1 left topological group. Let X be a local left group and let Xd be the partial semigroup X reendowed with the discrete topology. As in the case of ˇS, the partial operation of Xd can be naturally extended to ˇXd by pq D lim lim xy; x!p y!q

where x; y 2 X , making ˇXd a right topological partial semigroup. The product pq is deﬁned if and only if ¹x 2 X W ¹y 2 X W xy is deﬁnedº 2 qº 2 p; in particular, if the ultraﬁlter q converges to 1 2 X . Deﬁnition 8.2. Given a local left group X , Ult.X / D ¹p 2 Xd W p converges to 1 2 X º: It is straightforward to check that Ult.X / is a closed subsemigroup in Xd . If X is an open neighborhood of the identity of a left topological group .G; T /, we identify Ult.X / with Ult.T /.

Section 8.1 Local Left Groups and Local Homomorphisms

111

Deﬁnition 8.3. Let X and Y be local left groups. A mapping f W X ! Y is a local homomorphism if f .1X / D 1Y and for every x 2 X n ¹1º, there is a neighborhood U of 1 2 X such that f .xy/ D f .x/f .y/ for all y 2 U . We say that f W X ! Y is a local isomorphism if f is a local homomorphism and homeomorphism. Note that if f W X ! Y is a local isomorphism, so is f 1 W Y ! X . Lemma 8.4. Let f W X ! Y be a continuous local homomorphism. Deﬁne f W Ult.X / ! Ult.Y / by f D f jUlt.X/ , where f W ˇXd ! ˇYd is the continuous extension of f . Then f is a continuous homomorphism. Furthermore, if f is injective, f is a continuous injective homomorphism, and if f is a local isomorphism, f is a topological and algebraic isomorphism. Proof. Apply Lemma 7.22. Deﬁnition 8.5. Let X be a local left group and let S be a semigroup. A mapping f W X ! S is a local homomorphism if for every x 2 X n¹1º, there is a neighborhood U of 1 such that f .xy/ D f .x/f .y/ for all y 2 U n ¹1º. Lemma 8.6. Let T be a compact right topological semigroup and let f W X ! T be a local homomorphism such that f .X / ƒ.T /. Deﬁne f W Ult.X / ! T by f D f jUlt.X/ , where f W ˇXd ! T is the continuous extension of f . Then f is a continuous homomorphism. Furthermore, if for every neighborhood U of 1 2 X , f .U n ¹1º/ is dense in T , then f is surjective. Proof. That f is a homomorphism follows from Lemma 7.22. To check the second statement, let t 2 T . We have that for every neighborhood U of 1 2 X and for every neighborhood V of t 2 T , there exists x 2 U n ¹1º such that f .x/ 2 V . It follows from this that there exists p 2 Ult.T / such that f .p/ D t . L Deﬁnition 8.7. We say that an element a 2 ! Z2 is basic if supp.a/ is a nonempty L interval in !. Each nonzero a 2 ! Z2 can be uniquely written in the form a D a 0 C C ak where (a) for each i k, ai is basic, and (b) for each i k 1, max supp.ai / C 2 min supp.aiC1 /. We call such L a decomposition canonical . Endow ! Z2 with the group topology by taking as a neighborhood base at 0 the subgroups ° ± M Z2 W x.i / D 0 for all i < n Hn D x 2 !

where n < !.

112

Chapter 8 Finite Groups in ˇG

L Lemma 8.8. Let S be a semigroup and let f W ! Z2 ! S. The ﬁrst two of the following statements are equivalent and imply the third: (1) f .a/ D f .a0 / f .ak / whenever a D a0 C C ak is a canonical decomposition, (2) f .a C b/ D f .a/f .b/ whenever max supp.a/ C 2 min supp.b/, (3) f is a local homomorphism. Proof. .1/ ) .2/ Let max supp.a/ C 2 min supp.b/ and let a D a0 C C ak and b D b0 C C bl be the canonical decompositions. Since max supp.a/ D max supp.ak / and min supp.b/ D min supp.b0 /, one has max supp.ak / C 2 min supp.b0 /, so aCb D a0 C Cak Cb0 C Cbl is the canonical decomposition. Hence f .a C b/ D f .a0 C C ak C b0 C C bl / D f .a0 / f .ak /f .b0 / f .bl / D f .a/f .b/: .2/ ) .1/ If a D a0 C C ak is a canonical decomposition, then f .a/ D f .a0 C C ak1 /f .ak / D D f .a0 / f .ak /: L .2/ ) .3/ Let 0 ¤ a 2 ! Z2 and let n D max supp.a/ C 2. Then for every b 2 Hn n ¹0º, max supp.a/ C 2 min supp.b/, and so f .a C b/ D f .a/f .b/. We now come to the main result of this section. Theorem 8.9. Let X be a countable regular local left group and let ¹Ux W x 2 X n¹1ºº be L a family of neighborhoods of 1 2 X . Then there is a continuous bijection h W X ! ! Z2 with h.1/ D 0 such that (1) h1 .Hn / Ux whenever max supp.h.x// C 2 n, and (2) h.xy/ D h.x/ C h.y/ whenever max supp.h.x// C 2 min supp.h.y//. Notice that condition (2) in Theorem 8.9 may be rewritten as (20 ) h1 .ab/ D h1 .a/ C h1 .b/ whenever max supp.a/ C 2 min supp.b/, L so h1 W ! Z2 ! X is a local homomorphism, by Lemma 8.8. Since h is continuous, it follows that h also is a local homomorphism. Finally, if ¹Ux W x 2 X n ¹1ºº is a neighborhood base at 1 2 X , then h is open, and so h is a local isomorphism. To see that h is open, let x 2 X . Put n D max supp.h.x// C 2 (if x D 1, put n D 0). Since h is continuous, there is a neighborhood V of 1 such that h.V / Hn . Now let U be any neighborhood of 1 contained in V . Then h.xU / D h.x/ C h.U / by (2). Pick y 2 X n ¹1º such that Uy U and put m D max supp.h.y// C 2. Then h.Uy / Hm by (1). Hence, h.xU / h.x/ C Hm .

Section 8.1 Local Left Groups and Local Homomorphisms

113

Deﬁnition 8.10. Given m 2, let W D W .Zm / denote the set of all words on the alphabet Zm including the empty word ;. A nonempty word w 2 W is basic if all nonzero letters in w form a ﬁnal subword. In particular, every nonempty zero word (= all the letters are zeros) is basic. For every v; w 2 W such that jvj C 2 jwj and the ﬁrst jvj C 1 letters in w are zeros, deﬁne v C w 2 W to be the result of substituting v for the initial subword of length jvj in w. Each nonempty w 2 W can be uniquely written in the form w D w0 C C wk where (a) for each i k, wi is basic, and (b) for each i < k, wi is nonzero. We call such a decomposition canonical. Proof of Theorem 8.9. Enumerate X without repetitions as ¹1; x1 ; x2 ; : : :º and let W D W .Z2 /. We shall assign to each w 2 W a point x.w/ 2 X and a clopen neighborhood X.w/ of x.w/ such that (i) x.0n / D 1, X.;/ D X , and X.0n / Ux.v/ for all v 2 W with jvj n 2, (ii) X.w _ 0/ \ X.w _ 1/ D ; and X.w _ 0/ [ X.w _ 1/ D X.w/, (iii) x.w/ D x.w0 / x.wk / and X.w/ D x.w0 / x.wk1 /X.wk / where w D w0 C C wk is the canonical decomposition, and (iv) xn 2 ¹x.v/ W v 2 W and jvj D nº. Choose as X.0/ a clopen neighborhood of 1 such that x1 … X.0/. Put X.1/ D X n X.0/, x.0/ D 1 and x.1/ D x1 . Fix n 2 and suppose that X.w/ and x.w/ have been constructed for all w with jwj < n such that conditions (i)–(iv) hold. Notice that the subsets X.w/, jwj D n 1, form a partition of X . So, one of them, say X.u/, contains xn . Let u D u0 C C ur be the canonical decomposition. Then X.u/ D x.u0 / x.ur1 /.X.ur // and xn D x.u0 / x.ur1 /yn for some yn 2 X.ur /. If yn D x.ur /, choose as X.0n / a clopen neighborhood of 1 such that (a) X.0n / Ux.w/ for all w 2 W with jwj n 2, and (b) for each basic w with jwj D n 1, X.w/ n x.w/X.0n / ¤ ;. For each basic w with jwj D n 1, put X.w _ 1/ D X.w/ n x.w/X.0n / and pick as x.w _ 1/ any element of X.w _ 1/. Also put x.0n / D 1. If yn ¤ x.ur /, choose X.0n / in addition so that (c) yn … x.ur /X.0n / and put x.u_ r 1/ D yn . For all nonbasic w with jwj D n, deﬁne X.w/ and x.w/ by condition (iii). Then x.w/ D x.w0 / x.wk / 2 x.w0 / x.wk1 /X.wk / D X.w/

114

Chapter 8 Finite Groups in ˇG

and if xn … ¹x.w/ W jwj D n 1º, _ xn D x.u0 / x.ur1 /x.u_ r 1/ D x.u 1/ 2 ¹x.w/ W jwj D nº:

To check (ii), let jwj D n 1. If w is basic, X.w _ 0/ D x.w/X.0n / X.w/ and X.w _ 1/ D X.w/ n x.w/X.0n /. Suppose that w is nonbasic and let w D w0 C C wk be the canonical decomposition. If wk is zero, then w _ 0 D w0 C Cwk1 C0n is the canonical decomposition, and consequently, X.w _ 0/ D x.w0 / x.wk1 /X.0n / D x.w0 / x.wk1 /X.wk_ 0/: Otherwise w _ 0 D w0 C C wk1 C wk C 0n is the canonical decomposition, and then X.w _ 0/ D x.w0 / x.wk1 /x.wk /X.0n / D x.w0 / x.wk1 /X.wk_ 0/: In any case, X.w _ 0/ D x.w0 / x.wk1 /X.wk_ 0/: Next, since w _ 1 D w0 C C wk1 C wk_ 1 is the canonical decomposition, X.w _ 1/ D x.w0 / x.wk1 /X.wk_ 1/: It follows that X.w _ 0/ [ X.w _ 1/ D x.w0 / x.wk1 /ŒX.wk_ 0/ [ X.wk_ 1/ D x.w0 / x.wk1 /X.wk / D X.w/ and X.w _ 0/ \ X.w _ 1/ D ;. Now for every x 2 X , there is w 2 W with nonzero last letter such that x D x.w/, so ¹v 2 L W W x D x.v/º D ¹w _ 0n W n < !º. Consequently, we can deﬁne h W X ! ! Z2 by putting for every w D 0 n 2 W , h.x.w// D w D . 0 ; : : : ; n ; 0; 0; : : :/: Obviously, h.1/ D 0. It is clear also that h is bijective. Since for every z D . i /i

Section 8.1 Local Left Groups and Local Homomorphisms

115

To check (2), let max supp.h.x// C 2 min supp.h.y//. Pick w; v 2 W with nonzero last letters such that x D x.w/ and y D x.v/. Let w D w0 C C wk and v D v0 C C v t be the canonical decompositions. Then y 2 X.0jwjC1 /, consequently w C v D w0 C C wk C v0 C C v t is the canonical decomposition, and so xy D x.w0 / x.wk /x.v0 / x.v t / D x.w C v/: Hence, h.xy/ D h.x.w C v// DwCv DwCv D h.x.w// C h.x.v// D h.x/ C h.y/: Corollary 8.11. Let X be a countable nondiscrete regular L local left group. Then there is a continuous bijective local homomorphism f W X ! ! Z2 , and consequently, Ult.X / is topologically and algebraically isomorphic to a subsemigroup of H. If X is ﬁrst countable, f can be chosen to be a local isomorphism, and consequently, Ult.X / is topologically and algebraically isomorphic to H. Proof. It is immediate from Theorem 8.9 and Lemma 8.4. Corollary 8.12. Let X and Y be countable nondiscrete regular local left groups. Then there is a bijection f W X ! Y such that both f and f 1 are local homomorphisms. L L Proof. Let h W X ! ! Z2 and g W Y ! ! Z2 be bijections guaranteed by Theorem 8.9. Put f D g 1 ı h. We say that a homomorphisms of an ultraﬁlter semigroup is proper if it can be induced by a local homomorphism as in Lemmas 8.4 and 8.6. Corollary 8.13. Let X be a countable nondiscrete regular local left group and let Q and T be ﬁnite semigroups. Then for every local homomorphism f W X ! Q and for every surjective homomorphism g W T ! Q, there is a local homomorphism h W X ! T such that f D g ı h. Consequently, for every proper homomorphism ˛ W Ult.X / ! Q and for every surjective homomorphism ˇ W T ! Q, there is a proper homomorphism W Ult.X / ! T such that ˛ D ˇ ı . Proof. For every x 2 X n ¹1º, pick a neighborhood UL x of 1 2 X such that f .xy/ D f .x/f .y/ for all y 2 Ux n ¹1º, and let ' W X ! ! Z2 be a bijection guaranteed by by Theorem 8.9. It then follows that f ' 1 .a C b/ D f ' 1 .a/f ' 1 .b/

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L whenever max supp.a/ C 2 min supp.b/. For every basic a 2 L ! Z2 , pick .a/ 2 g 1 f ' 1 .a/ T , so g .a/ D f ' 1 .a/. Deﬁne W ! Z2 ! T by .a/ D .a0 / .ak / where a D a0 C C ak is a canonical decomposition. Let h D ı '. It is clear that h is a local homomorphism. To see that g ı h D f , let x 2 X n ¹1º and let '.x/ D a0 C C ak be the canonical decomposition. Then gh.x/ D g '.x/ D g .a0 C C ak / D g. .a0 / .ak // D g .a0 / g .ak / D f ' 1 .a0 / f ' 1 .ak / D f ' 1 .a0 C C ak / D f .x/: Finally, it follows from f D g ı h that f D g ı h . Let X be a local left group and let S be a ﬁnite semigroup. We say that a local homomorphism f W X ! S is surjective if for every neighborhood U of 1 2 X , f .U n ¹1º/ D S. Corollary 8.14. Let X be a countable nondiscrete regular local left group and let Q be a ﬁnite semigroup. Then for every local L homomorphism f W X ! Q and for every surjective local homomorphism g W ! Z2 ! Q, there is a continuous local L homomorphism h W X ! ! Z2 such that f D gıh. Consequently, for every proper homomorphism ˛ W Ult.X / ! Q and for every surjective proper homomorphism ˇ W H ! Q, there is a proper homomorphism W Ult.X / ! H such that ˛ D ˇ ı . Proof. For every x 2 X n ¹1º, pick a neighborhoodL Ux of 1 2 X such that f .xy/ D f .x/f .y/ for all y 2 Ux n ¹1º, and let ' W X ! ! Z2 be a bijection guaranteed by Theorem 8.9. For every n < !, pick ng > n such that g.a C b/ D g.a/g.b/ L whenever max supp.a/ n and min supp.b/ ng . Then for every basic a 2 ! Z2 , L pick a nonzero .a/ 2 g 1 f ' 1 .a/ ! Z2 such that the following condition is satisﬁed: If m < max supp.a/ and n D max¹max supp. .b// W b is basic and max supp.b/ mº; then min supp. .a// ng .

117

Section 8.2 Triviality of Finite Groups in ˇZ

Deﬁne

W

L !

Z2 !

L !

Z2 by

.a/ D

.a0 / C C

.ak /;

where a D a0 C C ak is a canonical decomposition, and let h D from the condition that g. .a0 / C C

ı '. It follows

.ak // D g .a0 / g .ak /;

and so f D g ı h. The condition also implies that min supp. .a// max supp.a/ for every basic a, which gives us that is continuous. To see this, suppose that max supp.a/ D m C 1 and the statement holds for all basic b with max supp.b/ D m. Pick any such b. Then by the inductive assumption, min supp. .b// m. It follows that n m. Now, applying the condition, we obtain that min supp. .a// ng n C 1 m C 1 D max supp.a/:

8.2

Triviality of Finite Groups in ˇZ

Lemma 8.15. Let G be a group and let Q be a group in ˇG. Then Q is contained either in G or in G . Proof. It is immediate from the fact that G is an ideal of ˇG (Lemma 6.8). Deﬁnition 8.16. Given a group G and a group Q in G , G.Q/ D ¹x 2 G W xQ D Qº: If x; y 2 G.Q/, then xy 1 Q D xy 1 yQ D xQ D Q, and so xy 1 2 G.Q/. Hence, G.Q/ is a subgroup of G. Also note that G.Q/ D ¹x 2 G W xu 2 Qº where u is the identity of Q. Indeed, if xu 2 Q, then xQ D x.uQ/ D .xu/Q D Q. Lemma 8.17. G.Q/ 3 x 7! xu 2 Q is an injective homomorphism. Proof. That this mapping is injective follows from Lemma 6.9. To see that this is a homomorphism, let x; y 2 G.Q/. Then .xy/u D x.yu/ D x.u.yu// D .xu/.yu/.

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Chapter 8 Finite Groups in ˇG

Theorem 8.18. Let G be a countable group and let Q be a ﬁnite group in G . If G.Q/ is trivial, so is Q. Proof. Assume on the contrary that Q is nontrivial while G.Q/ D ¹1º. Without loss of generality one may suppose that Q is a cyclic group of order n > 1. Let u be the identity of Q. Deﬁne C G by C D ¹x 2 G W xu 2 Qº Equivalently, C D ¹x 2 G W xQ D Qº: It is clear that C is a closed subsemigroup of G and Q is a minimal left ideal of C . Furthermore, C is left saturated. Indeed, let xy D z for some x 2 ˇG and y; z 2 C . Then xyQ D zQ, and so xQ D Q. Consequently, x 2 C [ G.Q/. Since G.Q/ D ¹1º, x 2 C [ ¹1º. By Theorem 7.13, there is a regular left invariant topology T on G such that Ult.T / D C . Since Q is a minimal left ideal of C , it follows that C has only one minimal right ideal. Consequently, T is extremally disconnected, by Proposition 7.7. Being regular extremally disconnected, T is zero-dimensional. (Note that we showed zero-dimensionality of T not using the fact that G is countable.) Next for every p 2 Q, let Cp D ¹x 2 C W xu D pº: It is easy to see that ¹Cp W p 2 Qº is a partition of C into closed subsets and p 2 Cp for each p 2 Q. Let Fp be the ﬁlter on G such that Fp D Cp . For every p 2 Q, choose Vp 2 Fp such that Vp \ Vq D ; if p ¤ q. We now show that for each p 2 Q, there is Wp 2 Fp such that Wp Cq Vpq for all q 2 Q. Indeed, let D u jˇGn¹1º . Then Cpq D 1 .pq/. It follows that there exists Apq 2 pq such that 1 .Apq / Vpq ; or equivalently, ¹x 2 ˇG n ¹1º W xu 2 Apq º Vpq : Since Cp q D Cp uq D pq and Q is ﬁnite, there is Wp 2 Fp such that Wp q Apq for all q 2 Q. Then Wp Cq u D Wp q Apq ; and consequently, Wp Cq Vpq .

119

Section 8.2 Triviality of Finite Groups in ˇZ

Choose the subsets Wp in addition so that Wp Vp and [ Wp [ ¹1º XD p2Q

is open in T . Then deﬁne f W X ! Q by f .x/ D p

if x 2 Wp :

The value f .1/ does not matter. We claim that f is a local homomorphism. To see this, let x 2 X n ¹1º. Then x 2 Wp for some p 2 Q. For each q 2 Q, choose Ux;q 2 Fq such that Ux;q Wq

and

xUx;q Vpq : This can be done because Wp Cq Vpq . Then choose a neighborhood Ux of 1 2 X such that [ Ux Ux;q [ ¹1º and xUx X: q2Q

Now let y 2 Ux n ¹1º. Then y 2 Ux;q for some q 2 Q. Since xUx;q Vpq , one has xy 2 Vpq . But then, since xUx X , xy 2 Wpq . Hence f .xy/ D pq D f .x/f .y/: Having checked that f is a local homomorphism, let ˛ D f . Then ˛ W Ult.X / ! Q is a proper homomorphism with the property that ˛jQ D idQ . Now let T be a cyclic group of order n2 and let ˇ W T ! Q be a surjective homomorphism. By Corollary 8.13, there is a proper homomorphism W Ult.X / ! T such that ˛ D ˇı . It follows that .Q/ T is a subgroup of order n. But T has only one subgroup of order n and this is the kernel of ˇ, so ˇ. .Q// D ¹0º, a contradiction. Corollary 8.19. Let G be a countable torsion free group. Then ˇG contains no nontrivial ﬁnite groups. Proof. By Lemma 8.15, every group in ˇG is contained either in G or in G . Let Q be a ﬁnite group in G . By Lemma 8.17, Q contains an isomorphic copy of G.Q/. Consequently, G.Q/ is ﬁnite. Since G is torsion free, it follows that G.Q/ is trivial. Then by Theorem 8.18, Q is trivial as well. As an immediate consequence of Corollary 8.19 we obtain that Corollary 8.20. ˇZ contains no nontrivial ﬁnite groups.

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Chapter 8 Finite Groups in ˇG

Corollary 8.20 and Corollary 8.11 give us the following. Corollary 8.21. Let X be a countable regular local left group. Then Ult.X / contains no nontrivial ﬁnite groups. Proof. Pick a nondiscrete ﬁrst countable group topology T on Z. By Corollary 8.11, Ult.T / is isomorphic to H and Ult.X / is isomorphic to a subgroup of H, and by Corollary 8.20, Ult.T / contains no nontrivial ﬁnite groups.

8.3

Local Automorphisms of Finite Order

Let X be a set and let f W X ! X . A subset Y X is invariant (with respect to f ) if f .Y / Y . We say that a family F of subsets of X is invariant if for every Y 2 F , f .Y / 2 F . For every x 2 X , let O.x/ D ¹f n .x/ W n < !º. Lemma 8.22. Let X be a space, let f W X ! X be a homeomorphism, and let x 2 X with jO.x/j D s 2 N. Let U be a neighborhood of x such that the family ¹f j .U / W j < sº is disjoint and suppose that there is n 2 N such that f n jU D idU . Then there is an open neighborhood V of x contained in U such that the family ¹f j .V / W j < sº is invariant. If X is zero-dimensional, then V can be chosen to be clopen. Proof. Clearly, n D sl for some l 2 N. Choose an open neighbourhood W of x such that f j Cis .W / f j .U / for all j < s and i < l, in particular, f is .W / U for all i < l. This can be done because f s .x/ D x. Now let V D

[

f is .W /:

i

Then V is an open neighborhood of x contained in U and f s .V / D

[

f .iC1/s .W /:

i

Since f ls D f 0 , f s .V / D

[

f is .W / D V:

i

It follows that ¹f

j .V /

W j < sº is invariant.

A bijection f W X ! X has ﬁnite order if there is n 2 N such that f n D idX , and the smallest such n is the order of f .

Section 8.3 Local Automorphisms of Finite Order

121

Corollary 8.23. Let X be a space with a distinguished point 1 2 X and let f W X ! X be a homeomorphism with f .1/ D 1. Then exactly one of the following two possibilities holds: (1) for every neighborhood U of 1 and n 2 N, there is x 2 U with jO.x/j > n, (2) there is an open invariant neighborhood U of 1 such that f jU has ﬁnite order. Deﬁnition 8.24. Let X be a space with a distinguished point 1 2 X and let f W X ! X be a homeomorphism of ﬁnite order with f .1/ D 1. Deﬁne the spectrum of f by spec.f / D ¹jO.x/j W x 2 X n ¹1ºº; and more generally, for any subset Y X , spec.f; Y / D ¹jO.x/j W x 2 Y n ¹1ºº: We say that f is spectrally irreducible if for every neighborhood U of 1, spec.f; U / D spec.f /: Also, a neighborhood U of a point x 2 X is spectrally minimal if for every neighborhood V of x contained in U , spec.f; V / D spec.f; U /: Corollary 8.25. Let X be a space with a distinguished point 1 2 X and let f W X ! X be a homeomorphism of ﬁnite order with f .1/ D 1. Then there is an open invariant neighborhood U of 1 such that f jU is spectrally irreducible. Corollary 8.25 tells us that we can restrict ourselves in the study of homeomorphisms of ﬁnite order in a neighborhood of a ﬁxed point to considering spectrally irreducible ones. Deﬁnition 8.26. A local automorphism of a local left group X is a local isomorphism of X onto itself. In other words, a mapping f W X ! X is a local automorphism if f is both a homeomorphism with f .1/ D 1 and a local homomorphism, that is, for every x 2 X n ¹1º, there is a neighborhood U of 1 such that f .xy/ D f .x/f .y/ for all y 2 U. The next lemma says that the spectrum of a spectrally irreducible local automorphism of ﬁnite order is a ﬁnite subset of N closed under taking the least common multiple lcm, that is, lcm.s; t / 2 spec.f / for all s; t 2 spec.f /.

122

Chapter 8 Finite Groups in ˇG

Lemma 8.27. Let X be a Hausdorff local left group and let f W X ! X be a spectrally irreducible local automorphism of ﬁnite order. Let x0 2 X n ¹1º with jO.x0 /j D s and let U be a spectrally minimal neighbourhood of x0 . Then spec.f; U / D ¹lcm.s; t / W t 2 ¹1º [ spec.f /º: Proof. For each x 2 O.x0 /, let Vx be a neighborhood of 1 such that Vx 3 y 7! xy 2 xVx is a homeomorphism and f .xy/ D f .x/f .y/ for all y 2 Vx . Choose a neighborhood V of 1 such that \ V Vx ; x2O.x0 /

x0 V U , and the subsets xV , where x 2 O.x0 /, are pairwise disjoint. Let n be the order of f . Choose a neighborhood W of 1 such that f i .W / V for all i < n. Then clearly this inclusion holds for all i < !, and furthermore, for every y 2 W , f i .x0 y/ D f i .x0 /f i .y/. Indeed, it is trivial for i D 0, and further, by induction, we obtain that f i .x0 y/ D ff i1 .x0 y/ D f .f i1 .x0 /f i1 .y// D f i .x0 /f i .y/: Now let y 2 W , jO.y/j D t and k D lcm.s; t /. We claim that jO.x0 y/j D k. Indeed, f k .x0 y/ D f k .x0 /f k .y/ D x0 y: On the other hand, suppose that f i .x0 y/ D x0 y for some i . Then f i .x0 /f i .y/ D x0 y. Since the subsets xV , x 2 O.x0 /, are pairwise disjoint, it follows from this that f i .x0 / D x0 , so sji . But then also f i .y/ D y, and so t ji . Hence kji . Now, given any ﬁnite subset of N closed under lcm, we produce a spectrally irreducible local automorphism of the corresponding spectrum. Example 8.28. Let S be a ﬁnite subset P L of N closed under lcm and let m D 1 C s. Consider the direct sum ! Zm of ! copies of the group Zm . Endow Ls2S Z with the group topology by taking as a neighborhood base at 0 the subgroups ! m ° ± M Hn D x 2 Zm W x.i / D 0 for all i < n !

where n < !. Write the elements of S as s1 < < s t . Deﬁne the permutation 0 on Zm by the product of disjoint cycles 0 D .1; : : : ; s1 /.s1 C 1; : : : ; s1 C s2 / .s1 C C s t1 C 1; : : : ; s1 C C s t /: L Let be the coordinatewise permutation on ! Zm induced by 0 , that is, .x/.n/ D 0 .x.n//. Then is a homeomorphism with .0/ D 0, spec.; Hn / D S for each

Section 8.3 Local Automorphisms of Finite Order

123

n < !, and .x C y/ D .x/ C .y/ whenever max supp.x/ < min supp.y/. Hence, is a spectrally irreducible local automorphism of spectrum S. We call the standard permutation of spectrum S. We now come to the main result of this section. Theorem 8.29. Let X be a countable nondiscrete regular local left group, let f W X ! X be a spectrally irreducible local automorphism of ﬁnite order, let P LS D spec.f /, and let m D 1 C s2S s. Let be the standard permutation on ! Zm L of spectrum S. Then there is a continuous bijection h W X ! ! Zm with h.1/ D 0 such that (1) f D h1 h, and (2) h.xy/ D h.x/ C h.y/ whenever max supp.h.x// C 2 min supp.h.y//. If X is ﬁrst countable, then h can be chosen to be a homeomorphism. Proof. The proof is similar to that of Theorem 8.9. W .Zm /. The permutation Enumerate X as ¹xn W n < !º with x0 D 1 and let W D L 0 on Zm , which induces the standard permutation on ! Zm , also induces the permutation 1 on W . If w D 0 n , then 1 .w/ D 0 . 0 / 0 . n /. We will write instead of 0 and 1 . We shall assign to each w 2 W a point x.w/ 2 X and a clopen spectrally minimal neighborhood X.w/ of x.w/ such that (i) x.0n / D 1 and X.;/ D X , (ii) ¹X.w _ / W 2 Zm º is a partition of X.w/, (iii) x.w/ D x.w0 / x.wk1 /x.wk / and X.w/ D x.w0 / x.wk1 /X.wk / where w D w0 C C wk is the canonical decomposition, (iv) f .x.w// D x..w// and f .X.w// D X..w//, and (v) xn 2 ¹x.v/ W v 2 W and jvj D nº. For this, we need the following Lemma 8.30. Let X be a countable nondiscrete regular space with a distinguished point 1 2 X and let f W X ! X be a homeomorphism of ﬁnite order with f .1/ D 1. Let U be a clopen invariant subset of X , let K be a ﬁnite invariant subset of U , and let P be a clopen invariant partition of U such that for each C 2 P , spec.f; K \ C / D spec.f; C /. Then there is a clopen invariant partition ¹U.x/ W x 2 Kº of U inscribed into P such that for each x 2 K, U.x/ is a spectrally minimal neighborhood of x. Proof. Enumerate U as ¹xn W n < !º with x0 2 K. For each x 2 K, we shall construct an increasing sequence .Un .x//n

124

Chapter 8 Finite Groups in ˇG

S inscribedSinto P and invariant, and xn 2 Un D x2K Un .x/. Then the subsets U.x/ D n 0 and suppose that we have constructed required Un1 .x/, x 2 K. Without loss of generality one may suppose also that xn … Un1 . Let jO.xn /j D s and let xn 2 Cn 2 P . Using Lemma 8.22, choose a clopen neighborhood Vn of xn such that for each j < s, f j .Vn / is a spectrally minimal neighborhood of f j .xn /, and the family ¹f j .Vn / W j < sº [ ¹Un1 .x/ W x 2 Kº is disjoint, inscribed into P and invariant. Pick zn 2 K \ Cn with jO.zn /j D s. For each j < s, put Un .f j .zn // D Un1 .f j .zn // [ f j .Vn /: For each x 2 K n O.zn /, put Un .x/ D Un1 .x/. Now write the elements of S as s1 < < s t . For each i D 1; : : : ; t , pick a representative i of the orbit in Zm n ¹0º of lengths si . Choose a clopen invariant neighborhood U1 of 1 2 X such that x1 … U1 and spec.f; X n U1 / D spec.f /. Put x.0/ D 1 and X.0/ D U1 . Then pick points S ai 2 X n U1 , i D 1; : : : ; t , with pairwise disjoint orbits of lengths si such that x1 2 tiD1 O.ai /. For each i D 1; : : : ; t and j < si , put x. j .i // D f j .ai /: By Lemma 8.30, there is an invariant partition ¹X. / W 2 Zm n ¹0ºº of X n U1 such that X. / is a clopen spectrally minimal neighborhood of x. /. Fix n > 1 and suppose that X.w/ and x.w/ have been constructed for all w 2 W with jwj < n so that conditions (i)–(v) are satisﬁed. Notice that the subsets X.w/, jwj D n 1, form a partition of X . So one of them, say X.u/, contains xn . Let u D u0 C C uq be the canonical decomposition. Then X.u/ D x.u0 / x.uq1 /X.uq / and xn D x.u0 / x.uq1 /yn for some yn 2 X.uq /. Choose a clopen invariant neighborhood Un of 1 such that for all basic w with jwj D n 1, (a) x.w/Un X.w/, (b) f .xy/ D f .x/f .y/ for all y 2 Un , and (c) spec.f; X.w/ n x.w/Un / D spec.X.w//.

Section 8.3 Local Automorphisms of Finite Order

125

If yn ¤ x.uq /, choose Un in addition so that (d) yn … x.uq /Un . Put x.0n / D 1 and X.0n / D Un . Let w 2 W be an arbitrary nonzero basic word with jwj D n 1 and let O.w/ D ¹wj W j < sº, where wj C1 D .wj / for j < s 1 and .ws1 / D w0 . Put Yj D X.wj / n x.wj /Un . Using Lemma 8.27, choose points bi 2 Y0 , i D 1; : : : ; t , with pairwise disjoint orbits of lengths lcm.si ; s/. If uq 2 O.w/, choose bi in addition so that t [ yn 2 O.bi /: iD1

For each i D 1; : : : ; t and j < si , put x. j .w _ i // D f j .bi /: Then, using Lemma 8.30, inscribe an invariant partition ¹X.v _ / W v 2 O.w/; 2 Zm n ¹0ºº into the partition ¹Yj W j < sº such that X.v _ / is a clopen spectrally minimal neighborhood of x.v _ /. For nonbasic w 2 W with jwj D n, deﬁne x.w/ and X.w/ by condition (iii). To check (ii) and (iv), let jwj D n 1 and let w D w0 C C wk be the canonical decomposition. Then X.w _ 0/ D x.w0 / x.wk /X.0n / D x.w0 / x.wk1 /x.wk /X.0n / and X.w _ / D x.w0 / x.wk1 /X.wk_ /; so (ii) is satisﬁed. Next, f .x.w// D f .x.w0 / x.wk1 /x.wk // D f .x.w0 //f .x.w1 / x.wk1 /x.wk // :: : D f .x.w0 // f .x.wk1 //f .x.wk // D x..w0 // x..wk1 //x..wk // D x..w0 / .wk1 /.wk // D x..w//; so (iv) is satisﬁed as well.

126

Chapter 8 Finite Groups in ˇG

To check (v), suppose that xn … ¹x.w/ W jwj D n 1º. Then xn D x.u0 / x.uq1 /yn D x.u0 / x.uq1 /x.u_ q / D x.u_ /: Now, for every x 2 X , there is w 2 W with nonzero last letter such that x D Lx.w/, so ¹v 2 W W x D x.v/º D ¹w _ 0n W n < !º. Hence, we can deﬁne h W X ! ! Zm by putting for every w D 0 n 2 W , h.x.w// D w D . 0 ; : : : ; n ; 0; 0; : : :/: Obviously, h.1/ D 0. It is clear also that h is bijective. Since for every z D . i /i

127

Section 8.3 Local Automorphisms of Finite Order

Proof. Consider an arbitrary relation p1 pk D q1 qs in Ult.X /, where pi ; qj 2 ¹p; f .p/º, pi ¤ piC1 and qj ¤ qj C1 . We prove that p1 D q1 and k D s. Without loss of generality one may suppose that f is spectrally irreducible. Let M hWX ! Zm !

be a bijection guaranteed by Theorem 8.29. Denote C the set of all nonﬁxed points in Zm (with respect to 0 ) and let Y D ¹x 2 X W there is a coordinate of h.x/ belonging to C º: (Equivalently, Y consists of all nonﬁxed points in X .) Note that Y \ Ult.X / is a subsemigroup containing p and f .p/. For every x 2 Y , consider the sequence of coordinates of h.x/ belonging to C and denote ˛.x/ and .x/ the ﬁrst and the last elements in this sequence. Then for every u; v 2 Y \ Ult.X /, ˛.uv/ D ˛.u/. Indeed, let ˛.u/ D c 2 C and let A D ¹x 2 Y W ˛.x/ D cº. Then A 2 u. For every x 2 A, put n.x/ D max supp.h.x// C 2 and Ux D h1 .Hn.x/ /. We have that S x2A xUx 2 uv and for every y 2 Ux , ˛.xy/ D ˛.x/ D c, so ˛.uv/ D c. Similarly, .uv/ D .v/, and if f .u/ ¤ u, then ˛.u/ ¤ ˛.f .u// and .u/ ¤

.f .u//. Applying ˛ and to the relation gives us that ˛.p1 / D ˛.q1 / and .pk / D

.qs /, so p1 D q1 and pk D qs . We now show that k D s. Deﬁne the subset F C 2 by F D ¹. .q/; ˛.q// W q 2 ¹p; f .p/ºº and let n max¹k; sº. For every x 2 X , consider the sequence of coordinates of h.x/ belonging to C and deﬁne .x/ 2 Z.n/ to be the number modulo n of pairs of neighbouring elements in this sequence other than pairs from F . Then for every u; v 2 Ult.X /, ´ .u/ C .v/ if . .u/; ˛.v// 2 F .uv/ D .u/ C .v/ C 1 otherwise. It follows from this that .p1 pk / D .p1 / C C .pk / C k 1 and .q1 qs / D .q1 / C C .qs / C s 1: Also we have that for every q 2 ¹p; f .p/º, .qq/ D 2.q/. Consequently, since q is an idempotent, .q/ D .qq/ D 2.q/. Hence, .q/ D 0. Finally, we obtain that .p1 pk / D k 1 and so k D s.

.q1 qs / D s 1;

128

8.4

Chapter 8 Finite Groups in ˇG

Finite Groups in G

Finite groups in G can be constructed in the following trivial way. Example 8.32. Let G be a group, let F be a ﬁnite subgroup of G, and let u be an idempotent in G which commutes with each element of F . Then F u is a ﬁnite subgroup of G isomorphic to F . The isomorphism is given by F 3 x 7! xu 2 F u: Indeed, that this mapping is injective follows from Lemma 6.9. To see that this is a homomorphism, let x; y 2 F . Then xyu D xyuu D xuyu. In this section we show that if G is a countable group, then all ﬁnite groups in G have such a trivial structure. Theorem 8.33. Let G be a countable group, let Q be a ﬁnite group in G with identity u, and let F D G.Q/. Then u commutes with each element of F and Q D F u. Proof. We ﬁrst show that u commutes with each element of F . Let a 2 F and assume on the contrary that a1 ua ¤ u. Note that both idempotents u and a1 ua belong to the semigroup C.u/ D ¹x 2 G W xu D uº. Indeed, since a 2 G.Q/, one has au 2 Q, so uau D au and then a1 uau D a1 au D u. By Theorem 7.17, there is a regular left invariant topology T on G with Ult.T / D C.u/. Consider the conjugation f W x 7! a1 xa on .G; T /. Clearly, f .u/ D a1 ua. We claim that f is a homeomorphism. To see this, let p 2 C.u/. Then a1 pau D a1 puau D a1 uau D a1 au D u; so a1 pa 2 C.u/. Consequently, f is continuous. Similarly, f 1 is continuous. It follows that f is a local automorphism of ﬁnite order. Since f .u/ ¤ u, idempotents u and f .u/ D a1 ua generate a free product by Theorem 8.31. But this contradicts the equality a1 uau D u. We now show that Q D F u. Without loss of generality one may suppose that Q is a ﬁnite cyclic group with a generator q. Then for every x 2 F , xq D qx. Indeed, since xu D ux and Q is Abelian, xq D xuq D qxu D qux D qx: Choose A 2 q such that xy D yx for all x 2 F and y 2 A. Let H be the subgroup of G generated by the subset A [ F . We have that Q H and F is central in H . Let L D H=F and let g W ˇH ! ˇL be the continuous extension of the canonical

Section 8.4 Finite Groups in G

129

homomorphism H ! L. Then g is a homomorphism and the elements of ˇL are the subsets of the form Fp where p 2 ˇH . Consider the group R D g.Q/ D Q=.F u/ in L . We claim that the subgroup L.R/ D ¹x 2 L W xR D Rº in L is trivial. To see this, let x 2 L.R/. Pick y 2 H with g.y/ D x. Then g.yQ/ D g.Q/. Since yQ and Q are complete preimages with respect to g, it follows that yQ D Q. Consequently y 2 F , and so x D 1 2 L. Since L.R/ is trivial, R is trivial as well by Theorem 8.18, and then Q D F u.

References The results of Sections 8.1 and 8.2 are from [85] (announced in [86]). Theorem 8.33 is due to I. Protasov [58]. Its proof is based on Theorem 8.18 and Theorem 8.31. The latter and Theorem 8.29 were proved in [87]. The exposition of this chapter is based on the treatment in [92].

Chapter 9

Ideal Structure of ˇG

In this chapter we show that for every inﬁnite group G of cardinality , the ideal U.G/ of ˇG consisting of uniform ultraﬁlters can be decomposed (D partitioned) into 22 closed left ideals of ˇG. We also prove that if G is Abelian, then ˇG contains 22 minimal right ideals and the structure group of K.ˇG/ contains a free group on 22 generators. We conclude by showing that if is not Ulam-measurable, then K.ˇG/ is not closed.

9.1

Left Ideals

Let G be an arbitrary inﬁnite group of cardinality . For every A G, let U.A/ denote the set of uniform ultraﬁlters from A. It is easy to see that the set U.G/ of all uniform ultraﬁlters on G is a closed two-sided ideal of ˇG. Deﬁnition 9.1. Let I.G/ denote the ﬁnest decomposition of U.G/ into closed left ideals of ˇG with the property that the corresponding quotient space of U.G/ is Hausdorff. Deﬁnition 9.1 can be justiﬁed by noting that the family of all such decompositions is nonempty (it contains the trivial decomposition ¹U.G/º) and considering the diagonal of the corresponding quotient mappings. Deﬁnition 9.2. For every p 2 U.G/, deﬁne Ip ˇG by \ Ip D cl.GU.A//: A2p

The next theorem is the main result of this section. Theorem 9.3. If is a regular cardinal, then I.G/ D ¹Ip W p 2 U.G/º. Before proving Theorem 9.3 we establish several auxiliary statements. Lemma 9.4. For every p 2 U.G/, Ip is a closed left ideal of ˇG contained in U.G/. Proof. Clearly, Ip is a closed subset of U.G/. In order to show that Ip is a left ideal of ˇG, it sufﬁces to show that for every x 2 G, xIp Ip . Since \ xcl.GU.A// and xcl.GU.A// D cl.xGU.A// D cl.GU.A//; xIp D A2p

it follows that xIp D Ip .

131

Section 9.1 Left Ideals

A decomposition D of aSspace X into closed subsets is called upper semicontinuous if for every open U X , ¹A 2 D W A U º is open in X . Equivalently, D is upper semicontinuous if for every A 2 D and for every neighborhood U of A X , there is a neighborhood V of A X such that if B 2 D and B \ V ¤ ;, then B U . Lemma 9.5. Let D be a decomposition of a compact Hausdorff space X into closed subsets and let Y be the corresponding quotient space of X . Then Y is Hausdorff if and only if D is upper semicontinuous. Proof. Let f W X ! Y denote the natural quotient mapping. Suppose that Y is Hausdorff and let U X be open. It then follows that f .X n U / Y is closed, so f 1 .Y n f .X n U // X is open. It remains to notice that [ f 1 .Y n f .X n U // D ¹A 2 D W A U º: Conversely, suppose that D is upper semicontinuous and let A1 ; A2 be distinct members of D.SPick disjoint neighborhoods U1 ; U2 of A1 ; A2 in X . For each i D 1; 2, let Vi D ¹B 2 D W B Ui º. Then Vi is a neighborhood of Ai X and Vi D f 1 .f .Vi //. It follows that f .V1 /; f .V2 / are disjoint neighborhoods of f .A1 /; f .A2 / 2 Y . Lemma 9.6. Let J be a decomposition of U.G/ into closed left ideals such that the corresponding quotient space of U.G/ is Hausdorff. Then for every J 2 J and p 2 J , Ip J . Proof. It sufﬁces to show that for every neighborhood V of J U.G/, Ip cl.V /. By Lemma 9.5, J is upper semicontinuous. Therefore, one may suppose that for every I 2 J, if I \ V ¤ ;, then I V . It follows from this that GV V . Since V is a neighborhood of p 2 U.G/, there is A 2 p such that U.A/ V . Consequently, GU.A/ V , so cl.GU.A// cl.V /. Hence, Ip cl.V /. Recall that given a set X and a cardinal , ŒX D ¹A X W jAj D º and

ŒX < D ¹A X W jAj < º:

Deﬁnition 9.7. For every p 2 U.G/, let Fp denote the ﬁlter on G with a base consisting of subsets of the form [ x.A n Fx / x2G

where A 2 p and Fx 2

ŒG<

for each x 2 G.

Lemma 9.8. For every p 2 U.G/, Ip D

\ C 2Fp

C:

132

Chapter 9 Ideal Structure of ˇG

T Proof. To see that Ip C 2Fp C , let A 2 ŒG and Fx 2 ŒG< for each x 2 G and S let C D x2G x.AnFx /. For every x 2 G, xU.A/ x.A n Fx / D x.A n Fx / C . Consequently, cl.GU.A// C . To see the converse inclusion, let B G and Ip B. It then follows that there is A 2 p such that cl.GU.A// B. (Indeed, G n B D G n B is compact and for every y 2 G n B, there is Ay 2 p such that y … cl.GU.Ay //.) For every x 2 G, one has xU.A/ B, consequently, there is Fx 2 ŒG< such that x.A n Fx / B. Let S C D x2G x.A n Fx /. Then C 2 Fp and C B. Lemma 9.9. Suppose that isSa regular cardinal. Let A 2 ŒG and Fx 2 ŒG< for every x 2 G and let B D G n x2G x.A n Fx /. Then there are Hx ; Kx 2 ŒG< for every x 2 G such that [ [ x.A n Hx / \ x.B n Kx / D ;: x2G

x2G

Proof. Enumerate G as ¹x˛ W ˛ < º. For every ˛ < , deﬁne Hx˛ ; Kx˛ 2 ŒG< by [ [ Hx˛ D Fx 1 x˛ and Kx˛ D x˛1 xˇ Fx˛1 xˇ : ˇ ˛

ˇ

ˇ ˛

Then for every ˛ < and ˇ ˛, xˇ1 x˛ .A n Hx˛ / \ B D ;

and

x˛1 xˇ A \ .B n Kx˛ / D ;;

x˛ .A n Hx˛ / \ xˇ B D ;

and

xˇ A \ x˛ .B n Kx˛ / D ;:

and so Consequently, for every ˛; ˇ < , x˛ .A n Hx˛ / \ xˇ .B n Kxˇ / D ;: Now we are in a position to prove Theorem 9.3. Proof of Theorem 9.3. Let I D ¹Ip W p 2 U.G/º. By Lemma 9.4, all members of I are closed left ideals of ˇG contained in U.G/. To show that I is an upper < semicontinuous decomposition S of U.G/, let p 2 U.G/, A 2 p, and Fx 2 ŒG for every x 2 G, and let B D x2G x.A n Fx /. By Lemma 9.9, there are Hx ; Kx 2 ŒG< for every x 2 G such that Q \ R D ; where [ [ QD x.A n Hx / and R D x.B n Kx /: x2G

x2G

By Lemma 9.8, Ip Q and for every r 2 U.B/, Ir R, consequently, Ir G n Q. This shows that I is a decomposition. It follows from this also that for every

133

Section 9.1 Left Ideals

q 2 U.Q/, Iq G n B, which shows that I is upper semicontinuous. Thus, I is a decomposition of U.G/ into closed left ideals such that the corresponding quotient space of U.G/ is Hausdorff. That I is the ﬁnest decomposition of this kind follows from Lemma 9.6. Now we consider decompositions of U.G/ with an additional property that for every member I of the decomposition, IG I . Deﬁnition 9.10. Let I 0 .G/ denote the ﬁnest decomposition of U.G/ into closed left ideals of ˇG with the property that the corresponding quotient space of U.G/ is Hausdorff and for every member I of the decomposition, IG I . Deﬁnition 9.11. For every p 2 U.G/, deﬁne Ip0 ˇG by \ Ip0 D cl.GU.A/G/: A2p

As in the proof of Lemma 9.8, one shows that \ Ip0 D C C 2Fp0

where Fp0 denotes the ﬁlter on G with a base consisting of subsets of the form [ x.A n Fx;y /y; x;y2G

where A 2 p and Fx;y 2 ŒG< for every x; y 2 G. The next lemma is the corresponding version of Lemma 9.9. Lemma 9.12. Suppose that is a regular cardinal. Let A 2 ŒG and Fx;y 2 ŒG< S for every x; y 2 G and let B D G n x;y2G x.A n Fx;y /y. Then there are Hx;y ; Kx;y 2 ŒG< for every x; y 2 G such that [ [ x.A n Hx;y /y \ x.B n Kx;y /y D ;: x;y2G

x;y2G

Proof. Enumerate G G as ¹.x˛ ; y˛ / W ˛ < º and for every ˛ < , deﬁne Hx˛ ;y˛ ; Kx˛ ;y˛ 2 ŒG< by [ [ Hx˛ ;y˛ D Fx 1 x˛ ;y˛ y 1 and Kx˛ ;y˛ D x˛1 xˇ Fx˛1 xˇ ;yˇ y˛1 yˇ y˛1 : ˇ ˛

ˇ

ˇ

ˇ ˛

Then for every ˛ < and ˇ ˛, xˇ1 x˛ .A n Hx˛ ;y˛ /y˛ yˇ1 \ B D ;

and

x˛1 xˇ Ayˇ y˛1 \ .B n Kx˛ ;y˛ / D ;;

134

Chapter 9 Ideal Structure of ˇG

so x˛ .A n Hx˛ ;y˛ /y˛ \ xˇ Byˇ D ;

and

xˇ Ayˇ \ x˛ .B n Kx˛ ;y˛ /y˛ D ;;

and consequently, for every ˛; ˇ < , x˛ .A n Hx˛ ;y˛ /y˛ \ xˇ .B n Kxˇ ;yˇ /yˇ D ;: It is easy to see that the corresponding versions of Lemma 9.4 and Lemma 9.6 also hold. Hence, we obtain the following analogue of Theorem 9.3. Theorem 9.13. If is a regular cardinal, then I 0 .G/ D ¹Ip0 W p 2 U.G/º. The next lemma will allow us to compute the cardinality of I 0 .G/. Lemma 9.14. Let A 2 ŒG . Then there are B 2 ŒA and Fx;y 2 ŒG< for every x; y 2 G such that whenever B0 ; B1 2 ŒB and B0 \ B1 D ;, one has [ [ x.B0 n Fx;y /y \ x.B1 n Fx;y /y D ;: x;y2G

x;y2G

Proof. Enumerate G G as ¹.x˛ ; y˛ / W ˛ < º. Construct inductively a -sequence .a /< in A such that for every < and ˛ , x˛ a y˛ … ¹xˇ aı yˇ W ˇ ı < º: Deﬁne B 2 ŒA and Fx˛ ;y˛ 2 ŒG< for every ˛ < by B D ¹a W < º

and

Fx˛ ;y˛ D ¹aˇ W ˇ < ˛º:

We claim that these are as required. Indeed, assume the contrary. Then x˛ a y˛ D xˇ aı yˇ for some ˛; ˇ < and some distinct ; ı < such that ˛ and ˇ ı. But this is a contradiction.

Corollary 9.15. If is a regular cardinal, then jI 0 .G/j D 22 , and for every I 2 I 0 .G/, I is nowhere dense in U.G/. Proof. Let A 2 ŒG and let B be a subset of A guaranteed by Lemma 9.14. Then jU.B/j D 22 and for any distinct p; q 2 U.B/, Ip \ Iq D ;. To see that I is nowhere dense in U.G/, suppose that U.A/\I ¤ ;. If U.B/\I D ;, we are done. Otherwise I D Ip for some p 2 U.B/. Pick C 2 ŒB such that C … p. Then U.C / \ I D ;.

135

Section 9.1 Left Ideals

The next theorem covers in some sense the case where is a singular cardinal. Theorem 9.16. If > !, then there is a decomposition J of U.G/ into closed left ideals of ˇG such that (1) the corresponding quotient space of U.G/ is homeomorphic to U. /, (2) for every J 2 J, J G J , and (3) for every J 2 J, J is nowhere dense in U.G/. The proof of Theorem 9.16 is based on the following lemma. Lemma 9.17. Let > !. Then there is a surjective function f W G ! such that (a) for every ˛ < , jf 1 .˛/j < , and (b) whenever x; y 2 G and f .x/ < f .y/, one has f .xy/ D f .yx/ D f .y/. Proof. Construct inductively a -sequence .G˛ /˛< of subgroups of G such that (i) for every ˛ < , jG˛ j < , (ii) for every ˛ < , G˛ G˛C1 , (iii) for every limit ordinal ˛ < , G˛ D S (iv) ˛< G˛ D G.

S ˇ <˛

Gˇ , and

Note that G is a disjoint union of nonempty sets G˛C1 n G˛ , where ˛ < , and G0 . Deﬁne f W G ! by ´ ˛ f .x/ D 0

if x 2 G˛C1 n G˛ if x 2 G0 :

Clearly, f is surjective and satisﬁes (a). To check (b), let x; y 2 G and f .x/ < f .y/. Then x 2 Gˇ and y 2 G˛C1 n G˛ for some ˇ ˛ < . It follows that both xy and yx also belong to G˛C1 n G˛ . Hence, f .xy/ D f .yx/ D f .y/. Proof of Theorem 9.16. Let f W G ! be a function guaranteed by Lemma 9.17 and let f W ˇG ! ˇ be the continuous extension of f . Then (i) f .U.G// D U. / and f

1

.U. // D U.G/,

(ii) f .qp/ D f .p/ for all p 2 U.G/ and q 2 ˇG, (iii) f .px/ D f .p/ for all p 2 U.G/ and x 2 G, and (iv) for every u 2 U. /, f

1

.u/ is nowhere dense in U.G/.

136

Chapter 9 Ideal Structure of ˇG

Indeed, (i) follows from surjectivity of f and condition (a). To see (ii), let A 2 p. For every x 2 G, let Ax D A n ¹y 2 G W f .y/ f .x/º. Then Ax 2 p and by condition (b), f .xy/ D f .y/ 2 f .A/ for all y 2 Ax . Consequently, B D S x2G xAx 2 qp and f .B/ f .A/. Hence, f .qp/ D f .p/. The check of (iii) is 1

similar. Finally, to see (iv), let A 2 ŒG and suppose that U.A/ \ f .u/ ¤ ;. Then E D f .A/ 2 u. Pick D 2 ŒE such that D … u and let B D f 1 .D/ \ A. Then 1 1 B A, U.B/ ¤ ;, but f .B/ … u, and so U.B/ \ f .u/ D ;. Hence, f .u/ is nowhere dense in U.G/. 1 Now let J D ¹f .u/ W u 2 U. /º. It then follows from (i)-(iv) that J is as required. Applying Theorem 9.16 in the case > ! and Corollary 9.15 in the case D !, we obtain the following result.

Theorem 9.18. jI 0 .G/j D 22 , and for every I 2 I 0 .G/, I is nowhere dense in U.G/. We conclude this section by the following consequence of Theorem 9.18. Corollary 9.19. ƒ.ˇG/ D G. Proof. We have to show that for every p 2 G , p W ˇG 3 x 7! px 2 ˇG is not continuous. Without loss of generality one may suppose that p 2 U.G/. By Theorem 9.18, there is a nontrivial decomposition I of U.G/ into closed left ideals of ˇG such that for every I 2 I, IG I . Let J be the member of I containing p. Pick any K 2 I different from J and any q 2 K. Then pq 2 K and cl.pG/ J . Consequently, p is discontinuous at q.

9.2

Right Ideals

In this section we prove the following result.

Theorem 9.20. For every inﬁnite Abelian group G of cardinality , ˇG contains 22 minimal right ideals.

The proof of Theorem 9.20 involves some additional concepts. The Bohr compactiﬁcation of a topological group G is a compact group bG together with a continuous homomorphism e W G ! bG such that e.G/ is dense in bG and the following universal property holds: For every continuous homomorphism h W G ! K from G into a compact group K there is a continuous homomorphism hb W bG ! K such that h D hb ı e. In the case where G is a discrete Abelian group, the Bohr compactiﬁcation can be naturally deﬁned in terms of the Pontryagin duality

Section 9.2 Right Ideals

137

as follows. Let GO be the dual group of G and let GO d be the group GO reendowed with the discrete topology. Then bG is the dual group of GO d . The mapping e W G ! bG is given by e.x/./ D .x/, where x 2 G and 2 GO d . It is injective. (See [34, 26.11 and 26.12].) Recall that ﬁlters F and G on a set X are incompatible if A \ B D ; for some A 2 F and B 2 G . A ﬁlter F on a space X is open if F has a base of open subsets of X . In order to prove Theorem 9.20, we show the following.

Theorem 9.21. For every inﬁnite Abelian group G of cardinality , there are 22 pairwise incompatible open ﬁlters on bG converging to zero. Before proving Theorem 9.21, let us show how it implies Theorem 9.20.

Proof of Theorem 9.20. Let T denote the Bohr topology on G, that is, the topology induced by the mapping e W G ! bG, and let S D Ult.T /. By Theorem 9.21, there are 22 pairwise incompatible open ﬁlters on bG converging to zero. Considering the restriction of the ﬁlters to e.G/, we conclude that there are pairwise incompatible open ﬁlters F˛ , ˛ < 22 , on .G; T / converging to zero. For each ˛ < 22 , let J˛ D F˛ n¹0º. Then by Lemma 7.3, each J˛ is a closed right ideal of S, and since the ﬁlters F˛ are pairwise incompatible, the ideals J˛ are pairwise disjoint. Furthermore, since .G; T / is a subgroup of a compact group, by Lemma 7.10, S contains all the idempotents of G , in particular, the idempotents of K.ˇG/. Consequently, S \ K.ˇG/ ¤ ;. But then, by Proposition 6.24, K.S / D K.ˇG/ \ S. It follows from this that every minimal right ideal R of S is contained in a minimal right ideal R0 of ˇG, and the correspondence R 7! R0 is injective. Consequently, the number of minimal right ideals of ˇG is greater than or equal to that of S, and so it is 22 . To prove Theorem 9.21, we need three lemmas. The ﬁrst of them is an elementary fact on inﬁnite Abelian groups. Lemma 9.22. Let G be an inﬁnite Abelian group of cardinality . Then G admits a homomorphism onto one of the following groups: L L (1) Z, ! Zp , Zp1 and p2Q Zp if D !, L L (2) Zp and Zp 1 if > ! and cf. / > !, L L L L L L (3) Zp , Zp 1 , p2Q p Zp and p2Q p Zp 1 if > ! and cf. / D !. Here, p is a prime number and Q is an inﬁnite subset of the primes. Zp1 denotes the quasicyclic p-group. If > ! and cf. / D !, . p /p2Q is an inﬁnite increasing sequence of uncountable cardinals coﬁnal in , that is, supp2Q p D .

138

Chapter 9 Ideal Structure of ˇG

Proof. If G is ﬁnitely generated, then D ! and G admits a homomorphism onto Z. Therefore, one may assume that G is not ﬁnitely generated. We ﬁrst prove that G admits a homomorphism onto a periodic group of cardinality . Let ¹ai W i 2 I º be a maximal independent subset of G L and let A D hai W i 2 I i be the subgroup generated by ¹ai W i 2 I º. Then A D i2I hai i, and for every nonzero g 2 G, one has hgi \ A ¤ ¹0º, so G=A is periodic. If jG=Aj D , we are done. Suppose that jG=Aj < . Then jAj D and jI j D , because G is not ﬁnitely generated. We show that Lthere is a subgroup H of G and a subset I1 I with jI1 j D such that G D H ˚ i2I1 hai i. To this end, choose a complete set S for representatives of the cosets of A in G, and let H0 D hSi \ A. Deﬁne I0 I by I0 D ¹i 2 I W x.i / ¤ 0 for some x 2 H0 º and put I1 D I n I0 . If G=A is ﬁnite, I0 is ﬁnite as well. If G=A is inﬁnite, jI0 j jG=Aj, because jhSij D jG=Aj and then jH0 j jG=Aj. In any case, jI0 j < , and consequently jI1 j D . Let A0 D hai W i 2 I0 i; A1 D hai W i 2 I1 i and H D hS [ A0 i: We claim that G D H ˚A1 . Indeed, since G D hS [A0 [A1 i, one has H CA1 D G. To see that H \A1 D ¹0º, let g 2 H \A1 . Then g D d Cc0 D c1 for some d 2 hSi, c0 2 A0 and c1 2 A1 . Consequently, d D c0 C c1 2 A. But then d 2 H0 A0 . Hence, c1 D 0, and g D 0. L Having established that G D H ˚ i2I1 hai i, we obtain that G admits a homoL morphism onto i2I1 hai i, and so onto a periodic group of cardinality . Now let G be a p-group. Then there is a so-called basic subgroup B of L G (see [28, Theorem 32.3]). We have that B is a direct sum of cyclic groups, say B D j 2J hbj i, L 1 and G=B is divisible, that is, isomorphic to Zp , where 0 . Suppose that jG=Bj D . Then > 0, and D if L> !. It follows that G admits a homomorphism onto Zp1 if D !, and onto Zp1 if > !. Now suppose that jG=Bj < . Then jBj D , and consequently jJ j D . It follows that G D L C ˚ j 2J1 hbj i for some subgroup C of G and a subset J1 J with L jJ1 j D (see the ﬁrst L part of the proof). Hence, G admits a homomorphism onto j 2J1 hbj i, and so onto Zp . L Finally, let G be periodic. Then G D p2M Gp , where M is the set of all primes p such that the p-primary component Gp of G is nontrivial. If jGp j D for some p 2 M , we are done, because then G admits a homomorphism onto Gp , a p-group of cardinality . Suppose that jGp j < for each p 2 M . Then M is inﬁnite and cf. / L D !. If D !, all Gp are ﬁnite, and so G admits a homomorphism onto p2M Zp . Suppose that > !. For each p 2 M , put p D jGp j. Clearly supp2M p D . Choose an inﬁnite subset N M such that . p /p2N is an increasing sequence of uncountable cardinals coﬁnal in . By the previous paragraph, for

Section 9.2 Right Ideals

139

each p 2 N , Gp L admits a homomorphism onto a group Kp of cardinality p which L is isomorphic to p Zp or p Zp1 . It L follows that there is an inﬁnite subset Q NL such that either Kp is isomorphic to p Zp forL all p 2 Q or Kp is isomor1 phic to Z for all p 2 Q. Then the group K D p2Q Kp is isomorphic to L p p L L L 1 p2Q p Zp or p2Q p Zp , jKj D , and G admits a homomorphism onto K. Now, using Lemma 9.22 and the Pontrjagin duality, we prove the following statement on bG. Lemma 9.23. For every inﬁnite discrete Q Abelian Q group G of cardinality , bG admits a continuous homomorphism onto 2 T or 2 Zp . Q Q Here, both products 2 T and 2 Zp are endowed with the product topology. Proof. The dual groups of continuous homomorphic images of bG are the subgroups of GO d and the dual groups of homomorphic images of G are the Q closed subgroups Q of GOL(see [34, Theorems 23.25 and 24.8]). The dual groups of 2 T and 2 Zp L are 2 Z and 2 Zp , respectively. Consequently, in order to prove the lemma, it sufﬁces to show that G admits onto a group whose dual group La homomorphism L contains an isomorphic copy of 2 Z or 2 Zp . Consider two cases. Case 1: L D !. Then G L admits a homomorphism onto one of Q the following 1 1 groups: p . Their dual groups are T , ! Zp , Z.p / Q Z, ! Zp , Zp and p2Q Z1 and p2Q Zp , respectively. Here, Z.p / denotes the group of p-adic integers. The L second group is algebraically isomorphic to 2! Zp . The others contain L torsion-free subgroups of cardinality 2! , and so contain an isomorphic copy of 2! Z. a homomorphism LCase 2:L > !. Then L L onto one of the following groups: L G admits L 1, 1 (the two latter groups apZ , Z Z and p p p p2Q p p2QQ p ZpQ Q Q 1 pear Q if cf. / D !). Their dual groups are Z , p Z.p /, p2Q p Zp Q 1 /, respectively. The ﬁrst group is algebraically isomorphic to and Z.p p2Q p L 2 Zp . The others contain L torsion-free subgroups of cardinality 2 , and so contain an isomorphic copy of 2 Z. The third lemma deals with products of topological spaces. Lemma 9.24. Let be an inﬁnite cardinal. For each ˛ < Q , let X˛ be a space having at least two disjoint nonempty open sets, and let X D ˛< X˛ . Then there are at least 2 many pairwise incompatible open ﬁlters on X converging to the same point. Q Before proving Lemma 9.24 note that if each factor in an inﬁnite product X D n

140

Chapter 9 Ideal Structure of ˇG

subsets of Xn , let xQ n 2 Vn , and let x D .xn /n m: S S It follows that U D m

Q jLj D , and for each ˛ 2 L, n

9.3

The Structure Group of K.ˇG /

In this section we prove the following result. Theorem 9.25. Let G be an inﬁnite group of cardinality embeddable into a direct sum of countable groups. Then the structure group of K.ˇG/ contains a free group on 22 generators.

Section 9.3 The Structure Group of K.ˇG/

141

Since every Abelian group can be isomorphically embedded into a direct sum of groups isomorphic to Q or Zp1 , we obtain from Theorem 9.25 as a consequence that Corollary 9.26. For every inﬁnite Abelian group G of cardinality , the structure group of K.ˇG/ contains a free group on 22 generators. Recall that if a semigroup S has a smallest ideal which is a completely simple semigroup, then for every p 2 E.K.S//, pSp K.S / is a maximal group in S with identity p. The ﬁrst step in the proof of Theorem 9.25 is the following result. L Theorem 9.27. Let A D ¹x 2 ! Z2 W jsupp.x/j D 1º and let p 2 E.K.H//. Then the elements p C q C p 2 K.H/, where q 2 A , generate a free group. Proof. Let q1 ; : : : ; qn be distinct elements of A , let G be the subgroup of p C H C p generated by p C qi C p, i D 1; : : : ; n, and let F be a free group on n generators, say x1 ; : : : ; xn . It sufﬁces to show that there exists a homomorphism of G into F sending p C qi C p to xi for each i D 1; : : : ; n. By Corollary 1.24, F can be algebraically embedded into a compact group K. Without loss of generality one may suppose that F K. Partition A into subsets Ai , i D 1; : : : ; n, such that Ai 2 qi . Deﬁne f0 W A ! K by f0 .a/ D xi if a 2 Ai : For every n < !, let an denote L the element of A with supp.an / D ¹nº. Extend f0 to a local homomorphism f W ! Z2 ! K by f .an1 C C ank / D f0 .an1 / f0 .ank / where 1 k < ! and n1 < < nk < !. Then f W H ! K is a homomorphism such that f .qi / D xi . Since f .p/ is the identity, it follows that f .p C qi C p/ D f .p/f .qi /f .p/ D f .qi / D xi : Lemma 9.28. Let G be a countable group, let T0 be a regular left invariant topology on G, and let .Un /1n

142

Chapter 9 Ideal Structure of ˇG

(3) Vn Vn1 , and (4) yk;m Vn Vk n VkC1 for all k; m < ! with k C m D n 2, where ¹yk;m W m < !º is an enumeration of Vk n VkC1 ﬁxed immediately after VkC1 has been chosen. T Then n

i D 1; : : : ; n;

generate a free group. Hence, the elements pqi p, i D 1; : : : ; n, also generate a free group. Given a set D and A D, let Q.A/ denote the set of countably incomplete ultraﬁlters from A ˇD.

Lemma 9.30. If jAj D !, then jQ.A/j D 22 .

Proof. We show that there are 22 uniform countably incomplete ultraﬁlters on . Let ¹An W n < !º be a partition of such that jAn j D for everySn < ! and let U be the set of uniform ultraﬁlters u on such that for every m < !, mn

Section 9.3 The Structure Group of K.ˇG/

143

Theorem 9.31. Let G be a group of cardinality > ! embeddable into a direct sum of countable groups. Then there is A G with jAj D such that for every p 2 E.K.ˇG//, the elements pqp 2 K.ˇG/, where q 2 Q.A/, generate a free group. L Proof. Let H D ˛< H˛ be a direct sum of countable groups and let G be a subgroup of H . Note that for every ˛ < , there is x 2 G n¹1º with min supp.x/ ˛. L Indeed, since j ˇ <˛ Hˇ j < , there are distinct y; z 2 G such that y.ˇ/ D z.ˇ/ for all ˇ < ˛. Put x D yz 1 . Then x ¤ 1 and for each ˇ < ˛, x.ˇ/ D 1ˇ . Choose inductively a -sequence .x /< in G n ¹1º such that max supp.xˇ / < min supp.x / L whenever ˇ < < . For every < , ˇ 2supp.x / Hˇ is countable. Therefore, without loss of generality one may suppose that jsupp.x /j D 1. Let A D ¹x W < º: We have to show that for any p 2 E.K.ˇG// and for any distinct q1 ; : : : ; qn 2 Q.A/, the elements pq1 p; : : : ; pqn p generate a free group. For every ˛ < , deﬁne the idempotent p˛ 2 ˇH˛ by p˛ D ˛ .p/, where ˛ W H ! H˛ is the projection and ˛ W ˇH ! ˇH˛ is the continuous extension of ˛ . Endow H˛ with a regular left invariant topology in which p˛ converges to 1˛ (Theorem 7.17). Now endow H with the topology induced by the product topology on Q H and let T denote the topology on G induced from H . Note that p 2 Ult.T / ˛ ˛< and A Ult.T /. Partition A into inﬁnite subsets Ai , i D 1; : : : ; n, such that Ai 2 qi , and for each i D 1; : : : ; n, partition Ai into inﬁnite subsets Aij , j < !, such that Aij … qi . Also partition the set ± ° M Z2 W jsupp.x/j D 1 BD x2 !

into inﬁnite subsets Bi , i D 1; : : : ; n, and enumerate Bi as ¹bij W j < !º without repetitions. L For every ˛ < , deﬁne a local homomorphism h˛ W H˛ ! ! Z2 as follows. If there is a 2 A with supp.a/ D ¹˛º, then pick a clopen neighborhood U˛ of 1˛ such that ˛ .a/ … U˛ , pick i; j such that a 2 Aij , and deﬁne h˛ by ´ 0 if x 2 U˛ h˛ .x/ D bij otherwise: If there is no a 2 A with supp.a/ D ¹˛º, deﬁne h ˛ by h˛ .x/ D 0. L Now deﬁne a local homomorphism h W H ! ! Z2 by h.x1 C C x t / D h˛1 .x1 / C C h˛ t .x t /;

144

Chapter 9 Ideal Structure of ˇG

where t 2 N, ˛1 < < ˛ t < , L and xs 2 H˛s for each s D 1; : : : ; t , and let f D hjG . Then f W .G; T / ! ! Z2 is a local homomorphism such that f .Aij / D ¹bij º for every i D 1; : : : ; n and j < !. It follows that M f W Ult.T / ! ˇ Z2 !

: : ; f .qn / are distinct elements from is a surjective homomorphism and f .q1 /; : L B . Since f is surjective, f .p/ 2 E.K.ˇ. ! Z2 /// (Lemma 6.25). Note that M E K ˇ Z2 D E.K.H//: !

Consequently, by Theorem 9.27, the elements f .p/ C f .qi / C f .p/ D f .pqi p/;

i D 1; : : : ; n;

generate a free group. Hence, the elements pqi p, i D 1; : : : ; n, also generate a free group.

9.4

K.ˇG / is not Closed

In this section we prove the following result. Theorem 9.32. Let G be an inﬁnite group of cardinality and assume that is not Ulam-measurable. Then both K.ˇG/ and E.cl K.ˇG// are not closed. Pick p 2 E.K.ˇG// and let T D C.p/ D ¹x 2 G W xp D pº. By Theorem 7.17, there is a regular extremally disconnected left invariant topology T on G such that Ult.T / D T . Note that K.T / is a right zero semigroup consisting of the idempotents from the minimal right ideal p.ˇG/ of ˇG, that is, K.T / D E.p.ˇG//. It follows that in order to prove Theorem 9.32, it sufﬁces to show that Theorem 9.33. There are elements in cl K.T / which are not in .cl K.T //T . Indeed, let q 2 .cl K.T // n ..cl K.T //T /. Then clearly, q 2 cl K.ˇG/. To see that q … K.ˇG/, assume the contrary. Then q 2 K.ˇG/ \ T D K.T /, and so q is an idempotent. Consequently, q D qq 2 .cl K.T //T , a contradiction. To see that q … E.cl K.ˇG//, assume the contrary. Then q D qq 2 .cl K.T //T , a contradiction. To prove Theorem 9.33, we need the following lemma. Lemma 9.34. Let X be a nondiscrete regular extremally disconnected space and assume that jX j is not T Ulam-measurable. Then there is a sequence .Un /n

145

Section 9.4 K.ˇG/ is not Closed

T Proof. Choose a family B of clopen sets in X such that Y D B is not open and D jBj is as small as possible. Since X is extremally disconnected and Y is closed, it follows that cl int Y D int Y , that is, int Y is clopen.TLet C D ¹U n int Y W U 2 Bº. Then jC j D , all members of C are clopen, and C D Z is a nonempty closed nowhere dense set. Enumerate C as ¹U˛ W ˛ < º. Deﬁne a decreasing T -sequence .W˛ /˛< of clopen T subsets of X by putting W0 D X and W D ˛ ˇ <˛ Uˇ for ˛ > 0. Then T W˛ D ˇ <˛ Wˇ if ˛ is a limit ordinal and ˛< W˛ D Z. Deﬁne f W X n Z ! by f .x/ D ˛ if x 2 W˛ n W˛C1 : Pick x 2 Z and let F denote the ﬁlter on with a base consisting T of subsets of the form f .U n Z/ where U is a neighborhood of x 2 X . Clearly, F D ;. We claim that F is a -complete ultraﬁlter, and so D !. To see that F isS an ultraﬁlter, let A . Then X nZ is a disjoint union of open sets VA D f 1 .A/ D ˛2A W˛ n W˛C1 and V nA D f 1 . n A/. Since X is extremally disconnected, there is a neighborhood U of x such that either U \ V nA D ; or U \ VA D ;, so either U n Z VA or U n Z V nA . It follows that either A 2 F or n A 2 F . To see that F is -complete, let < , and for every ˛ < , let A˛ 2 F . Then 1 .A / [ Z is a neighborhood of x. Consequently, by for every ˛ < , V˛ D fT ˛ the minimality of , V D T˛< V˛ is also a neighborhood of x. Deﬁne A 2 F by A D f .V n Z/. Then A ˛< A˛ . Proof of Theorem 9.33. By Theorem 9.18, there is an inﬁnite decomposition I of U.G/ into closed left ideals of ˇG. Pick a sequence .In /n

S Then by Corollary 2.24, either qm 2 cl n

146

Chapter 9 Ideal Structure of ˇG

Remark 9.35. In the case D ! Theorem 9.33 can be strengthened as follows: There are elements in cl K.T / which are not in T 2 . Indeed, inTthis case the sequence .Wn /n

References

It has long been known that U.G/ can be decomposed into 22 left ideals [14]. That it can be decomposed into 22 closed left ideals was established by I. Protasov [61] in the case where is a regular cardinal and by M. Filali and P. Salmi [24] for all . Theorems 9.3, 9.13, and 9.16 are from [109]. Corollary 9.19 is a partial case of the result of A. Lau and J. Pym [44]. Our proof of Corollary 9.19 is from [24]. Theorem 9.20 is from [104]. An introduction to the Pontryagin duality can be found in [55] and [34]. See also [50] and [17]. Theorem 9.25 is from [111]. Corollary 9.26 was proved also independently by S. Ferri, N. Hindman, and D. Strauss [23]. Theorem 9.27 is due to N. Hindman and J. Pym [36]. Theorem 9.32 complements the result from [106] which says that, for every inﬁnite semigroup S embeddable algebraically into a compact group, both K.ˇS / and E.cl K.ˇS // are not closed.

Chapter 10

Almost Maximal Topological Groups

In this chapter almost maximal topological groups and their ultraﬁlter semigroups are studied. A topological (or left topological) group is said to be almost maximal if the underlying space is almost maximal. We show that the ultraﬁlter semigroup of any countable regular almost maximal left topological group is a projective in the category F of ﬁnite semigroups. L Assuming MA, for every projective S in F, we construct a group topology T on ! Z2 such that Ult.T / is isomorphic to S. We show that every countable almost maximal topological group contains an open Boolean subgroup and its existence cannot be established in ZFC. We then describe projectives in F. These are certain chains of rectangular bands. We conclude by showing that the ultraﬁlter semigroup of a countable regular almost maximal left topological group is its topological invariant.

10.1

Construction

Deﬁnition 10.1. An object S in some category is an absolute coretract if for every surjective morphism f W T ! S there exists a morphism g W S ! T such that f ı g D idS . Let C denote the category of compact Hausdorff right topological semigroups. The next lemma gives us some simple examples of absolute coretracts in C. Lemma 10.2. Finite left zero semigroups, right zero semigroups and chains of idempotents are absolute coretracts in C. Proof. Let S be a ﬁnite left zero semigroup, let T be a compact Hausdorff right topological semigroup, and let f W T ! S be a continuous surjective homomorphism. Pick a minimal left ideal L of T . For each e 2 S, f 1 .e/ is a right ideal of T , so pick a minimal right ideal Re f 1 .e/ of T , and let g.e/ be the identity of the group Re \ L. Then ¹g.e/ W e 2 S º is a left zero semigroup and f ı g D idS . The proof for ﬁnite right zero semigroups is similar. Finally, suppose that S is a ﬁnite chain of idempotents, say e1 > > en . Construct inductively a chain of idempotents g.e1 / > > g.en / in T such that f .g.ei // D ei for each i D 1; : : : ; n. As g.e1 / pick any idempotent in T1 D f 1 .e1 /. Now ﬁx k < n and assume that we have constructed idempotents g.e1 / > > g.ek / in T such that f .g.ei // D ei . Let TkC1 D f 1 .¹e1 ; : : : ; ekC1 º/. Pick

148

Chapter 10 Almost Maximal Topological Groups

a minimal right ideal RkC1 g.ek /TkC1 of TkC1 . Note that RkC1 f 1 .ekC1 / (Lemma 6.25), so g.ek / … RkC1 . And pick a minimal left ideal LkC1 TkC1 g.ek / of TkC1 . Deﬁne g.ekC1 / to be the identity of the group RkC1 \ LkC1 . Deﬁnition 10.3. Let S be a ﬁnite semigroup. We say that S is an absolute Hcoretract if for every surjective proper homomorphism ˛ W H ! S there exists a homomorphism ˇ W S ! H such that ˛ ı ˇ D idS . Clearly, every ﬁnite absolute coretract in C is an absolute H-coretract. Theorem 10.4. Assume p DL c. Let S be a ﬁnite absolute H-coretract. Then there exists a group topology T on ! Z2 such that Ult.T / is isomorphic to S. The proof of Theorem 10.4 is based on the following lemma. Lemma 10.5. Let G be an inﬁnite group, let S D ¹pi W i < mº be a ﬁnite semigroup in G , and let F be the ﬁlter on G such that F D S. For every i < m, let Ai 2 pi , and for every x 2 G, let Bx 2 F . Then there is a sequence .xn /n

\

¹Bx W x 2 FP..xn /n

Choose C0 2 p0 such that C0 A0 and for each p 2 S, one has C0 p Ai where i is deﬁned by p0 p D pi . Pick x0 2 C0 . Now ﬁx l > 0 and assume that we have constructed a sequence .xn /n

149

Section 10.1 Construction

(v) whenever k < l, n0 < < nk < l and p 2 S, one has xn0 xnk Cl p Ai where i is deﬁned by pi D pin0 pink pj p, T (vi) whenever p 2 S, Cl p ¹Bx W x 2 FP..xn /n

\

¹Bx W x 2 FP..xn /n

Then pick xl 2 Cl . Proof of Theorem 10.4. Let G D basis of G, that is,

L !

yn .m/ D

Z2 and let Y D ¹yn W n < !º be the standard ´

1 0

if m D n otherwise.

Let T0 denote the group topology on G with a neighborhood base at 0 consisting of subgroups FS..yn /mn

150

Chapter 10 Almost Maximal Topological Groups

As .x0;n /n

10.2

Properties

Deﬁnition 10.7. An object S in some category is a projective if for every morphism f W S ! Q and for every surjective morphism g W T ! Q there exists a morphism h W S ! T such that g ı h D f . Note that every projective is an absolute coretract. In many categories these notions coincide but not in all.

151

Section 10.2 Properties

Deﬁnition 10.8. Let S be a ﬁnite semigroup. We say that S is an H-projective if for every homomorphism ˛ W S ! Q of S into a ﬁnite semigroup Q and for every surjective proper homomorphism ˇ W H ! Q there exists a homomorphism W S ! H such that ˛ D ˇ ı . Clearly, every H-projective is an absolute H-coretract. Let F denote the category of ﬁnite semigroups. Lemma 10.9. Every H-projective (absolute H-coretract) is a projective (absolute coretract) in F. Proof. Let S be an H-projective, let Q and T be ﬁnite semigroups, let ˛ W S ! Q be a homomorphism, and let ˇ W T ! Q be a surjective homomorphism. Pick a surjective proper homomorphism ı W H ! T . Then ˇ ı ı W H ! Q is a surjective proper homomorphism. Consequently, since S is an H-projective, there is a homomorphism " W S ! H such that ˛ D ˇ ı ı ı ". Deﬁne the homomorphism W S ! T by

D ı ı ". Then ˛ D ˇ ı . The proof for an absolute H-coretract is similar. Theorem 10.10. The ultraﬁlter semigroup of a countable regular almost maximal left topological group is an H-projective. Before proving Theorem 10.10 we establish the following fact. Lemma 10.11. Let .G; T / be an almost maximal T1 left topological group and let S D Ult.T /. Then for every homomorphism ˛ W S ! Q, there are an open neighborhood X of the identity of .G; T / and a local homomorphism f W X ! Q such that f D ˛. Proof. For each p 2 S, choose Ap 2 p such that Ap \ Aq D ; if p ¤ q. Then, for each p 2 S, choose Bp 2 p such that Bp q Apq for all q 2 S. This can be done because the mapping ˇG 3 x 7! xq 2 ˇG is continuous and S is ﬁnite. Choose the subsets Bp in addition so that Bp Ap and XD

[

Bp [ ¹1º

p2S

is open in .G; T /. Deﬁne f W X ! Q by putting for every p 2 S and x 2 Bp , f .x/ D ˛.p/. The value f .1/ does not matter. We claim that f is a local homomorphism and f D ˛. It sufﬁces to check the ﬁrst statement. Let x 2 X n ¹1º. Then x 2 Bp for some p 2 S . For each q 2 S, choose Cq 2 q such S that Cq Aq and xCq Apq . Choose a neighborhood U of 1 2 X such that U q2S Cq [ ¹1º and xU X . Now let

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Chapter 10 Almost Maximal Topological Groups

y 2 U n ¹1º. Then y 2 Cq for some q 2 S, so y 2 Aq and then y 2 Bq . Hence f .x/f .y/ D ˛.p/˛.q/. On the other hand, xy 2 Apq , then xy 2 Bpq , and so f .xy/ D ˛.pq/ D ˛.p/˛.q/: Hence f .xy/ D f .x/f .y/. Proof of Theorem 10.10. Let .G; T / be a countable regular almost maximal left topological group and let S D Ult.T /. Let Q be a ﬁnite semigroup, let ˛ W S ! Q be a homomorphism, and let ˇ W H ! Q be a surjective proper homomorphism. By Lemma 10.11, there are an open neighborhood X of the identity of .G; T / and a local homomorphism f W X ! Q such that f D ˛, so ˛ is proper. Now by Corollary 8.14, there is a proper homomorphism W Ult.T / ! H such that ˛ D ˇ ı . Hence, S is an H-projective. Recall that a band is a semigroup of idempotents. Theorem 10.12. The ultraﬁlter semigroup of a countable regular almost maximal left topological group is a band. Proof. Let .G; T / be a countable regular almost maximal left topological group and let S D Ult.T /. Assume on the contrary that S is not a band. By Corollary 8.21, S contains no nontrivial ﬁnite groups. Consequently, S contains a 2-element null subsemigroup, that is, there are distinct p; q 2 S such that q 2 D qp D pq D p 2 D p: It follows from q 2 D p 2 that for any A 2 p and B 2 q, Bq \ Ap ¤ ;, and hence by Corollary 2.23, either Bq \ Ap ¤ ; or Bq \ Ap ¤ ;. Consider two cases. Case 1: Bq \ Ap ¤ ; for some A 2 p and B 2 q. Then yq D p 0 p for some y 2 B and p 0 2 A, so q D y 1 p 0 p D y 1 p 0 pp D qp D p, a contradiction. Case 2: Bq \ Ap ¤ ; for all A 2 p and B 2 q. Then qA;B q D xA;B p for some 1 q 1 qA;B 2 B and xA;B 2 A, so p D xA;B A;B q D rA;B q where rA;B D xA;B qA;B . Since T is regular, it follows from p D rA;B q that rA;B 2 S. And since S is ﬁnite, it then follows from qA;B D xA;B rA;B that qB D prB for some rB 2 S and qB 2 B. Consequently, q D pr for some r 2 S. But then q D pr D ppr D pq D p, again a contradiction. Theorem 10.12, Theorem 10.10 and Theorem 10.4 raise the question of characterizing ﬁnite bands which are H-projectives. We address this question in Section 10.4. Now we consider the structure of a countable almost maximal topological group. Let X be a space with a distinguished point 1 2 X and let f W X ! X be a homeomorphism with f .1/ D 1. We say that f is trivial if the set of ﬁxed points of f is a neighborhood of f .

Section 10.2 Properties

153

Lemma 10.13. Let X be a countable regular local left group and suppose that K.Ult.X // is ﬁnite. Then every local automorphism on X is trivial. Proof. Assume on the contrary that there is a nontrivial local automorphism f W X ! X . For every n 2 N, let Xn D ¹x 2 X W jO.x/j > nº. Note that Xn is open, f n .x/ ¤ x for all x 2 Xn , and XnC1 Xn . By Corollary 8.23, it sufﬁces to consider the following two cases. Case 1: for every n 2 N, 1 2 cl Xn . Let F be the ﬁlter on X with a base consisting of subsets U \ Xn , where U runs over neighborhoods of 1 2 X and n 2 N, and let R D F . Then for every p 2 R, all elements p; f .p/; .f /2 .p/; : : : are distinct. Indeed, otherwise .f /n .p/ D p for some n 2 N, and consequently by Corollary 2.18, ¹x 2 X W f n .x/ D xº 2 p, which contradicts Xn 2 p. Next, by Corollary 7.3, R is a right ideal of Ult.X /, so there is p 2 R \ K.Ult.X //. Since f is an automorphism on Ult.X /, it follows that .f /n .p/ 2 K.Ult.X // for every n. Hence, K.Ult.X // is inﬁnite, a contradiction. Case 2: f has ﬁnite order. Since f is nontrivial, 1 2 cl X1 . Let R1 D X1 \Ult.X /. Then R1 is a right ideal of Ult.X /, so there is an idempotent p 2 R1 \ K.Ult.X //. We have also that f .p/ ¤ p and clearly f .p/ 2 K.Ult.X //. But then, applying Theorem 8.31, we obtain that the structure group of K.Ult.X // is inﬁnite, again a contradiction. Lemma 10.14. Let G be a countable group endowed with an invariant topology. Suppose that for every a 2 G, the conjugation G 3 x 7! axa1 2 G is a trivial local automorphism. Then the inversion W G 3 x 7! x 1 2 G is a local automorphism. Proof. To see that is a local homomorphism, let a 2 G n ¹1º. Since G 3 x 7! axa1 2 G is a trivial, U D ¹x 2 G W axa1 D xº is a neighborhood of 1. Pick a neighborhood V of 1 such that V D V 1 U . Then for every x 2 V , .ax/ D .ax/1 D x 1 a1 D a1 ax 1 a1 D a1 x 1 D .a/.x/: Theorem 10.15. Every countable almost maximal topological group contains an open Boolean subgroup. Proof. Let G be a countable almost maximal topological group. By Lemma 10.13, the conjugations of G are trivial. Then by Lemma 10.14, the inversion W G 3 x 7! x 1 2 G is a local automorphism. Consequently by Lemma 10.13, is trivial, so ¹x 2 G W .x/ D xº is a neighborhood of 1. Since ¹x 2 G W .x/ D xº D B.G/, G contains an open Boolean subgroup by Lemma 5.3. We conclude this section by showing that the existence of a countable almost maximal topological group cannot be established in ZFC.

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Chapter 10 Almost Maximal Topological Groups

Theorem 10.16. The existence of a countable almost maximal topological group implies the existence of a P -point in ! . Proof. Let .G; T / be a countable almost L maximal topological group. By Theorem 10.15, one may suppose that G D ! Z2 . Let F be the neighborhood ﬁlter of 0 in T and let T denote the topology on G induced by the product topology on 0 Q Z . By Theorem 10.12 and Lemma 7.10, T0 T . Then by Theorem 5.19, each ! 2 point of .F / is a P -point. Combining Theorem 10.16 and Theorem 2.38 gives us that Corollary 10.17. It is consistent with ZFC that there is no countable almost maximal topological group. Remark 10.18. It is easy to see that Theorem 10.15 and Theorem 10.16 remain to be true if to replace ‘countable almost maximal topological group’ by ‘countable topological group .G; T / with ﬁnite K.Ult.T //’.

10.3

Semilattice Decompositions and Burnside Semigroups

This is a preliminary section for the next one. A partially ordered set is a semilattice if every 2-element subset ¹a; bº has a greatest lower bound a^b. Obviously, every semilattice is a commutative band with respect to the operation ^. Conversely, every commutative band is a semilattice with respect to the standard ordering on idempotents. That is, a b if and only if ab D ba D a, and then a ^ b D ab. We shall identify semilattices with commutative bands. A semilattice decomposition of a semigroup S is a homomorphism f W S ! of S onto a semilattice . Equivalently, a semilattice decomposition of S is a partition ¹S˛ W ˛ 2 º of S into subsemigroups with the property that for every ˛; ˇ 2 there is 2 such that S˛ Sˇ S and Sˇ S˛ S . A semigroup which is a union of groups is called completely regular. Theorem 10.19. Every completely regular semigroup decomposes into a semilattice of completely simple semigroups. Proof. Let S be a completely regular semigroup. For every a 2 S, let J.a/ D SaS . We ﬁrst note that if H is a subgroup of S containing a, then H J.a/ and for every h 2 H , J.h/ D J.a/. For h D haa1 2 J.a/ and a D ahh1 2 J.h/. It follows from J.a2 / D J.a/ that J.ab/ D J.ba/ for all a; b 2 S. For ab ab D a ba b 2 J.ba/, so ab 2 J.ba/, and similarly ba 2 J.ab/. Now we show that J.ab/ D J.a/ \ J.b/. That J.ab/ J.a/ \ J.b/ is obvious. Conversely, let c 2 J.a/ \ J.b/. Write c D uav D xby for some u; v; x; y 2 S.

Section 10.3 Semilattice Decompositions and Burnside Semigroups

155

Then c 2 D xbyuav 2 J.byua/ D J.abyu/. Consequently, c 2 J.abyu/ J.ab/, and so J.a/ \ J.b/ J.ab/. Hence J.ab/ D J.a/ \ J.b/. It follows that D ¹J.a/ W a 2 Sº is the semilattice of principal ideals of S under intersection and S 3 a 7! J.a/ 2 is a surjective homomorphism. For every a 2 S, the preimage of the element J.a/ 2 is the set Ja D ¹b 2 S W J.b/ D J.a/º. Being the preimage of an idempotent, Ja is a subsemigroup of S . To see that Ja is simple, let b 2 Ja . Write a D ubv for some u; v 2 S. Then a D eae D eubve where e is the identity of a group containing a. It follows that J.a/ J.eu/ J.e/ D J.a/, so eu 2 Ja . Similarly ve 2 Ja . Hence a 2 Ja bJa . Finally, to show that Ja is completely simple, it sufﬁces to show that Ja e is a minimal left ideal of Ja . Let b 2 Ja e. Write b D ue for some u 2 Ja and, since Ja is simple, e D vbw for some v; w 2 Ja . Then e D vuew D xew where x D vu 2 Ja . Let f be the identity of a group containing x and x 1 the inverse of x with respect to f . Then f e D f xew D xew D e and e D f e D x 1 xe D x 1 vue D x 1 vb 2 Ja b. A band is a completely simple semigroup if and only if it is rectangular, that is, isomorphic to the direct product of a left zero semigroup and a right zero semigroup. Corollary 10.20. Every band decomposes into a semilattice of rectangular bands. The next lemma tells us that a semilattice decomposition of a completely regular semigroup is in fact unique. Lemma 10.21. Let S be a completely regular semigroup and let ¹S˛ W ˛ 2 Y º be a decomposition of S into a semilattice of completely simple semigroups. Then the semigroups S˛ are precisely the maximal completely simple subsemigroups of S. Proof. Let f W S ! Y be a homomorphism of S onto a commutative band Y and let T be a simple subsemigroup of S. Then f .T / is a simple subsemigroup of Y . But simple commutative bands are trivial. Hence, T f 1 .˛/ for some ˛ 2 Y . Maximal completely simple subsemigroups of a completely regular semigroup S are called completely simple components of S. If S is a band, we say rectangular components instead of completely simple components. Recall that Green’s relations R; L; J on any semigroup S are deﬁned by aRb , aS 1 D bS 1 ; aLb , S 1 a D S 1 b; aJb , S 1 aS 1 D S 1 bS 1 : The proof of Theorem 10.19 shows that J-classes of a completely regular semigroup are its completely simple components, R-classes and L-classes are minimal right and left ideals of the components. Deﬁnition 10.22. Let 1 k < ! and 0 m < n < !. The Burnside semigroup B.k; m; n/ is the largest semigroup on k generators satisfying the identity x m D x n .

156

Chapter 10 Almost Maximal Topological Groups

Note that B.k; 1; 2/ is the free band on k generators and B.k; 0; n/ is the Burnside group B.k; n/ (see Example 1.25). Theorem 10.23. For every ﬁxed n 2, the following statements are equivalent: (1) the semigroups B.k; 1; n/ are ﬁnite for all k 1, and (2) the groups B.k; 0; n 1/ are ﬁnite for all k 1. Proof. The implication .1/ ) .2/ is obvious. We need to prove .2/ ) .1/. Let B D B.k; 1; n/, let F be the free semigroup on a k-element alphabet A, and let h W F ! B be the canonical homomorphism. Note that for any u; v 2 F , h.u/ D h.v/ if and only if v can be obtained from u by a succession of elementary operations in each of which a subword w of a word is replaced by w n , or vice versa. For every w 2 F , let ct.w/ denote the set of letters of A appearing in w. Lemma 10.24. Let u; v 2 F . Then h.u/Jh.v/ if and only if ct.u/ D ct.v/. Proof. If h.u/Jh.v/, then h.u/ D h.v1 vv2 / and h.v/ D h.u1 uu2 /. Since the function ct is invariant under elementary operations, ct.v1 vv2 / D ct.u/, so ct.v/ ct.u/. Similarly, ct.u/ ct.v/. Hence, ct.u/ D ct.v/. Conversely, suppose that ct.u/ D ct.v/ D ¹x1 ; : : : ; xn º. Since B is a semilattice of its J-classes, it then follows that h.u/Jh.x1 / h.xn / and h.v/Jh.x1 / h.xn /. Hence, h.u/Jh.v/. By Lemma 10.24, the mapping B 3 h.w/ 7! ct.w/ A gives us the semilattice decomposition of B into completely simple components (D J-classes). The semilattice involved is the set of nonempty subsets of A under union. We need to show that J-classes of B are ﬁnite. For every nonempty C A, let JC denote the J-class of B corresponding to C . Clearly, for every a 2 A, J¹aº is just the group ¹a; a2 ; : : : ; an1 º (with identity an1 ). For every w 2 F , let .w/ denote the letter of A which has the latest ﬁrst appearance in w, 0 .w/ the subword of w that precedes the ﬁrst appearance of .w/, and .w/ the subword 0 .w/ .w/. Let .w/ denote the letter of A which has the earliest last appearance in w, 0 .w/ the subword of w that follows the last appearance of .w/, and .w/ the subword .w/0 .w/. Lemma 10.25. Let C A with jC j > 1 and let u; v 2 F with ct.u/ D ct.v/ D C . Then (1) h.u/Rh.v/ if and only if .u/ D .v/ and h.0 .u// D h.0 .v//, (2) h.u/Lh.v/ if and only if .u/ D .v/ and h.0 .u// D h.0 .v//.

Section 10.3 Semilattice Decompositions and Burnside Semigroups

157

Proof. (1) If h.u/Rh.v/, then h.u/ D h.vw/ for some w 2 F with ct.w/ D C . Clearly, .vw/ D .v/. It is easy to see that the functions and h0 are invariant under elementary operations. Consequently, .u/ D .v/ and h.0 .u// D h.0 .v//. Conversely, suppose that .u/ D .v/ and h.0 .u// D h.0 .v//. Then h..u// D h..v//. Consequently, in order to ﬁnish the proof it sufﬁces to show that for every w 2 F , h.w/Rh..w//. We proceed by the induction on the length jwj of w. The statement is obviously true if jwj D 1. Let jwj > 1 and suppose that the statement holds for all words of length < jwj. If w D .w/, there is nothing to prove. Let w ¤ .w/. Then w D w1 aw2 a for some words w1 ; w2 , possibly empty. Consequently, .w/ D .w1 aw2 / and by the inductive assumption, h..w1 aw2 //Rh.w1 aw2 /, so h..w//Rh.w1 aw2 /. But h.w1 aw2 / D h.w1 .aw2 /n / D h.w1 aw2 a/h.w2 .aw2 /n2 / D h.w/h.w2 .aw2 /n2 / and h.w/ D h.w1 aw2 /h.a/, so h.w1 aw2 /Rh.w/. Hence h..w//Rh.w/. The proof of (2) is similar. Lemma 10.25 gives us a one-to-one correspondence between R-classes (L-classes) of the J-class JC and the pairs .x; a/ where a 2 C and x 2 JC n¹aº . Lemma 10.26. Let S be a ﬁnitely generated completely regular semigroup and let C be a be a completely simple component of S. Suppose that C contains a ﬁnite number of minimal left (right) ideals. Then the structure group of C is ﬁnitely generated. Proof. Let T be the subsemigroup of S consisting of all completely simple components of S over C , let X be a ﬁnite generating subset of S, and let Y D X \ T . Then Y is a generating subset of T and C D K.T /. Suppose that C contains a ﬁnite number of minimal left ideals. Let H be a maximal subgroup of C , let R be the minimal right ideal of C containing H , and let E D E.R/. Then T e, where e 2 E, are the minimal left ideals of C and R \ .T e/, where e 2 E, the maximal subgroups of R. Let e1 2 E be the identity of H . We claim that ¹exe1 W x 2 Y [ Y 1 ; e 2 Eº is a generating subset of H . To see this, let h 2 H . Write h D x1 xn for some x1 ; : : : ; xn 2 Y [ Y 1 . Then h D e1 he1 D e1 x1 xn e1 . Deﬁne inductively e2 ; : : : ; en 2 E by ei xi 2 T eiC1 . Then ei xi D ei xi eiC1 D ei xi e1 eiC1 , and so h D .e1 x1 e1 /.e2 x2 e1 / .en xn e1 /. Now for each i D 1; : : : ; k, pick Ci A with jCi j D i and let ci D jJCi j. Clearly, c1 D n 1. Fix i > 1 and assume that ci1 is ﬁnite. Then by Lemma 10.25, the number of minimal right (left) ideals of JCi is i ci1 , and by Lemma 10.26, the cardinality gi of the structure group of JCi is ﬁnite, so ci D .i ci1 /2 gi is also ﬁnite.

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Chapter 10 Almost Maximal Topological Groups

From Theorem 10.23 we obtain the following. Corollary 10.27. The semigroups B.k; 1; 2/ and B.k; 1; 3/ are ﬁnite for all k 1. Proof. Since B.k; 0; 2/ is the Boolean group on k generators, jB.k; 0; 2/j D 2k . Hence, the result follows from Theorem 10.23.

10.4

Projectives

In this section we describe ﬁnite bands which are H-projectives. Let V denote the set of words of the form i1 i2 ip p p1 1 where p 2 N and iq ; q 2 ! for each q D 1; : : : ; p. Deﬁne the operation on V by 8 ˆ if p D q

159

Section 10.4 Projectives

Lemma 10.29.

(a) For every x 2 S, x ? x D xx ?? D x.

(b) If x 2 Sp and y 2 Sq , where q < p, then .xy/?? D x ?? y and .yx/? D yx ? . (c) If x; y 2 Sp , then x ?? y ? D ep1 . Proof. (a) is obvious. (b) Let x D i1 ip p 0 0s 1 and y D j1 jq q 1 , where s ¤ 0. Then x ?? D 1 1s 1 and xy D i1 ip p qC1 q 1 . If s D q, then x ?? y D 1 1q 1 and .xy/?? D 1 1q 1 . If s > q, then x ?? y D 1 1s qC1 q 1 and .xy/?? D 1 1s qC1 q 1 . If s < q, then jsC1 D D jq D 1, so x ?? y D 1 1q 1 , and .xy/?? D 1 1q 1 . The proof that .yx/? D yx ? is similar. (c) Let x D i1 ip p 0 0s 1 and y D j1 j t 0 0jp p 1 , where s ; j t ¤ 0. Then x ?? D 1 1s 1 , y ? D j1 j t 1 1, so 8 ˆ ˆ <1 11 1 ?? ? x y D 1 1s tC1 1 1 ˆ ˆ :1 1j j 11 sC1

t

if s D t if s > t if s < t:

Now, if s > t , then s D D tC1 D 1, and if s < t , then jsC1 D D j t D 1. Furthermore, s D p 1 or t D p 1. If x D i1 ip p 1 2 Vp , we put x 0 D i1 ip and x 00 D p 1 , and for each q D 1; : : : ; p, put xq0 D iq and xq00 D q . If x 2 Sp , we put R.x/ D ¹y 2 Sp W y 0 D x 0 º and L.x/ D ¹y 2 Sp W y 00 D x 00 º. Note that these are respectively minimal right and minimal left ideals of Sp containing x. p1 Deﬁne h on S0 by h.;/ D 1T . Suppose that h has been deﬁned on S0 . We shall show that h can be extended to Sp . Let Ip D ¹xp0 W x 2 Sp º. For each i 2 Ip , choose zi 2 Sp such that .zi /p0 D i and min¹q 2 Œ0; p 1 W .zi /0t D 0 for all t 2 Œq C 1; p 1º is as small as possible. Then choose a minimal right ideal Rp .i / in g 1 .f .Sp // with g.Rp .i // f .R..zi ///. Note that for any x 2 Sp with xp0 D i , one has x ? R..zi // R.x/, so g.h.x ? /Rp .i // f .x ? /f .R..zi /// f .R.x//. Consequently, for any x 2 Sp , one has g.h.x ? /Rp .xp0 // f .R.x//. We deﬁne minimal left ideals Lp ./ in g 1 .f .Sp // in the dual way. Now for every x 2 Sp , we deﬁne h.x/ to be the idempotent of the group h.x ? /Rp .xp0 /Lp .xp00 /h.x ?? /. Since gh.x/ 2 g.h.x ? /Rp .xp0 //g.Lp .xp00 /h.x ?? // f .R.x//f .L.x/ D f .¹xº/; we have that gh.x/ D f .x/.

160

Chapter 10 Almost Maximal Topological Groups p

Now we shall show that h.x/h.y/ D h.xy/ for every x 2 Sp and y 2 S0 . The proof that h.y/h.x/ D h.yx/ is similar. p1 First let y 2 S0 . We have that h.x/h.y/ 2 h.x ? /Rp .xp0 /Lp .xp00 /h.x ?? /h.y/. ? ? But x D .xy/ , xp0 D .xy/p0 , xp00 D .xy/p00 , and x ?? y D .xy/?? by Lemma 10.29, so h.x ?? /h.y/ D h.x ?? y/ D h..xy/?? /. It follows that h.x/h.y/ and h.xy/ belong to the same group in g 1 .f .Sp //. Therefore, it sufﬁces to show that h.x/h.y/ is an idempotent. We show this by proving that h.x/h.y/h.x/ D h.x/. Write h.x/ D h.x ? /zh.x ?? / for some z 2 Rp .xp0 /Lp .xp00 /. Then h.x/h.y/h.x/ D h.x ? /zh.x ?? /h.y/h.x ? /zh.x ?? / D h.x ? /zh.x ?? yx ? /zh.x ?? /: Since x ?? yx ? D .xy/?? x ? D ep1 D x ?? x ? by Lemma 10.29, h.x/h.y/h.x/ D h.x ? /zh.x ?? x ? /zh.x ?? / D h.x ? /zh.x ?? /h.x ? /zh.x ?? / D h.x/h.x/ D h.x/: Now let y 2 Sp . Again, we have that h.x/h.y/ 2 h.x ? /Rp .xp0 /Lp .yp00 /h.y ?? / and also x ? D .xy/? , xp0 D .xy/p0 , yp00 D .xy/p00 , and y ?? D .xy/?? . So h.x/h.y/ and h.xy/ belong to the same group. We again show that h.x/h.y/ is an idempotent by proving that h.x/h.y/h.x/ D h.x/. We know that either zp00 D 1 for all z 2 Sp or zp0 D 1 for all z 2 Sp . Suppose that the ﬁrst possibility holds (considering the second is similar). Then h.y/ 2 Lp .1/h.y ?? / and h.x/ 2 Lp .1/h.x ?? /. Consequently, h.y/h.x ? / and h.x/h.y ? / belong to the same minimal left ideal Lp .1/h.ep1 / in g 1 .f .Sp //. We have seen that these elements are idempotents, so h.x/h.y ? /h.y/h.x ? / D h.x/h.y ? /. Hence, h.x/h.y/h.x/ D h.x/h.y ? /h.y/h.x ? /h.x/ D h.x/h.y ? /h.x/ D h.x/h.x ?? /h.y ? /h.x/ D h.x/h.ep1 /h.x/: This statement holds with y replaced by x, and so h.x/ D h.x/h.ep1 /h.x/ D h.x/h.y/h.x/: Theorem 10.30. Let S be a ﬁnite band. If S is an absolute coretract in F, then S is isomorphic to some semigroup from P. Proof. Let k D jSj and B D B.k; 1; 3/. By Corollary 10.27, B is ﬁnite. We can deﬁne a surjective homomorphism f W B ! S. Then, since S is an absolute coretract in F, there exists a homomorphism g W S ! B such that f ı g D idS . Identifying S and g.S /, we may suppose that S is a subsemigroup of B and f jS D idS . Let F be the free semigroup on a k-element alphabet A and let h W F ! B be the canonical homomorphism. Note that h.u/ D h.v/ if and only if v can be obtained from u by

161

Section 10.4 Projectives

a succession of elementary operations in each of which a subword w of a word is replaced by w 3 , or vice versa. By Lemma 10.24, h.u/ and h.v/ belong to the same completely simple component of B if and only if ct.u/ D ct.v/. Recall that ct.v/ denotes the set of letters from A appearing in v. For any w 2 F and C A, let wjC denote the word obtained from w by removing all letters from A n C and let ˛.w; C / and ˇ.w; C / denote the ﬁrst and the last letters in wjC , respectively. It is easy to see that if h.u/ D h.v/, then ˛.u; C / D ˛.v; C / and ˇ.u; C / D ˇ.v; C /. For any w 2 F , C A and C 2 , let .w; C; / denote the number of pairs of neighboring letters in wjC belonging to . Lemma 10.31. If h.u/ D h.v/, then .u; C; / .v; C; / .mod 2/. Proof. It sufﬁces to consider the case where u D w1 ww2 , v D w1 w 3 w2 . Put .t / D .t; C; /. Then ´ .u/ C 2.w/ C 2 if wjC ¤ ; and .ˇ.w; C /; ˛.w; C // 2 .v/ D .u/ C 2.w/ otherwise. Lemma 10.32. S is a chain of its rectangular components. Proof. Assume the contrary. Then there exist u; v 2 h1 .S / with a 2 ct.u/ n ct.v/ and b 2 ct.v/ n ct.u/. Put .w/ D .w; ¹a; bº; ¹.a; b/º/. Then .uv/ D 1 and .uvuv/ D 2, although h.uvuv/ D h.uv/, a contradiction. Let S1 > S2 > > Sl be the rectangular components of S and for each p 2 Œ1; l, let Ap D ¹a 2 A W f h.a/ 2 Sp º: Observe that for any u 2 h1 .S /, h.u/ 2 Sp if and only if p D max¹q l W ct.u/\Aq ¤ ;º. Indeed, if u D a1 an , then h.u/ D f h.u/ D f h.a1 / f h.an /. Also if 1 p q l, let Apq D

q [

Ar

and

Spq D

rDp

q [

Sr ;

rDp

and let Mp D ¹˛.u; Apl / W u 2 h1 .Spl /º

and

Np D ¹ˇ.u; Apl / W u 2 h1 .Spl /º:

Observe that Mp \ Ap ¤ ; and Np \ Ap ¤ ;. Lemma 10.33. For every p 2 Œ1; l, at least one of the sets Mp , Np is a singleton.

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Chapter 10 Almost Maximal Topological Groups

Proof. Choose u 2 h1 .Sl /. Let a D ˛.u; Apl / and b D ˇ.u; Apl /. Put .w/ D .w; Apl ; ¹.b; a/º/. Since .uu/ D 2.u/ C 1 .u/ .mod 2/, .u/ is odd. Suppose that there exist v1 ; v2 2 h1 .Spl / with ˛.v1 ; Apl / ¤ a and ˇ.v2 ; Apl / ¤ b. Put v D v1 v2 . Since .vv/ D 2.v/ .v/ .mod 2/, .v/ is even. Then .uvu/ D 2.u/ C .v/ is also even. On the other hand, in S, as in any chain of rectangular bands, the following statement holds: if x; z 2 Sq ; y 2 Sr , and r q, then xyz D xz. Therefore h.uvu/ D h.uu/ D h.u/, and so .uvu/ .u/ .mod 2/, a contradiction. Lemma 10.34. If x 2 Sp ; y 2 Sq ; z 2 Sr , and q p; r, then xyz D xz. Proof. Adjoin identities ;, 1B D 1S to F; B; S and to extend h; f in the obvious way. Also put S0 D ¹1S º. Then the lemma is obviously true if q D 0. Fix q > 0 and assume that the lemma holds for all smaller values of q. Pick u 2 h1 .x/, v 2 h1 .y/ and w 2 h1 .z/. By Lemma 10.33, one of the sets Mq , Nq is a singleton. Suppose that Nq D ¹aº. Then we can write u D u1 au2 and v D v1 av2 , where q1 ct.u2 /; ct.v2 / A1 . Since x D f h.u/ and y D f h.v/, it follows from this that x D x1 sx2 and y D y1 sy2 , where s D f h.a/ 2 Sq , x2 D f h.u2 /; y2 D f h.v2 / 2 q1 q S0 and y1 D f h.v1 / 2 S0 . So xyz D x1 sx2 y1 sy2 z and xz D x1 sx2 z. It is clear that sx2 y1 s D s. By our inductive assumption, sy2 z D sz and sx2 z D sz. Hence xyz D x1 sz and xz D x1 sz. The case jMq j D 1 is similar. Enumerate sets Mp \ Ap and Np \ Ap without repetitions as ¹api W 1 i mp º and ¹bp W 1 np º so that ap1 D ˛.u; Ap / and bp1 D ˇ.v; Ap / for some u; v 2 h1 .Sp /. Deﬁne functions 'p and p on Spl as follows. Let x 2 Spl . Pick u 2 h1 .x/ and put ´ ´ 0 if ˛.u; Apl / … Ap 0 if ˇ.u; Apl / … Ap and .x/ D 'p .x/ D p i if ˛.u; Apl / D api if ˇ.u; Apl / D bp : We now deﬁne the mapping W S ! V by putting for every x 2 Sp , .x/ D '1 .x/'2 .x/ 'p .x/

p .x/ p1 .x/

1 .x/:

It is clear that both 'p .x/ ¤ 0 and p .x/ ¤ 0. By Lemma 10.33, either 'p .y/ D 1 for all y 2 Spl or p .y/ D 1 for all y 2 Spl . Lemma 10.35. is injective. Proof. Let x 2 Sp and pick u 2 h1 .x/. Let p1 < p2 < < ps D p be all r 2 Œ1; p with 'r .x/ ¤ 0, q1 < q2 < < q t D p all r 2 Œ1; p with r .x/ ¤ 0, 'pj .x/ D ij and qk .x/ D k . Then u D ap1 i1 u1 ap2 i2 u2 us1 aps is wbq t t v t v2 bq2 2 v1 bq1 1 ;

163

Section 10.4 Projectives p

q

where ct.uj / A1j and ct.vk / A1k . But then, by Lemma 10.34, x D f h.ap1 i1 ap2 i2 aps is bq t t b 2 q2 bq1 1 /; and consequently, x is uniquely determined by .x/. That is a homomorphism follows from the next lemma. Lemma 10.36. Let x 2 Sp and y 2 Sq . Then (a) 'r .xy/ D 'r .x/ if r p, (b) 'r .xy/ D 'r .y/ if p < r q, (c)

r .xy/

D

r .y/

if r q, and

(d)

r .xy/

D

r .x/

if q < r p.

Proof. Let u 2 h1 .x/, v 2 h1 .y/, and w D uv. If r p, then ˛.w; Alr / occurs in u, because ct.u/ \ Ap ¤ ;, so 'r .xy/ D 'r .x/. If p < r q, then ˛.w; Alr / occurs in v, because ct.v/ \ Alr D ;, so 'r .xy/ D 'r .y/. The check of (c) and (d) is similar. It remains to verify that the semigroup .S/ satisﬁes condition (2) in the deﬁnition of the family P. Let x 2 Sp and let 'q .x/ D a ¤ 0 for some q 2 Œ1; p (the case .x/ ¤ 0 is similar). Pick u 2 h1 .x/ and write it in the form u D u1 au2 , q1 where ct.u1 / A1 . Deﬁne y 2 Sq by y D f h.u1 a/. Since x D f h.u1 au2 /, yx D x. By Lemma 10.36 (a), 'r .x/ D 'r .y/ for each r q. By our choice of br1 , there exists vr 2 h1 .Sr / such that ˇ.vr ; Ar / D br1 . Let v D vq vq1 v1 and deﬁne z 2 Sq by z D h.v/. Then yz 2 Sq and by Lemma 10.36, .yz/ D '1 .x/ 'q .x/1 1. From Theorem 10.28 and Theorem 10.30 we obtain the following result. Theorem 10.37. Let S be a ﬁnite band. Then the following statements are equivalent: (1) S is isomorphic to some semigroup from P, (2) S is an H-projective, (3) S is an absolute H-coretract, (4) S is a projective in C, (5) S is an absolute coretract in C, (6) S is a projective in F, (7) S is an absolute coretract in F.

164

Chapter 10 Almost Maximal Topological Groups

Proof. We proceed by the circuits .1/ ) .4/ ) .2/ ) .6/ ) .7/ ) .1/ and .1/ ) .4/ ) .5/ ) .3/ ) .7/ ) .1/. The implications .1/ ) .4/ and .7/ ) .1/ are Theorem 10.28 and Theorem 10.28, .2/ ) .6/ and .3/ ) .7/ is Lemma 10.9, and the remaining implications are obvious. Using Theorem 10.37, we can summarize Theorem 10.10, Theorem 10.12 and Theorem 10.4 as follows: Theorem 10.38. The ultraﬁlter semigroup of any countable regular almost maximal left topological group is isomorphic to some semigroup from L P. Assuming p D c, for every semigroup S 2 P, there is a group topology T on ! Z2 such that Ult.T / is isomorphic to S. From Theorem 10.38 and Proposition 7.7 we obtain the following. Corollary 10.39. Assuming p D c, for each combination of the properties of being extremally disconnected, irresolvable, and nodec, except for the combination .; ; C/, there is a corresponding almost maximal topological group. There is no countable regular almost maximal left topological group corresponding to the combination .; ; C/. Proof. A maximal topological group corresponds to the combination .C; C; C/. For the combinations .; C; C/, .C; ; C/ and .C; C; /, pick topological groups whose ultraﬁlter semigroups are the 2-element left zero semigroup, right zero semigroup and chain of idempotents, respectively. For the combinations .C; ; / and .; C; /, pick the semigroups ¹11; 1111; 1121º and ¹11; 1111; 1211º in P. These are 3-element semigroups consisting of 2 components with the second components being the 2-element right zero semigroup and left zero semigroup, respectively. For the combination .; ; /, pick the semigroup ¹11; 1111; 1110; 1211; 1210º in P. This is a 5-element semigroup consisting of 2 components with the second component being the 2 2 rectangular band. Finally, every rectangular band in P is either a right zero semigroup or a left zero semigroup. Consequently, if a countable regular almost maximal left topological group is nodec, it is either extremally disconnected or irresolvable. As a consequence we also obtain from Theorem 10.37 the following result. Theorem 10.40. Let G be any inﬁnite group, let Q 2 P, and let S be a subsemigroup of Q. Then there is in ZFC a Hausdorff left invariant topology T on G such that Ult.T / U.G/ and Ult.T / is isomorphic to S. Proof. By Theorem 7.26, there is a zero-dimensional Hausdorff left invariant topology T0 on G such that T D Ult.T0 / is topologically and algebraically isomorphic to

Section 10.5 Topological Invariantness of Ult.T /

165

H and T U.G/. Pick a surjective continuous homomorphism g W T ! Q (Theorem 7.24). By Theorem 10.37, Q is an absolute coretract in C. Consequently, there is an injective homomorphism h W Q ! T (such that g ı h D idQ ). By Proposition 7.8, there is a left invariant topology T on G such that Ult.T / D h.S /. Since T0 T , T is Hausdorff. It turns out that Theorem 10.37 remains to be true with ‘ﬁnite band’ replaced by ‘ﬁnite semigroup’. Let FR denote the category of ﬁnite regular semigroups. Theorem 10.41. Let S be a ﬁnite semigroup. Then the following statements are equivalent: (1) S is isomorphic to some semigroup from P, (2) S is an H-projective, (3) S is an absolute H-coretract, (4) S is a projective in C, (5) S is an absolute coretract in C, (6) S is a projective in F, (7) S is an absolute coretract in F, (8) S is a projective in FR. Proof. See [95].

10.5

Topological Invariantness of Ult.T /

In this section we show that the ultraﬁlter semigroup of a countable regular almost maximal left topological group is its topological invariant. We start by pointing out a complete system of nonisomorphic representatives of P. l Let M denote the set of all matrices M D .mp;q /p;qD0 without the main diagonal l .mp;p /pD0 , where l 2 N and mp;q 2 !, satisfying the following conditions for every p 2 Œ1; l: (a) m0;p m1;p mp1;p 2 N and mp;0 mp;1 mp;p1 2 N, (b) either mp1;p D 1 and mp1;pC1 D D mp1;l D 0 or mp;p1 D 1 and mpC1;p1 D D ml;p1 D 0.

166

Chapter 10 Almost Maximal Topological Groups

These are precisely matrices of the form 0

BC B B B B :: B: B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @

1 C :: :

0 1 :: :

:: :

0 0 0 :: :

1 C C 1 0 1 :: :: 0 : :

0

1

0 0 0 :: :

0

C :: :

:: :

0

:: :

:: :

:: :

0 0 0 C 0 0 0 1 C :: : 0

1 C :: :

0 0 1 :: :

:: :

0 0 0 :: :

C 0

0

C C C C C C C C C C C C C C C C 0 C C C C C 0 C C 0 C C 0 C C :: : 0 C C C 1 C C C C 1 C C C 0 1 C A :: :: :: : : : : : : 0

0

and their transposes, where C is a positive integer, is a nonnegative integer, and all rows and columns are nondecreasing up to the main diagonal. l 2 M, let V .M / denote the subsemigroup of V Now, for every M D .mp;q /p;qD0 consisting of all words i1 i2 ip p p1 1 , where p 2 Œ1; l, such that (i) both ip ¤ 0 and p ¤ 0, (ii) for every q < r p, if i t D 0 for all t 2 Œq C 1; r 1, then ir mq;r , and dually, if t D 0 for all t 2 Œq C 1; r 1, then r mr;q . It is obvious that for every M 2 M, V .M / 2 P. Proposition 10.42. For every S 2 P, there is a unique M 2 M such that S is isomorphic to V .M /. Proof. Let l D max¹p 2 N W S \ Vp ¤ ;º and, for each p 2 Œ1; l, let Sp D S \ Vp . For every q < p l, let Iq;p D ¹ip W i1 iq 0 0ip p 1 2 Sp º ƒq;p D ¹p W i1 ip p 0 0q 1 2 Sp º

and

167

Section 10.5 Topological Invariantness of Ult.T /

and let mq;p D jIq;p j and mp;q D jƒp;q j. For every p 2 Œ1; l, choose bijections fp W Ip1;p ! Œ1; mp1;p

and

gp W ƒp;p1 ! Œ1; mp;p1

fp .Iq;p / D Œ1; mq;p

and

gp .ƒp;q / D Œ1; mp;q

such that for each q p 1. Also put fp .0/ D 0 and gp .0/ D 0. An easy check shows that l M D .mp;q /p;qD0 2 M and S 3 i1 ip p 1 7! f1 .i1 / fp .ip /gp .p / g1 .1 / 2 V .M / is an isomorphism. Next, adjoin an identity ; to S and put S0 D ¹;º. For every p 2 Œ0; l, let rp denote the number of minimal right ideals of Sp , and for every different p; q 2 Œ0; l, let ´ Sq1 Sp Sq n kDpC1 Sk Sq if p < q Sp;q D Sp1 Sp Sq n kDqC1 Sp Sk if p > q: Then the uniqueness of M follows from the next lemma. Lemma 10.43. For every different p; q 2 Œ0; l, one has mp;q D

jSp;q j rq : rp jSq j

Proof. To compute jSp;q j, one may suppose that S D V .M /. Then ´ if p < q ¹i1 ip 0 0iq q 1 2 Sq W ip ¤ 0º Sp;q D ¹i1 ip p 0 0q 1 2 Sp W q ¤ 0º if p > q: Since j¹i1 ip W i1 ip p 1 2 Sp ºj D rp j¹q 1 W i1 iq q 1 2 Sq ºj D

and jSq j ; rq

it follows that jSp;q j D rp mp;q

jSq j : rq

Note that it also follows from Lemma 10.43 that the matrix M is uniquely determined by the numbers l and rp and the sets Sp and Sp Sq , where p; q 2 Œ1; l and p ¤ q.

168

Chapter 10 Almost Maximal Topological Groups

Theorem 10.44. If countable regular almost maximal left topological groups are homeomorphic, then their ultraﬁlter semigroups are isomorphic. Proof. Let .G; T / be a countable regular almost maximal left topological group and let S D Ult.T /. By Theorem 10.38, S is isomorphic to some semigroup from P. Let S1 > > Sl be the rectangular components of S. For every p 2 Œ1; l, let Tp be the left invariant topology on G with Ult.Tp / D Sp . For every different p; q 2 Œ1; l, let Tp Tq be the left invariant topology on G with Ult.Tp Tq / D Sp Sq . By Lemma 7.3, the number rp of minimal right ideals of Sp is equal to the number of maximal open ﬁlters on .G; Tp / converging to the identity. Then by Lemma 10.43, in order to show that Ult.T / is a topological invariant of .G; T /, it sufﬁces to show that topologies Tp and Tp Tq are determined purely topologically. O O SpFor every p D Œ1; l, let Tp be the left invariant topology on G with Ult.Tp / D S . Then T D T and by Proposition 7.7, for p < l, a nonprincipal ultraﬁlter l kD1 k U on G converges to a point x 2 G in TOp if and only if U converges to x in TOpC1 and U is nowhere dense in TOpC1 . Consequently, topologies TOp , p D l; l 1; : : : ; 1, are determined purely topologically. But then this holds for topologies Tp as well, since a nonprincipal ultraﬁlter U on G converges to a point x 2 G in Tp if and only if U converges to x in TOp and U is dense in TOp . Finally, a neighborhood base at a point x 2 G in the topology Tp Tq consists of subsets of the form [ ¹xº [ Vy n ¹yº y2U n¹xº

where U is a neighborhood of x in Tp and Vy is a neighborhood of y in Tq . Hence, topologies Tp Tq are also determined purely topologically. In Section 12.1 we will see that every countable homogeneous regular space admits a structure of a left topological group (Theorem 12.5). Deﬁnition 10.45. For every countable homogeneous regular space X , pick a group operation on X with continuous left translations and let Ult.X / denote the ultraﬁlter semigroup of the left topological group .X; /. By Theorem 10.44, Ult.X / does not depend, up to isomorphism, on the choice of the operation , so Ult.X / is a topological invariant of X .

References Theorem 10.4 is a result from [99] and Corollary 10.6 from [84]. Theorem 10.10 was proved in [99], and Theorem 10.12 in [90]. Theorem 10.15 and Theorem 10.16 are from [87].

Section 10.5 Topological Invariantness of Ult.T /

169

Theorem 10.19 is due to A. Clifford [9] and Corollary 10.20 to D. McLean [49]. Theorem 10.23 is a result of J. Green and D. Rees [31]. The deﬁnition of the family P, Theorem 10.28 and Theorem 10.30 are from [93]. Theorem 10.41 is a result from [95]. Its proof is based on Theorem 10.28, Theorem 10.30 and the fact that every projective in FR is a band. The latter is a result of P. Trotter [74, 75] who also characterized projectives in FR. Theorem 10.41 tells us among other things that the semigroups from P are the same that those characterized by Trotter. Theorem 10.44 was proved in [99]. The exposition of this chapter is based on the treatment in [92].

Chapter 11

Almost Maximal Spaces

In this chapter we show that for every inﬁnite group G and for every n 2 N, there is in ZFC a zero-dimensional Hausdorff left invariant topology T on G such that Ult.T / is a chain of n idempotents and Ult.T / U.G/. As a consequence we obtain that for every inﬁnite cardinal and for every n 2 N, there is a homogeneous zero-dimensional Hausdorff space of cardinality with exactly n nonprincipal ultraﬁlters converging to the same point, all of them being uniform. In particular, for every inﬁnite cardinal , there is a homogeneous regular maximal space of dispersion character .

11.1

Right Maximal Idempotents in H

Recall that given anL inﬁnite cardinal , H D Ult.T0 /, where T0 denotes the group topology on H D Z2 with a neighborhood base at 0 consisting of subgroups H˛ D ¹x 2 H W x. / D 0 for each < ˛º, ˛ < . When working with H , the following two functions are also usuful. Deﬁnition 11.1. Deﬁne functions ; W H n ¹0º ! by .x/ D min supp.x/ and

.x/ D max supp.x/

and let ; W ˇH n ¹0º ! ˇ denote their continuous extensions. The main properties of these functions are that for every x 2 ˇH n ¹0º and y 2 H , .x C y/ D .x/ and

.x C y/ D .y/:

In this section we show that for every !, there is a right maximal idempotent p 2 H such that C.p/ D ¹x 2 ˇH n¹0º W xCp D pº is a ﬁnite right zero semigroup, and if is not Ulam-measurable, every right maximal idempotent p 2 H enjoys this property. Note that H is left saturated in ˇH , so for every p 2 H , one has C.p/ H . The proof of the result about right maximal idempotents in H involves right cancelable ultraﬁlters in H . An element p of a semigroup S is called right cancelable if whenever q; r 2 S and qp D rp, one has q D r. Equivalently, p is right cancelable if the right translation by p is injective.

Section 11.1 Right Maximal Idempotents in H

171

Theorem 11.2. For every ultraﬁlter p 2 H , the following statements are equivalent: (1) p is right cancelable in ˇH , (2) p is right cancelable in H , (3) there is no idempotent q 2 H for which p D q C p, (4) there is no q 2 H for which p D q C p. (5) H C p ˇH is discrete, (6) H C p ˇH is strongly discrete, (7) p is strongly discrete. Proof. .1/ ) .2/ is obvious. .2/ ) .3/ Assume on the contrary that there is an idempotent q 2 H for which p D q C p. Clearly q 2 H . For every ˛ < , deﬁne e˛ 2 H by supp.e˛ / D ¹˛º, and let E˛ D ¹eˇ W ˛ ˇ < º. Pick any ultraﬁlter r on H extending the family of subsets E˛ , where ˛ < . Then r; r C q 2 H and r ¤ r C q. Indeed, Y D

[

.e˛ C H˛C1 n ¹0º/ 2 r C q

˛<

and jsupp.y/j > 1 for all y 2 Y , but jsupp.x/j D 1 for all x 2 E0 . On the other hand, it follows from p D q C p that r C p D r C q C p and, since p is right cancelable in H , we obtain that r D r C q, a contradiction. .3/ ) .4/ Assume on the contrary that there is q 2 H for which p D q C p. Then C.p/ ¤ ;. Since C.p/ is a closed subsemigroup of H , it has an idempotent, a contradiction. .4/ ) .5/ Assume on the contrary that H C p ˇH is not discrete. Then there is a 2 H such that a C p 2 cl..H n ¹aº/ C p/. Since cl..H n ¹aº/ C p/ D .ˇH n ¹aº/ C p; we obtain that there is r 2 ˇH n ¹aº such that a C p D r C p, so a C r C p D p. Let q D a C r. Then q C p D p, and since r ¤ a, q 2 C.p/ H H , a contradiction. .5/ ) .6/ Since H C p ˇH is discrete, for every x 2 H , there is Bx 2 p such that y C p … x C Bx for all y 2 H n ¹xº, that is, x C Bx … y C p for all y 2 H n ¹xº. For every x 2 H n ¹0º, let Fx D ¹0º [ ¹y 2 H n ¹0º W .y/ < .x/ and

supp.y/ supp.x/º:

Put A0 D B0 and inductively for every ˛ < and for every x 2 H with .x/ D ˛, choose Ax 2 p such that

172

Chapter 11 Almost Maximal Spaces

(i) Ax Bx \ H.x/C1 , and (ii) .x C Ax / \ .y C Ay / D ; for all y 2 Fx . This can be done because Fx is ﬁnite and y C Ay y C By … x C p for all y 2 Fx . We now claim that .x C Ax / \ .y C Ay / D ; for all different x; y 2 H . Indeed, without loss of generality one may suppose that x ¤ 0 and .y/ .x/ or y D 0. If y 2 Fx , the statement holds by (ii). Otherwise supp.y/ n supp.x/ ¤ ; or .y/ D .x/, in any case .y C H.y/C1 / \ .x C H.x/C1 / D ;; so the statement holds by (i). For every x 2 H , x C Ax is a neighborhood of x C p 2 ˇH , and all these neighborhoods are pairwise disjoint. Hence H C p ˇH is strongly discrete. .6/ ) .7/ Since H C p ˇH is strongly discrete, for every x 2 H , there is Ax 2 p such that the subsets x C Ax ˇH , where x 2 H , are pairwise disjoint. Then the subsets x C Ax H , where x 2 H , are pairwise disjoint. It follows that p is strongly discrete. .7/ ) .1/ Since p is strongly discrete, for every x 2 H , there is Ax 2 p such that the subsets x C Ax are pairwise disjoint. Let q; r 2 ˇH and q ¤ r. Choose disjoint Q 2 q and R 2 r and put [ [ x C Ax and B D x C Ax : AD x2Q

x2R

Then A 2 q C p, B 2 r C p and A \ B D ;, so q C p ¤ r C p. Hence p is right cancelable. Recall that given a group G and p 2 ˇG, T Œp is the largest left invariant topology on G in which p converges to 1, and Cp is the smallest closed subsemigroup of ˇG containing p. Corollary 11.3. Let p be a right cancelable ultraﬁlter in H . Then (1) the topology T Œp is zero-dimensional, and (2) there is a continuous homomorphism W Ult.T Œp/ ! ˇN such that .p/ D 1 and .Cp / D ˇN. Proof. By Theorem 11.2, p is a strongly discrete ultraﬁlter on H . Then apply Theorem 4.18 and Theorem 7.29. We now turn to the right maximal idempotents in H . Proposition 11.4. For every right maximal idempotent p 2 H , C.p/ is a right zero semigroup.

Section 11.1 Right Maximal Idempotents in H

173

Proof. Let C D C.p/ and let q 2 C . Suppose that q is not right cancelable in H . Then by Theorem 11.2, there is an idempotent r 2 H such that r Cq D q. It follows that r C q C p D q C p, and so r C p D p. Thus, p R r, and since p is right maximal, r R p, that is, p C r D r. From this we obtain that pCq DpCr Cq Dr Cq Dq and q C q D q C p C q D p C q D q; so q is an idempotent. Hence, p R q and q R p. It then follows that the elements of C which are not right cancelable in H form a right zero semigroup. Now we claim that no element of C is right cancelable in H . Indeed, assume on the contrary that some q 2 C is right cancelable in H . Then by Corollary 11.3, Cq admits a continuous homomorphism onto ˇN. Taking any nontrivial ﬁnite left zero semigroup in ˇN, we obtain, by the Lemma 10.2, that there is a nontrivial left zero semigroup in Cq C , a contradiction. Proposition 11.5. Let C be a compact right zero semigroup in H and let .C / D ¹uº. If u is countably incomplete, then C is ﬁnite. Note that for every right zero semigroup C H , .C / is a singleton. Indeed, if x; y 2 C , then y D x C y, and so .y/ D .x C y/ D .x/: Proof of Proposition 11.5. Assume on the contrary that C is inﬁnite. Pick any countably inﬁnite subset X C and pick p 2 .cl X / n X . Put Y D .H n ¹0º/ C p. Since cl Y D .ˇH n¹0º/Cp and p D p Cp, p 2 cl Y . Consequently, .cl X /\.cl Y / ¤ ;. Also we have that for every x 2 X , x … cl Y . Indeed, otherwise x D y C p for some y 2 ˇH and then x C p D y C p C p D y C p D x: But x C p D p ¤ x, since x 2 X C , p 2 .cl X / n X C and C is a right zero semigroup. Hence, in order to derive a contradiction, it sufﬁces, by Corollary 2.24, to construct a partition ¹An W n < !º of H n¹0º such that .cl X /\.cl Yn / D ; where Yn D An Cp. Since u is countably incomplete, there is a partition ¹Bn W n < !º of such that Bn … u for all n < !, equivalently u … Bn . Put An D 1 .Bn /. Then for every x 2 cl X , .x/ D u, and for every y 2 Yn , .y/ 2 Bn , so for every y 2 cl Yn , .y/ 2 Bn . Hence, .cl X / \ .cl Yn / D ;. Combining Proposition 11.4 and Proposition 11.5, we obtain the following result. Theorem 11.6. Let p be a right maximal idempotent in H . Then C.p/ is a compact right zero semigroup, and if .p/ is countably incomplete, C.p/ is ﬁnite.

174

11.2

Chapter 11 Almost Maximal Spaces

Projectivity of Ult.T /

Theorem 11.7. Let T be a translation invariant topology on H such that T0 T and let X be an open zero-dimensional neighborhood of 0 in T . Then for every homomorphism g W T ! Q of a semigroup T onto a semigroup Q and for every local homomorphism f W X ! Q, there is a local homomorphism h W X ! T such that f D g ı h. The proof of Theorem 11.7 is based on the following notion. Deﬁnition 11.8. A basis in X is a subset A X n ¹0º together with a partition ¹X.a/ W a 2 Aº of X n ¹0º such that for every a 2 A, X.a/ is a clopen neighborhood of a 2 X n ¹0º and X.a/ a X \ H.a/C1 . Lemma 11.9. Whenever ¹Ux W x 2 X n ¹0ºº is a family of neighborhoods of 0 2 X , there is a basis A in X such that for every a 2 A, X.a/ a Ua . Proof. Without loss of generality one may suppose that for every x 2 X n ¹0º, Ux is a clopen neighborhood of 0 2 X and x C Ux X n ¹0º. For every x 2 X n ¹0º, let Fx D ¹y 2 X n ¹0º W .y/ < .x/ and

supp.y/ supp.x/º:

Note that Fx is ﬁnite. For every ˛ < , let X˛ D ¹x 2 X n ¹0º W .x/ D ˛º: Now put A1 D ; and inductively, for every ˛ < , deﬁne a subset A˛ X˛ and for every a 2 A˛ , a clopen neighborhood X.a/ of a 2 X n ¹0º by S S (i) A˛ D X˛ n b2B˛ X.b/, where B˛ D ˇ <˛ Aˇ , and S (ii) X.a/ D .a C Ua \ H.a/C1 / n b2Fa X.b/. S We put A D ˛< A˛ . S It follows from (i) that for every a 2 A˛ , a … b2B˛ X.b/. Then, since Fa is ﬁnite, we obtain from (ii) that X.a/ is indeed a clopen neighborhood of a 2 X n ¹0º. It is clear also that X.a/ a Ua \ H.a/C1 and that the subsets X.a/, a 2 A, cover X n ¹0º. To see that they are disjoint, let a 2 A˛ , b 2 A˛ [ B˛ and a ¤ b. If b 2 Fa , then X.a/ \ X.b/ D ; by (ii). Otherwise supp.b/ n supp.a/ ¤ ; or b 2 A˛ , in any case .a C H.a/C1 / \ .b C H.b/C1 / D ;, so again X.a/ \ X.b/ D ;. Lemma 11.10. Let A be a basis in X . Then (1) every x 2 X n ¹0º can be uniquely written in the form x D a1 C C an , where a1 ; : : : ; an 2 A and ai C C an 2 X.ai / for each i D 1; : : : ; n 1, and (2) every mapping h0 W A ! S of A into a semigroup S extends to a local homomorphism h W X ! S by h.a1 C C an / D h0 .a1 / h0 .an /, where a1 ; : : : ; an 2 A and ai C C an 2 X.ai / for each i D 1; : : : ; n 1.

Section 11.2 Projectivity of Ult.T /

175

Proof. (1) Let x 2 X n ¹0º. Then x 2 X.a1 / for some a1 2 A. If x D a1 , we are done. Otherwise x D a1 C x1 , where x1 D x a1 2 X n ¹0º, and .a1 / < .x1 /. Suppose that we have written x as x D a1 C C ai C x i where a1 ; : : : ; ai 2 A, xi 2 X n ¹0º, and (i) .aj / < .aj C1 / for each j D 1; : : : ; i 1 and .ai / < .xi /, and (ii) aj C C ai C xi 2 X.aj / for each j D 1; : : : ; i . Then xi 2 X.aiC1 / for some aiC1 2 A. Since .aiC1 / D .xi /, .ai / < .aiC1 /. If xi D aiC1 , we are done: x D a1 C C aiC1 and aj C C aiC1 2 X.aj / for each j D 1; : : : ; i . Otherwise xi D aiC1 C xiC1 where xiC1 D xi aiC1 , then x D a1 C C aiC1 C xiC1 and (i) and (ii) are satisﬁed with i replaced by i C 1. After jsupp.x/j steps we obtain the required decomposition. Now suppose that x has two such decompositions, say x D a 1 C C an D b 1 C C b m : We show that n D m and ai D bi for each i D 1; : : : ; n. We proceed by induction on min¹n; mº. Let min¹n; mº D 1, say n D 1. We have that a1 2 X.a1 / and a1 D b1 C C bm 2 X.b1 /: Since ¹X.a/ W a 2 Aº is disjoint, it follows that a1 D b1 . But then also m D 1. Indeed, otherwise b2 C C bm D 0 which contradicts b2 C C bm 2 X.b2 /. Now let min¹n; mº > 1. Again we have that a1 C C an 2 X.a1 / and a1 C C an D b1 C C bm 2 X.b1 /; and so a1 D b1 . But then a2 C C an D b2 C C bm and we can apply the inductive assumption. (2) Let x 2 X n ¹0º. Write x D a1 C C an , where a1 ; : : : ; an 2 A and ai C C an 2 X.ai / for each i D 1; : : : ; n. Deﬁne the neighborhood U of 0 2 X by n \ U D .X.ai / .ai C C an // \ X: iD1

Now let y 2 U n ¹0º. Then ai C C an C y 2 X.ai / for each i D 1; : : : ; n. Write y D anC1 C C anCm , where anC1 ; : : : ; anCm 2 A and anCj C C anCm 2 X.anCj / for each j D 1; : : : ; m 1. We obtain that ai C C anCm 2 X.ai / for each i D 1; : : : ; n C m 1 and h.x C y/ D h.a1 C C an C anC1 C anCm / D h0 .a1 / h0 .an /h0 .anC1 / h0 .anCm / D h.x/h.y/:

176

Chapter 11 Almost Maximal Spaces

Now we are in a position to prove Theorem 11.7. Proof of Theorem 11.7. For every x 2 X n ¹0º, pick a neighborhood Ux of 0 2 X such that f .xy/ D f .x/f .y/ for all y 2 Ux n ¹0º. By Lemma 11.9, there is a basis A in X such that for every a 2 A, X.a/ a Ua . By Lemma 11.10, every x 2 X n ¹0º can be uniquely written as x D a1 C C an where a1 ; : : : ; an 2 A and ai C C an 2 X.ai / for each i D 1; : : : ; n 1. Then f .x/ D f .a1 / f .an /. Indeed, a1 C C an 2 X.a1 / and X.a1 / a1 C Ua1 , consequently a2 C C an 2 Ua1 . It follows that f .a1 C a2 C C an / D f .a1 /f .a2 C C an / and then by induction f .a1 C C an / D f .a1 / f .an /. Now for every a 2 A, pick ta 2 T such that g.ta / D f .a/. Deﬁne h W X ! T by h.a1 C C an / D ta1 tan where a1 ; : : : ; an 2 A and ai C C an 2 X.ai / for each i D 1; : : : ; n 1. By Lemma 11.10, h is a local homomorphism, and since g.ta1 tan / D g.ta1 / g.tan / D f .a1 / f .an /; it follows that g ı h D f . As a consequence we obtain from Theorem 11.7 the following result. Corollary 11.11. Let T be a locally zero-dimensional almost maximal translation invariant topology on H such that T0 T . Then Ult.T / is a projective in F. Proof. Let T and Q be ﬁnite semigroups, let ˛ W Ult.T / ! Q be a homomorphism, and let ˇ W T ! Q be a surjective homomorphism. By Lemma 10.11, there are an open zero-dimensional neighborhood X of zero of .H; T / and a local homomorphism f W X ! Q such that f D ˛, so ˛ is proper. Now by Theorem 11.7, there is a local homomorphism h W X ! T such that f D ˇ ı h. Deﬁne W Ult.T / ! T by

D h . Then ˛ D ˇ ı . We will need also the following proposition. Proposition 11.12. Every projective S in F is a chain of rectangular bands satisfying the following conditions: (i) whenever x; y; z 2 S and yRz, xy D xz implies y D z, and dually (ii) whenever x; y; z 2 S and yLz, yx D zx implies y D z. Proof. By Theorem 10.41, S is isomorphic to some semigroup from P. It is easy to see that every subsemigroup of V possesses these properties. We conclude this section by noting that Theorem 11.7 can be used also to prove the following result.

177

Section 11.3 The Semigroup C.p/

Theorem 11.13. For every inﬁnite cardinal , the semigroup H contains no nontrivial ﬁnite groups. Proof. Similar to the proof of Theorem 8.18 with Corollary 8.13 replaced by Theorem 11.7.

11.3

The Semigroup C.p/

Lemma 11.14. Let p 2 H and let C.p/ be ﬁnite. Then for every q; r 2 ˇG, the equality q C p D r C p implies that q 2 r C C 1 or r 2 q C C 1 , where C 1 D C 1 .p/. Proof. Assume the contrary. Then, since C 1 is ﬁnite, there exist A 2 q and B 2 r such that A \ .B C C 1 / D ; and B \ .A C C 1 / D ;: By Theorem 7.18, there is a zero-dimensional translation invariant topology T on H with Ult.T / D C.p/. It follows that for every x 2 A [ B, there exists a clopen neighborhood U of 0 2 H in T such that A \ .x C U / D ; if x 2 B;

and

B \ .x C U / D ; if x 2 A:

Enumerate A [ B as ¹x˛ W ˛ < º so that the sequence ..x˛ //˛< is nondecreasing. For each ˛ < , choose inductively a clopen neighborhood U˛ of 0 in T so that the following conditions are satisﬁed: (i) U˛ H.x˛ /C1 , (ii) A \ .x˛ C U˛ / D ; if x˛ 2 B, and B \ .x˛ C U˛ / D ; if x˛ 2 A, and (iii) .x˛ C U˛ / \ .x C U / D ; for all < ˛ such that supp.x / supp.x˛ / and elements x˛ ; x belong to different sets A; B. To this end, ﬁx ˛ < and suppose that we have already chosen U for all < ˛ satisfying (i)–(iii). Without loss of generality one may suppose also that x˛ 2 A. Let F D ¹ < ˛ W supp.x / supp.x˛ / and x 2 Bº: It follows from (ii) that x˛ …

[

.x C U /:

2F

Since F is ﬁnite and each U is closed, there is a clopen neighborhood U˛ of 0 such that [ .x˛ C U˛ / \ .x C U / D ;; 2F

which means that (iii) is satisﬁed. Obviously, one can choose U˛ to satisfy also (i) and (ii).

178

Chapter 11 Almost Maximal Spaces

We now claim that .x˛ C U˛ / \ .x C U / D ; whenever < ˛ < and elements x˛ ; x belong to different sets A; B. Indeed, if supp.x / supp.x˛ /, then .x˛ C U˛ / \ .x C U / D ; by (iii). If supp.x / n supp.x˛ / ¤ ;, then .x˛ C H.x˛ /C1 / \ .x C H.x /C1 / D ;, and consequently, .x˛ C U˛ / \ .x C U / D ; by (i). Thus, we have that [ [ .x˛ C U˛ / \ .x C U / D ;; x˛ 2A

x 2B

so q C p ¤ r C p, which is a contradiction. Theorem 11.15. Let p 2 H and let C.p/ be ﬁnite. Then (1) C.p/ is a projective in F, and (2) C.p/ is a chain of right zero semigroups. Proof. (1) By Theorem 7.18, there is a zero-dimensional translation invariant topology T on H such that Ult.T / D C.p/. Then apply Corollary 11.11. (2) Let C D C.p/. By (1) and Proposition 11.12, C is a chain of rectangular bands. We have to show that for every x; y 2 C , xLy implies x D y. Let K D K.C /. Pick any z 2 K. Then x C z; y C z 2 K and .x C z/L.y C z/. We have also that x C z C p D y C z C p. It follows from this and Lemma 11.14 that either x C z 2 y C z C C 1 or y C z 2 x C z C C 1 , where C 1 D C 1 .p/. Both y C z C C 1 and x C z C C 1 are R-classes of K. Therefore in any case, .x C z/R.y C z/. Since also .x Cz/L.y Cz/, we obtain that x Cz D y Cz and then, by Proposition 11.12 (ii), x D y. In the rest of this section, we show that for every n 2 N, there is an idempotent p 2 H such that C.p/ is a chain of n ﬁnite right zero semigroups. We ﬁrst prove several auxiliary statements. Lemma 11.16. Let p 2 H and let C.p/ be ﬁnite. Then for every q 2 H , j¹x 2 ˇH W x C p D qºj jC 1 .p/j: Proof. Let X D ¹x 2 ˇH W x C p D qº and let C 1 D C 1 .p/. Choose y 2 X with maximally possible jy C C 1 j. For every z 2 C 1 , one has y C z C p D y C p D q, so y C C 1 X . We claim that X D y C C 1 . To see this, let x 2 X . We have that x C p D y C p. Then by Lemma 11.14, either x 2 y C C 1 or y 2 x C C 1 . The ﬁrst possibility is what we wish to show. The second implies that y C C 1 x C C 1 . Since jy C C 1 j is maximally possible, we obtain that y C C 1 D x C C 1 , so again x 2 y C C 1 . It follows from X D y C C 1 that jX j jC 1 j.

179

Section 11.3 The Semigroup C.p/

Lemma 11.17. Let p; q 2 H and let C.p/; C.q/ be ﬁnite. Then jC 1 .p C q/j jC 1 .p/j jC 1 .q/j: Proof. We have that C 1 .p C q/ D ¹x 2 ˇH W x C p C q D p C qº D ¹x 2 ˇH W x C p 2 ¹y 2 ˇH W y C q D p C qºº: Let Y D ¹y 2 ˇH W y C q D p C qº and for each y 2 Y , let X.y/ D ¹x 2 ˇH W x C p D yº. Then [ C 1 .p C q/ D X.y/ y2Y

and, by Lemma 11.16, jY j

jC 1 .q/j

and jX.y/j C 1 .p/.

Lemma 11.18. Let G be a group, let p 2 G , and let Q D Ult.T Œp/. Then Q D .Qp/ [ ¹pº. Proof. Clearly .Qp/ [ ¹pº Q. We have to show that for every q 2 G n ..Qp/ [ ¹pº/, one has q … Q. Pick A 2 q such that 1 … A and A \ ..Qp/ [ ¹pº/ D ;. It sufﬁces to construct a neighborhood U of 1 in T Œp such that U \ A D ;. To this end, we use Theorem 4.8. Since A \ ..Qp/ [ ¹pº/ D ;, there is an open neighborhood V of 1 in T Œp such that A \ .V p/ D ;. For every x 2 V , pick M.x/ 2Sp such that xM.x/ V and .xM.x// \ A D ;. Put U D ŒM 1 . Then U ¹1º [ x2V .xM.x//. It follows that U \ A D ;. Lemma 11.19. For every p 2 H and q 2 Ult.T Œp/, one has .q/ D .p/. Proof. Let A 2 p. Choose M W H ! p such that M.0/ A and for every x 2 H n ¹0º and y 2 M.x/, .x/ < .y/. Then whenever 0 ¤ z 2 ŒM 0 , .z/ 2 .A/. Proposition 11.20. For every idempotent p 2 H with ﬁnite C.p/, there is a right maximal idempotent q 2 H with ﬁnite C.q/ such that for each x 2 C.q/, x

180

Chapter 11 Almost Maximal Spaces

We claim that r C p is right cancelable and p … .ˇH / C r C p. To see that r C p is right cancelable, suppose that u C r C p D v C r C p for some u; v 2 ˇH . Then by Lemma 11.14, either u C r 2 v C r C C 1 or v C r 2 u C r C C 1 , where C 1 D C 1 .p/. But .u C r/ D .v C r/ D .r/ and for every x 2 C.p/, .u C r C x/ D .v C r C x/ D .x/ 2 .C.p//: It follows that u C r D v C r and, since r is right cancelable, u D v. To see that p … .ˇH / C r C p, assume on the contrary that p D u C r C p for some u 2 ˇH . Then u C r 2 C.p/, which is a contradiction, since .u C r/ D .r/ and .r/ … .C.p//. Now let Q D Ult.T Œr C p/. Pick any right maximal idempotent q 2 Q. By Corollary 11.3 and Lemma 7.12, Q is left saturated. Consequently, C.q/ Q and q is right maximal in H . By Lemma 11.19, .q/ D .r C p/ D .r/, so .q/ is countably incomplete. Then by Theorem 11.6, C.q/ is ﬁnite. By Lemma 11.18, Q .ˇH / C r C p. For every x 2 .ˇH / C r C p, one has x C p D x. Indeed, x D u C r C p for some u 2 ˇH and then x C p D u C r C p C p D u C r C p D x. It follows that for every x 2 C.q/, x C p D x, so x L p. But p C x ¤ p, since p C x 2 .ˇH / C r C p and p … .ˇH / C r C p. Hence x

Section 11.4 Local Monomorphisms

181

We now come to the main result of this section. Theorem 11.22. For every right maximal idempotent q 2 H with ﬁnite C.q/ and for every n 2 N, there is an idempotent p 2 H such that q 2 C.p/ and C.p/ is a chain of n ﬁnite right zero semigroups. Proof. If n D 1, put p D q. By Theorem 11.6, C.p/ is a ﬁnite right zero semigroup. Now let n > 1 and suppose that we have found an idempotent p 0 2 H such that q 2 C.p 0 / and C.p 0 / is a chain of n 1 ﬁnite right zero semigroups, say C1 > > Cn1 . By Proposition 11.20, there is a right maximal idempotent q 0 2 H with ﬁnite C.q 0 / such that for each x 2 C.q 0 /, x

11.4

Local Monomorphisms

Given a local left group X and a semigroup S with identity, a local monomorphism is an injective local homomorphism f W X ! S with f .1X / D 1S .

182

Chapter 11 Almost Maximal Spaces

Lemma 11.24. Let X be a local left group, let S be a left cancellative semigroup with identity, and let f W X ! S be a local monomorphism. Then there is a left invariant T1 -topology T f on S with a neighborhood base at s 2 S consisting of subsets sf .U /, where U runs over neighborhoods of 1X . Furthermore, let Y D f .X / .S; T f / and let f W ˇXd ! ˇS be the continuous extension of f . Then f homeomorphically maps X onto Y and f isomorphically maps Ult.X / onto Ult.T f /. Proof. Let B be an open neighborhood base at 1X . For every U 2 B and x 2 U , there is V 2 B such that xV U and f .xy/ D f .x/f .y/ for all y 2 T V . Then f .x/f .V / D f .xV / f .U /. Since f is injective, we have also that f .B/ D ¹1º. Consequently by Corollary 4.4, there is a left invariant T1 -topology T f on S in which for each s 2 S , sf .B/ is an open neighborhood base at s. To see that f homeomorphically maps X onto Y , let x 2 X . Choose a neighborhood U of 1X such that xU X and f .xy/ D f .x/f .y/ for all y 2 U . Then whenever V is a neighborhood of 1X and V U , one has f .xV / D f .x/f .V /. Finally, by Lemma 8.4, f isomorphically maps Ult.X / onto Ult.T f /. Deﬁnition 11.25. Let T be a translation invariant topology on H such that T0 T and let X be an open neighborhood of 0 in T . Denote by P .X / the set of all x 2 X n ¹0º which cannot be decomposed into a sum x D y C z where y; z 2 X n ¹0º and .y/ < .z/. Note that jP .X /j D . We say that X satisﬁes the P -condition if there is a neighborhood W of 0 2 X such that jP .X / n W j D . It follows from the next lemma that P .X / is a strongly discrete subset of X with at most one limit point 0. Lemma 11.26. Let x 2 P .X / and y 2 X n ¹0º. If x 2 y C H.y/C1 \ X , then x D y. Proof. Otherwise x D y Cz for some z 2 H.y/C1 \.X n¹0º/ and then .y/ < .z/, which contradicts x 2 P .X /. Suppose that X has the property that, whenever D is a strongly discrete subset of X with exactly one limit point 0, there is A D such that jAj D and 0 is not a limit point of A. Then, obviously, X satisﬁes the P -condition. In particular, X satisﬁes the P -condition if T is almost maximal. In fact, the P -condition is always satisﬁed in the following sense. Lemma 11.27. Let x 2 X n ¹0º, let Fx D ¹y 2 X n ¹0º W supp.y/ supp.x/º, and let Y D X n Fx . Then Y satisﬁes the P -condition. Proof. Choose a subset A Y with jAj D such that whenever y; z 2 A and y ¤ z, one has supp.y/ \ supp.z/ D supp.x/. For every y 2 A, there is zy 2 P .Y / such that supp.zy / supp.z/ and supp.zy / \ supp.x/ ¤ ;. Since Fx \ Y D ;, we have

Section 11.4 Local Monomorphisms

183

that supp.zy / n supp.x/ ¤ ; for all y 2 A. Put B D ¹zy W y 2 Aº. Then B P .Y /, B \ H.x/C1 D ; and jBj D . We now come to the main result about local monomorphisms. Theorem 11.28. Let T be a translation invariant topology on H such that T0 T , let X be an open neighborhood of 0 in T , and let G be a group of cardinality . Suppose that X is zero-dimensional and satisﬁes the P -condition. Then there is a local monomorphism f W X ! G such that the topology T f is zero-dimensional. If G D H , then f can be chosen to be continuous with respect to T0 . Proof. Using the P -condition, choose a clopen neighborhood W of 0 2 X such that jP .X / n W j D . By Lemma 11.9, there is a basis A in X such that (1) for each a 2 A, X.a/ a W , and (2) for each a 2 A n W , X.a/ \ W D ;. Let F be the free semigroup on the alphabet A including the empty word ;. Deﬁne h W X ! F by putting h.0/ D ; and h.a1 C C an / D a1 an where a1 ; : : : ; an 2 A and ai C Can 2 X.ai / for each i D 1; : : : ; n1. By Lemma 11.10, h is a local monomorphism. By Lemma 11.24, h induces a left invariant T1 topology T h on F . We have that Y D h.X / is an open neighborhood of the identity of .F; T h / and h homeomorphically maps X onto Y , so Y is zero-dimensional. Lemma 11.29. T

h

is zero-dimensional.

Proof. It sufﬁces to show that Y is closed in T h . Let a1 an 2 F n Y . Then ai C C an … X.ai / for some i D 1; : : : ; n 1, so aiC1 C C an … X.ai / ai . Taking the biggest such i we obtain that aiC1 C Can 2 X.aiC1 / X . It follows that there is a neighborhood U of 0 2 X such that .aiC1 C C an C U / \ .X.ai / ai / D ;, so .ai C C an C U / \ X.ai / D ;: We claim that .a1 an h.U // \ Y D ;. Indeed, assume on the contrary that a1 an h.y/ 2 Y for some y 2 U n ¹0º. Let h.y/ D anC1 anCm 2 Y . Since a1 an h.y/ D a1 an anC1 anCm 2 Y , we obtain that ai C C an C y D ai C C an C anC1 C C anCm 2 X.ai /; which is a contradiction.

184

Chapter 11 Almost Maximal Spaces

Denote I D A n W . Then jI j D and for every a1 an 2 Y , a 2 ¹a1 ; : : : ; an º \ I

implies a D a1 :

Indeed, by the construction of A and Lemma 11.26, P .X / A, and by the choice of W , jP .X / n W j D , so the ﬁrst statement holds. And since ai C C an 2 X.ai1 / ai1 W for each i D 2; : : : ; n, the second one holds as well. Now let Z denote the subset of F consisting of all words a1 an such that .ai / < .aiC1 / for each i D 1; : : : ; n 1 and a 2 ¹a1 ; : : : ; an º \ I

implies a D a1 :

Clearly Z is a neighborhood of the identity of .F; T h / containing Y . Furthermore, for every ˛ < , Z˛ D ¹b1 bm 2 h.W / W .b1 / ˛º [ ¹;º is a neighborhood of the identity, and for every a1 an 2 Z, a1 an Z.an /C1 Z; so Z is open. In addition, and as distinguished from Y , Z has the property that, whenever a1 an 2 Z and i D 1; : : : ; n 1, one has a1 ai 2 Z. Lemma 11.30. There is a bijective local monomorphism g W Z ! G. If G D H , then g can be chosen to be continuous with respect to T0 . Proof. We shall construct a bijection g W Z ! G such that g.;/ D 1 and g.a1 an / D g.a1 / g.an / for every a1 an 2 Z. That such g is a local homomorphism follows from the last but one sentence preceding the lemma. It sufﬁces to deﬁne g on A so that (i) whenever a1 an and b1 bm are different elements of Z, g.a1 / g.an / and g.b1 / g.bm / are different elements of G, and (ii) for each s 2 G n ¹1º, there is a1 an 2 Z such that g.a1 / g.an / D s. To this end, enumerate A without repetitions as ¹c˛ W ˛ < º so that if a; b 2 A, .a/ < .b/, a D c˛ and b D cˇ , then ˛ < ˇ. This deﬁnes W A ! by .c˛ / D ˛. Note that whenever a1 an 2 Z, one has .a1 / < < .an /. Also enumerate G n ¹1º as ¹s˛ W ˛ < º.

Section 11.4 Local Monomorphisms

185

Fix ˛ < and suppose that values g.cˇ / have already been deﬁned for all ˇ < ˛ so that, whenever a1 an and b1 bm are different elements of Z with .an /; .bm / < ˛, g.a1 / g.an / and g.b1 / g.bm / are different elements of G. Let G˛ D ¹g.a1 / g.an / 2 G W a1 an 2 Z and .an / < ˛º [ ¹1º: Consider two cases. Case 1: c˛ … I . Pick as g.c˛ / any element of G n .G˛1 G˛ /. This can be done because jG˛1 G˛ j jG˛ j2 < . Then whenever a1 an 2 Z and an D c˛ , one has g.a1 / g.an / … G˛ . Indeed, otherwise g.c˛ / D g.an / 2 .g.a1 / g.an1 //1 G˛ G˛1 G˛ : Also if a1 an and b1 bm are different element of Z with an D bm D c˛ , then g.a1 / g.an / ¤ g.b1 / g.bm /, by the inductive hypothesis. Case 2: c˛ 2 I . Then whenever a1 an 2 Z and an D c˛ , one has n D 1. Put g.c˛ / to be the ﬁrst element in the sequence ¹sˇ W ˇ < º n G˛ . It is clear that the mapping g W A ! G so constructed satisﬁes (i), and since jI j D , (ii) is satisﬁed as well. If G D H , the construction remains the same with the only correction in Case 1: we pick g.c˛ / so that .g.cˇ // < .g.c˛ // for all ˇ < ˛. To see that such g is continuous, let ˛ < be given. Choose ˇ < such that .g.cˇ // ˛ and put U˛ D ¹a1 an 2 h.W / W .a1 / > ˇº [ ¹;º: Then U˛ is a neighborhood of identity of Z and g.U˛ / H˛C1 . The bijective local monomorphism g W Z ! G induces a left invariant topology T g on G. Since g homeomorphically maps Z onto .G; T g /, T g is zero-dimensional and Hausdorff. Let f D g ı h. Then f W X ! G is a local monomorphism and T f D T g . If G D H , choose g to be continuous with respect to T0 , then so is f . We will use Theorem 11.28 in a very special situation when the topology T is almost maximal. But it is also interesting in the general case. Corollary 11.31. Let T1 be a locally zero-dimensional translation invariant topology on H such that T0 T1 and let G be a group of cardinality . Then there is a zero-dimensional left invariant topology T on G such that (1) .G; T / and .H; T1 / are locally homeomorphic, (2) Ult.T / is topologically and algebraically isomorphic to Ult.T1 /, and (3) Ult.T / is left saturated in ˇG. If G D H , then T can be chosen to be stronger then T0 .

186

Chapter 11 Almost Maximal Spaces

Proof. Let X be an open zero-dimensional neighborhood of zero of .H; T1 /. By Lemma 11.27, one may suppose that X satisﬁes the P -condition. Then by Theorem 11.28, there is a local monomorphism f W X ! G such that the topology T f on G is zero-dimensional. Put T D T f . By Lemma 11.24, conditions (1) and (2) are satisﬁed, and by Lemma 7.12, (3) is satisﬁed as well. If G D H , choose f to be continuous with respect to T0 . Now, using Theorem 11.23 and Theorem 11.28, we prove the main result of this chapter. Theorem 11.32. For every inﬁnite group G and for every n 2 N, there is a zerodimensional Hausdorff left invariant topology T on G such that Ult.T / is a chain of n idempotents and Ult.T / U.G/. Proof. Let G be a group of cardinality D jH j and let n 2 N. By Theorem 11.23, there is a locally zero-dimensional translation invariant topology T1 on H such that T0 T1 and Ult.T1 / is a chain of n idempotents. Pick an open zero-dimensional neighborhood X of 0 2 H . By Theorem 11.28, there is a local monomorphism f W X ! G such that the topology T f on G is zero-dimensional. Put T D T f . The dispersion character of a space is the minimum cardinality of a nonempty open set. Corollary 11.33. For every inﬁnite cardinal and for every n 2 N, there is a homogeneous zero-dimensional Hausdorff space of cardinality with exactly n nonprincipal ultraﬁlters converging to the same point, all of them being uniform. In particular, for every inﬁnite cardinal , there is a homogeneous regular maximal space of dispersion character . Remark 11.34. If G D H , the topology T in Theorem 11.32 can be chosen to be stronger than T0 , and if G D R, stronger than the natural topology of the real line. To see the second, apply Theorem 11.32 to the circle group T . This gives us a left saturated chain of n uniform idempotents in ˇTd . Since T is a compact group, every idempotent converges to 1 2 T (Lemma 7.10). Then identifying T with the subset Œ 12 ; 12 / R, we obtain a left saturated chain of n uniform idempotents in ˇRd converging to 0 2 R. One can show also that if G D R, the topology T in Theorem 11.32 can be chosen to be stronger than the Sorgenfrey topology.

References The question of whether there exists a regular maximal space was raised by M. Katˇetov [42]. A countable example of such a space was constructed by E. van Douwen

Section 11.4 Local Monomorphisms

187

[78, 80] and that of arbitrary dispersion character by A. El’kin [19]. The ﬁrst consistent example of a homogeneous regular maximal space was produced by V. Malykhin [46]. Right cancelable and right maximal idempotent ultraﬁlters on a countable group G have been studied by N. Hindman and D. Strauss [37, Sections 8.2, 8.5, and 9.1]. In particular, they showed that for every right cancelable ultraﬁlter p on G, the semigroup Cp admits a continuous homomorphism onto ˇN [37, Theorem 8.51], and for every right maximal idempotent p in G , C.p/ is a ﬁnite right zero semigroup [37, Theorem 9.4]. That every countably inﬁnite group admits in ZFC a regular maximal left invariant topology was proved by I. Protasov [60]. Theorem 11.2 and Theorem 11.6 are from [107]. Theorem 11.7 and Theorem 11.13 were proved in [101]. The results of Section 11.3 are from [108] and Theorem 11.28 is from [107]. Theorem 11.32 was proved in [107] for n D 1 and in [108] for any n.

Chapter 12

Resolvability

In this chapter we prove a structure theorem for a broad class of homeomorphisms of ﬁnite order on countable regular spaces. Using this, we show that every countable nondiscrete topological group not containing an open Boolean subgroup is !resolvable. We also show that every inﬁnite Abelian group not containing an inﬁnite Boolean subgroup is absolutely !-resolvable, and consequently, can be partitioned into ! subsets such that every coset modulo inﬁnite subgroup meets each subset of the partition.

12.1

Regular Homeomorphisms of Finite Order

Deﬁnition 12.1. Let X be a space with a distinguished point 1 2 X and let f W X ! X be a homeomorphism with f .1/ D 1. We say that f is regular if for every x 2 X n ¹1º, there is a homeomorphism gx of a neighborhood of 1 onto a neighborhood of x such that fgx jU D gf .x/ f jU for some neighborhood U of 1. Note that if a space X admits a regular homeomorphism, then for any two points x; y 2 X , there is a homeomorphism g of a neighborhood of x onto a neighborhood of y with g.x/ D y, and if in addition X is zero-dimensional and Hausdorff, then g can be chosen to be a homeomorphism of X onto itself. Hence, a zero-dimensional Hausdorff space admitting a regular homeomorphism is homogeneous. The notion of a regular homeomorphism generalizes that of a local automorphism on a local left group. To see this, let X be a local left group and let f W X ! X be a local automorphism. For every x 2 X n ¹1º, choose a neighborhood Ux of 1 such that xy is deﬁned for all y 2 Ux , xUx is a neighborhood of x and x W Ux 3 y 7! xy 2 xUx is a homeomorphism, and put gx D x . Clearly gx .1/ D x. Choose a neighborhood Vx of 1 such that Vx Ux , f .Vx / Uf .x/ and f .xy/ D f .x/f .y/ for all y 2 Vx . Then for every y 2 Vx , fgx .y/ D f .xy/ D f .x/f .y/ D gf .x/ f .y/. We now show that, likewise in the case of a local automorphism, the spectrum of a spectrally irreducible regular homeomorphism of ﬁnite order is a ﬁnite subset of N closed under taking the least common multiple. Lemma 12.2. Let X be a Hausdorff space with a distinguished point 1 and let f W X ! X be a spectrally irreducible regular homeomorphism of ﬁnite order. Let x0 2 X n ¹1º with jO.x0 /j D s and let U be a spectrally minimal neighbourhood of x0 .

Section 12.1 Regular Homeomorphisms of Finite Order

189

Then spec.f; U / D ¹lcm.s; t / W t 2 ¹1º [ spec.f /º: Proof. The proof is similar to that of Lemma 8.27. For each x 2 O.x0 /, let gx be a homeomorphism of a neighborhood Ux of 1 onto a neighborhood of x such that fgx jVx D gf .x/ f jVx for some neighborhood Vx Ux of 1. Choose a neighborhood V of 1 such that \ V Vx ; x2O.x0 /

gx0 .V / U , and the subsets gx .V /, where x 2 O.x0 /, are pairwise disjoint. Let n be the order of f . Choose a neighborhood W of 1 such that f i .W / V for all i < n. Then clearly this inclusion holds for all i < !, and furthermore, for every y 2 W , f i gx0 .y/ D gf i .x0 / f i .y/. Indeed, it is trivial for i D 0, and further, by induction, we obtain that f i gx0 .y/ D ff i1 gx0 .y/ D fgf i1 .x0 / f i1 .y/ D gf i .x0 / f i .y/: Now let y 2 W , jO.y/j D t and k D lcm.s; t /. We claim that jO.gx0 .y//j D k. Indeed, f k .gx0 .y// D gf k .x0 / .f k .y// D gx0 .y/: On the other hand, suppose that f i .gx0 .y// D gx0 .y/ for some i . Then gf i .x0 / .f i .y// D gx0 .y/: Since the subsets gx .V /, x 2 O.x0 /, are pairwise disjoint, it follows from this that f i .x0 / D x0 , so sji . But then also f i .y/ D y, as gx0 is injective, and so t ji . Hence kji . The next theorem is the main result of this section. Theorem 12.3. Let X be a countable nondiscrete regular space with a distinguished point 1 2 X , let f W X ! X be a spectrally irreducible P regular homeomorphism of ﬁnite order, let S D be the stanL spec.f /, and let m D 1 C s2S s. LetL dard L permutation L on Z of spectrum S, and for every a 2 m ! ! Zm , deﬁne a W Z ! Z by .x/ D a C x. Then there is a continuous bijection m m a !L ! h W X ! ! Zm with h.1/ D 0 such that (1) f D h1 h, and (2) for every x 2 X , x D h1 h.x/ h is a homeomorphism of X onto itself. Furthermore, if X is a local left group and f is a local automorphism, then h can be chosen so that (3) x .y/ D xy, whenever max supp.h.x// C 1 < min supp.h.y//.

190

Chapter 12 Resolvability

L Recall that the topology of ! Zm is generated by taking as a neighborhood base at 0 the subgroups ° ± M Hn D x 2 Zm W x.i / D 0 for all i < n !

where n < !. The conclusion of Theorem 12.3 can be rephrased as follows: L One can deﬁne the operation of the group Z ! m on X in such a way that 0 D 1, L the topology of ! Zm is weaker than that of X and (1) D f , and (2) for every x 2 X , x W X 3 y 7! x C y 2 X is a homeomorphism. Furthermore, if X is a local left group and f is a local automorphism, then the operation can be deﬁned so that (3) x C y D xy, whenever max supp.x/ C 1 < min supp.y/. Actually, Theorem 12.3 characterizes spectrally irreducible regular homeomorphisms of ﬁnite order on countable regular spaces. If f W X ! X is a spectrally irreducible homeomorphism and for some m there is a continuous bijection h W X ! L Z with h.1/ D 0 such that ! m L (1) hf h1 is a coordinatewise permutation on ! Zm , and (2) for every x 2 X , h1 h.x/ h is a homeomorphism of X onto itself, then f is regular. To see this, let D hf h1 . For every x 2 X n¹1º, let n.x/ D max supp.h.x//C1, Ux D h1 .Hn.x/ / and gx D h1 h.x/ hjUx . Then for every y 2 Ux , f .y/ 2 Ux D Uf .x/ and fgx .y/ D h1 hh1 h.x/ h.y/ D h1 h.x/ h.y/ D h1 .h.x/ C h.y// D h1 ..h.x// C .h.y/// D h1 .h.f .x// C h.f .y/// D h1 h.f .x// h.f .y// D gf .x/ f .y/: Proof of Theorem 12.3. Let W D W .Z Lm /. The permutation 0 on Zm , which induces the standard permutation on ! Zm , also induces the permutation 1 on W . If w D 0 n , then 1 .w/ D 0 . 0 / 0 . n /. We will write instead of 0 and 1 . For each x 2 X n ¹1º, choose a homeomorphism gx of a neighborhood

Section 12.1 Regular Homeomorphisms of Finite Order

191

of 1 onto a neighborhood of x with gx .1/ D x such that fgx D gf .x/ f jU for some neighborhood U of 1. Also put g1 D idX . If X is a local left group and f is a local automorphism, choose gx so that gx .y/ D xy. Enumerate X as ¹xn W n < !º with x0 D 1. We shall assign to each w 2 W a point x.w/ 2 X and a clopen spectrally minimal neighborhood X.w/ of x.w/ such that (i) x.0n / D 1 and X.;/ D X , (ii) ¹X.w _ / W 2 Z.m/º is a partition of X.w/, (iii) x.w/ D gx.w0 / gx.wk1 / .x.wk // and X.w/ D gx.w0 / gx.wk1 / .X.wk //, where w D w0 C C wk is the canonical decomposition, (iv) f .x.w// D x..w// and f .X.w// D X..w//, (v) xn 2 ¹x.v/ W v 2 W and jvj D nº. Enumerate S as s1 < < s t and for each i D 1; : : : ; t , pick a representative i of the orbit in Zm n ¹0º of length si . Choose a clopen invariant neighborhood U1 of 1 such that x1 … U1 and spec.f; X n U1 / D spec.f /. Put x.0/ D 1 and X.0/ D U1 . Then choose points Sai 2 X n U1 , i D 1; : : : ; t , with pairwise disjoint orbits of lengths si such that x1 2 tiD1 O.ai /. For each i D 1; : : : ; t and j < si , put x. j .i // D f j .ai /: By Lemma 8.30, there is an invariant partition ¹X. / W 2 Zm n ¹0ºº of X n U1 such that X. / is a clopen spectrally minimal neighborhood of x. /. Fix n > 1 and suppose that X.w/ and x.w/ have been constructed for all w 2 W with jwj < n so that conditions (i)–(v) are satisﬁed. Note that the subsets X.w/, jwj D n 1, form a partition of X . So one of them, say X.u/, contains xn . Let u D u0 C C uq be the canonical decomposition. Then X.u/ D gx.u0 / gx.uq1 / .X.uq // and xn D gx.u0 / gx.uq1 / .yn / for some yn 2 X.uq /. Choose a clopen invariant neighborhood Un of 1 such that for all basic w with jwj D n 1, (a) gx.w/ .Un / X.w/, (b) fgx.w/ jUn D gf .x.w// f jUn , and (c) spec.f; X.w/ n gx.w/ .Un // D spec.X.w//. If yn ¤ x.uq /, choose Un in addition so that (d) yn … gx.uq / .Un /. Put x.0n / D 1 and X.0n / D Un . Let w 2 W be an arbitrary nonzero basic word with jwj D n 1 and let O.w/ D ¹wj W j < sº, where wj C1 D .wj / for j < s 1 and .ws1 / D w0 . Put Yj D X.wj / n gx.wj / .Un /. Using Lemma 12.2, choose points bi 2 Y0 , i D 1; : : : ; t ,

192

Chapter 12 Resolvability

with pairwise disjoint orbits of lengths lcm.si ; s/. If uq 2 O.w/, choose bi in addition so that t [ yn 2 O.bi /: iD1

For each i D 1; : : : ; t and j < si , put x. j .w _ i // D f j .bi /: Then, using Lemma 8.30, inscribe an invariant partition ¹X.v _ / W v 2 O.w/; 2 Zm n ¹0ºº into the partition ¹Yj W j < sº such that X.v _ / is a clopen spectrally minimal neighborhood of x.v _ /. For nonbasic w 2 W with jwj D n, deﬁne x.w/ and X.w/ by condition (iii). To check (ii) and (iv), let jwj D n 1 and let w D w0 C C wk be the canonical decomposition. Then X.w _ 0/ D gx.w0 / gx.wk / .X.0n // D gx.w0 / gx.wk1 / .gx.wk / .X.0n /// and X.w _ / D gx.w0 / gx.wk1 / .X.wk_ //; so (ii) is satisﬁed. Next, f .x.w// D fgx.w0 / gx.wk1 / .x.wk // D gf .x.w0 // fgx.w1 / gx.wk1 / .x.wk // :: : D gf .x.w0 // gf .x.wk1 // f .x.wk // D gx..w0 // gx..wk1 // .x..wk /// D x..w0 / .wk1 /.wk // D x..w//; so (iv) is satisﬁed as well. To check (v), suppose that xn … ¹x.w/ W jwj D n 1º. Then xn D gx.u0 / gx.uq1 / .yn / D gx.u0 / gx.uq1 / .u_ q / D x.u_ /:

193

Section 12.1 Regular Homeomorphisms of Finite Order

Now, for every x 2 X , there is w 2 W with nonzero last letter such that x D Lx.w/, so ¹v 2 W W x D x.v/º D ¹w _ 0n W n < !º. Hence, we can deﬁne h W X ! ! Zm by putting for every w D 0 n 2 W , h.x.w// D w D . 0 ; : : : ; n ; 0; 0; : : :/: It is clear that h is bijective and h.1/ D 0. Since for every z D . i /i

L !

Zm ,

h1 .z C Hn / D X. 0 n /; h is continuous. To see (1), let x D x.w/. Then h.f .x.w/// D h.x..w/// D .w/ D .w/ D .h.x.w///: To see (2), let x D x.w/, w D w0 C C wk and n D max supp.h.x// C 1. We ﬁrst show that x jh1 .Hn / D gx.w0 / gx.wk / jh1 .Hn / : Let y 2 h1 .Hn /, y D x.v/ and v D v0 C C vl . Then hgx.w0 / gx.wk / .y/ D hgx.w0 / gx.wk / gx.v0 / gx.vl1 / .x.vl // D h.x.w C v// DwCv DwCv D h.x.w// C h.x.v// D h.x/ h.y/: It follows from (iii) that gx.w0 / gx.wk / homeomorphically maps X.0n /, a neighborhood of 1, onto X.w _ 0/, a neighborhood of x, and so x does. Now, to see that x homeomorphically maps a neighborhood of an arbitrary point y 2 X onto a neighborhood of z D x .y/, it sufﬁces to check that x D z .y /1 . Indeed, z D h1 h.x/ h.y/ D h1 .h.x/ C h.y// and then z .y /1 D h1 h.x/Ch.y/ h.h1 h.y/ h/1 D h1 h.x/Ch.y/ hh1 .h.y/ /1 h D h1 h.x/Ch.y/ h.y/ h D h1 h.x/ h D x :

194

Chapter 12 Resolvability

To see (3), let x D x.w/ and w D w0 C Cwk . If k D 0, then x .y/ D gx .y/ D xy. Continuing, by induction on k, we obtain that x .y/ D gx.w0 / gx.wk / .y/ D gx.w0 / gx.wk1 / .x.wk / y/ D x.w0 C C wk1 / .x.wk / y/ D .x.w0 C C wk1 / x.wk // y D .gx.w0 / gx.wk1 / .x.wk /// y D x.w/ y: The second part of Theorem 12.3, the case where X is a local left group and f is a local automorphism, is Theorem 8.29. The ﬁrst part with f D idX tells us that Corollary 12.4. Every countably inﬁnite homogeneous regular space admits a Boolean group operation with continuous translations. We conclude this section by deducing from Corollary 12.4 and Corollary 8.12 the following result. Theorem 12.5. Let X be a countably inﬁnite homogeneous regular space and let G be a countably inﬁnite group. Then there is a group operation on X such that .X; / is a left topological group algebraically isomorphic to G. Proof. One may suppose that X is nondiscrete. By Corollary 12.4, there is a Boolean group operation C on X with continuous translations. Endowing G with any nondiscrete regular left invariant topology and applying Corollary 8.12, we obtain that there is a bijective local homomorphism f W .X; C/ ! G. For any x; y 2 X , deﬁne x y D f 1 .f .x/f .y//. Obviously, .X; / is a group isomorphic to G. Now given any x 2 X , we can choose a neighborhood U of 0 such that f .x C z/ D f .x/f .z/ for all z 2 U , and then x z D f 1 .f .x/f .z// D f 1 .f .x C z// D x C z. It follows from this that the left translations of .X; / are continuous and open at the identity. Consequently, the left translations of .X; / are continuous.

12.2

Resolvability of Topological Groups

Theorem 12.6. If a countable regular space admits a nontrivial regular homeomorphism of ﬁnite order, then it is !-resolvable. Proof. Let X be a countable regular space with a distinguished point 1 2 X and let f W X ! X be a nontrivial regular homeomorphism of ﬁnite order. By Corollary 8.25 and Corollary 3.27, one may suppose that f is spectrally irreducible. Let h W X !

Section 12.2 Resolvability of Topological Groups

195

L

! Zm be a bijection guaranteed by Theorem 12.3. Pick an orbit C in Zm (with respect to 0 ) of a smallest possible length s > 1. For every x 2 X , consider the sequence of coordinates of h.x/ belonging to C and deﬁne .x/ to be the number of pairs of distinct neighbouring elements in this sequence. Denote also by ˛.x/ and

.x/ the ﬁrst and the last elements in the sequence (if nonempty). Then whenever x; y 2 X and max supp.h.x// C 1 < min supp.h.y//, ´ .x/ C .y/ if .x/ D ˛.y/ .x .y// D .x/ C .y/ C 1 otherwise:

We deﬁne a disjoint family ¹Xn W n < !º of subsets of X by Xn D ¹x 2 X W .x/ 2n .mod 2nC1 /º: To see that every Xn is dense in X , let x 2 X and let U be an open neighbourhood of 1. We have to show that x .U / \ Xn ¤ ;. Put k D 2nC1 and choose inductively x1 ; : : : ; xk 2 U such that (i) jO.xj /j D s, (ii)

max supp.h.xj // C 1 < min supp.h.xj C1 //, and if x ¤ 0, then max supp.h.x// C 1 < min supp.h.x1 //,

(iii) y1 yk .1/ 2 U whenever yj 2 O.xj /. Without loss of generality one may suppose that .xj / D ˛.xj C1 /, and that if x 2 X , then .x/ D ˛.x1 /. For every l D 0; 1; : : : ; k 1, deﬁne zl 2 U by zl D x1 f .x2 / f l .xlC1 / f l .xlC2 / f l .xk / .1/ (in particular, z0 D x1 x2 xk .1/). Then h.x .zl // D h.x/Ch.x1 /Ch.x2 /C C l h.xlC1 /C l h.xlC2 /C C l h.xk /: It follows that .x .z0 // D .x/ C .x1 / C C .xk / and .x .zl // D .z0 / C l. Hence, for some l, .x .zl // 2n .mod 2nC1 /, so x .zl / 2 Xn . The next proposition says that every nondiscrete topological group not containing an open Boolean subgroup admits a nontrivial regular homeomorphism of order 2. Proposition 12.7. Let G be a nondiscrete topological group not containing an open Boolean subgroup. Suppose that for every element x 2 G of order 2, the conjugation G 3 y 7! xyx 1 2 G is a trivial local automorphism. Then the inversion G 3 y 7! y 1 2 G is a nontrivial regular homeomorphism.

196

Chapter 12 Resolvability

In order to prove Proposition 12.7, we need the following lemma. Lemma 12.8. Let X be a homogeneous space with a distinguished point 1 2 X and let f W X ! X be a homeomorphism of ﬁnite order n with f .1/ D 1. Suppose that for every x 2 X n ¹1º with jO.x/j D s < n, there is a homeomorphism gx of a neighborhood U of 1 onto a neighborhood of x with gx .1/ D x such that f s gx .y/ D gx f s .y/ for all y 2 U . Then f is regular. In particular, if for every x 2 X n ¹1º, jO.x/j D n, then f is regular. Proof. Consider an arbitrary orbit in X distinct from ¹1º and enumerate it as ¹xi W i < sº, where xiC1 D f .xi / for i D 0; : : : ; s 2 and f .xs1 / D x0 . If s D n, choose as gx0 any homeomorphism of a neighborhood U of 1 onto a neighborhood of x0 with gx0 .e/ D x0 . If s < n, choose gx0 in addition such that f s gx0 .y/ D gx0 f s .y/ for all y 2 U . For every i D 1; : : : ; s 1, put gxi D f i gx0 f i jU . Then for every i D 0; : : : ; s 1 and y 2 U , fgxi .y/ D ff i gx0 f i .y/ D f iC1 gx0 f .iC1/ f .y/: If i < s 1, then f iC1 gx0 f .iC1/ f .y/ D gxiC1 f .y/, so fgxi .y/ D gxiC1 f .y/. Hence, it remains only to check that fgxs1 .y/ D gx0 f .y/. If s D n, then fgxs1 .y/ D f s gx0 f s f .y/ D idX gx0 idX f .y/ D gx0 f .y/: If s < n, then fgxs1 .y/ D f s gx0 f s f .y/ D gx0 f s f s f .y/ D gx0 f .y/: Proof of Proposition 12.7. Let f denote the inversion and let B D B.G/. We have that f is a homeomorphism of order 2 and B is the set of ﬁxed points of f , in particular, f .1/ D 1. By Lemma 5.3, B is not a neighborhood of 1, so f is nontrivial. To see that f is regular, let x 2 G n ¹1º and jO.x/j < 2. Then x 2 B. But then there is a neighborhood U of 1 such that xyx 1 D y for all y 2 U , that is, xy D yx. Deﬁne gx W U ! xU by gx .y/ D xy. We have that fgx .y/ D .xy/1 D .yx/1 D x 1 y 1 D xy 1 D gx f .y/: Hence, by Lemma 12.8, f is regular. Combining Theorem 12.6 and Proposition 12.7, we obtain that Theorem 12.9. Every countable nondiscrete topological group not containing an open Boolean subgroup is !-resolvable. Note that in the Abelian case Theorem 12.9 can be proved easier. If a topological group is Abelian, then the inversion is a local automorphism. Therefore, it sufﬁces to use Theorem 8.29 instead of Theorem 12.3. Also in the Abelian case the restriction ‘countable’ is redundant.

Section 12.2 Resolvability of Topological Groups

197

Theorem 12.10. Every nondiscrete Abelian topological group not containing a countable open Boolean subgroup is !-resolvable. Since every Abelian group can be isomorphically embedded into a direct sum of countable groups, Theorem 12.10 is immediate from the Abelian case of Theorem 12.9 and the following result. Theorem 12.11. Let > !. ForLevery ˛ < , let G˛ be a countable group and let G be an uncountable subgroup of ˛< G˛ . Then G can be partitioned into ! subsets dense in every group topology of uncountable dispersion character. The proof of Theorem 12.11 involves the functions 2 ; 2 W N ! ! (see Deﬁnition 6.35). Lemma 12.12. Let m 2 N and n < !. Let I denote the integer interval Œ2 .m/ C 1; 2 .m/ C 2nC1 : Then for every k 2 N, there is i 2 I such that 2 .2 .m C k 2i // D n. Proof. For every i 2 .m/ C 1, one has 2 .k 2i / 2 .m/ C 1. It follows that 2 .m C k 2i / D 2 .k 2i / D 2 .k/ C i: Consequently, J D ¹2 .m C k 2i / W i 2 I º is an integer interval of length 2nC1 , and so there is j 2 J such that j 2n .mod 2nC1 /. Proof of Theorem 12.11. Deﬁne a disjoint family ¹Yn W n < !º of subsets of G by Yn D ¹x 2 G W 2 .2 .jsupp.x/j// D nº: Let G be endowed with any group topology of uncountable dispersion character, let U be a neighborhood of 1 2 G, and let x 2 G. We show that .xU / \ Yn ¤ ;. Without loss of generality one may suppose that x ¤ 1. Let m D jsupp.x/j. Put nC1 s D 22 .m/C2 and pick a neighborhood V of 1 such that V s U . Since G has uncountable dispersion character, it follows that for every countable A , there is y 2 V n ¹1º such that supp.y/ \ A D ;. To L see this, pick a neighborhood W of 1 such that W W 1 V . Since jW j > ! and j ˛2A G˛ j !, there exist distinct v; w 2 W such that v.˛/ D w.˛/ for each ˛ 2 A. Put y D vw 1 . Then 1 ¤ y 2 W W 1 V and supp.y/ \ A D ;. Now construct an !1 -sequence .y˛ /˛

198

Chapter 12 Resolvability

Passing to a coﬁnal subsequence, one may suppose that there is k 2 N such that jsupp.y˛ /j D k for all ˛ < . Let i be a natural number guaranteed by Lemma 12.12 i and let y D y1 y2i . Then y 2 V 2 U , jsupp.xy/j D m C k 2i , and 2 .2 .jsupp.xy/j// D n. We conclude this section by showing that the existence of a countable nondiscrete !-irresolvable topological group cannot be established in ZFC. Theorem 12.13. The existence of a countable nondiscrete !-irresolvable topological group implies the existence of a P -point in ! . Proof. Let .G; T / be a countable L !-irresolvable topological group. By Theorem 12.9, one may suppose that Q G D ! Z2 . Let T0 denote the topology on G induced by the product topology on ! Z2 . Then T _ T0 is a nondiscrete !-irresolvable group topology. Therefore, one may suppose that T0 T . By Theorem 3.33, every discrete subset of .G; T / is closed. Let F be the neighborhood ﬁlter of 0 in T . It is easy to 1 see that ¹ .u/ W u 2 .F /º is a partition of Ult.T / into right ideals. It then follows from Theorem 3.35 and Proposition 7.7 that .F / is ﬁnite. Hence by Theorem 5.19, each point of .F / is a P -point. Combining Theorem 12.13 and Theorem 2.38 gives us that Corollary 12.14. It is consistent with ZFC that there is no countable nondiscrete !-irresolvable topological group.

12.3

Absolute Resolvability

Deﬁnition 12.15. Let G be a group. A subset A G is absolutely dense if A is dense in every nondiscrete group topology on G. Given a cardinal 2, G is absolutely -resolvable (absolutely resolvable if D 2) if G can be partitioned into absolutely dense subsets. Lemma 12.16. Let G be an Abelian group and let A G. If A is absolutely dense, then G n A contains no coset modulo inﬁnite subgroup. Proof. Suppose that there exist an inﬁnite subgroup H of G and g 2 G such that g C H G n A. Pick any nondiscrete group topology TH on H and extend it to the group topology T on G by declaring H to be an open subgroup. Then g C H is an open subset of .G; T / disjoint from A, so A is not dense. Hence, A is not absolutely dense. The next proposition is a consequence of Hindman’s Theorem.

Section 12.3 Absolute Resolvability

199

Proposition 12.17. Whenever an inﬁnite Boolean group is partitioned into ﬁnitely many subsets, there is a coset modulo inﬁnite subgroup contained in one subset of the partition. Proof. Let B be an inﬁnite Boolean group and let ¹Ai W i < rº be a ﬁnite partition of B. By Hindman’s Theorem, there exist i < r and a one-to-one sequence .xn /n

In this section we prove the following result. Theorem 12.20. Let G be an Abelian group and let A D G n B.G/ be inﬁnite. Then there is a disjoint family ¹Am W m < !º of subsets of A such that whenever .xn /n

200

Chapter 12 Resolvability

Corollary 12.21 and Lemma 12.16 give us in turn that Corollary 12.22. Every inﬁnite Abelian group not containing an inﬁnite Boolean subgroup can be partitioned into ! subsets such that every coset modulo inﬁnite subgroup meets each subset of the partition. Note that the cardinal number ! in Theorem 12.20, Corollary 12.21 and Corollary 12.22 is maximally possible. We ﬁrst prove Theorem 12.20 in the case where G is a direct sum of ﬁnite groups, not necessarily Abelian. Theorem 12.23. LetL !. For each ˛ < , let G˛ be a ﬁnite group written additively, let G D ˛< G˛ , and let A D G n B.G/ be inﬁnite. Then there is a disjoint family ¹Am W m < !º of subsets of A such that whenever .xn /n d . Proof. Without loss of generality one may suppose that cnC1 cn 5 for all n < !. For each n < !, pick an integer dn 2 such that dn2 C dn 1 cnC1 cn and dn ! 1. To deﬁne ', let k < ! and let a ckC1 . Choose the largest integer l 0 for which ckC1 C ldk2 a, then choose the largest integer i 0 for which ckC1 C ldk2 C idk a, and then put j D a ckC1 ldk2 idk . Thereby we write a in the form a D ckC1 C ldk2 C idk C j;

201

Section 12.3 Absolute Resolvability

where l is a nonnegative integer and i; j 2 ¹0; : : : ; dk 1º. Put ak D ck C jdk C i: Since ak 2 Œck ; ck C dk2 /, the function ' so deﬁned satisﬁes the condition (1). To check (2), let d 2 N be given. Choose n0 < ! such that dn d C 2 for all n n0 and put c D cn0 . Now let a; b 2 N, 0 < ja bj d , u 2 '.a/, v 2 '.b/ and u; v c. Since ckC1 ak D ckC1 ck jdk i dk2 C dk 1 jdk i dk ; one may suppose that u D ak and v D bk for some k n0 . Let a D ckC1 C ldk2 C idk C j; b D ckC1 C l 0 dk2 C i 0 dk C j 0 : Then ak D jdk C i; bk D j 0 dk C i 0 : Thus, we have that a b D Œ.l l 0 /dk C .i i 0 /dk C .j j 0 /; ak bk D .j j 0 /dk C .i i 0 /: Notice that ji i 0 j < dk and jj j 0 j < dk . Then it follows from the second equality that if jj j 0 j > 1, jak bk j > dk . Therefore one may suppose that jj j 0 j 1. But then it follows from the ﬁrst equality that a b differs from a multiple of dk by 0, 1 or 1. Since ja bj d dk 2, this multiple of dk must be zero, so .l l 0 /dk C .i i 0 / D 0: This gives us l D l 0 and i D i 0 . Consequently, jj j 0 j D 1, and we obtain that jak bk j D dk . Let ' W N ! ŒN

202

Chapter 12 Resolvability

and deﬁne .x/ to be the number of pairs of distinct neighbouring elements in the sequence sgn.x.˛l0 //; : : : ; sgn.x.˛ln1 //: (If l 2 Œc0 ; c1 /, then '.l/ D ;, and then supp' .x/ D ; and .x/ D 0.) We deﬁne the disjoint family ¹Am W m < !º of subsets of A by Am D ¹x 2 A W .x/ 2m .mod 2mC1 /º: Now let .xn /n

\

FS..xn /mn

m

(Proposition 6.39) and pick an idempotent p 2 T (Theorem 6.12). We then have that FS..xn /mn

\ ik

The sequence .yn /n

Uyi :

203

Section 12.3 Absolute Resolvability

By Lemma 12.25, one may suppose that supp.g/ \ supp.xi / D ;

and

supp.xi / \ supp.xj / D ;

for all i < j < !. Also one may suppose that the sequence .min supp0 .xn //n

D sup¹min supp0 .xn / W n < !º: Every x 2 G can be uniquely written in the form x D x 0 C x 00 , where x 0 ; x 00 2 G, supp.x 0 / D supp.x/ \

and

supp.x 00 / D supp.x/ n supp.x 0 /:

Put k D 2mC1 1. Choose inductively a sum subsystem .yn /n

(b) each of the intervals .jsupp0 .g 0 /j; jsupp0 .g 0 C y00 /j and 0 /j; .jsupp0 .g 0 C y00 C C yi0 /j; jsupp0 .g 0 C y00 C C yi0 C yiC1

where i < k 1, contains some interval Œcn ; cnC1 /. P Let h D g C y0 C C yk1 , yk1 D n2H xn and n0 D max H C 1. We claim that there is yk 2 FS..xn /n0 n

and

for all z 2 FS..zn /n

Put d D jsupp0 .h C z0 /j and let c be a constant guaranteed by Lemma 12.16. Pick z 2 FS..zn /1n

and

204

Chapter 12 Resolvability

Let a D jsupp0 .h C z0 C z/j and b D jsupp0 .h C z/j. Then 0 < a b D jsupp0 .z0 /j jsupp0 .h C z0 /j D d: Let u D jsupp0 .h C z0 C z/ \ .ı C 1/j, v D jsupp0 .h C z/ \ .ı C 1/j, w D jsupp0 .h C z0 / \ .ı C 1/j and t D jsupp0 .h/ \ .ı C 1/j. Then u D v C w t , and so u v D w t; which is a contradiction, since u 2 '.a/, v 2 '.b/, u; v c and w t w d . Now consider the element g0 D g C y0 C y1 C C yk : By the construction, supp' .g0 / \ supp0 .yk / ¤ ;; and for each i k 1, ; ¤ supp' .g0 / \ supp0 .yi / supp0 .yi0 /: Therefore we have that ˇ0 < ˛1 ˇ1 < ˛2 ˇ2 < < ˛k1 ˇk1 < ˛k ; where ˛i D min.supp' .g0 / \ supp0 .yi //; ˇi D max.supp' .g0 / \ supp0 .yi //: Put "0 D 1 and, by induction on i D 1; : : : ; k, choose "i 2 ¹1; 1º so that "i1 yi1 .ˇi1 / D "i yi .˛i /: Without loss of generality one may suppose that all "i D 1. For each i k, deﬁne gi 2 g C FSI..yn /nk / by gi D g C y0 y1 C C .1/i yi C .1/i yiC1 C C .1/i yk : Then .gi / D .g0 / C i: Hence, there is j k D 2mC1 1 such that .gj / 2m and so gj 2 Am .

.mod 2mC1 /;

Section 12.3 Absolute Resolvability

205

In order to to prove Theorem 12.20 in the general case, we need in addition the following result. Theorem 12.26. Let G be an inﬁnite Abelian group endowed with the largest totally bounded L group topology and let jGj D . Then there is a continuous injection f W G ! Z4 such that (1) f .x/ D f .x/ for all x 2 G, (2) f .x C y/ D fL .x/ C f .y/ for all x; y 2 G with S.f .x// \ S.f .y// D ;, where for each a 2 Z4 , S.a/ D supp.a/ [ .supp.a/ C 1/. L Q Here, Z4 is endowed with the topology induced by the product topology on Z4 . Theorem 12.26 Lallows us to identify an arbitrary Abelian group G of cardinality with a subset of Z4 so that L (a) for every x 2 G, x 2 Z4 is the inverse of x in G, L (b) for every x; y 2 G with S.x/ \ S.y/ D ;, x C y 2 Z4 is the sum of x and y in G, and (c) the largest L totally bounded group topology on G is stronger than that induced from Z4 . And that is all what we need to extend the proof of Theorem 12.23 to Theorem 12.20. We deduce Theorem 12.26 from the following fact. Proposition 12.27. Let G be a countable Abelian topological group. Suppose that B.G/ is neither L discrete nor open. Then there is a continuous bijection f W G ! ! Z4 such that (i) f .x/ D f .x/ for all x 2 G, (ii) f .x C y/ D f .x/ C f .y/ for all x; y 2 G n ¹0º with max supp.f .x// C 2 min supp.f .y//. Proposition 12.27 is an immediate consequence of Theorem 8.29. Before proving Theorem 12.26 we show that condition (ii) in Proposition 12.27 can be replaced by (2) from Theorem 12.26. L Indeed, every nonzero element a 2 ! Z4 has a unique canonical decomposition a D a1 C C a n ; where for each i D 1; : : : ; n, supp.ai / is a nonempty interval in !, and for each i D 1; : : : ; n 1, max supp.ai / C 2 min supp.aiC1 /. And it follows from (ii) that if xi D f 1 .ai / for each i D 1; : : : ; n, then f .x1 C C xn / D a1 C C an :

206

Chapter 12 Resolvability

Let a D f .x/, b D f .y/ and S.a/ \ S.b/ D ;. Let a D a1 C C an and b D b1 C C bm be the canonical decompositions. Since S.a/ \ S.b/ D ;, there is a permutation c1 ; : : : ; cnCm of a1 ; : : : ; an ; b1 ; : : : ; bm such that a C b D c1 C C cnCm is the canonical decomposition. Let xi D f 1 .ai /, yj D f 1 .bj / and zk D f 1 .ck /. Then x D x1 C C xn , y D y1 C C ym and, since G is Abelian, x C y D z1 C C znCm . Finally, we obtain that f .x C y/ D f .z1 C C znCm / D c1 C C cnCm D .a1 C C an / C .b1 C C bm / D f .x/ C f .y/: Proof of Theorem 12.26. Without loss of generality one may suppose that M GD G˛ ; ˛<

where for each ˛ < , jG˛ j D jB.G˛ /j D jG˛ W B.G˛ /j D !: For each ˛ < , endow G˛ with a totally bounded group topology and let M f˛ W G˛ ! Z4 !

be a continuous bijection guaranteed by Proposition 12.27. We deﬁne M M f WG! Z4 ˛< Œ˛;˛C!/

by f D

M

f˛ :

˛<

References The study of resolvability for topological groups was initiated by W. Comfort and J. van Mill [13]. They showed that if G is an inﬁnite Abelian group not containing an inﬁnite Boolean subgroup, then every nondiscrete group topology on G is resolvable. Theorem 12.3 and Theorem 12.6 are results from [105]. Theorem 12.5 was proved in [96]. Theorem 12.9 is a result from [94] and Theorem 12.10 from [89]. Theorem 12.11 is due to I. Protasov [57]. He also proved Theorem 12.13 in the Abelian case [59].

Section 12.3 Absolute Resolvability

207

The question of characterizing absolutely resolvable groups was raised in [13]. Corollary 12.18 is due to I. Protasov [57]. Theorem 12.20 was proved in [102]. That every inﬁnite Abelian group not containing an inﬁnite Boolean subgroup is absolutely resolvable was proved in [91]. Corollary 12.22 and Proposition 12.17 complement the Graham-Rothschild Theorem [29] (see also [30, Section 2.4] ) which can be stated as follows. If an inﬁnite Abelian group of ﬁnite exponent is partitioned into ﬁnitely many subsets, then there are arbitrarily large ﬁnite cosets contained in one subset of the partition.

Chapter 13

Open Problems

In this chapter we list several open questions related to the topic of this book. (1) (Old question) Is there any element of ﬁnite order in ˇN other than idempotents? L (2) Let > ! and Q let T denote the topology on Z2 induced by the product topology on Z2 . Are ﬁnite groups in Ult.T / trivial? (3) Is the structure group of K.H/ a free group? (4) (Old question) Does every point of Z lie in a maximal proper principal left ideal of ˇZ? (5) (Old question) Is there an inﬁnite increasing sequence of principal left ideals of ˇZ? (6) (N. Hindman and D. Strauss) Is there an inﬁnite increasing (L -increasing) sequence of idempotents in ˇZ? (7) (N. Hindman and D. Strauss) Is there in ZFC a left maximal idempotent in Z ? (8) Can every compact Hausdorff right topological semigroup be topologically and algebraically embedded into a compact Hausdorff right topological semigroup with a dense topological center? (9) Let bZ denote the Bohr compactiﬁcation of the discrete group Z and let T be the topology on N induced by bZ. Is the largest semigroup compactiﬁcation of .N; T / different from bZ? (10) (A. Arhangel’ski˘ı, 1967) Is there in ZFC a (countable) nondiscrete extremally disconnected topological group? (11) (W. Comfort and J. van Mill) Is there in ZFC a nondiscrete irresolvable (!irresolvable, almost maximal) topological group? (12) (V. Malykhin) Is there an irresolvable (!-irresolvable, almost maximal) topological group of uncountable dispersion character? (13) Is there a (countable) nondiscrete !-irresolvable topological group different from almost maximal? (14) (I. Protasov and Y. Zelenyuk) Does every maximal (almost maximal) topological group have a neighborhood base at zero consisting of subgroups? (15) (A. Arhangel’ski˘ı and P. Collins) Is there in ZFC a countable nondiscrete nodec topological group?

209 (16) (I. Protasov) Is there in ZFC a countable nondiscrete topological group in which every discrete subset is closed? (17) Is it true that for every projective S in F, there exists in ZFC a countable homogeneous regular almost maximal space X with Ult.X / isomorphic to S? (18) Let .G1 ; T1 / and .G2 ; T2 / be countable regular left topological groups and suppose that they are homeomorphic. Are Ult.T1 / and Ult.T2 / topologically and algebraically isomorphic?

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Index

Symbols Ap0 , 103 A , 29 B.G/, 70 B.k; m; n/, 155 C.p/, 102 C 1 .p/, 102 Cp , 109 E.S /, 88 G.Q/, 117 Ip0 , 133 Ip , 130 K.S /, 87 S 1 , 86 Sn , 8 U.A/, 130 U.D/, 33 V .M /, 166 W .Zm /, 113 Xd , 110 Y 0 , 60 ŒM a , 55 ŒX , 131 ŒX < , 131 ƒ.T /, 83 ˇD, 29 ˇS, 84 FP..xn /1 nD1 /, 23 FSI..xn /n

M, 165 P, 158 I 0 .G/, 133 I.G/, 130 M.GI I; ƒI P /, 89 P .X /, 4 Pf .X /, 23 T .F /, 7 T ŒF , 54 T f , 182 C, 147 F, 151 p, 14 A, 29 A, 34 f , 31 , 170 2 .x/, 93 , 170 2 .x/, 93 a1 U , 52 bG, 136 f , 111 A absolute H-coretract, 148 absolute coretract, 147 B band, 152 rectangular, 155 basic mapping, 56 Bohr compactiﬁcation, 136 Burnside semigroup, 155 C canonical decomposition, 111 character, 14 D dispersion character, 186

218 F F -syndetic, 94 ﬁlter, 4 closed, 41 dense, 41 Fréchet, 57 image, 28 locally Fréchet, 61 nowhere dense, 41 open, 41 strongly discrete, 58 trace, 28 ﬁlter base, 6 ﬁnite intersection property, 26 G Green’s relations, 155 group absolutely -resolvable, 198 absolutely resolvable, 198 Boolean, 70 topologizable, 11 H H-projective, 151 homeomorphism of ﬁnite order, 120 regular, 188 spectrally irreducible, 121 trivial, 152 I ideal, 85 left (right), 85 minimal left (right), 87 smallest, 87 idempotent, 86 minimal, 91 right (left) maximal, 91 invariant subset (family), 120 L left saturated subsemigroup, 101 left topological group, 52 left topological semigroup, 52 local automorphism, 121 local homomorphism, 111 surjective, 116

Index local isomorphism, 111 local left group, 110 local monomorphism, 181 P preorderings on the idempotents, 90 projective, 150 proper homomorphism, 115 pseudo-intersection number, 14 R right cancelable, 170 right topological semigroup, 83 S semigroup cancellative, 54 completely simple, 88 left (right) zero, 88 semigroup compactiﬁcation, 84 semilattice, 154 semilattice decomposition, 154 space -irresolvable, 46 hereditarily -irresolvable, 46 open-hereditarily -irresolvable, 46 -resolvable, 46 -compact, 33 almost maximal, 46 extremally disconnected, 32 ﬁrst countable, 3 homogeneous, 2 locally regular, 105 maximal, 45 nodec, 42 strongly extremally disconnected, 43 submaximal, 45 zero-dimensional, 2 spectrum, 121 standard permutation, 123 ˇ Stone–Cech compactiﬁcation of a discrete semigroup, 82 of a discrete space, 29 strong ﬁnite intersection property, 14 structure group, 90 subset of a space locally maximal discrete, 59 perfect, 44

Index strongly discrete, 48 syndetic, 94 T topological center, 83 topological group, 1 almost maximal, 147 maximal, 71 maximally nondiscrete, 79 metrizable, 3 totally bounded, 12 topology group, 1 invariant, 63 left invariant, 52 U Ulam-measurable cardinal, 39 ultraﬁlter, 26 P -point, 38 countably complete, 39 nonprincipal, 27 principal, 27 Ramsey, 36 uniform, 28 ultraﬁlter semigroup, 98 upper semicontinuous decomposition, 131

219

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