Algebra in the Stone-Čech compactification: theory and applications

de Gruyter Expositions in Mathematics 1 The Analytical and Topological Theory of Sernigroups. K H. H o f m m . J. D. Luw...

Author: Neil Hindman | Dona Strauss

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de Gruyter Expositions in Mathematics 1 The Analytical and Topological Theory of Sernigroups. K H. H o f m m . J. D. Luwson, J. S. Pym (Eds.) 2 Combinatorial Homotopy and &Dimensional Complexes. H. J. Baues 3 The Stefan Problem, A. M. Meimtonov 4 Finite Soluble Groups. K Doerk. 7: 0.Hawkes 5 The Riemann Zeta-Function, A. A. Karatsuba, S. M. Voronin 6 Contact Geometry and Linear Differential Equations, CI E Nazaikinskii, K E. Shatalov, B. fir. Sternin 7 Infinite Dimensional Lie Superalgebras, Yu A. Bahturin, A. A. Mikhalev, K M. Petrogradsky, M.CI Zaicev 8 Nilpotent Groups and their Automorphisms. E. I. Khukhra 9 Invariant Distances and Metrics in Complex Analysis, M. Jarnicki, J? Pflug 10 The Link Invariants of the Chern-Simons Field Theory, E. Guadognini I I Global Affine DiKerential Geometry of Hypersurfam, A.-M Li, U. Simon. G. Zhao 12 Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and Theta Functions, K. Hulek. C.Kahn, S. H. Weintraub 13 Elliptic Problems in Domains with Piecewise Smooth Boundaries, S. A. Na:arov, B. A. Plamenevsky 14 Subgroup Lattices of Groups, R. Schmidt I5 Orthogonal Decompositions and Integral Lattices, A. I. Kosrrikin. J? H. Rep 16 The Adjunction Theory of Complex Projective Varieties, M. C. Belrramefti, A. J. Sommese 17 The Restricted 3-Body Problem: Plane Periodic Orbits, A. D. Bruno 18 Unitary Representation Theory of Exponential Lie Groups, H. Leptin, J. Ludwig 19 Blow-up in Quasilinear Parabolic Equations, A.A. Samarskii. V.A. Galaktionov, S. J? Kurdyumov, A. J? Mikhailov 20 Semigroups in Algebra, Geometry and Analysis, K H. Hofmann. J. D. Lawson, E. B. Vinberg (Eds.) 21 Compact Projective Planes, H. Salzmann. D. Betten, Z Grundhifer, H. Hiihl. R L5wen. M. Stroppel 22 An Introduction to Lorentz Surfaces, Z Weinstein 23 Lectures in Real Geometry, E Broglia (Ed.) 24 Evolution Equations and Lagrangian Coordinates. A. M. Meirmanov, K V Pukhnachov, S. I. Shmarev 25 Character Theory of Finite Groups, B. Huppert 26 Positivity in Lie Theory: Open Problems, J. Hilgert. J. D. Lawson. K-H. Neeb. E. B. Vinberg (Eds.)

Algebra in the stone-tech Compactification Theory and Applications

Neil Hindman Dona Strauss

I

I

UNIVERSITA~SBIBLIOTHEK HANNOVER TECHNISCHE INFORMATIONSBIBLIOTHEK

Walter de Gruyter . Berlin New York 1998

.

de Gruyter,Expositions in Mathematics 27

Editors

0.H.Kegel, Albert-Ludwig-UnivelsitZt,Freiburg V. P. Maslov, Academy of Sciences, Moscow W. D. Neumann, The University of Melbourne, Parkville R. 0.Wells, Jr., Rice University, Houston

Authors Neil Hindman Department of Mathematics Howard University Washington. D C 20059 U S

KE

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Dona S t r a w Mathematics Department University of Hull Hull H U 6 7RX United Kingdom

1991 Mathematics Subject Classi/ication: 22-02; 22A15, OSDIO. 54D35 Krywordr. Compactilication, semigroup, Stonc-tech compactifieation, Ramsey theory. combinatorics, nght topological remigroup, idempotent. ,deal minimal ideal, smallest ideal, minimal idempotent

8 Rialed on acid-frssppr which falls within t k guidelines of the ANSI to ensure prmanena and durability.

Hindman, Neil, 1943Algebra in the Stone-tech compactification : theory and applications / b y Neil Hindman. Dona Strauss. cm.- (Dc Gruyler expositions in mathematics : 271 p. Includes bibliographical references and index. ISBN 3-11-015420-X (alk. paper) I. S t o n b h mmpactilicatian. 2. Topological migroups. 1. Strauss, Dona. 1934-. 11. Title. Ill. Serin QA611.23.H56 1998 514.32-dc21 98-29957

CIP

Hlndmn, Neil: Algebra in the Stone-Ckh compactifition :theory and applications I by Neil Hindman ; Dona Strausr. - Berlin ; New York : de Gmyter.

.,,"

100R

(De Gnryter expositions in mathematics ; 27) ISBN 3-1 I-015420-X

OCopyright 1998 by Walter dc Gruytcr GmbH & Co.. D-I0785 krlin. All rights resewed, including thov of translation into fomgn languages. No part of tbk book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy. recording, or any information s t o r a s or rctricval system. without permission in writing from the publisher. Typeset using the authors' TEX liln:1. Zimmmann, Freiburg Printmn: WB-Druck GmbH & Co.. RiedtnlAllxku. - Binding: Liideritz & Bauer GmbH. Bmlin. Cover &sign: Thomas Bonnie. Hamburg.

Preface

The semigroup operation definedpn a discrete semigroup (S, .) has a natural extension. also den& by ., to the Stone-Cech compactification pS of S. Under the extended operation. j3S is a compact right topological semigroup with S contained in its topological center. That is. for each p € OS, the function pp : BS -L ,9S is continuous and for each s E S, the function A, : j3S s j3S is continuous, where pp(q) = q . p and A&) = s .q. In Part I of this book, assuming only the mathematical background standardly provided in the first year of graduate school, we develop the basic background information about compact right topological semigroups, the ~ t o n e d e c compactification h of a dis. cnte space, and the extension of the semigroup operation on S to 0s. In Pas II. we study in depth the algebra of the semigroup (BS. .) and in Part III present some of the powerful applicationsof the algebraof p S to the part of cambinatorics known as R a m q Theory. We conclude in Part N with connections with Topological Dynamics, Ergodic Theory, and the general theory of semigroup compactifications. The study of the semigroup (BS. .) has interested several mathematicians since it was first defined in the late 1950's. As a glance at the bibliography will show, a large number of research papers have been devoted to its properties. There are several reasons for an interest in the algebra of pS. It is inhiwically interesting as being a natural extension of S which plays a special role among semigroup compactifications of S. It is the largest possible compactification of this kind: If T is a compact right topological semigroup. is a continuous homomorphism from S to T. q[S] is dense in 7 , and &(, is continuous for each s E S, then T is aquotient of BS. We believe that pW is interesting and challenging for its own sake. as well as for its applications. Although it is a natural extension of the most familiar of all semigroups, it has an algebraic structure of extraordinary complexity, which is constantly surprising. For example, j3N contains many copies of the freegroup on 2' generators [1521. Algebraic questions about pN which sound deceptively simple have remained unsolved for many years. It is. for instance, not known whether j3N contains any elements of finite order, other than idempotents. And the corresponding question about the existence of nontrivial finite groups was only very recently answered by E.Zelenuk. (His negative answer is presented in Chapter 7.) The semigroup p S is also interesting because of its applications to combinatorial number theory and to topological dynamics.

. Algebraic properties of j3S have been a useful tool in Ramscy theory. Results in Ramsey Theory have a twin beauty. On the one h a d they are rep&zntaxives of pure mathematics at its purst: simple statements easy for almost anyom to understand (though not necessarily to prove). On the other hand, the area has becn widely applied from its beginning. In fact a perusal of the titles of several of the original papers reveals that many of the classical results were obtained with applications in mind. (Hilbert's Theorem Algebra. Schur's Theorem Number Theory; Ramsey's Theorem Logic; the Hales-lewett Theorem Game Theory). The most striking example of an application of the algebraic suuenm of j3S to Ramsey Theory is perhaps provided by the Finite Sums Theorem. This theorem says that whenever N is partitioned into finitely many classes (or in the terminology common withinRamsey Theory, isfinitelycolored), there isasequence (x.)z, with FS((x,,)zl) contained in one class (or monochmme). (Here FS((X.)F=~) 1(X,,,n x. : F i s a finite nonempty subset of PI).) This theorem had b a n an open problem for some decades, even though several mathematicians (including Hilben) had worked on it. Although it was initially proved without using BPI. the first proof given was one of enormous complexity. In 1975 F. Galvin and S. Glazer provided a brilliantly simple proof of the Finite Sums Theorem using the algebraic structure of BPI. Since this time numerous strong combinatorialresults have been obtained using the algebraic structure of BS, whcre S is an arbitrary discrete semigroup. In the process. more detailed knowledge of the algebra ofBS has been obtained. Other f m w s combinataid theonms, such asvander Waerden'sTheoremorRado's Theorem, have elegant proofs b a d on the algebraic properties of j3N. These proofs have in common wjth the Finite Sums Theorem the fact that they w m initially estab lished by combinatorial methods. A simple extension ofthe Finite Sums Theorem was first established using thealgebraof j3W. Thisextension says that wheneverN is finitely ~ (y,,)zl such that FS((xn)z,)UFP((y.)Z,) colored there exist sequences ( x , , ) ~and is monochrome, where FP((y,)g,) = ( l l n e yn ~ : F is a finite nonempty subset of N]. This combined additive and multiplicative result was first proved in 1975 using the algebraic structure of j3N and it was not until 1993 that an elementary p m f was found. Other fundamental results have been established f a which it seems unlikely that elementary proofs will be found. Among such results is a density version of the Finite Sums Theorem, which says roughly that the sequencc (x,,),~=, whose finite sums are monochrome can be chosen inductively in such a way that at each stage of the induction the set of choices for the next term has positive upper density. Another such result is the Central Set Theorem. which is a common generalization of many of the basic results of Ramsey Theory. Significant progress continues to be made in the combinatorid applications. The semigroup j3S also has applications in topological dynamics. A semigroup S of continuous functions acting on a compact Hausdorff space X has a closure in X~ (the space of functions mapping X to itself with the product topology). which is a compact right topological semigroup. This semigmup. called the enveloping semigroup, was first studied by R. Ellis [86]. It is always a quotient of the Stonedech compactification.

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i

flS, as is every semigroupwmpactification of S, and is. in some important cases, equal to #IS. In this Framework, the algebraic propenies of flS have implications for the dynamical behavior of the system. The interaction with topological dynamics works both ways. Several notions which originated in topological dynamics, such as syndetic and piecewise syndetic sets, are important in describing the algebraic suucture of pS. For example. a point p of flS is in the closme of the smallest ideal of p S if and only if for wery neighborhood U of p. U 17S is piecewise syndetic. This last statement can be made more concise when one notes the particular wnshuction of flS that we use. That is. BS is the set of all ultrafillers on S, the principal ultrafilters being identified with the points of S. Under this construction, any point p of flS is precisely {U n S : U is a neighborhood of p). Thus p is in the closure of the smallest ideal of fZS if and only if every member of p is piecewise syndetic. In this book,we develop the algebraic theory of pS and present several of its combinatorial applications. We assume only that the reader has had graduate courses in algebra, analysis, and general topology as well as a familiarity with the basic facts about ordinal and cardinal numbers. In patticular we develop the basic structure of compact right topological semigroups and provide an elementary construction of the . S t o n d e c h compactification of a discrete space. With only three exceptions, this book is self contained for those with that minimal background. The three cases where we appeal to non elementary rsults not p v e d here are Theorem 6.36 (due to M. Rudin and S. Shelah) which assens the existence of a collection of 2' elements of pN no two of which are comparable in the Rudin-Keisler order. Theorem 12.37 (due to S. Shelah) which states that the existence of P-points in pN\N cannot be established in ZFC, and Theorem 20.13 (due to H. Furstenberg) which is an ergodic theoretic result that we use to derive Szemeddi's Theorem. All of our applicationsinvolve Hausdorff spaces, so we will be assuming throughout. except in Chapter 7. that all hypothesized topological spaces are Hausdorff. The first five chapters are meant to provide the basic preliminary material. The concepts and theorems given in the first three of these chapters are also available in other books. The remaining chapters of the book contain results which. for the most pa& can only be found in research papers at present. as well as several previously unpublished results. Notes on the historical development are given at the end of each chapter. Let us make a few remarks about organization. Chapters are numbered consecutively throughout the bmk. regardless of which of the four parts of the bwk contains them. Lemmas, theorems, comllaries, examples, questions, comments, and remarks are numbered consecutively in one group within chapters (so that Lemma 2.4 will be found after Theorem 2.3, for example). There is no logical distinction between a theorem and a remark. The difference is that proofs are never included for remarks. Exercises come at the end of sections and are numbered consecutively within sections. The authors would like to thank Andreas Blass. Karl Hofmann, Paul Milnes, and Igor Pmtasov for much helpful correspondence and discussions. Special thanks go to John Pym for a careful and critical reading of an early version of the manuscript.

viii l-he authors also wish m single out Igor Rotasov for special tlianks, as he has conhibuted several new theorems to the book 'Ihey would like to thank Arthur Grainga Amir Maleki. Dan Tang, Elaine Terry, and Wen Jio Woan f a participating in a seminar where much of the material in this book was presented. and David Gundasw for presenting lectures based on the early material in the book. Acknowledgement is also due to our collaborators whose efforts are fcarured in this book. These collaborators include John Baker, Vitaly Bergelson, John Berglund, Andreas Blass, Dennis Davenport, Walter Deuber. Ahmed El-Mabhouh, Hillel Furstenberg. Salvador Garcia-Ferreira, Yitzhak Kaunelson. Jimmie Lawson, Amha Lisan, Imre Leader. Hanno Lefmann, Amir Maleki. Jan van Mill, Paul Milnes, John Pym Pea Simon. Benjamin Weiss, and Wen-jin Woan. The authors would like to acknowledge suppon of a c,onferencc on the subject of this book in March of 1997 by DFG Sonderforxhungsbereich 344, Diskrete Shukturen in der Mathematik, Universitat Bielefeld, and they would like to thank Walter Deuber for organizing this conference. Both authors would Lie to thank the EPSRC (UK)for support of a visit and the first author acknowledges suppnt received from the National Science Foundation (USA) under grant DMS 9424421 during the preparation of this book. Finally. the authos would like to thank their spousu, Audrey and Ed, for their patience throughout the writing of this book as well as hospitality extended to each of us during visits with the other

April 1998

Contents

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I Background Development

1

2

Notation 1 Semigroup and Their Ideals 1.1 Semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Idempotenu and Subgroups . . . . . . . . . . . . . . . . . . . . . . 1.3 Powers of a Single Element . . . . . . . . . . . . . . . . . . . . . . . 1.4 Ideals 1.5 ldempotents and Order . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Minimal L A Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Minimal Left Ideals with ldempotents . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I2 15 19 23 30

Right Topological Semigroups 2.1 Topological Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Compact Right Topological Semigroups . . . . . . . . . . . . . . . . 2.3 Closures and Roducts of Ideals . . . . . . . . . . . . . . . . . . . . 2.4 Semitopological Semigroups . . . . . . . . . . . . . . . . . . . . . . 2.5 Ellis'Theorern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 33 38 41 43 47

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2

3 Po

3.1 Ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Topological Space BD . . . . . . . . . . . . . . . . . . . . . . . 3.3 Stonezech Compactification . . . . . . . . . . . . . . . . . . . . . . 3.4 More Topology of fl D 3.5 Uniform Limiu via Ultrafilten . . . . . . . . . . . . . . . . . . . . . 3.6 The Cardinality of fl D . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 13s 4.1 4.2

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3 3

8 11

48 48 53 55 58 62

66 69 70 72

Extending the Operation to @S , . . . . . . . . . . . . . . . . . . . 72 Commutativity in BS . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 K(,9 S) and Its Closure Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82 85 88

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90

5 flsandR.nseymeOry 5.1 RamseyTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Idempotmts and F i t e Roducts 5.3 Sums and Products in N . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Adjacent F i t e Unions . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Compactness Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

102

IL Algebra of flS

105

90 92

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%

97

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6 Id& and Commuta~tyin flS 107 6.1 Thesemigroup W . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.2 Intersecting Left l&als . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.3 Numbers of ldempoterits and Idcals . . . . . . . . . . . . . . . . . . 114 6.4 Weakly Left Cancellative Semigroups . . . . . . . . . . . . . . . . . 122 6.5 Semiprincipal Left Ideals 125 6.6 Principal Ideals in BZ . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.7 Ideals and Density 133 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135

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7 Groups in flS 7.1 Zelenuk's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Semigroups Isomorphic to EX 7.3 Free Semigroups and Free Gmups in pS Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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136 136 148 153 157

8 Cancellalion 158 158 8.1 Cancellation Involving Elements of S 8.2 Right Cancelable Elements in BS . . . . . . . . . . . . . . . . . . . 161 8.3 Right Cancellation in BW and BZ . . . . . . . . . . . . . . . . . . . 170 8.4 Left Cancelable Elements in BS . . . . . . . . . . . . . . . . . . . . 174 8.5 Compact Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . 178 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

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9 Idempotenb 9.1 Right Maximal ldempotents . . . . . . . . . . . . . . . . . . . . . . 9.2 Topologies Defined by ldempotents . . . . . . . . . . . . . . . . . . 9.3 Chains of Idempotenu . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 IdenritiesinpS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

186 186 194 199 202 204

f i

10 Homomorpbm?l 205 206 10.1 Homomorphisms to rhe Circle Group 10.2 Homomorphisms from 87 into S* . . . . . . . . . . . . . . . . . . . 210 214 10.3 Homomorphisms from T* into S* 10.4 Isomorphisms on Principal Ideals 218 221 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11.The Rudin-Kelsler Order 11.1 Connections with Right Cancelability . . . . . . . . . . . . . . . . . 11.2 Connections with Left Cancelability in N* 11.3 Further Connections with the Algebra of SS . . . . . . . . . . . . . . 11.4 The Rudin-Frolik Order . . . . . . . . . . . . . . . . . . . . . . . . Notes: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

222 223 229 231 232 234

12 Ullralilters Generated by Finite Sums 12.1 Martin's Axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Strongly Summable Ulbafilters Existence . . . . . . . . . . . . . 12.3 Strongly Summable Ultrafilters - Independence 12.4 Algebraic Propties . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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236 236 240 245 248 256

13 Multiple Structures In @S 13.1 Sums Equal to Products in Sa: . . . . . . . . . . . . . . . . . . . . . 13.2 The Distributive Laws in /3Z . . . . . . . . . . . . . . . . . . . . . . 13.3 Ultrafilters on R Near 0 . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Left and Right Continuous Extensions . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

258 258 264 268 272 275

III.Combinatorial Applications

277

14 The Central Sets Theorem 14.1 Van der Waerden's Theorem . . . . . . . . . . . . . . . . . . . . . . 14.2 The Hales-Jewett Theorem . . . . . . . . . . . . . . . . . . . . . . 14.3 The Commutative Central Sets Theorem . . . . . . . . . . . . . . . . 14.4 The Noncommutative Central Sets Theorem . . . . . . . . . . . . . . 14.5 Combinatorial Characterization . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

279 279 281 283 286 288 294

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296 15 Partition Regularity of Matrices 15.1 Image Partition Regular Matrices . . . . . . . . . . . . . . . . . . . 2% 15.2 Kernel Partition Regular Matrices . . . . . . . . . . . . . . . . . . . 301 15.3 Kernel Partition Regularity Over N . . . . . . . . . . . . . . . . . . 304 15.4 Image Partition Regularity Over N . . . . . . . . . . . . . . . . . . . 308 15.5 Matrices with Entries from Fields . . . . . . . . . . . . . . . . . . . 315 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

Contents 16 IP. IP.Ccotral. and Central* S e b 16.1 Sets in Arbitrary Semigroups . . . . . . . . . . . . . . . . . . . . . . 16.2 IP* and Central Sets in PI . . . . . . . . . . . . . . . . . . . . . . . 16.3 IP* Sets in Weak Rings . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Spectra and Iteratad Spectra . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

320 320 324 332 336 338

17 Stuns and Pmdoets 339 17.1 Ultrafilters with Rich Smtcturr . . . . . . . . . . . . . . . . . . . . . 339 17.2 Painvise Sums and Products . . . . . . . . . . . . . . . . . . . . . . 17.3 Sums of Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Linear Combinations of Sums . . . . . . . . . . . . . . . . . . . . . 17.5 Sums aod Products in (0 I) . . . . . . . . . . . !. . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

341 346 354 361 367

18 Moltidimensional Ramey Theory 18.1 Ramsey's Theorem and Generalizations . . . . . . . . . . . . . . . . 18.2 IP* Sets in Product Spaces . . . . . . . . . . . . . . . . . . . . . . . 18.3 Spaces of Variable Words . . . . . . . . . . . . . . . . . . . . . . . 18.4 Carlson's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

369 369 376 381 386 393

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N Connections With Other Stroeturn

395

19 Relatfom With Topological Dynamics 19.1 Minimal Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . 19.2 Enveloping Semigroups . . . . . . . . . . . . . . . . . . . . . . . . 19.3 Dynamically Central Sets . . . . . . . . . . . . . . . . . . . . . . . 19.4 Dynamically Generated IP* Sets . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

397 397 400 404 407 411

20 Density -- Connections with Ergodic Tbcory 20.1 Upper Density and Banach Density . . . . . . . . . . . . . . . . . . 20.2 The Correspondence Ptinciple . . . . . . . . . . . . . . . . . . . . . 20.3 A Density Version of the Finite Sums Theorem : . . . . . . . . . . . Nous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

412 412 417 419 424

21 Other Semigmup Compactifications 21.1 The L M C WAP. AP.and 8APCompactifications . . . . . . . . . 21.2 Right Topological Compactifications. . . . . . . . . . . . . . . . . . 21.3 Periodic Compactifications as Quotients . . . . . . . . . . . . . . . . 21.4 Spaces of Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Uniform Compactifications . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

425 425 429 432 441 445 453

.

BibUopphy List of Symbols

Index

Part I Background Development

Notation

.

We write N for the set (1.2.3. . ..] of positive integers and o = (0. 1.2,. .) for the nonnegative integers. Also o is the first infinite ordinal. and thus the first infinite cardinal. Each ordinal is the set of all smaller ordinals. W+=(XEW:X>O). Given a function f and a set A contained in the domain of f . we write f [ A ] = [f(x) : x E A)andgivenany setB wewrite f - ' [ E l = [x E Domain(f) : f(x) E B). Given a set A, .Pf(A) = IF : 0 # F G A and F is finite). Definitions of additional unfamiliar notation can be located by way of the index.

~. Chnpter 1

Semigroups and Their Ideals

We assume that the reader has had an introductory modern algebra course. This assumption is not explicitly used in this chapter beyond the fact that we expect a cenain amount of mathematical malurity.

1.1 Semigroups Definidon 1.1. A semigmup is a pair (S. *) wherc S is a nonempty set and * is a binary associative o p t i o n on S.

Formally a b i ~ r operation y on S is a function * : S x S -r S and the operation is ass~ciativeifandonlyif *(*(x, y ) , z ) = * ( x , * ( y , z ) ) forallx, y , andz i n s . However. we customarily write x * y instead o f * ( x , y ) so the associativity requirement becomes the more familiar (x * y ) * z = r * ( y * 2). The statement that * : S x S -+ S. i.e.. that x * y E S whencverx, y E S is commonly referred to by saying that "S is closed under *". Example 1.2. Each of thefollowing is a semigroup.

Clearly, thecompoaition of twohomomorphism,if itexists. is also ahomomorphism.

me reader who is familiar with the concept of a category will recognize that there is a category of semigroups. in which the objects an semigroups and the morphisms an homomorphisms. The free semigrwp S o n the alphabet A has t k following property. Suppose that T is an arbitrary semigroup and that g : A + T is any mapping. Then there is a unique honmmorphism h : S + T with the property that h(a) = g(a) for every a E A. ( T k proof of this assertion is Exercise 1.1.1.) Delinition 1.5. Let (S, *) be a semigroup and let a E S. (a) The element a is a Ieji identify for S if and only if a x = x for every x E S. (b) The element a is a right idenriry for S if and only if x + a = x for every x e S.

*

(c) The element a is a two sided idenriry (or simply an idenriry) for S if and only if a is both a left identity and a right identity. Nole that in a "free semigroup with identity" the element 0 is a two sided identity (SOthe terminology is appropriate). Note also that in a left zero semigroup. every element is a right identity and in a right zero semigroup, every element is a left identity. On the other hand we have the following simple fact.

Remark 1.6. Let (S. *) be a semigmup. Ife is a lefi identity for S and f is a right identifyfor S. then e = f . In particular: a semigmup can have ar most one two sided identity. Given a collection of semigroup ((Si, *i));,~, the Cartesian pmduct XierSi is nahnally a semigroup with the coordinatewise operations. M u i t i o n 1.7. (a) Let ((Si, *i))jE, be an indexed family of semigroup and let S = X ;GI S;. With the operation defined by (.? a F); = x; *i yi, the semigroup (S. 8 ) is

*

called the direcrpmducr of the semigroup (Si, *i). (b) Let ((S;, *;))i~rbe an indexed family of semigroups where each Si has a two sided identity ei. Then the direct sum of the semigroups (S;. g i ) is Si = I.? 6 X;,l S; : [ i E I :X; # ei] is finite).

el,,

We leave to the reader the easy verificarion that the & i t product opetation is associative as well as the verification that if i .3 E Sirthen f * E Si.

eiEl

eiEI

D e M t i o n 1.8. Let (S. *) be a semigroup and let a . b, c E S. (a) The element c is a left a-inverse for b if and only if c b = a .

*

(b) The element c is a righr a-inverse forb if and only if b * c = a. (c) The element c is an a-inverse forb if and only if c is both a left a-inverse forb and a right a-inverse forb.

The terms left a-inverse, righr a-inverse. and a-inverse are usually replaced by left inverse. right inverse, and inverse respectively. We introduce the more precise notions because one may have many left or right identities.

Definition 1.11. Let S be a semigroup. (a) S is commutarive if and only if xy = yx for all x , y E S. (b) The center of S is (x E S : for all y E S, xy = yx]. (c)Givenx E S.Ax : S + Sisdefincdby AAy) =xy. (d) Given x € S, p, : S -+ S is defined by pAy) = yx. (e) U S ) = {A, : x € S). (f) R(S) = (p, :x E S).

R e d 1.12. k t S be a scmigmup. 7'hen (L(S), o) ond (R(S),o) a n semigmups. Since our semigroups are not necessarily commutative we need to specify what we xi. There are 2 reasonable interpretations (and n ! 2 unreasonable mean by ones). We choose it to mean the pmduct in increasing order of indices because that is the order that naturally arises in our applications of right topological xmigroups. More formally we have the following.

-

n:=I

Definition 1.13. Let S be a semigroup. We define inductively on n E N.

n:=Ixi for [xt

.

xz, .. .,x.) g S

Definition 1.14. Let S be a semigroup. (a) An element x E S is right cancelable if and only if whenever y. z G S and yx = zx. one has y = z. (b) An elemmtx E S is leftcancelable if and only ifwhenever y. z E Sandxy = xz, one has y = z. (c) S is right cmcellarive if and only if every x E S is right cancelable. (d) S is lefr cancellative if and only if every x 6 S is left cancelable. (e) S is cancellative if and only if S is both left cancellative and right cancellative. Theorem 1.15. L a S be a semigmup. (a) 7kJunction A : S + L(S) is a homomorphism onro L(S). (b) mefunction p : S + R(S) is an anti-homomorphism onto R(S). (c)IfS is right cancellarive, then S and L(S) are isomorphic. . (d) IfS is lefr cancellative, then Sand R(S) a n anri-isomorphic. Pmof: (a) Given x, y, and z in S one has (A, o A,)(z) = A,(yz) = x(yz) = (xy)z = A.&) so Ax o Ay = AKy. (c) This is part of Exercise 1.1.4. 0

Right cancellation is a far stronger requirement than is needed to have S == L(S). See Exercise 1.1.4.

-

Exercise 1.1.1. Let S be the free semigroupon the alphabet A and let T be an arbitrary T is any mapping. Prove that there is a unique semigmup. Assume that g : A homomorphism h : S + T with the property that h ( a ) = g ( a ) for every a E A.

8

1 Semigmups and Thei~Ideals

Exercise 1.1.2. Rove that statements (b). (d), and (e) of 'lheorem 1.10 are equivalent.

1.13. R a r e th& in the semigroup ( 'x. o), the letl cancelable elements are the tnjective functions and the right cancelableelements arc the sujectlve functions.

En*

Exercise 1.1.4. (a) Rove Theorem 1.15 (c). (b) Give an example of a semipup S which is not right eanceUative such that S f ; L(S).

Exercise 1.15. Let S be a right cancellative semigroup and let a E S. Rove that if there is some b E S such that ab = b, then a is a right identity for S. Exercise 1.1.6. Ron thal'4f S does not have an identity, one may be adjoined" (and in fact one may be adjoinedeven if S already has an identity). m t is, Let S be a semigroup and let e be an element not in S. Define an operation * on S u [el by x * y = x y if x . y E Sand x * e = e c x = x . Rove that (S U [el. *) is a s e m i p u p with identity e. (Note that if S has an identity f , it is no longer the identity of S U(eJ.) Exercise 1.1.7. Suppose that S is a cancellative semigroup which does not have an identity. Rove that an identity can be adjoined to S so that the extended s e m i p u p is also cancellative. Exercirc 1.11 Let S be a commutative cancellative semigroup. We define a relation E on S x S by stating that (a,* (c, d ) if and only if a d = bc. Rove that this is an equivalence relation. Let (a. b) denote the equivalence class which contains the element (a, b) E S x S. and let G denote the set of all these equivalence classes. We define a binary relation t on G by stating that (a, b) (c. d ) = (ac, bd). Rove that this is well defined, that (G. .) is a group and that it contains an isomorphic copy of S. (The p u p G is called the gmup of quotients of S. If S = (N. +), G = (Z, +); if S = (N, .), G = (Q+, .),when Q+ = {x E Q : x z 01.)

.

13 Idempotents and Subgroups Our next subject is "idempotents". They will be very i m p l a n t to us throughout this book. Definition 1.16. Let S be a semigroup. (a) An element x E S is an idempormr if and only if x x = x . (b) E(S) = [ x e S :x is an idempotent]. (c) T is a subsemigmup of S if and only if T S and T is a semigroup under the restriction of the operation of S. (d) T is a subgroup of S if and only if T G Sand T is a p u p under the resaiction of the operation of S. (e) Let e E E(S). Then H(e) = UIG : G is a subgroup of S and c E GI.

=

1

I

I

1

I

I

I

1

!

1.2 Idemp~enuand Subgmups

9

Lemma 1.17. k t G be a gmup with identity e. Then E ( G ) = (e). p m f Assume f E E(G). Then f f = f = f e. Multiplying on the left by the inverse 0 off. one gets f = e. As a consequence of Lemma 1.17 the statement "e E G" in the definition of H ( e ) is synonymous with "e is the identity of G". Note that it is quite possible for H(e) to q u a i [e),but H(e) is never empty.

Theorem 1.18. Let S be a smigmup and let e subgroup of S with e as idmtiv.

E

E(S). Then H ( e ) is the largest

P m f :It sufficesto show that H ( e )is a group since e is trivially an identity for H(e) and H ( e )containsevery group with e as identity. Forthis it in turn suffices to show that H(e) is closed. So let x. y E H ( e ) and pick subgroups G I and G2 of S with e E G I r l G2 and x E G I and y E Gz. Let G = (ny=,x; : n E N and ( ~ 1 ~...., x 2x.) c G I U G2). Then x y E G and e f G so it suffices to show that G is a group. For this the only requirement that is not immediate is the existence of inverses. So let n:=,x; ~ G . F o r i€11.2 ,...,n),picky;suchthatx.+~-iy; = e . T h e n n ~ = l yE~ G and xi) . y;) = e. 0

(n:,,

The groups H ( e )are referred to as maximalgmups. Indeed, given any group G g S, G has an identity e and G c H(e). Lemma 1.19. Let S & a semigroup. let e E E(S), and let x E S. Then the fallowing sratements are equivalent.

(a) x E H(e). (b) x e = x a n d t h e n i s s o m e y ~ S s u c h t h a t y e = y a n d x y = y x = e .

Pmof We show the equivalence of (a) and (b): the equivalence of (a) and (c) then follows by a left-right switch. The fact that (a)implies (b) is immediate. (b) implies (a). Let G = ( x E S : xe = x and there is some y E S such that ye = y and x y = yx = el. It suffces to show that G is a group with identity e. To establish closure, let x , z E G. Then xze = xz. Pick y and w in S such that ye = y , w e = w , x y = yx = e. and zw = wz = e. Then wye = w y and xzwy = i e y = x y = e = w z = wez = wyxz. Trivially, e is a right identity for G w, it suffice^ to show that each element of G has a right e-inverse in G. Let x E G and picky E S such that ye = y and yx = x y = e. Note that indeed y does satisfy the requiremenu to he in G. 0 Example 120. Let X be any set Then the idempotents in ( x X , o) a n Ihefunctions f E XX with the propem that f ( x ) = x for every x E f [XI. We next define the concept of a free group on a given set of generators. The underlying idea is simple, but the rigorous definition may seem a little troublesome. The

1 Semigroupsand Their Ideals

10

...

basic idea is that we want to construct all expressions of theform a;'@ a f , where each ai E A and each exponent ei E Z, and to combine them in the way that we are forced to by the group axioms.

Definition 1.21. Let S be the free semigmup with identity on the alphabet A x ( I , -1 ] and let

+

G = (g E S : there do notexist t.r 1 idomain(g). a E A and i E (1. -1) for which g(t) = (a, i) andg(r + 1) = (a. -i)]. Given f, g E G\f0) with domain(f) = (0, I..

..,n - 1)

and domaiin(g) =, (0.1..

...m - 11.

define f .g = f-g unless there exist a E A and i E (1, -1) with f (n - 1) = (a. i) and g(0) = (a. 4 ) . In the latter case, pick the largest k E N such that f a all r E (1.2, .. . k), there e ~ sbt E A and j E (1. -1) such that f (n t) = (b, j) and g(t 1) = (b. - j ) . If k = m = n , t h e n f . g = 0 . OtheM.ise.domain(f . g ) = ( O , l , ....n + m - 2 k - 1 ) andforre(1.2 n+m-2k-1).

-

-

.

.....

Then (G, .) is thefree gmup generated by A. It is not hard to prove that, with the operation defined above, G is a group. We customarily write a in lieu of (a. 1) and a-' in lieu of (a, -1). Then in keeping with the notation to be introduced in the next section (Section 1.3) we shall write the ~ .illustration, we have w o r d a b - ~ b - ~ b - ~ aa-- ~1 bb, for example, as ~ b - ~ a - ~Asb an ( ~ b - ~ a - ~ b(b-2a3b-4) ~). =~ b - ~ a b - ~ . We observe that the free group G generated by A has a universal property given by the following lemma.

Lemma 1.22. k t A be a set, let G be the* gmup generated by A. let H be an a_rbirmrygmup, and lek# : A + H be any mapping. There is a unique homomorphism 4 : G + H f o r w h i c h # ( g ) = # ( g ) f o r e v e r y g € A. PmoJ This is Exercise 1.2.1. We shall need the following result later.

Theorem 133. Ler A beaset. Ier G berhefreegmu~generatedbyA. andkt g E G\(0]. There exist ofinire gmup F and a homomorphism 4 : G -+ F such that @(g)is nor the identify of F.

1.3 Powan of a Single Element

11

pmfi Let n be the let@ of g, let X = (0.1,. ..,n), and let F = (f E X~ : f is one-to-one and onto X ) . (Since X is finite, the "'onto" requirement is redundant.) men (F,o) is a group whose identity is r, the identity function from X to X. Given a E A, let D(a) = (i E (0, I, ....n 1) : g(i) = a-I) and let E(a) r (i E (1.2.. ..,n) : g(i - 1) = a). Note that since g E G. D(a) n E(a) = 0. Define )(a) :D(a) U E(a) + X by

-

and note that, because g e G, )(a) is one-to-one. Extend @(a)in any way to a member of F. Let ) : G + F be the homomorphism extending $ which was guaranteed by Lemma 1.22. Supposethatg=aoi~al'l...am-1'"-l.wherea, E Aandi, E (-1,l)foreachr E (0.1.2,. . . n - 1). We shall show that, for each k E (1.2.. ,n). $(~&-1"-')(k) = k-1. To see Whis. first suppose that ik-I = 1. Then k E E(ax-I) and so $ ( ~ k - ~ ) ( k= ) k-1. Now s u p p t h a t i r - 1 = -1. Thenk 1 E D(ak-l)andso@(at-l)(k - 1) = k. Thus F ( ~ k - ~ - ' ) ( k= ) )(ak-l)-'(k) = k ;I. easy to see that $(g)(n) = $(ao")T(al'l). ..&(a.-l'n-l)(n) = 0 and It is o hence that $(g) is not the identity map.

.

..

-

-

Exercise 12.1. Rove Lemma 1.22.

1 3 Powers of a Single Element Suppose that x is a given element in a semigmup S. For each n E N. we define an element xn in S. We do this inductively. by stating that x1 = x and that xu+' = xxn if x" has already been defined. It is then straightforward to prove by induction that xmx" = xm+" for every m, n E N. Thus (xn : n E W) is a commutative subsemigroup of S. We shall say that x hasfinite onfer if this subsemigroup is finite; otherwise we shall say that x has infinite order. If S has an identity e, we shall define x0 for every x E S by stating that x0 = e. If x has an inverse in S,we shall denote this inverse by x-'. and we shall define x-" for every n E N by stating that x-" = (I-')". If x does have an inverse, it is easy to prove that xmx" = xm+" for every m, n E 2. Thus (x" : n E Z) forms a subgroup of S. If additive notation is being used, x" might be denoted by nx instead. The index law mentioned above would then be written as: mx nx = (m n)x.

+

+

Theorem 1.24. Suppose that S is a semigmup and that x E S has infinite order: Then he subsemigmup T = (x" : n E N) of S is isomorphic to (N, +).

Proof:The mapping n I+ x" from (N.+) onto T is a Juject@e homomorphism, and so it will be sufficient to show that it is one-to-one. Suppse then that xm = x" for some m, n E N satisfying m < n. Then x"-" is an identity for xm. and the same statement holds forx4'"-"'I, when q denotes any positive integer. Suppose that s is any integer satisfying s z m. We can writes m = q(n m) r where q and m arc non-negative integers and r < (n m). So x h xXS-"'x" = X ~ ( ~ + " ) + ' X ~= xrxm. It follows that (x' : s > m} is hnite and hence that T is finite. contradicting our assumption that x has infinite order. 0

-

-

- +

Theorem 135. AnyJbrite semigmup S contains on idempotent

Proof This statement is obviously me if S contains only one element. We shall prove it by induction on the number of elements in S. We make the inductive assumption that the theorem is hue for all semigroups with fewer ekdents than S. Choose any x E S. There are positive integers rn and n satisfying xm = xn and m c n. Then xn-mxnt = xm. Consider the subsemigroup ( y E S : xn-"'y = y) of S. If this is the whole of S it contains x''-~ and so xn-" is idempotent. If it is smaller than S.it contains an idempotent. by our inductive assumption. 0

k r e i s e 13.1. Prove that any fidite cancellative semigroup is a group.

1.4 Ideals The terminology "id& is borrowed from ring theory. Given subsets A and B of a semigroup S. by AB we of course mean {ab : a E A and b E B].

=

DeAaftion 126. Let S be a semigroup. (a) L i s a I e ~ i & u l o f S i f a n d o n l yi f @ # L E S a n d S L L. (b)Risarightidealof SifandonlyifOf R E S a n d R S S R . (c) I is an ideal of S if and only if I is both a left ideal and a right ideal of S. An ideal I of S satisfying I # S is called apmper ideal of S. Sometimes for emphasis an ideal is called a '2wo sided ideal". We oftm deal with semigroups in which the operation is denoted by In this case the tenninology may seem awkward for someone who is accustomed to working with rings. That is, a left ideal L satisfies S + L c L and a right ideal R satisfies R S C R. Of special importance for us is the notion of minimal left and right ideals. By this we mean simply left or right ideals which are minimal with respect to set inclusion.

+.

+

Deflnltion 137. Let S be a semigroup. (a) L is a minim1 lefr ideal of S if and only if L is a left ideal of S and whenwer J is a left ideal of S and J g L one has J = L. (b) R is a minimal right ideal of S if and only if R is a right ideal of S and whenever J is a right ideal of S and J c R one has J = R.

(c) S is Iej? simple if and only if S is a minimal left ideal of S. (d) S is right simple if and only if S is a minimal right ideal of 3. (e) S is simple if and only if the only ideal of S is S.

.~.

We do not deb a minimal ideal. As a cansequence of Lemma 1.29 below. we shall see that there is at most one minimal two sided ideal of a semigroup. Consequently we use the term "smallest" to refer to an ideal which docs not properly contain another ideal. Observe that S is left simple if and only if it has no proper left ideals. Similarly, S is right simple if and only if it has no proper right ideals. Whenever one has a t h m m about left ideals, there is a comsponding theorem about right ideals. We shall not usually state bothresults. Clearly any semigroup which is either right simple or left simple must be simple. The following simple example (pun intended) shows that the convene fails. Example 138. Lct S = { a .b , c , d ] where a, b, c, and d are any distinct objects and let S have rhe following multiplication table. Then S is simplc bur is neither lef?simple nor right simple.

b

a

b

a

b

d

c

d

c

d

One can laboriously verify'tiiat the table docs define an associative operation. But 128 computations (of (xy)z and x(yz)) are required. somewhat fewer if one is clever. It is usually much easier to establish as&iativiG by representing the new semigroup as a subsemigroup of one with which we are already familiar. In this casc, we can represent S as a semigroup of 3 x 3 matrices, by putting:

To verify the assmioris of the example, note that { a ,b ) and { c ,d l are right ideals of S and ( a , c ) and { b ,d ) are left ideals of S. Lemma 1.29. Lcr S be a semigroup. (a) Lct L I and L2 be lef?ideals of S. Then LIn L2 is a I& ideal of S ifand only if L,n L z + 0 . @) Let L be a I& ideal of S and let R be a right ideal of S. Then L n R # 0. Pmof: Statement (a) is immediate. To see (b). let x E because x E L and yx E R because y E R.

L and y E R. Then yx

E

L 0

1 Semigroupsaud Their I d 4 s

14

.Lemma 130. Let S be a semigmup. (a) k t x E S. Then x S is a right idcal , S x is a l@ ideal and SxS is on ideal. (b) Let e E E(S). Then e is a l q i identity for eS, a right identity for Se, and an identityfor eSe. let e E EfS). To see that e is a left identity Proof Statement (a) is immediate. For 6). for eS, let x E eS and pick t E S such that x = et. Then e x = eet = et = x. Likewise e is a right identity for Se. 0

Theorem 131. Let S be a semigmup. (a) I f S is I$. simple and e E E(S), then c is a right identityfor S. @)IfLisaleftidealofSandsE L, thenSs G L. (c) k t 0 # L C S. Then L is a minimal lefl ideal of S if@ only iffor each s E L. Ss = L.

Pmof (a) By Lemma 1.30 (a). Se is a left ideal of S. so Se = S so Lemma 1.30 (b) applies. (b) This follows immediately from the definition of left ideal. (c) Necessity. By Lemma 1.30 (a) Ss is a left ideal and by (b) Ss G L so, since L is minimal, Ss = L. Sufficiency.Since L = Ss for some s E L, L is a left ideal. Let J be a left ideal of S w i t h J G L a n d p i c k s ~J.Thenby(b),SsG J s o J E L = S s c J. We shall observe at the conclusion of the following definition that the objects defined rhne exist.

Definition132. Let S be a semigroup. (a) The smallest ideal of S which contains a given element x E S is called the principal ideal generated by x. (b) The smallest left ideal of S which contains x is called the principal l q i ideal of S generated by x. fc)The smallest right ideal of S which contains x is called the principal right ideal generated by x. Theorem 133. Let S be a semigmup and let x E S. (a) The principal ideal generated by x is SxS U x S U Sx U (x).

(b) I f S has an identity, then the principal ideal generated by x is SxS. (c) Theprincipal left ideal generated by x is Sx U 1x1 and the principal right ideal generated by x is x S u 1x1.

Proof This is Exercise 1.4.1.

0

Exercise 1.4.1. Prove Theorem 1.33, Exertisc 1.43.. Describe the ideals in each of the following semigroups. Also describe the minimal left ideals and the minimal right ideals in the cases in which these exist.

(i) 0%+). (ii) (P(X), V). where X is any set (iii) (P(X), fl).w h m X is any set. (iv) ([O, 11. .). when . denotes multiplication. (v) The set of real-valued functions defined on a given set, with pointwise multiplication as the semigroup opration. (vi) A left z m semipup. (vii) A right zero semigroup.

Exerdse 1A.3. Let X be any set. Describe the minimal left and right ideals in XX. Exercise 1.44. Let S be a commutative semigroup with an identity e. Prove that S has a proper ideal if and only if there is some s E S which has no e-inverse. In this case. prove that (s E S :s has no e-inverse) is the unique maximal proper ideal of S.

1.5 Idempotents and Order Intimately related to the notions of minimal Iefl and minimal right ideals is the notion of minimal idempotents. Definition 1.34. Let S be a semigroup and let e, f (a)esLf ifandonlyife=ef. (b) es,q f if and only if e = f e, and (c) e f f if and only if e = ef = fe.

€

E(S). Then

InthesemigroupofExample 1.28.onesetsthat c c L a , a 5 ~ cb, f ~ d d. c ~ b , o S ~ b , b s R a ,cZRd,and d j ~ c while . the relation 5 is simply equality on this semigroup.

Remark 1.35. Let S be a semigroup. Then f ~f .~and, 5 am mnsitive and reflexive relations on E(S). In addition. 5 is antisymmem'c. When we say that a point e is minimal with respect to a (not necessarily antisymmetric) relation 5 on a set B, we mean that iff E B and f 5 e, then e 5 f (so if j is antisymmehic, the conclusion becomes e = f ) .

Theorem 136. k t S be a semigroup and Ier e E E(S). 7lrefollowing statementsore equivalent. (a) The element e is minimal with respect fo 5 (b) The element e is minimal with respect to f R. (c) The element e is minimal with respect ro f ~

.

'

1 Semigmup and Their Ideals

P m f (b)implies (a). Assume that e is minimal with respect to S R and let f

j e. Then f = e f so f ~ n e s o e s ~ f . T h e n e f=e z f . We show that (a) implies (b). (Then the equivalence of (a) and (c) follows by a I&-right switch.) Assume that e is minimal with respect to 5 and let f jRe. Let g = fe.Thengg= f e f e = f f e = f e = g s o g ~ E ( S ) . A l s o . g = f e = e f e s o eg = eef e = ef e = g = efee = ge. Thus g 5 e so g = e by the minimality o f e. That is. e = f e so ~ S f R as required. 0

-.

As a consequence o f Theorem 1.36, we are justified in making Ibe following definition. Dehilion 137. Let S be a semigroup. Then e is a minimal idemporenr i f and only i f

e

E(S) and e is minimal with respect to any (hence all) o f the orders 5. +R, or SL.

We see that the notions o f "minimal idempotent" and "minimal left ideal" and "minimal right ideal" are intimately related We remind the reader that there is a corresponding "right" vmion o f the following theokm.

Theorem W& Let S be a semigroup and let e l E(S). (a)Ife is a member of some minimal Ief ideal (equivalentlyif Se is a minimal lef ideal), then e is a minimal idempotent. (b)If S is simple and e is minimal, then Se is a minimal I& ideal. (c) I f every leJi ideal of S contains an idempotent and e is minimal, then Se is a minimal I& ideal. (d) I f S is simple or every I& ideal of S hns an idempotenf then the following statements a n equivalent. ( i ) e is minimal. (ii) e is a member of some minimal lef ideal of S. (iii) Se is a minimol lefr ideal of S.

P m f (a) Let L be a minimal left ideal with e E L. m e existence of a set L with this propeny is equivalent to Se being minimal, by Theorem 1.31 (c).) Then L = Se. Let f E E ( S ) with f 5 c. Then f = f e so f E L so (by Theorem 1.31(c)) L = Sf so e ~ S sobyLemma1.30(b).e=ef f soe=ef =f . (b)Let L be a left ideal with L Se. W e show that Se g L (and hence Se r L). Pick some s E L. Then s E Se so by Lemma 1.30(b), s = se. Also. since S is simple SsS = S. so pick u and v in S with e = vsu. Let r = eue and t = ev. Then tsr = evseue = eusue = eee = e and er = eeue = cue = r. Let f = rrs. Then f f = rtsrrs = r(tsr)ts = r e f s = reevs = revs = rts = f , so f E E(S). Also. f e = r f s e = rts = f and ef = errs = rrs = f so f 5 c so f = e. Thus Se = Sf = Srts g Ss g L. ( c )Let L be a left ideal with L g Se. We show that e E L (so that Se E L and hence Se = L). Pick an idempotent r E L, and let f = et. Then f E L. Since t E Se. t = re. Thus f = et = ere. Therefore f f = etet = ett = et = f so f E E(S). Also ef = e e t e = e t e = f and f e = e t e e = e t e = f so f i e s o f =eandh'enceeEL.

i

1.5 Idmpotents and Order

17

(d) This follows from (a), @). and (c). We now obtain several chara~ferizationsof a group.

Theorem 139. Let S be a semigmup. Thefollowing statements are equivalent. (a) S is cancellative and simple and E(S) # 0. (b) S is both lefi simple and right simple. (c) For all a and b in S. the eqdonr ax = b and ya = b hove solutionr x, y in S. (d) S is a group.

P m f (a) implies @). Pick an idempotent e in S. We show first that e is a (two sided) identity for S. Let x E S. Then ex = eex so by left cancellation x = ex. Similarly. x = xe. To see that S is left simple, let L be a left ideal of S. Then LS is an ideal of S so LS = S, so pick t E.L and s E S such that e = Is. Then srs = s e = s = es so cancelling on the right one has s t = e. Thus e E L so S = SL E L. Consequently S is left simple, and similarly S is right simple. @)implies (c). Let a , b E S. ThenaS = Ssothere is somex E S suchthatax = b. Similarly. since Sa = S, there is some y E S.such that ya = b. (c) implies (d). Pick a E S and pick e G S such that ea = a. We show that e is a left identity for S. Let b E S. We show that eb = b. Pick some y E S such that ay = b. Then eb = eay = ay = b. Now given any x E S there is some y E S such that yx = e so every element of S has a left e-inverse. (d) implies (a). Trivially S is cancellative and E(S) # 0. To x e that S is simple, let IbeanidealofSandpickxeI.Letybetheinvnseofx.~enxyEIsoI =S. As promised earlier, we now see that any semigroup with a left identity e such that every element has a right e-inverse must be (isomorphic to) the Cartesian product of a p u p with a right zero semigroup.

Theorem 1.40. Let S be a semigmupand let e be a lcfr identifyfar S such tharfor each x E S t h e m i s s o m e y ~ S w i t h x y= e . LetY = E(S)andIetG=Se. 7henYisa right zem semigmup and G is a group and S = GY 2: G x Y. Pmof: We show first that:

To establish (:), let x E Y and y E S be given. Pick z E S such that xz = e. Then xe = xxz = xz = e. Thereforexy = x(ey) = (xe)y = ey = y, as required. From (*) it follows that for all x. y G Y. xy = y, and Y # 0 because e E Y. so to see that Y is a right zem semigroup. it suffices to show that it is a semigroup. that is that Y is closed. But this also follows from (*) since. given x , y E Y one has xy = y E Y. Now we establish that G = Se is a group. By Lemma 1.30i.b). e is a right identity for G. Now every element in S has a right e-inverse in S. So every element of G has

18

I Semipups and lhcir Ideals

a right e-inverse in S. By Theom 1.10 we need only to show that every clement of G has a right e-inverse in G. To this end let x E G be given and picky E S such that xy = e. Then ye E G and xye = ee = e so ye is as required. Since we also have GG = SeSe G SSSe Se = G. it follows that G is a group. Now define rp : G x Y + S by ~ ( gy), = gy. To see that rp is a homomorphism. E G x Y. Then l e t h . ~ ~(gz,yz) ).

To see that rp is sujstive. let s E S be given.. Then s e E Se = G, and so there exists x E Se such that x(se) = (se)x = e. We claim that x5 E Y = E(S). Indeed, xsxs = xsexs = xes = xs

(since x E G. ex = X) (since x

E G.

xe = x).

Thus (se, xs) E G x Y and (p(se, xs) = sexs = es = s. Since p is onto S. we have established that S = GY. Finally to see that rp is one-to-one, let (g, y) E G x Y and lets = &-, y) We show that g = se and y = xs where x is the (unique) inverse of s e in Se. Now s = gy so se = gye = ge =g

(by (*) ye = e) (since g E Se).

Also

We know that the existence of a left identity e for a semigroup S such that every element of S has a right e-inverse does not suffice to make S a group. A right zero semigmup is the standard example. Theorem 1.40 tells us that is essentially the only example. Corollary 1.41. Let S be a semigroup and assum rhot S hos a unique leff identity e and thnt every element of S has 4 right e-inverse. Then S is a group.

Pmof: This is Exercise 1.5.1.

Exercise 1.5.1. h v e Corollary 1.41. (Hint: Consider IYI in Theorem 1.40.)

0

1.6 Minimal Left Ideals

19

1.6 Minimal Left Ideals We shall see in this section that many significant fonsequences follow fromthe existence of minimal left (or right) ideals, especially those with idempotents. 'Ihis is impanant for us. because, as we shall see in Corollary 2.6, any compact right topological semigroup has minimal left ideals with idempotents. We begin by establishing an easy consequence of Theorem 1.40.

Tl~eorem1.42. Let S be a semigmup and assume that there is a minimal k$ ideal L of S which has an idempotent e. Then L = XG X x G where X is the (I& zem) semigmup of idempotem of L,and G = e L = eSe is a gmup. All maximal gmups in L are isomorphic to G. Pmof: Given x E L , Lx is a left ideal of S and Lx 5 L so Lx = L and hence thm is some y E L such that yx = e. By Lemma 1.30(b). e is a right identity for Le = L. Therefore the right-left switch of Theorem 1.40 applies (with L replacing S). It is a routine exercise to show that the maximal gmups of X x G are the sets of the form ( x ) x G.

Lemma 1A3. Let S be a semigmup, let L be a leji ideal of S, and let T be a leji ideal of L. (a) For all r E T . Lt is a leji ideal of S and Lt G T . (b) L is a minimal lefr ideal of S, then T = L. (So minimal I& ideals are leji simple.) (c) IfT is a minimal lefi ideal of L, then T is a left idrol of S. Pmof (a) S(Lr) = (SL)t G Lr and Lr G LT C T . (b)Pickanyt E T. By(a),LtisaleftidealofSandLr S T g L s o L t = L s o T = L. (c) Pick any r E T. By (a), Lt is a left ideal of S. so Lt is a left ideal of L. Since Lt G T , Lt = T . Therefore, ST = S(Lt) = ( S L ) t Lr = T . 0

As a consequence of Lemma 1.43, if L is a left ideal of S and T is a left ideal of L andeither L is minimal in S or T is minimal in L, then T is a left ideal of S. Of course, the right-left switch of this statement also holds. That is. if R is a right ideal of S and T is a right idea1 of R and either R is minimal in S or T is minimal in R. then T is a right ideal of S. We see now that without some assumptions, T need not be a right ideal of S. EXaInple 1.44. Let X = i0.1.2) and let S = *x. Let R = { f E X : Range(f) G 0-0 10,l I ) and let T = 16. a ) where 6 is the conrrantfunction and a : 1 -r 0. Then R is 2+ 1 a righr ideal of S and T is a right ideal of R, but T is not a righr ideal of S.

Lemma 1.45. Let S be a semigmup. Icf I be an ideal of S and let L be a minimal lefi ideal of S. Then L G I.

We now see that all minimalleft ideals of a scmigmupare intimately connectedwith each other.

Theomn 1.46. Let S be a semigmup. let L be a minimal Iep ideal of& andler T c S. Then T is a minimal I@ ideal of S i f a d o n l y ifthere u some a E S such that T = La. Pm$ Necessity. Picka € T. T h e n S L a S L a a n d L a E S T g T s o L a i s a I e f t ideal of S contained in T so La = T. Sufficiency. Since SLa G La. La is a left ideal of S. Assume that B is a left ideal o f S a n d B E La. L e t A = ( s ~L : s a € B].ThenA S L a n d A #0. Weclaimthat A is a left ideal of S, so lets E A and let t E S. Then sa f B so rsa E B and. since sEL,ts€L.sots€Aasrrquhd.ThusA=LsoLa~BsoLa=B. 0

Comllaj 1A7. Let S be a semi8roup. ? f S has a minimal Lq? ideal, d m every lefr ideal of S contains a minimal Iefr ideal. Pmof: Let L be a minimal left ideal of S and let J be a left ideal of S. Pick a E 1. Then by Theorem 1.46. La is a minimal left ideal which is contained in J . 0

Theorem 1.48. Let S be a semigmup and let e € E(S). Statements (a) thmugh (0a n equivalent and imply statement (g). Ifeither S is simple or every left ideal of S has an idempotem, then all statements a n equivalent. (a) Se is a minimal I& ideal, (b) Se is lep simple.

(c) eSe is a group. (d) eSe = H(e). (e) eS is a minimal right ideal. (0 eS is right simple. (g) e b a minimal idempotem.

Pmof By Theomm 1.38(a), we have that (a) implies (g) and by Theonm 1.38(d), if either S is simple or every left ideal of S has an idempotent, then (g) implies (a). (a) (b) (e) (f) Weshowthat f from which f 0 follows by left-right (dl (c) (d) (c) duality and the fact that (c) and (d)arc two sided statements. That (a) implies (b) follows from Lemma 1.43(b). (b)implies (c). Trivially eSe is closed. B y Lemma 1.34 e is a two sided identity for eSe. Also let x = ese E eSe be given. One has x E Se so Sx is a left ideal of Se and consequently Sx = Se. since Se is left simple. Thus e E Sx, so pick y E S such that e = yx. Then eye E eSe and eyex = eyx = ee = e so x has a left e-inverse in eSe.

*

*

I

I

II I

I

1.6 Minimal Lcft Idcab

21

(c) implies (d). Since eSe is a group and e E eSe. one has eSe c H(e). On the othm hand, by Theorem 1.18, e is the identity of H(e) so given x E H(e). one has that x = exe E eSe, so H(e) E eSe (d) implies (a). Let L be a lee ideal of S with L E Se and pick t 6 L. Then t E Se w e t E eSe. Pickx E eSesuchthatx(er) = e . Thenxt =(xe)r =x(cr) = e s o e E L soSe E SL E L.

We note that in the semigmup (N. .), 1 is the only idempotent, and is consequently minimal, while N1 is not a minimal let? ideal. Thus Theorem 1.48(a) - does not in aeneral imply the other statements of Theorem 1.48. We m a l l that in a ring there may be many minimal two sided ideals. This is because a "'minimal ideal" in a ring is an ideal minimal among all ideals not equal to (0). and one may have ideals I, and 12 with I , n12 = (0). By conhast. we see that a semigroup can have at most one minimal two-sided ideal.

-

Lemma 1.49. Let S be a semigmup and let K be an ideal of S. If K is mini& (J : J is an ideal of S) and I is an ideal of S, then K g I.

in

Prwf: ByLemma1.29(b),KnI#BsoKnIisaniddcontainedinKsoKnI = K. 0

The terminology "minimal ideal" is widely used in the literature. Since, by Lemma 1.49. there can be at most one minimal ideal in a semigroup, we prefer the terminology "smallest ideal". Dcanitinn 1.50. Let S be a semigroup. If S has a smallest ideal, then K(S) is that . . smallest ideal. We see that a simple condition guarantees the existence of K(S).

Theorem 151. L a S be a semigroup. IfS has a minimal lefi ideal, fin K(S) uisrs and K(S) = U [ L : L is a minimal left ideol of S). Pmof Let I = U ( L : L is a minimal left ideal of 3 ) . By Lemma 1.45, if J is any ideal of S, then I c J , so it suffices to show that I is an ideal of S. We have that I # 0 by assumption, so let x 6 I and lets e S. Pick a minimal left ideal L of S such that x E L. Then sx E L S I . Also, by Theorem 1.46. Ls is a minimal left ideal of S so Ls E I while xs E Ls. Observe, however, that many common semigroups do not have a smallest ideal. This is true for example of both (N. +) and (W. .).

Lemma 152. La S be a semigmup. (a) Let L be a lcfr ideal of S. %n L is minimal if and only if Lx = L for every x E L. (b) Let I be an ideal of S. Then I is the smallest ideal ifand only if 1x1 = I for

I Scmigmups and Their Ideals

22

L so L x = L . NowassumeLx=Lforeveryx E LandletJbealeftidedofSwith J G L . P ~ C ~ X E J . T ~ ~ ~ L = L X ~ L J ~ J ~ L . (b)This is Exercise 1.6.2. a

PmoJ (a) If L is minimal and x E L, then Lx is a left ideal of S and Lx

Theorem 1.53. Let S be a semigroup. If L is a minimal lefi ideal of S and R is a minimal right ideal of S, then K(S) = LR. P m f Clearly LR is an ideal of S. We use Lemma 1.52 to show that K(S) = L R. So. letx E LR. Thm LRxL isaleftidealof S whichiscontainedin L so LRxL = Land hence LRxLR = LR. 0

Theorem 1.54. Let S be a semigroup and (1sswne that K(S) exists and e E E(S). lXe following statements are equivalent and a n implied by any Wthe equivalent statements (a) through (0 of Theorem 1.48.

01) e E KG). (i) K(S) = SeS.

Pmof: By Theorem 1.51, it follows that Theorem 1A8(a) implies 0. (h) implies (i). Since SeS is an ideal, we have K ( S ) E SeS. Since e E K(S), we have SeS G K(S). (i) implies (h). We have e = ere E SeS = K(S). 0 .nKo naflval questions arc raised by Theorems 1.51 and 1.54. First, if K(S) exists, is it the union of all minimal Left ideals or at least is it either the union of all minimal left ideals or be the union of all minimal right ideals? Second. given that K(S) exists and e is an idempotent in KG). must Se be a minimal left ideal. or at least must e be a minimal idempotent? The following example, known as the bicyclic semigroup. answers the weaker versions of both of these questions in the negative. Recall that w=NU(O)=(O.1,2 ....I.

.

Example 155. Let S = w x wand &fie an operation on S by

Then S isasimplesemigmup(so K(S) = S), S harnominimllefiideaIsand~minima1 rightideals. E(S)=((n,n) : n ~ w t , a n d f o r e a c h n~ w . ( n + l . n + 1) 5 (n,n).

One may verify direstly that the operation in Example 1.55 is associative. It is ~ S is isomorphic to a subsemigroup of NN. probably easier. however. to O ~ S G Nthat Specifically define f. g E

NW by f (t) = t + 1 and g(t) =

given n, r E o one has g" o f' =

ifn c r

I 1

ii ff ' f1 - I '

. Consequently. one has

Then

I

1.7 Minimpl Ldt Ideals with Idempotents

23

To see that the semigmup in Example 1.55 is simple, note that given any ( m , n), € S, (k, m ) .tm. n ) . ( n , r ) = ( k , r). To see thzt S has no minimal left ideals, let L be a left ideal of S and pick ( m , n ) € L. Then ( ( k ,r ) € S : r s n ) is a left ideal of S which is properly contained in L. Similarly, if R is a right ideal of S and ( m ,n) G R then ((k,r ) E S :k r m ) is a right ideal of S which is properly contained in R. It is routine to vcnfj' the assertions about the idempotents in Example 1.55.

(k,r)

Exemise 1.6.1. Rove Lemma 1.45. E x e m k 1.6.2. h v e Lemma 1.520).

Exerdse 1.63. Let S = (f

is ininhnite). Rove € "N : f is onetwone and N\f that (S, o) is left simple (so S is a minimal left ideal of S ) and S has no idempotents.

Exercise 1.6.4. Suppose that a minimal left ideal L of a semigroup is commutative. Rove that L is a group.

Exerch 1.65. Let S be a semigroup and assume thatthere is a minimal left ideal of S. Rove that, if K ( S ) is commutative, then it is a group.

1.7 Minimal Left Ideals with Idempotents ~ e ~ n s e n t h eseveral r e results that have as hypothesis '*Let S bea semigroupand assume that there is a minimal left ideal of S which has an idempotent". These are important to us because. as we shall see in Corollary 2.6, this hypothesis holds in any compact right topological semigroup. (See Exercise 1.6.3 to show that the reference to the existence of an idempotent cannot be deleted from this hypothesis.)

Theorem 1.56. Let S be a semigmup and assume that t h e n is a minimal Iql ideal of S which has an idempotent. Then every minimal llfr ideal has an idempotenr.

P m J Let L be a minimal Iefl ideal with an idernpotent c and let J be a minimal left ideal. By Theorem 1.46, there is some x G S such that J = Lx. By Theorem 1.42, e L = eSe is a group, so let y = eye be the inverse of exe in this group. Then yx E Lx = J and yxyx = (ye)x(ey)x= y(exe)yx = eyx = yx. 0 We shall get left and right conclusions from this one sided hypothesis. We see now that in fact the right version follows from the left.

Lcmma 1.57. Let S be a semigmup and assume that there is a minimal I@ ideal of S which has an idempotent. Then there is a minimal right ideal of S which has an idempotent.

24

I Semi-

and 'It&Ideals

P m f Pick a minimal left ideal L of S and an idempotenta E L. By Thumm 1.31(c) Se is a minimal left ideal of S so by Theorem 1.48 eS is a minimal right ideal o f S and e is an idempotent in eS.

n

Thcorrm 158. Let S be a semigroup and assume that there is a minimal lrft i&al of S which has an idempotmr Let T C S. (a) T is a minimal 1 4 ideal of S ifand only ifthere is some e E E ( K ( S ) ) such thal T = Se. (b) T is a minimal right ideal o f S ifand only iffhere is some e E E ( K ( S ) ) such rlunT =eS.

P m j Pick a minimal left ideal L o f S and an idempotent f E L. (a) Necessity. S i n a S f is a left ideal contained in L, Sf = L. Thus b y Theorem 1.48. f S f is a group. Pick any a E T . Then f a f E f S f so pick x E f S f such that x ( f a f ) = f . Then xaxa = ( x f ) a (f x b = (xfaf)xa = fxa = xu. Consequently, x a is an idempotent. Also x u E T while T g K ( S ) by Theorem 1.51 so xu E E ( K ( S ) ) . Finally. Sxa is a left ideal contained in T , so T = Sxa. Sufficiency. Since e E K ( S ) , pick by Theorem 1.5 1 a minimal left ideal I o f S with e E I . Then Se = I by Theorem 1.31(c). 0 @)As a consequence o f Lemma 1.57 this follows by a I&-right switch.

Theorem 159. Let S be a semigroup, assume that there is a minimal IefI ideal of S which has an idempotmf,and let e E E(S). Thefollowing statements are equivalmt. (a) Se is a minimal lefl ideal. @) Se is lefi simple. (c) eSe is a gmup. (d) eSe = H(e). (e) eS is a minimal right ideal. (f) eS is right simple. ( g ) e is a minim1 idernpotent. Q~EK(S). (i) K ( S ) = SeS. Pmof: By CoroUary 1.47 and Theorem 1.56 every left ideal o f S contains an idempotent so by T h e m 1.48 statemenm (a)Ulrough (g) are equivalent. By Theorems 1.51 and 1.54 we need only show that (h) implies (a). But this follows fromTheorem 1.58.

Theorem 1.60. Let S be a semigmup, assume that them is a minimal lcfr ideal of S which has an idemporent, and let e be an idemporent in S. Thcn is a minimal idempotent f of S such that f 5 e.

j

1.7 Minimal Left Ideals wirh Idempotam

25

p m f : Se is a left ideal which thus contains a miniaal left ideal L with an idunpotmt g by Corollary 1.47 and Theorem 1.56. Now g E Se so ge = g by Lemma 1.30. Let f = eg. Then f f = cgeg = egg = eg = f so f is an idempotent. Also f E L so L = Sf so by Theorem 1.59 f is a minimal idempotent Emally e f = eeg = eg = f and f e = e g e = e g = f so f i e .

Tbeamn 1.61. Let S be a semigroup and assume that there is a minimal left ideal of S which his an idempatent. Given ony minimal lep ideal L of S and any minimal right ideal R of S, r h m is an idempotent e E R n L such that R n L = R L = eSe and eSe is a group. Pmof: Let R and L be given. Pick by Theorem 1.58 an idempotent f E K ( S ) such that L = S f . By Theorem 1.48. f S f is a group. Pick a E R and let x be the inverse of fafinfSf.Thenx~Sf=Lsoax~RnL.ByThoorrm1.51axeK(S).Also axax = a ( x f ) a ( f x ) = a(xfaf)x =afx = ax..

Let e = ax. Then eSe C SX G L and eSe g US E R so eSe G R ll L. TO see that R n L g eSe. let b E R n L. By Theomn 1.31 L = Se and R = eS so by Lemma 1.30, b = eb = be. Thus b = eb = ebe E eSe. Now R L = eSSe 2 eSe g RL. so RL = eSe. As we have observed, e E K(S). SO by Theorem 1.59 eSe is a ~~OUP. Lemma 1.62. Let S be a semigroup and assum thnt them is a minimal left ideal of S which has an idempatent. Then all minimal lcfr ideals of S o m isomorphic.

P m f : Let L be a minimal left ideal of S with an idempotent e. Then L = Se so by Theorem 1.59 eSe is a group. We claim lint that given any s E K ( S ) and any t E S. s(ese)-' = st(esre)-I, where the inverses are taken in eSe. Indeed, using the fact that (ese)-le = e(ese)-' = (ese)-I,

andsimilarlyst(este)-' is an idempotent By Lemma 1.57 and'l%~heorrm1.51, K ( S ) = U { R : R is a minimal right ideal of S ) . Pick a minimal right ideal R of S such that s E R. Then s(ese)-' and st(este)-I areboth idempotenu in R n L, which is a group by Theorem 1.61. Thuss(ese)-' = st(este)-' as claimed. Now let L' be any other minimal left ideal of S. By Theorem 1.59. eS is a minimal right ideal of S so by Theomn 1.61 L'neS is agroup so pick anidempotentd E L'neS. Notice that L' = Sd and d S = eS. In particular, by Lemma 1.30(b). de = e. ed = d. andforanys E L1.sd = s .

26

1 Semigmups and Their ldcals

Define @ : Sd + Se by @(s) = s(ese)-'dse, whercthe inverse is in the group eSe. We claim first that @ is a homomorphism. To this end, lets, t E Sd. Then

Now define y : Se -+ Sd by y ( r ) = t(dtd)-'etd where the inverse is in dSd, which is a group by Theorem 1.59.. We claim that y is the inverse of @ (and hence takes Sd one-to-one onto Se). To this end, lets E L'. Then ds E L' so Sds is a left ideal contained in L' and thus L' = Sds. So pick x E S such that s = xds. Then

+

Y (@W)= +(s)(d@(s)d)-'e@(s)d = s(ese)-'dse(ds(ese)-'dsed)-'es(ese)-'dsed = xds(ese)-'dsed(ds(ese)-'dsed)-'ese(ese)-'dsed = xdedsed = xddsd = xds = S. Similarly. if r E L, then @ (y ( t ) ) = r .

0

We now analyze in some detail the structure of a particular semigroup. Our motive is that this allows us to analyze the structure of the smallest ideal of any semigroup that has a minimal left ideal with an idempotent.

Theorem 1.63. Let X be a lefr zem semigmup. let Y be a right zem semigmup. and let G beagmup. Lcte betheidentityof G,Jiru E X and v E Y andlet[. ] : Y x X + G be E YandaNx E X. L c t S = X x G x Y anddefine on operation . on S by (x, g , y ) . (x', g', y') = ( x , g[y. x'lg', y'). Then S is a simple semigmup (so that K ( S ) = S = X x G x Y J and each of the followhg starements holds. (a) For every (x, y ) E X x Y. ( x , [y, X I - ' , y ) is an idempotent (when the inverse is taken in G ) and all idemporents are of t h i s f o n I n particular, the idempotenrs in X x G x { v ] are of theform (x, e. v ) and the idempotenrs in { u ) x G x Y a n of the form (u. e, y). (b) For every y E Y. X x G x { y 1 is a minimal left ideal of S and all minimal lefr ideals of S are of this form. (c) For every x E X. { X I x G x Y is a minimal right ideal of S and all minimal right ideals of S are of this form.

afuncrionsuchthar[y,u]=[u,x]=efornlly

1.7 Minimal Ldt Ideals with Idempotcnts

27

(d)Forcvery(x,y) E X x Y . [ x ) x G x [ y ] i s a m a r i m a l g r o u p i n S o n d a l l m ~ ' d p u p irz S are of thisform. (e) The minimal I@ ideol X x G x (v) u the direct pmduct of X. G, and (v) and rhe minimal riglu ideol [u] x G x Y is the directproduct of (u). G, and Y . (0All maximal gmups in S an isomorphic to G. @)All minimal left ideals of S are isomorphic to X x G ondall minimal right ideals of S are isomorphic to G x Y .

Proof. T k associativity of . is immediate. To see that S is simple. I n (x, g, y), (x', s', y') E S. By Lemma 1.52(b), it suffices to show that (x', g', y') E S(x, g, y)S. Toseethis,leth = g'g-'[y.x]-lg-'[y,x]-'. Then(x1,g'. Y') = (x'. h, y).(x.g. y). (x*8 , Y'). (a) That (x, [y,x]-', y) is an idempotent is immediate. Given an idempotent (x. g, y), one has that g b . xlg = g so g = [y,xI-'. (b) W y E Y. Trivially X x G x {y1 is a left ideal of S. To see that it is minimal, let (x, g), (x'. g') E X x G . It suffices by Lemma 1.52(a) to note that (x', g', y) = (x', g1g-'Ly. XI-', y).(x. g, y). Givenany minimalleftideal LofS,pick(x, g, y) E L. Then L f l ( X x G x [y)) # 0 so L = X x G x {y). Statement (c) is the right-left switch of statement (b). (d) By the equivalence of (h)and (d) in Theorem 1.59 we have that the maximal groups in S are precisely the setsof thefonn f Sf where f is an idempotent of K (S) = S. That is. by (a), where f = (x, [y. XI-', y). Since

we are done. (e) We show that X x G x (v) is a direct product. the other statement being similar. Let (x, g), (x', g') E X x G . Then

(0Trivially [u] x G x [v) is isomorphic to G. Now. let (x. y) E X x Y . Then (x) x G x [v) and {u) x G x [u] arcmaximal gmups in theminimal left ideal X x G x { v ) . hence are isomorphic by Theorem 1.42. Also (x) x G x [ v ) and 1x1 x G x [y) are maximal groups in the minimal right ideal { x ) x G x Y . hence are isomorphic by the left-right switch of Theorem 1.42. (g) By Lemma 1.62 all minimal left ideals of S are isomorphicand by (e) X x G x [u) is isomorphic to X x G. The other conclusion follows similarly. Note that in Theorem 1.63. the set S is the cartesian product of X. G,and Y . but is not the direct pmduct unless [y, XI = e for every (y, x ) E Y x X. Observe that as a consequence of Theorem 1.63(g) we have that for any y E Y, X x G x (y 1 X x G . However. there is no transparent isomorphism unless [ y . x] = e for all x E X. such as when y = u.

lheorem 1.63 spells out in detail the stmchnc of X x G x Y . We see now that this is in fact the sauehnc of the smallest ideal of any semigroup which has a minimal left ideal with an idempotent S m ?hUXem). Let S k a semigmup and assume that there T h c o ~ m1.64 is a minimal I@ ideal of S which has an idempotent M R be a minimal right ideal of S, IetL beaminimalleftideolofS, letX = E(L). IetY =: E(R), andletG = RL. DEfne anoperation . on X x G x Y by ( x , g. y).(xf,g', y') = ( x , gyx'g', y'). Then X x G.x Y satispes the conclusions of Theorem 1.63 (when [ y , x ] = y x ) and K (S)-" X x G x Y. In parricular: (a) The minimal right ideok of S partition K ( S ) and fhe minimal I@ ideals of S panition K (S). (b)The maximal p u p s in K (S) panition K(S). (c) All minimal right ideok of S a n isomorphic andall minimal I@ ideok o f S are isomorphic. gmups in K ( S ) o n isomrphic. (d) All

P m d After noting that. by Lemma 1.43 andTheorem 1.51. the minimal left ideals of S and of K(S).are identical (and the minimal right ideals of S and of K ( S ) are identical). the "in particular" conclusions follow immediately from ' l h e m m 1.63. So it suffices to show that X x G x Y satisfies the hypotheses of Theorem 1.63 with Ly. x ] = yx and t h a t K ( S ) z X x G x Y. We know by Lemma 1.57 that S has a minimal right ideal with an idempotent (so R exists) and hence by Theorem 1.56 R has an idempotent We know by 'lheorem 1.61 that RL is a group and we know by T h e m m 1.42 that X is a left zem semigroup and Y is a right zero semigmup. Let e bi: the identity of R L = R n L and let u = v = e. Given y E Y one has, since Y is a right zero semigroup, that [ y ,u] = yu = u = e. Similarly, given x E X. [ v , x ] = e. Consequently the hypotheses of Theorem 1.63 are satisfied. Defineg : X x G x Y + Sby g(x. g, y ) = xgy. Weclaimthatgisanisomorphism onto K(S). From the definition of the operation in X x G x Y we see immediately that 9 is a homomorphism. By Theorem 1.42 we have that L = XG and R = G Y . By Theorem 1.53, K(S) = LR = XGGY = XGY = p[X x G x Y ] . Thus it suffices to produce an inverse for p. For each r E K(S), let y ( t ) be the invrrsc of ere in eSe = G. Then ry(r) = t y ( t ) e E Se = L and t y ( t ) t y(0 = t y ( W e y(0 = fey([) = w (0. s o t y ( t ) E X . Similarly. y(t)r E Y . Define + : K ( S ) -r X x G x Y by r ( t ) = (ty(r),ere, y(t)t). We claim that r = 9-'.So let ( x ,g. y) E X x G x Y . Then

Now

Similarly y(xgy)xgy = y. Since also exgye = ege = g. we have that r = (p-1 as required.

The following theorem enables us to identify the smallest ideal of many scmigmups that arise in topological applications. T h e o m 1.65. Let S be a semigmup and assume that there is a minimal J c f t idral of S which has an idempotent. k t T be a subsemigmup of S and assume also that T has a minimal ideal with an idempotmt. i f K ( S ) n T # 0, then K ( T ) = K ( S ) n T . Pmof: By Theorem 1.51, K ( T ) exists so,since K ( S ) h T is an ideal of T, K ( T ) E K ( S )nT . For the mcrse inclusion, let x E K ( S ) fl T be given. Then T x is a left ideal of T so by Corollary 1.47 and Theorem 1.56 T x contains a minimal left ideal T e of T for some idempotent e 6 T . Now x 6 K ( S ) so by Theorem 1.51 pick a minimal left idealLofSwithx E L. ThenL = S x a n d e € T x G S x s o L = S e s o x € Sesoby Lemma 1.30. x = xe E T e c K(T). We know from the Structure Theorem (Thwrcm 1.64) that maximal groups in the smallest ideal are isomorphic. It will be convenient for us later to know an explicit isomorphism between them.

Theorem 1.66. Lcr S be a semigmup and assume that there is a minim01 lef? ideal of S which has an idempatent. k t e , f E E ( K ( S ) ) . Ifg is the invcrse of e f e in eSe. then thefunction (p : eSe -t f S f drfined by ~ ( x=) f x g f is an isomorphism. Pmof: To see that (p is a homomorphism, let x , y E eSe. Then

To see that (p is one-to-one, let x be in the kernel of (p. Then

fxgf = f efxgfe = efe efexgefe = e f e efexe = e f e efex = efee x=e

(ex = x , ge = g ) (left cancellation in eSe).

I Semigroups and Their Ideals

30

To see that p is onto f Sf, let y E f Sf and let h and k be the inverses of fgf and f ef respectively in f Sf. Then ekyhe E eSe and v(ekyhe) = f e W w f = f ef kyhgf = fyh8f = fyhfgf

= fuf

= y.

(f k = k, eg = g) (fefk=f) (h = h f ) (hfgf = f 0

We conclude the chapter with a theorem characterizing arbitrary elements of K(S).

Theorem 1.67. k t S be a semigroup and assume that there is a minimal left ideal of S which has an idemporent. Let s E S. Thefollowing statements are equivalent. (a) (b) (c) (d)

s E K(S). For all t E S, s E Sts. For all r E S, s E st S. For all t E S, s E SISn 3s.

PmoJ (a) implies (d). Pick by Theorem 1.51 and Lemma 1.57 a minimal left ideal L of S and a minimal right ideal R of S with s € L n R. Lett E S. Then ts E L so Sts is a left ideal contained in L so Sts = L. Similarly s t S = R. The facts that (d) implies (c) and (d) implies (b)are trivial. (b)implies (a). Pick r E K(S). Thens E Sts g K(S). 0 Similarly (c) implies (a).

Exercise 1.7.1. Let S be a semigroup and assume that there is a minimal left ideal of S which has an idempotent. Rove that if K(S) # Sand x E K(S), then x is neither left nor right cancelable in S. (Hint: If x is a member ofthe minimal left ideal L, then L = Sx = Lx.) Exercise 1.72. Identify K(S) for those semigroups S in Exercises 1.4.2 and 1.4.3 for which the smallest ideal exists. Exercise 1.73. Let S and T be semigroups and let h : S -+ T be a surjective homomorphism. If S has a smallest ideal, show that T does as well and that K(T) = h[K(S)].

Notes Much of the material in this chapter is b a d on the treatment in [39, Section 11.11. The presentationofthe SbuctureTheorem was suggested to us by I. Pym and is based on his treatment in [202]. The Structure Theorem (Theorem 1.64) is due lo A. Suschkewitsch [23i]in the case of finite semigroups and to D. Rees [2101 in the general case.

Chapter 2

Right Topological (and Semitopological and Topological) Semigroups

In h i s (andsubsequent) chapters, we assume that the reader bas masteredm introductory course in general topology. In particular, we expect familiarity with the notions of continuous functions, nets. and compactness.

2.1 Topological Hierarchy

.

DeMtion 21. (a) A right topological semigroup is a hiple (S. , 7 ) where (S. .) is a semigroup. (S, 7 )is a topological space, and for all x S,p, : S + S is continuous. (b) A Iefr topological semigroup is a triple (S. .,7 ) where (S. .) is a semigroup, (S. 7) is a topological space, and for all x S. A, :S -i S is continuous. (c) A ~emiropologicalsemigroup is a right topological semigroup which is also a left topological semigroup. (d) A topologicalsemigmup is ahiple (S.. 7 ) where ( S , .)is a d g r o u p . (S.7) is a topological space. and . : S x S + S is continuous. (e) A topological group is a hiplc (S. . 7 ) such that (S. .) is a group, (S, 7 ) is a topological space, . : S x S + S is continuous, and In : S + S is continuous (where In(x) is the inverse of x in S).

.

.

We did not include any separation axioms in the definitions given above. However. all of our applications involve Hausdorff spaces. So we shall be assuming throughout, except in Chapter 7.that all hypothesized topological spaces are Hausdorff. In a right topological semigroup we say that the operation "." is 'kight continuous". We should note that many authors use the term "left topological" for what we call "right topological" and vice versa One may reasonably ask why somwne would refer to an operation for which multiplication on the right is continuous as "left continuous". The people who do so ask why we refer to an operation which is continuous in the left variable as "right continuous". We shall customarily not mention either the operation or the topology and say something like "let S be a right topological semigroup".

(

1

2 Right Topological Smigmup

Note that hivially each topological group is a topological semigroup, each topological semigroup is a semitopological semigroap and each semitopological semigroup is both a kft and right topological semigroup. Of c o m e any semigroup which is not a group provides an example of a topological semigroup which is not a topological group simply by providing it with the discrete topology. It is the content of Exercise 2.1.1 to show that there is a topological semigroup which is a group but is not a topological group. It is a celebrated theorem of R. Ellis 1841, that if S is a locally compact semiropological semigroup which is a group then S is a topological group. That is. if S is locally compact and a group, then separate continuity implies joint continuity and continuity of the inverse. We shall prove this theorem in the last section of this chapter. For an example of a semitopological semigroup which is a group but is not a topological semigroup see Exercise 9.2.7. It is the content of Exercise 2.1.2 that there is a semitopalogical semigroup which is not a topological semigroup. Recall that given any topological space ( X , 7).the product topology on ' X is the topology with subbasis [n;'[Ul:x E X and U E 7 ) .where for f E ' X andx E X. n,( f ) = f ( x ) . Whenever we refer to a "basic" or"subbgsic" open set in 'x, we mean sets d e h c d in terms of this subbasis. The pmduct topology is also often referred to as the topology of poinhvise convergence The w o n for this terminology is that a net (f,),,,converges to f in ' X if and only if (f,(x)),,l converges to f (x) for every x E X.

.

Theorem 2.2. Let ( X , T)be any ropological space and k r V be the pmduct topology on 'x. (a) ('X, o. V ) is a right wpological semigmup. (b)For each f E 'x. Af is continuous if and only i f f is continuous.

Prwf: La f E 'x. Suppose that the net ( g , ) , , ~mwcrges to g in ('X, V ) . Tlen, for any x 6 X. k ( f ( x ) ) ) , ~C IO ~ V ~ ~10~gC( fS(x))in X. n u s (s,0 f ),EI converges to g o f in ('X, V),and so pf is continuous. This establishes (a). Now Af is continuous if and only if ( f ( g , ( ~ ) ) ) , Econverges l to f (g(x)) for every net (g,),,~converging to g in ('x. V ) and every x E X. This is obviously the case if f is continuous. Conversely, suppose that A, is continuous. Let ( x , ) , ~be I a net converging to x in X. We define g, = 5,the fumtion in 'X which is constantly equal to x, and g = f . Then (g,),,, converges to g in ' X and so ( f o &),,I converges to f o g. This means that (f( x , ) ) , ~converges ! to f (x). Thus f is continuous, and we have established (b). Corollary 23. Ler X be o topologicalspace. Thefollowing statements a n equivolmr.

(a) ' X is o topological semigmup. (b) XX is a semitopological semigmup. (c) For 011 f E ' X I f is continuous. (d) Xis discrete.

23 Compaa Right Topologicnl Scmigmups

33

If X is any nondiscrete space, it follows from Thconm 2.2 and Corollary 2.3 that *X is a right topological s e m i p u p which is not left topological. Of come. reversing (he ordw of operation yields a left topological semigroup which is not right topological.

Deenition 2.4. Let S be a right topological semigroup. The ropological center of S is the set h(S)= [x E S :A, is continuous).

Thus a right topological semigroup S is a semitopological semigroup if and only if h(S) = S. Note that hivially the algebraic center of a right topological semigroup is contained in its topological center. Exercise 2.1.1. Let 7 be the topology on B with basis 3 = { ( a ,b] : a , b e R and a c b). Rove that (R, 7) is a topological semigroup but not a topological group.

+.

Exerdse 2.13. Let S = R U (w).let S have the topology of the one point wmpactification of R (with its usual topology), and define an operation * on S by

(a) Prove that (S. *) is a semitopological semigroup. (b) Show that * : S x S -r S is not continuous at (w. w ) . Exerdse 2.1.3. R m e Corollary 2.3.

2.2 Compact Right Topological Semigroups We shall be concerned throughout this book with certain compact right topological semigroups. Of fundamental importance is the foUowing theorem Theorem 2.5. Let S be a compact right ropological semigroup. Then E(S) # 0.

P m f L e t A = { T E S : T #0.Tiscompact,andT.T GT].Thatis,Aistheset of compact subsemigroups of S. We show that A has a minimal member using Zorn's Lemma. Since S E A, A # 0. Let C'be achain in A. Then C is a collection of closed subsets of the compact space S with the finite intersection property, so C # 0 and C is hivially compact and a semigroup. Thus C E A, so we may pick a minimal

n

n

n

member A of A. Pickx E A. We shall show that xx = x. (It will follow that A = 1x1, but we do not need this.) We start by showing that Ax = A. Let B = Ax. Then B # 0 and since B = p, [ A ] . B is the continuous image of a compact space, hence compact. Also

34

2 Right Topological Semigroups

B B = A X A ~ ~ A A A ~ G A X = B . ~ U ~ B € A . S ~ ~ ~ ~ B is minimal, B = A. Let C = (y E A : yx = XI. Since x E A = Ax, we have C # 0. Also. C = A n p;'[[x)]. so C is closed and hence compact. Given y, r E C one has y z ~ A A E A a n d y z x = y x = x s o y z € C . T b u s C ~ A .S i n c e C E A a n d A i s minimal.wehaveC = A s o x E Candsoxx =xasrequired. 0

In Section 1.7 there we're several results which had as part of their hypotheses "Let S be a semigroup and assume there is a minimal left ideal of S which has an idempotent."

Because i f the folIowing coro~ary,we are able to incorporate

of these results.

Corollaq 2.6. Let S be a compact righr topological semigmup. Then every I@ ideal of S contains a minimal I& idea1:Minimal lefr ideals a n ~losed,and each minimal lefr ideal has an idempotent. P w J : I f L is any left ideal L of S andx € L, then Sx is acompact left ideal contained in L. (It is compact because Sx = pAS1.) Consequently any minimal left ideal is closed and by Theorem 2.5 any minimal left ideal contains an idempotent. Thus we need only show that any left ideal of S contains a minimal left ideal. So let L be a left ideal of S and let +4 = ( T : T is a closed left ideal of S and T G L). Applying Zom's Lemma to A, one gets a left ideal M minimal among all closed left ideals contained in L. But since every left ideal contains a closed left ideal, M is a minimal left ideal. 0

We now deduce some consequences of Corollary 2.6. Note that these consequences apply in particularto any finite semigroup S. since S is acompaa topological semigoup when provided with the discrete topology. Theorem 2.7. Let S be a compact right topological semigmup. (a) Every righl ideal of S contains a minimal right ideal which has an idempotent. (b) Let T E S. Then T is a minimal lefr ideal of S if and only if there is some e E E(K(S)) such that T = Se. (c) Lcr T G S. Then T is a minimal right ideal of S if and only if t h e n is some e E E(K(s)) such that T = eS. (d) Given any minimal lefr ideal L of S andany minimal righr ideal R of S, then is on idempotent e E R f l L such that R n L = eSe and eSe is a group.

Pmoj (a) Corollary 2.6. Lemma 1.57. Corollary 1.47. and Theorem 1.56. (b) and (c). Corollary 2.6 and Thwrem 1.58. (d) Corollary 2.6 and Theorem 1.61.

D

Theorem 2.8. Let S be a compacr right topological semigmup. Then S has a smallest (rwosided)ideal K (S) which is the union ofallminimallefr ideals of S anda/sothe union ofanminimalrightidealsofS.Eachof (Se : e E E(K(s))). (eS :e E E(K(s))), and [eSe : e E E(K(S))) ampanifionsof K(S). p m f . Corollary 2.6andTheorems 1.58. 1.61, and 1.64.

36

2 Right Topological Scmigmups

see that Q is continuous. we show that Q is the maiction of p,f to eSe. To this end, k t x E eSe. Then Q(x) = f x g f =fexgf (x-ex) = exgf ( fe =e) = xgf (ex = x). Now let h and k be the illverses in f S f of f g f and f ef nspectively. We showed in the proof of T h e m m 1.66 that if y E f S f , then Q-' ( y ) = ekyhe. Thus Q-'(y)= ekyhe = fefkyhe = fyhe = yhe

(fk=kand f e r e ) (fefk=f) ( f =~ Y ) '

So Q-' is the restriction of phr to f S f and hence is continuous. (c) Let L and L' tc minimal left ideals of S and let r E L'. By Theom 2.7W pick e E E ( K ( S ) ) such that L = Se. Then p Z ! is ~ a continuous function hom Se to Sz = L' and p,[Se] = L' because Set is a left ]deal of S which is contained in L'. To see that & is one-to-one on Se. let g be the imerse of ere in eSe. We show that for x E Se. & ( h ( x ) ) = x. so let x E Se be given.

xzg = xezeg = xe

( x = x e and g = eg)

= X. Since f i l ~is one-to-one and contkuous and L is compact. p,lr is continuous.

Recall that given any idempoten6 e , f in a semigroup S , e 5~ f if and only if fe=e.

Theorem 2.1% Let S be a compact right topological semigmup and let e E E(S). There is a iR-maximal idempotent f in S wirh e 9 f .

P m g Let A = ( x E E ( S ) : e 5~ x ) . Then A # 0 because e E A. Let C bc a sR-chain in A. Then ( c t ( r E C :x SRr ) : x E C ) is a collection of closed subsets CC(I E C :x CR r ] # 0. Since S is of S with the finite intersection property, so ce(r E C :x i~r ) g ( t E S :forallx E C,rx = x ) . Consequently. Hausdorff. ( t E S : for all x E C. t x = x ) is a compact subsemipupof S and hence by Theorem 2.5 there is an idempotent y such that for all x E C, yx = x. This y is an upper bound o for C, so A has a maximal member.

nxEc

nxec

Given e. f E E ( K ( S ) ) and an assignment to find an isomorphism from eSe onto f S f . most of us would try first the function r : eSe + f S f defined by r ( y ) = f y f . In fact, if eS = f S, this works (Exercise 2.2.1). We see now that this n a N d function need not be a homomorphism if eS # f S and Se # S f

2.2 Compact Right Topalogical S r m i ~ p s

37

E-ple 213. Let S be the semigmup c o ~ ' ~ of n the g eight disrinct elements e, f . e f , f e, efe. f e f , e f e f . and f e f e. with thefollowing multiplication table.

Then S with the discrete topology is a compaci topological semigmup and f ef is not an idempotent so the function r : eSe -r f Sf d@ned by r ( x ) = f x f is not a homomorphism

Once again. the simplest way to see that one has in fact defined a semigmup is to produce a c o n m e representation. In this case the 2 x 2 integer matrices e =

):

the semiW S. = (;I The topological center o f a compact right topological semigroup is important for many applications. The following lemma will be used later in this book.

and

Lemma Z.14. Let S and T be compct right ropological semigmups, h D be a dense subsemigmup of S such that D 5 A(S), and let q be a continuousfunctionfmm S to T such that ( 1 ) q[Dl E A ( T ) and (2) q p is e homomorphism

Then q is a homomorphism, Pmof. For each d E D, q o Ad and A,(d) o q an continuous functions agreeing on the dense subset D o f S. Thus for all d € D and all y E S. M Y )= v(d)q(y).Therefore. for all y E S, q o py and P , , ~ ,o rl arc continuous functions agreeing on a dense subset o f S so for all x and y in S, q ( x y ) = q(x)q(y). 0 Exercise 22.1. Let S be acompact right topological s e m i p u p a n d e f E E ( K ( S ) ) such that eS = f S. Let g be the inverse o f ef e in the group eSe and define (p :eSe + f S f b y q ( x ) = fxgf.Definer:eSe-* f S f b y r ( x ) = fxf.P~ovethatr=(p.

Exercise 22.2. Let S be a compact right topological s e m i p p , let T be a semigmup with topology. and let (p : S + T be a continuous homomorphism. Prove that (p[S]is

a compact right topological semigroup.


~.

Em&

2.2.3. Rove that a compact cancellative. right topological semigroup is a

group.

23 Closures and Products of Ideals We investigate briefly the closures of right ideals. Ieft ideals. and maximal subgroups of right topological semigroups. We also consider their Cartesian product.

Theorem 215. Let S be a right topological semigmup and let R be a right ideal of S. Then ct R is a right ideal MS. P m j This is Exercise 2.3.1.

0

On the other hand, the closure of a left ideal need not be a left ideal. We leave the verification of the details in the following example to the reader.

Example 2.16. Let X be any compuct space with a subset D such that cL D'# X and I cC D\DI >_ 2. DeJine an operation. on X by

Then X is a compact right tapological semigmup, A ( X ) = 8, and the set d l e p ideais of X is (XI U [ B :0 # B G Dl. In paniculaz D is a lep ideal of X and c t D is nat a Ieft ideal. Often. however. we see that the closure of a left ideal is an ideaL

Theorem 2.17. Let S be a compocr right ropological semigmup and assume that A(S) is dense in S. Let L be a Ieft ideal of S. Then c t L is a I e f t ideal of S. Pmoj Let x E c! Land let y E S. To see that y x E cL L,let U bean opm neighborhood of y x . Pick a neighborhood V of y such that Vx = pJV1 E U and pick r E A ( S ) n V. Then tx = l , ( x ) G U so pick a neighborhood W of x such that zW g U. Pick W E WnL.Thenrw~UnL. 0

In our applications we shall often be concerned with semigroups with dense center. ( I f S is d i t e and commutative. then pS has dense center- see Theorern 4.23.) LMma 2.18. k t S be a compact right topological semigmup. Thefollowing statements a n equivalent. (a) The (algebmic)center of S is dense in S. (b) T k n is a dense commutative subset A of S with A G A ( S )

2 Right Topological Semigroup

40

We have J a n that the Cartesian product of semigroups is itself a semigroup under the coordinatewise operation. If the semigroPpsarc also topological spaces the product is nanrrally a topological space.

Theorem 2.22. Let (Si)i,l be afanrily of right topological semigmups and let S = X i,rSi. With the product topology and c w r d i ~ t m i s eoperations, S is a right topological semigroup. I f each Si is compact, then so is S. If? E S and for each i E I. 1 , : Si + Si is conrhuous, then 1~: S s S is continuous. Pmof: This is Exercise 2.3.5.

0

Note that the followingtheorem is entirely algebraic. We shall only ncedit. however. in the case in which each Si is a compact right topological semigroup. In this case. each Si does have a smallest ideal by Theorem 2.8. Theorem 2.23. Let (Si)iel be a family of semigroups and let S = X i,l Si. Suppose that. for each i E I. Si hm a smallest ideal. 7 k n S also has a smallest ideal and K ( S ) = X i e l K(Si).

P m J We fist note that X i e l K(Si) is an ideal of S. Let ;E K(S). Then ( X i e ! K ( S i ) ).ri. ( X i , l K ( S i ) ) = X i , ~ ( K ( S i ) - u i .K ( s ~ ) = ) Xi,rK(Si) so by Lemma 1.52(b), K ( S ) = Xi,lK(Si).

While we are on the subject of products. we use Cartesian products to establish the following result which will be useful lam.

Theorem 224. Let A be a set and let G be t h e f i e gmup generated by A. Then G can be embedded in a compact topological gmup. This means that there is a compact topological gmup H and a one-to-one homomorphism I# : G s H. Pro@. R e d that 0 is the identity of G. For each g E G\{0) = G', pick by Theorem 1.23 a finite gmup F8 and a homomorphism : G + Fs such that #,(g) is no( , is the identity of F,. Let each F, have the discrete topology. Then H = X b c ~ Fg a compact topological group. Define a homomorphism (p : G -+ H by stating that (p(h), = e8(h). Then, if g E G', we h o w that ( ~ ( gis)not ~ the identity of F,. So the 0 kernel of (p is (0) and hence I# is one-to-one as required.

eg

Exe&

23.1. Rove Theorem 2.15.

Exercise 23.2 Let S be a right topological semigroup, and let T be a subset of the topological center of S. Rove that cC T is a semigroup if T is a semigroup. Exercise 23.3. Verity the assertions in Example 2.16.

Exerdsc 23.4. Let X and D be as in Example 2.16 except that D isopen and c l D\D = [ z ) . Rovethat A(X) = (2). Exercise 23.5. Rove Theorem 2.22.

2.4 Semitopological and Topological Semigroups we only scratch the surface of the theory of semitopological&groups and the theory of topological semigroups, in order to indicate the kinds of m u l a that hold in these settings but not in the setting of right topological semigroups. We shall see many examples of right topological semigroups that have no closed minimal right ideals, including our favorite (pW. +). (See Theorems 6.9 and 2.20.) By way ofcontrast we have the following theorem.

Theorem 225. Ln S be a compact semitopological semigroup and let a E S. Then aS, Sa, andaSa a m closed. Pmof. Exercise 2.4.1.

0

C o d W 2.26. Let S bcocomptsemitopologica[snnigroup with densecentec Then S is commutative and K ( S ) is a group. Proof: For any x E S, A, and p, are continuousfunctions agreeing on a dense subspace of S and therefore on the whole of S. So S is commutative. By Corollary 2.6 and Exercise 1.6.5. K ( S ) is a group. 0

In fact a stronger conclusion holds. See Corollary 2.40.

Theomm 2.27. Let S be a semitopological semigroup and let T be a subsemigmup of S. Then c t T is a subsemigroup of S. Proof: This follows fmm Exercise 2.3.2.

By way of contrast we shall now sez that in a right topological semigroup, the closure of a subsemigroup need not be a semigroup. (This also holds. as we shall see in 'Ihcorrm 8.21, in (BN. +).)

Theorem 238. Let X be the one point compactiification of W and let T = (f E X '

:

f is one-to-one). Then c l T is not o semigmup. PmoJ For each n

E W,

define g, : X + X by

Let g : X +. X be the constant function E5, Then each g. E tog inXX sog E cCT. Nowdefine f : X + X by

T and ( g . ) z , converges

42

2 Right Topological Sanigmups

Then f E T while f o g $ ce T because [h 'EX ' : h(1) = h(2) = 1) is aneighborhood o f f o g in X ' which misses T . 0 We see, however, that many subsemigroups of X ' do have closures that are semigroups.

Theorem 2.29. Lrr X be a topological space and let S 2 X ' be a semigroup such rhat for each f E S, f is continuous. Then cC S is a semigroup. Pmof This follows fmmThcorem 2.2 and Exercise 2.3.2.

0

The closed semigroups of Theorem 2.29 are important in topological dynamics and we will have occasion to refer to them often. Definition 2.30. Let X be a topological space and let S be a semigroup contained in ' X such that for all f E S,f is continuous. Then ct S is the enveloping semigmup of s. If T E X ' is continuous and S = {T" : n E N), then c t S is the enveloping semigmup of T . By Theorem2.29, if T E X ' is continuous, then c l ( T n : n E N)is asemigroup. We see that one cannot add a single point of discontinuity a d expect the same conclusion. Example 2.31. Lrr X = [O,11 with the usual topology and definc f E X ' by

Theorem 2.32. Ler S be a compoetropologicalsemigmup. Thenall manmlsubgmups of S are closed and are ropological gmups. P m f Let e E E(S). By Lemma 1.19, an element x E eSe is in H(e) if and only if there is an element y E eSe for which yx = xy = e. Now eSe is closed as it is the continuous image of S under the mapping p, o I.. Suppose that x is a limit point of a net (x,),,, in H(e). Foreach i E I there is anelement y, E H(e) forwhichx,y, = y,x, = e. By joint continuity any limit point y of the net (y,),,~satisfies xy = yx = e and so x E H(e). To see that the inverse function In : H ( e ) + H(e) is continuous. let x E H ( e ) and let (&),,I be a net in H ( e ) converging to x. As observed above, any limit pointy of the net (1,-'),,I satisfies xy = yx = e and sox-' is the only limit point of (x,-'),,I. That is. (x,-'),,I converges tox-I. 0 We sac that wecannot obtain the conclusion of Theorem 2.32 in an arbitray compact semitopological semigroup.

Eumpk 233. Let S r B U {m)with the topology and smtigroyp operation given in fiexisc 2.1.2 Tlun S isasemitopologicdsemigmup,E ( S ) = (0. oo)and H(0) t R Thus H(0) is not closed. Notethat in ExampIe2.33. H(m)= (m],which isclosed. In fact. by Theorern2.ZS. in any compact semitopological semipup, dl maximal gmups in K(S) are closed E x e h 2.4.1. Prove Theorem 2.25

Excrdsc 2.42. Let X be any set. We can identify P ( X ) w i t h X [ ~I), by identirying each Y E P ( X )with its chamercrisricfuncrion X y , whne.

We can u s this identification to give P ( X ) a compact topology. Prow that ( P ( X ) . U ) and ( P ( X ) ,17 are topological semigroups. Prove also that ( P ( X ) ,A ) is a topological group which can be identified topologically and algebraically with X ~ (where 2 Y AZ = (Y U Z)\(Y n Z ) , the symmetric diffemnce of Y and Z).

c

Exercise 2.4.3. Givenaset A R" andx E R" defined($, A) = inf{d(x, y) :y e A ] Let X(1Rn)denote the set of non-empty compact subsets of Rn.The Hausdorfmetric on X ( W n )is defined by

Rove that with the topology delined by this metric. (X(Wn),U) and (X(IRn),+) are topological semigroups, where A B = (a b : a E A and b E B ] .

+

+

2.5 Ellis' Theorem We now set out lo show, in ComUary 2.39. that if S is a locally compact semitopological group. then S is in fact a topological group. While this result is of fundarrsental importance to the theory of semitopological semigroups, it will not be used again in this book until Chapter 21. Consequently, this section may be viewed as "optional". In a metric space (X,d ) , given x E X and E > 0 we write N ( x . E ) = { y E X : d(x, Y ) < €1.

Lemma 2.34. Let X be a compact metric spaceand let g : X x X + W be a separately continuousfunction. m e n is a dense Ga subset D ofX such thatg isjointly continuous at each point of D x X. Pmd Let d be the mebic of X. For every 6 r 0 and every 8

w 0, let

44


. We fust obsene that each Ea, is closed t We next obsmre that for each B s 0,X = Elln.. Indeed, for each x the function y I+ g(x, y) is u ~ f d continuous, y so for some n, x E Ell.,. Now let U beanonempty opensubseof X. Foreachn E N.if Urlint,y(Et/.,.) = 0. then UrlEI/,., is nowhere dense. Thus, by theBairc Category Theorem,thereexists n E N such that U r l intx(El/,,,) # 0. Let H, = intx(Eu., ) I8 . We shall inductively choose sequences ( x , , ) ~and ~ (y,)zl in S. We hrst choose xl and yl arbitrarily. We then assume that m E PI and that xi and yi have been chosen forevetyi E (1.2 ,.... m). Put

-

46

2 Right Topological Scmigmups

We note that C, is finite and that for each u, v E C,. y o A,, o pu is continuous. Wecan therefore choose *,+I and ym+l satisfying ly(ux,,,+l u) y ( U X U c )~ and foreveryu. IJ E Cm.while I Y ( X ~ + I Y ~ + I ) - Y ( X >Y8.) ~ Iy(uy,+~v)-~(u~v)l< Let C = U = :, C, . Then C is asubscmigmup of S. because, if u E Ck and u E C,. lhen uu E Let X = cC C. Then X is a subsemigroup of S by Theorem 2.27. Let @ = vx and let Y be the quotient of X as described in Lemma 2.37. Then Y is a compact meaizable semitopological semigroup with identity n(e). and n(x) has the inverse n(x-') in Y. So,by Lemma 2.35.. is jointly continuous at (n(x), n(y)). We claim that x(x.) 4 n(x). Suppose instead that there is some z E X such that x(z) # x(x) and n(z) is an accumulation point of the sequence (n(x,,))z,. Choose by Lemma 2.36 some u, v E C such that @(uzvp f @ ( m u ) and pick k E W such that I@(uzv)- @(uxu)l z Pick rn 2 2k such that u, v E C,. Let V = (w E X : I@(uwv)-@(urv)l < &I. Then V = n-' [ ~ [ v I ] son[^] isaneighbo~oodofn(z)so choosen r rnsuchlhatI@(ux.v)-@(uzv)l c SmcealsoI@(ux,u)-@(uxu)l < f, we have that I@(urv)- @(uxv)l .c a contradiction. Similarly, n(y.) -+ n(y). Son(x.)x(y.) + n(xy) in Y. Let W = ( w E X : I@(w) @(xy)l i6). Then n[W] is a neighborhood of n(xy) in Y. so pick n such that n(xny.) E n[W]. Then ly (x.y.) - y (xy)l < 6, a contradiction. This establishes that. is jointly continuous at every point of (x) x S. By symmetry, it is alsojointly continuous at e v q point of S x { x ) . 0

-

i.

i.

&.

-

We can now prove Ellis' Theorem. Cordlpry 2 3 9 (Ellis' Theorem). Let S be a locally compact semitopological semigmup which is algebraically a gmup. Then S is a topological gmup.

Proof: If S is compact, let ? = S. Otbewise, let ? = S U ( m ) denote the onepoint compactification of S. We exte_ndthe semigroup operation of S ? S by putting s . m = m s = m for every s E S. It is then simple to check that S is a compact semitopologic_al semigroup with an identity. So, by T h e ~ m 2.38, the semigroup operation on S is jointly continuous at every point of (S x S) U (S x S). In particular, S is a topological semigroup. To see that the inverse function is continuous on S, I d x E Sand let (x,),,~ be a net in S wnverging to x. _We claim that ( X ~ - ' ) ~converges ~ I to x-'. Let z be any cluster point of (x,-'),,I in S and choose a subnet (xs-')se, which converges to z. Then by the continuity o f . at (x. z), (xax~-')s.j converges to xz. Therefore z

.

=I-'.

Cornnary 2.40. LetS be a compact smtitopologicalsemigmup with e

c e n t Then

K(S) is a compact topological gmup.

P m f : By CoroUary 2.26. K(S) is a semilopulogical group. Thus K(S) has a unique idempotent e and so K(S) = Se by Theorem 2.8. Thus K(S) is compact and so by Corollary 2.39, K(S) is a topological group. 0 Exercke 2.5.1. Rove Lemma 2.36.

Notes

47

Notes Theomn2.5 was proved fortopological semigroups by K. Numskura [I871and A. Wallace [MO, 241,2421. The first proof using only one sided continuity seems to be that of R. Ellis 186, Corollary 2.101. Theorem 2.9 is due to W. Ruppcn [2151. Example 2.13 is from [38]. a result of cuUaboration with 1. Berglund. Example 2.16 is due to W. Rupp& [215], as is Theorem 2.19(a) [218]. Lemma 2.34 is a special casc of a thcmm of J. Christensen [631. As we have already remark4 CamUary 2.39 is a result of R. Ellis [84]. The proof given here involves essentially the same ideas as that given by W. Ruppen in [216]. (An alternate proof of comparable length to that given here is presented by J. Auslander in [5. pp. 57631.) Theorem 2.38 was proved by J. Lawson 11731. For additional information see [2l6] and [174].

/3D -Ultrafilters and the ~ t o n d e c h Compactification of a Discrete Space

Tbere are many different mnstructionsof the hetondechwmpactification of atopological space X. In the case in which the space is discrete. the ~ t o n d e c compactification h of X may be viewed as the set of ulnafilters on S and it is this approach that we adopt.

Definition 3.1. Let D be any set. Afilter on D is a nonzmpty set U of subsets of D with the following properties:

A classic example of a filter is the set of neighborhoods of a point in a topological space. (We remark that a neighborhood of a point is a set containing an open set containing that point. That is neighborhoods do not, in au view, have to be open.) Another example is the set of subsets of any infinite set whose complements are finite. We observe that. if U is any filter on D, then D E U. DefiniUon 32. Let D be a set and let U be a filter on D. A family A is afilrer bare for 41ifandonlyifAE UandforeachB E UthereissomtA E AsuchthatA'E B.

Thus. in a topological space, the open neighborhoods of a point form a filter base for the filter of neighborhoods of that point. Definition 33. An uhmfiltcr on D is a filter on D which is not pmperiy contained in any other filter on D. We record immediately the following very simple but also very useful fact about ultrafilters.

Those more familiar with measures may find it helpful to view an ultrafiltcr on D

a a (0.1)-valued finitely additive measure on P ( D ) . (The members of the ultrafilter an the "big" sets. See Exercise 3.1.2.) Lemma 3.5. Lef 91 be afilrer on the set D and let A C D. Either (a) t h e n i s s o m e B ~U s u c h t h a t A n B = 0 o r (b) ( C G D : r h e m i s s o m e B ~ 9 1 w i t h A ~ B G C ) i s a f i l r e r o n D .

pmof 'Ibis is Exeercise 3.1.3. Recall that a set A of sets has thefinite intersectionproperly i f and only if whenever 3 is a finite nonempty subset of A, 3 0.

n +

Theortm 3.6. Lcr D be a set and let U g P(D). The following statemenrs are equivalent. (a) 91 is an ulmfilter on D. (b) PL has fhefvlire intersection property and for each A f P(D)\U then is some B E UsuchtharAflB = O . (c) 91 is ~ i m awith l tzspect to thefinite inrersecrion properly (Thar is, U is a -mar member o f ( V E P ( D ) : V has thefvlite intersectionproperly].) (d) 91 is afilreron D und for all 3 E Pf(P(D)). if U 3 E 91. then 3n 91 # 0. (e) U is afilter on D and for all A E D either A E 91 or D\A E U .

P m J (a) implies (b). By conditions (a) and (c) of Delinition 3.1. U has the finite intersection property. Let A E P ( D ) \ U and let V = (C E D :them is some B E U with A (7 B C ) . Then A E V so 91 $ V so V is not a filter on D. Thus by Lemma 3.5. there is some B E 91 such lhalA flB = 0. (b) implies (c). If PL V C P(D), pick A E V\U and pick B E 91 such that A n B = 0. Then A, B E V so V does not have the finite intersection property. (c) implies (d). Assume that e( is maximal with respect to the finite interseaion property among subsets of P ( D ) . Then one has immediately that 91 is a nonempty set of subsea of D. Since U U ( D )has the finite intersection property and 91 E U U (D), one has 91 = e( U (Dl. That is. D E 91. Given A, B E U.91 U ( A n B ) has the finite intersection property so A n B E U . Given A and B with A E U and A B G D. 91 U ( B }has the finite intersection property so B E U . Thus U is a filter. Nowlet3 E Pf ( P ( D ) ) w i t h U 3 E U,andsupposethatfoteachA E 3 . A 6 91. Then given A E 3 , 91 P U U ( A }so 91 U ( A )does not have the finite intersection property so there exists $A E P f ( W such that A fl $ A = 0. Let X = UAE9gA. Then X U [U3 ) G 91 while ( U7 )n X = 0 , a conmdiction. (d) implies (e). Let F = [ A .D\A).

n

n

(e) implies (a). Assume that U is a filtu on D and for all A D either A E U or D\A s U . Let V be a filter with U G and suppose that U # V. Pick A E V\U. Then D\A E P1 G V while A n (D\A) = 0, a contradiction. 0

v

If a E D. (A E P ( D ) : a E A) is easily seen to be an ulhafilter on D. This ultrafilter is called the principal ultmfilter defined by a . Theomn 3.7. Let D be a set andlet U bean ulhafilter on D. Tlu/ollowing statements are equivalmi. (a) U is a principal uItm~Iter: F E P f ( D ) such that F G U . (c) The set [ A D : D\A i s f i i t e ) is not contained in U. (a u # 0. (e) Then is some x E D such that P1- {XI. (b) Then is some

c

n

n

Pmof: (a) implies (b). Pick x E D such that 41 = {A G D : x E A). Let F = 1x1. (b) implies (c). Given F E PI(D) n P1 one has D\F $ U. (c) implies (d): Pick A G D such that D\A is finite and A $ U. Let F = D\A. Then F G U and F = U[{x): x E F ] so by Theorem 3.6, we may pick x E F such that{x)e U . ThenforeachB E U , B n ( x ) # 0 s o x G

nU.

n

n

(d) implies (e). Assume that U f 0 and pick x € U. Then D\(x) $ U so {XI E u s o n u G { X I . (e) implies (a). Pick x E D such that U = (XI. lben U and {A G D : x E A) arc both ultra6Iters so by Remark 3.4 it suffices to note that U 2 (A G D :x € A).

n

It is afactthat principal ultrafiltersarethe only ones whose members can be explicitly defined. There are no others which can be defined within Zmelc-Fraenlrel set theory. (See the notes to this chapter.) However, the axiom of choice produces a rich set of nonprincipal ultrafilters on any infinite set

Theorem 38. Let D be a set and let A be a subset of P ( D ) which has thejinite intersection pmperty. Then then is on ultmplter U on D such that A c U. PmoJ Let

Then A

E

r = (8 c P ( D ) : A c 51 and 2I has UK finite intersection pmpxty). r so r # 0. Given a chain C in r one has immediately that A 5 UC. .

GivenP ~ ~ ~ ( U C ) t h e r r i s s o r n e ~ E C g w iBt sh o~ n P $ 0 . ThusbyZorn's Lemma we may pick a maximal member P1 of r. Xvially U is not only maximal in r. but in fact U is maximal with respect to the finite intersection property. By Theorem 3.6. U is an ulaafilter on D. 0

Corollary 3.9. Let D be a set, let A be ajilter on D. and let A g D. Then A $ A if andonly ifthere is some ultmflter U with A U (D\A) P1.

PmoJ Since one cannot have. a filter U with A E U and D\A wivial.

E

U . the sufficiency is

.

For the necessity it suffices by Theorem 3.8 to show that A U (D\A) has the finite intersection property. So suppose instead that there is some finite nonempty 3 G A s u c h t h a t ( D \ A ) n n 3 = 0 . T h e n n T s AsoA E A . 0

.

The following concept will be important in the combimatorial applications of ultrafilters. Dellnition 3.10. Let R be a nonempty set of sets. We say that R is partition regular if and only if whenever 9 is a finite set of sets and U 3 E R,there exist A E 3 and B E R such that B G A. Given a propeay 4, of subsets of some set D, 0.e. a statement about these sets), we say the propmy is partition regular provided (A G D : * ( A ) ] is partition regular. Notice that we do not require that a partition regular family be closed under supersets (in some set D). (The reason is that there are comhinatorial families, such as the sets of finite pmducts from infinite sequences. that are partition regular under our requirement (by Corollary 5.15) but not under the strongerrequirement.) See however Exercise 3.1.6

Theorem 3.11. k t D be a set md let R g P ( D ) be nonempfy and ossume fhat 0 f R. Let Rt = (B E P ( D ) : A c B for some A E 31. Thefollowing statements are equivalent. (a) R is partition regulox @) Whenever A G B(D) has thepmpeny that evcryj5nitc nonempty subfamily of A has an intersection which is in Rt, there is an uhmfrlter U on D such fhat

AEUGR~. (c) Whenever A E R,there is some ultmfilter U on D such thnt A E U g .Ut.

Pmof: (a) implies (b). Let a) = (A c D : for all B E R.A n B # 0) and note that B # 0 since D E a). Note also that we may assume that A # 0.since (Dl has the hypothesized property. Let C = A U a). We claim that C has the finite intersection property. To see this it suffices (since A and B are nonempty) to let 3 E 5PJ (A) and $ E Q ( B ) and show that 3n $ # 0. So suppose instead that we have such F and$Withn3nn$=:0.PickB~RsuchthatB~n3.ThenBnn$=0and so B = U,,s(B\A). Pick A 6 $ and C E R such that C E B\A. Then A n C = 0. conbadicting the fact that A E 51. By Theorem 3.8 there is an ultrafilter ei on D such that C g U. Given C E U, D\C f S (sinceCn(D\C) = 0 f U). Sopicksome B E R suchthat Bfl(D\C) = 0. 'Ihat is. B G C. 0)implies (c). Let A = [A). (c) implies (a). Let 3he a finite set of sets with UP E R and let U be an ultrafilter on D such that U3 E U and for each C E e( there is some B E R such that B G C. Pick by Theorem 3.6 some A E T r l U. 0

n

n

Definition 3.12. Let D be a set and let U be an ultrafilter on D. The norm of U is IlUll = min{lAl : A E U).

Note that. by T h m m 3.7, if U is an dtra6lter. then Il Ull is either 1or infinite. DeBniIion 3.13. Let D be a set and let*. be an infinite cardinal. A r-uniform ulmfiltrr on D is anuluafilter U on D such that llUll 2 r. The set V,(D) = [ U : U is a r-uniform ultrafiller on D). A umyonn ulmfilter on D is a r-uniform ulhafilter on D. w h e ~= r IDI. Coroilam 3.14. Let D beanyserandlcr A beafamily ofsubsets@D. Iftheintersection of everyhire subfanu'ly of A is inhire, then A is contained in an ultrafilter on D all of whose members are infiire. More gmemlly, ifw is an infinite cardinal and if the intersectionof everyfinire subfmily of A has cardinality at lemtx, then A is contained in a r-unifonn ulnafirer on D.

Pmof: m e property of being infiniteis partition ngular. and so is the properly of having cardinality at least r. 0

If the intmection of every finite subfamily of A is infinite, A is said to have the infinitefinite intersectionpmpcny. Thus. Corollary 3.14 says that if A has the infinite finite intersection property, then thm is a nonprincipal ultrafilter U on D with A E PL. 3.1.1. Let A be a set of sets. Rove that there is some filter U o n D = UA such that A is a filter base for U if and only if A # 0, 0 4 A, and for every finite nonempty subset 3 of A, there is some A E A such that A S 3.

Ex&

n

Exercise 3.13. Let D be any set. Show that the ulhafilters on D are in one-to-one comspondence with the finitely additive mcasws defined on P ( D ) which take values in (0. 1) and are not identically zero. Exercise 3.13. Rove Lemma 35. Exercise 3.1.4. Let D be any set Show that the ulpafilters on D are in one-to-one comspondenccwith theBoolean algebra homomorphisms mapping (P(D), U, n)onto the Boolean algebra ([0, I}. v, A). Exercise 3.15. Let D be a set with cardinality r. Show that there is no nonprincipal ultrafilter U on D with the prop* that every countable subfamily of U has an intersection which belongs to PL. (Hint: D can be taken to be the i n t d 10.1: in B.If an ultrafilter U with these properties did exist. every a E [O. I] would have aneighborhood which did not belong to U.) Exercise 3.1.6. Let X be a set of sets and let D be a set such that UR D. Let R t = {B G D : there is some A E R with A g B ) . Rove that the following statements are equivalent. (a) $3 is partition regular. E R with C existA~3andB~RwithBCA. (c) R t is partition regular. (d) If 3 is a finite set of sets and U3 E Rt.then 3n R t # 0.

(b) If 3 is a finite set of sets and there is some C

U3.then there

i

.

!

3.2 'Ihe Topological Space BD

3.2 The Topological Space B D ~nthis seetion we define a topology on the set of all ulrrafiltem on a set D, and establish m e of the proprtics of the resulting space. m t l o n 3.15. Let D be a discrete topological space. (a) p D = ( p : p isT ulnafilter on D). (b) Given A S D. A = ( p E p D : A E p). The m n for the notation P D will become clear in the next section. We shall hence f o d use lower case letters to denote ulnafilters on D, since we shall be thinking of ulnafilters as points in a topological space. Dcfinitlan3.16. LetDbeasetandleta e D . T h e n e ( a ) = [ A E D : a €A). Thus for each a E D, e(a) is the principal ulmfilter cnrrsponding to a.

-

Lcmna3.17. LetDbeasctandIetA. B c D. (a) ( ~ B)n= Xn 5; (b)(A u B) = XU E; (c) p \ ~ =) BD\Z (d) A = 0 ifand only ifA = 0: (e) e D ifandonly ifA = D; (f)A=BifandonlyifA=B.

z=

Pmof This is Exercise 3.2.1.

0

. .

We obsm2at

the sets oft& form Tare closed under finite intmectiow, because

Xn B = (A n B). Consequently, (A : A S D) forms a basis for a topology on pD.

We define the topology of g~ to be the topology which has these sets as a basis. The following theorem describes some of the basic topological pmpesties of PD.

Theorem 3.18.

Lcr D be any set. (a) PD is a compact Hayfoifspace. (b) The sersofthe form-A are the clopen subsets of PD. (c) For aery A 2 D, A = c e a e[Al. ~ ~ ifandonly ifA E p. (d) Forany A c D andany p E D, p E c t g e[Al (e) The mapping e is injective and e[D] is a dense subset of BD whoJe points am pmcisely the isolatedpoints of PD. (0IfU is an open subset of PD, c L p U is also open.

PmoJ (a) Supposethat p andq aredistinctelernentsofBD. If A e p\q, then D\A E q. So Xand (D\A) are disjoint open subsets of PD containing p andq respectively. Thus BD is Hausdorff. We obseeat the sets of the form A^ are also a base for the c l o d sets, because BD\A = (D\A). 'Ihus, to show that BD is compact, we shall consider a family A

of sets of the f m zwith rhe finite inteDcctlion propmy.and show that A has anonempty intersection.-Let 8 = [A S D : A E A]. If 9 E P f ( 8 ) . then there is A and so 7 E p and thus 3 # 0. That is, 8 has the finite some p E intersection property. so by Theorem 3.8 pick q E p D S q. Then q E A. @) We pointed out in the proof of (a) that each set _A was closed as-well as open. Suppose that C is any clopen subset off3D. Let A = [A : A D and A & C). Since C is open, A is an open cover of C.Since C is closed, it is compact by (a) so picka finite subfamily 7 of 9@ such that C = UdaP Then by Lemma 3.17(b). C = (c) Clearly, for each a fi A. eja) E A and therefon CCBD e[Al S 2 To p m e the revme inclusion. let p E A. If B denotes a basic neighborhood of p.@n A E I and B EpandsoAnBf0.Chooseanya EAnB. Sincee(a) ~ e [ A ] n 6 , e [ A ] n B$ 0 and thus p E c L p e[A]. (d) By (c) and the definition of A?

nAeg

n

n

n

with

c

A?

3.

(e);a, b c D aredistinct, D\[a) E e(b)\e(a) and so e(a) # e(b). If A is a no:-empty basic opzn subset of flD, then A # 0. Any a E A satisfies e(a) E e[D] n A and so e[D] n A # 0. Thus e[D] is dense in pD. For any a E D, e(a) is isolated in p D because @ is an open subset of ,9D whose only member is e(a). Conversely if p is an isolated point of pD. then (p) n e[D] # 0 and so p E e[Dl. (1)If U = 0, the conclusionis hivialandsoweassumetb~U# 0. Put A = e-'[U]. We claim k t that U S c L p e[A]. So let p E U and let B be_a basic neighborhood of p. Then U n B is a nonempty o p n set and so byp). U n B n e[D] # 0. So pick b E B with e(b) E U.Then e(b) E B n e [ A ] and so B f l e[A]~.f 0. Alsoe[Al S U and hence U G c t # ~ e [ A5 I C C ~ DU . ThuscCBDU = cfgoe[A] = z ( b y (c)). and so c L g ~ u is open in BD. 0

i

We next establish a characterization of the closed subsetsof & D which will be useful later. Definition3.19. Let D be

a set and let A be a filter on D. Then 2 = { p E flD :

A S PI.

Theorem 320. Let D be a set.

(a) IfA is afilter an D, then 2 is a closed subset of PD. (b)IfA S p ~ a n d A = n then ~ , Aisajlreron ~ a n d x = c f ~ .

Pmof: (3Let p E P D \ ~ Pick . B E A\p. Then D\B is a neighborhood of p which misses A. (b) A is the i$ersection of a set of filterJ,so A is a filterAFurther, for each p-E A. A S p s o A SA%dthusby(a).ctA E A . T o s e e t h a t A S c L A . l e t p ~Aand I e t B E p . SupposeBnA=O.Thenforeachq E A.D\B ~ q s o D \ B e A S p . a contradiction.

It is customary to identify a principal ultn6ltcr c'(*) with the point x. and we shall adopt that practice ourselves after we have proved that fiD is the ~tone&ch compactification of the disnete space! in Section 3.3. Once this is done, we can write the following definition as A* = A\A. For the moment we shall continue to maintain the distinction between x and e(x). Definition 3.21. Let D be a set and let A E D. Then A* = & e [ ~ ] . The following theorem is simple but useful. Once e ( x ) is identified with x the conclusion becomes "U f l D E p".

Theorem 322. La p

E p D and In U be a subset of BD. I f

U is a neighborhoodof p

in fiD, then e-'[U] E p. PmoL If U is a neighborhwd of p. there is a basic open subset of fiD for which a p E A G U . This implies that A E p and so ed'[U1€ p, because A e-'[u]. Recall that a space is zem dimensional if and only if it has a basis of clopen sets.

Theorem 3.23. Let X be a zem diWIenrio~1space and let Y be a compact subset of X. The clopen subsets of Y a m the sers of theform C n Y where C is clopen in X . In particular, if D is an infinite set. then the nonempry clopen subsets of D* am the sers of theform A* where A is an infinite subset of D. Proof: Trivially, if C is clopen in X, then C n Y is clopen in Y . For the converse, let B be clopen in Y and let A = [A n Y : A is clopen in X and A n Y E B). Since X is zero dimensional. A is an open cover of 6. Since Y is compactand hence B is compact, pick a finite set 7 of clopen subsets of X such that B = UAEF(A n Y ) and letC=UF. Tbe "in particular" conclusion follows from Corollary 3.14 and Theorem 3.18@). a

Exemise 32.1. Rove Lemma 3.17.

3.3 ~ t o n d e c hCompactification In this section we show that /3D is the ~tone-&ch compactification of the discrete space D. Recall that by an embedding of a topological space X into a topological space 2. one means a function rp : X -+ Z which defines a homeomorphism from X onto pl[X]. We remind the reader that we are assuming that all hypothesized lopological spaces are Hausdorff. Definition 3.24. Let X be a topological space. A compactificationof X is a pair (rp, C) such that C is acompact space. rp is an embedding of X into C. and rp[X]is dense in C.

3 PJ

56

~ n complaely y regular space X hasalargest compactification called its ~to&ch compactification.

Definitm 335. Let X be acompletely regular topological space. A S d e d compactificmr'on of X is a pak (9. Z) such that (a) Z is a compact space, @) (p is an embedding of X into Z, (c) q[X] is dense in Z, and (d) given any compact space Y aad any continuaus function f : X -+ Y then exists a continuous function g : Z + Y such that g o (p = f . ( T b t is the diagram

commutes.) One customarily refm to the StonbCech compactification of a space X tather than a ~ t o n d e c compaetification h of X. 'Ihc Feason is made clear by the following r e d If one views (p as an inclusion map, then Remark 3.26 may be viewed as saying: 'The ~ t o n d e c hcompactififation of X is unique up to a homeomaphism leaving X pointwise fixed."

.

Remark 3.26. Let X be acompletely rrguIa~spaceandLet ((p. Z) and (r. W )bestone&ch compact~canionrof X. Then there is a homeomorphism y : Z + W such t h yo*=r. Theorem 3.27. Let D be a discrete space. Then (e, BD) is a ~ t o n d e c h compact$cation of D.

P w f : Conditions (a), (b), and (c) of Definition 3.25 hold by Theorem 3.18. It remains for us to verify condition (d). Let Y be a compact space and let f : D + Y. For each p E flD let Ap = [c!~ f [ A ] : A E p ) . Then for each p E flD. Ap has the finite intersection prop* (Exercise 3.3.1) and so has a nonempty intersection. Choose g(p) E 4.Then we

n

have the following diagram.

We need to show that the diagram c a m s and that g is continuous. For the fnst assenion. let x E D. Then (XI E e(x) so g(e(x)) E e l y f[{x]] = cLy[(f (x)ll= if (x)) SO g o e = f a s n q u i r d To see that g is continuous. let p E BD and let U be a neighborhod of g(p) in Y. S i e Y is regular. pickaneighborhood V of g(p) withcty V & U and let hr&.-'[V]. We claim that A E p so suppose instead that D\A E p. Then g(p) E kyf [D\A] and V is a neighbo$ocd of g(p) so V fl f [D\Al # 0, c o e i c t i n g the fact that [ ~U,soletq ] E zand A = f-'[V]. Thus A isamigbborhoodof p . { W e ~ l a i ~ t h a t g E supposethatg(q) $ U. Then Y\cCr V isaneighborhoodof g(q) andg(q) E cLy f [A] so (Y\ cty V )fl f [A] # 0, again contradicting the fact that A = f -'[V].

n

Although we have not used that fan. each of the sets A, is a singleton. (See Exercise 3.3.2.) We have shown in Theorem 3.27 that BD is the S t o n d e c h compactification of D. This explains the nason for using the notation BD for this space: if X is any completely tegular space, BX is the standard notation for its S t o n d e c h compactification. X is usually regarded as being a subspace of BX.

IdentifySng D with e [ D ] It is common practice in dealing with BD to identify the points of D with the principal ultratilters generated by those points, and we shall adopt this practice from this point on. Only rarely will it be necessq to remind the reader that when we writes we sometimes mean e(s). Once we have identified s e S with e(s) E.BD. we shall suppose that D G BD and gall write D' = BD\D, r a t h e r b P = BD\e[DI. Furtkr, with thiJ identification. A = ctgD A for every A E P D , by Theorem 3.18. So the notations A and Ti become interchangeable. We illustrate the conversion process by restating Theorem 3.27.

Theorem 3.28 ( S t o d e c h Compactification -restated). Lcr D be an injinire discrere space. Then (a) BD is a compaci space, (b)D EBD. (c) D is dense in BD. and (d)givm any cofnpactspace YmdanyfWIctionf :D -t Y thew exisrsa continuous ~ f. fwtcrion g : BD -t Y such thul g l =

Identifying PTwith ^T for T

S

In a similar fashion. if T E S, we shall identify p s T (which is an ultrafilteron S) with the ultrafilter (A fl T : A E p] (which is an ulaafilter on T) and thus we shall pretend mat BT BS.

Exerdse 33.1. Let D be a discrete space. let Y be acompact space, and let f : D 4 Y. For each p E BD let Ap = (cCr f [A1 : A E p). Rove that for each p E BD. Ap has the finite intersection propny. Exercise3.33. Let the sets Ap be as defined in the pmof of Thmrem 3.27. Rove that for each p E PD. fl Ap is a singleton. (Hinc Consider the fact that hato continuous functions agreeing on e[D] must be equal.) Exerrisc 333. Let D be any discrete space and let A identified with BA.

D. Rove that cLgo A can be

3.4 More Topology of D Iff is acontinuous_mappingfroti a completely regular space X into acompact space Y. we shall often use f to denote the continuous mapping from BX to Y which extends f. although in some cases we may use the same notation for a function and its extension. (Notice that there can be only one continuous extension, since any two extensions agree on a dense subspace.) Defhitjnn 3.29. Let Dbeadiscretespace, let Y beacompac~pace,andlet f : D Then f is the continuous function from BD to Y such that f i D =: f .

+ Y.

If f : X + Y is a continuous function between completely regular spaces. it has a continuous extension ffl : BX + BY. The reader with an interest in category theory might like to know that this defines a functor from the category of completely regular spaces to that of compact HausdortT spaces, and that this is an adjoint to the inclusion. functor embedding the second category in the first. Lemma 330. k t D and E be discrete spaces and let f : D + E g BE. Forzach p E BD. = [A C E : f -'[A1 E p). In panicuhr; ifA E p, rhm f [A] E f (p); ond i f B E f (p). then f [ B ] E p.

-'

Proof: It is routine to veriQ that (A G E : f -'[A] E p ) is an ultrafilter on E. For each p E BD. let g(p) = ( A G E : f - ' [ ~ ] E p). Now, given x E D we have g(x) = (A G E : f (x) E A). Recalling that we arc identifyingx with_e(x)(and f (x) withe(f (x))) we have g(x) = f (x). To see that g is continuous. let A be a basic open set in BE. Then g-'[A^] = f

f =s.

[A]. Since g is a continuous extension o f f . we have 0

Lemma_J.31. L a D and E be discrete spaces, Ier f, g : D + E,and let p E pD satisfv f (p) = Z(p). Then for each A E p. {x E D : f (x) E g[A]J E p. Pmof: By Lemma 3.30. gIA1 E ?(p) = j ( p ) so. again applying Lemma 3.30. f -'[8[~1] E P.

I I

3.4 M o n Topology of SD

.

59

umma 3.32 Suppo;~ that D is any discrete space and that f is a mappingfmn D to itself Themapping f :pD -+pD has afrrcdpoint ifand only ifeveryfinitepanition

p m f Suppose Grst that D has a finite partition in which every cell C satisfies C n f[C] = 0. Let p _E pD and pick a c e 5 C satisfying C E p. This implies by Lemma 3.30 that f [Cl E f ( P ) and hence that f (PI # P. Conversely, suppose that f ha%nZfixekpinls. Por each p E pD pick Ap E p l T ( p ) , pick Bp E p such that f [B,] n A, = 0. and let C, = A,, nABp. Then (C, : p E pD) is an open cover of pD so pickfinite F G pD such that [Cp : p E F ] coven BD. Then (Cp : p E F ] covers D and can thus be relined to a finite partition F of D such that each C E F satisfies C n f [C]= 0. 0

Lemma 333. Suppose that D is a set and that f : D + D is afunction with nofirrd points. Then D can bepanirionedinto three sets Ag, Al, and A2 with thepmperry rhar A i n f [ A , l = @ f o r e a c h i E [0,1,2).

P m f We consider the set

We observe that 8, is non-empty, because the function 0 is a member of $. Then $ is p d a l l y orderedby setinclusionand, if C isachainin $, then UC E $.so Zom's Lemma impliesthat $ has a maximal element g. We shall show that dom(g) = D. We assume the contrary and suppose that b E D\ dom(g). We then define a function h extending g. To do so,we put h(a) = g(a) for every a € dom(g). We choose h(b) to be a value in (0. 1,2),choosing it so that h(b) # h(f ( b ) )if h( f ( b ) )has already been defined. Now suppose that h( f '"(b))has been defined for each m E [O. 1,2. . n). (Where f o ( b ) = band fl+l(b) = f (ft(b)).) I f h(f "+'(b)) has not yetbcen defined, we choose h (f"+ ( b ) ) to be a value in [O,1.2) different from h( f ( b ) ) and from h(f"+'(b)). if the latter has already been defined. In this way, we can inductively define afunction h E $ with dom(h) = dom(g) U 1f n ( b ) : n E w). Since g P; h, we have contradicted our choice of g as being maximal in $. Thus dom(g) = D. We now define the sets Ai by putting Ai = g-'[[ill for each i E (0.1.2). 0

. ..

Theorem 3.34; k t D be a discmte space and let f : D -+ D. I f f hcu no jiredpoints, neifherdoes f : pD + pD. P m f This follows from Lemmas 3.32 and 3.33. Theorem33$. L e t D b e a s e f a n d l e t f : D + D. I f f " : p ~ + p ~ a n d i f ~ e p ~ . = p ifandonlyif[a E D : f ( a ) =a1 E p. fhen

T(p)

P m f Let E denote ( a E D : f (a) = a ) . We first assume that 7(p) = p. We want to show that E E p so suppose instead that D\E G p. We choose any b E D\E, and define g : D + D by stating that g ( a ) = f ( a ) if a 6 D\E_and g(a) = b if a E E. Then g has no fixedpoints. However. f = g on D\E and so f = Zon cf (D\E). Since p E cL(D\E). a ( p ) = f ( p ) = p. This cotmadicts Theorem 3.34 and hence E E p. Conversel~,supsuppose that E E p. Let r : g + D be the identity function. Since f = I on E, f =Son c t E. Since p E CL E, f ( p ) = i ( p ) = p. 0 Theorem 336. If D is any infinite set, every non-mtpfy Ga-subset of D* has a nonempty interior in D*.

ng,

Pmof: Suppose tha&for each n E N. U , is an open subset of p D and that (U. n D') # 0. C h w a n y p E (UnfiD*). Foreachn 6 wecanchoose asubset A. of D for which p E A,' E U,,. We may assume that t h s e sets are decreasing. because we can replace each A. by A,. Observe that each A, is infinite, for otherwise we would have Ana = 0. We can thus choose an infinite sequence ( a n ) L Iof distinct points_D forwhicha. E A, foreachn € N. Put A = (a, :n E N ) . If q E A*.then q E A, for every n E N. because A\A. is finite. Thus the m e m p t y open subset A* of D* is contained in (Un r l D*). 0

P.

ny,I

nzl

Corollary 337. k t D be any set. Any countable union of nowhere d m subsets of D* is again nowhere dense in D*.

Ug,

P m f For each n E N, let A, be a nowhm dense subset of D*. To see that A. is nowhm dense, it suffices to show that B 3 Uz,cL A,, is nowhere dense. Suppose instead one has U = in* c l B # 0. By the Baire Category Theorem D*\B is dense in D' so U\B # 0. Thus U\B = (U\ c t A.) is a nonempty G6 which thus; by Theorem 3.36. has nonempty interior, say V. But then V r l cL B = 0 while V U , a 0 contradiction.

nzI

A point p in a topological space is said to be a P-point if the interseaion of any countable family of neighborhoods of p is also a neighborhood of p. Thus an ulmfilter pEN ' is a P-point of N* if and only if whenever (An& is a sequence of members of p, t h m is some B E p such that IB\A.I < o for all n E W. We see now that the Continuum Hypothesis implies that P-points exist in N*. It is a fact that their existence cannot be proved in ZFC. (See the notes to this chapter.)

Theom 3.38. The Conrinuum Hypothesis implies rhm P-points exist in W.

Pm$ Assume the Continuum Hypothesis and enumrratc P ( N ) as (C,),,,,

with

Co = N. Let A. = (Co).Inductively let 0 c a < ol and assume that we have chosen A, for all a c a such that (I) A, has the infinite finite intersection pmpity,

(2) either C, E A, or N\C, G A,. ( 3 ) if 8 < u,then As S A..

,

3.4 M n c Topology of BD

Let (B& I i t &c elements of U,, JI, (with repetition if necessary). If there is some n such that (C, nn:=I B,,J < o,lqt D = NC, and otherwise let D = C, . Then for each n E N,ID n BmI = w so we may choose a one-to-one sequence (x.)zl A,. with each& E D n G z l Em. LetA = [x,, :n E N)andlet A, = [A. D)uU,,, m e n the induction hypotheses are satisfied. Let p = U ,,,, A,. Then by hypolhcses (1) and (2), p E N*.Given a sequence (&)Elof members of p. pick a < ol such that {En : n E N) g [C, :a c a). By hypothesis (S), there is some A E A, c p such that IA\E. I < a, for each n E N.

mm=I

Definitton 339. A topological space is said to beammally disconnecled if theclosure of every open subset is open. We showed in Theorem 3.180 that BD isan exmmally disconnected space. Since we often work with D'. the reader should be cautioned that D' is not exmmally disconnected [I04 Exercise 6Wl. The following theorem will be vcryuscful in our algebraic investigations of countable semigroups.

Theorem 3dQ. Lcr D be a discrrte space A let A A B be a-compocr subsem of /3D. r f A n c ~ B = c e A n B = 0 , t h e n c e A n c l B = 0 .

Pmof: Write A =

Uz,An and B = c=, B. where A. and B. are compact for each For each n E N.A. and cC B

rl. Since @ Dis acompact (HausdoB) space, it is normal.

are disjoint closed sets and CC A and 8, are disjoint closed sets so pick open sets T,. (I,. V.,andW.suchthatT.nU. = V,nW, =0.A. ET..cLBGU.,ceAc V.,and B. 2 W. For each n E N.let G . = T. n Vk and let H . = W,, n Uk. Then for eachn. one has A, E G . and B. C H,,. Furthcr, given any n, rn E N,G. n H, = 0. . and let D = H.. Then C and D arr disjoint open sets Let C = Ug,G (SO D n cC C = 0). A c C, and B c D. By Theorem 3.18 Q ce C is open, so

n;=,

a=,

Uzl

c e ~ n c e c = 0 .SinceceA ~ c C C w e h a v e c C A n c e B=0asrequind.

o

The followingequivalent (see Exercise 3.4.2) vemion of Theorem 3.40is also useful. Corollary 3.41. L a D be a discrete space A Ief A and B be a-compact subsets of BD. ~ h r ncc A n cc B E ce(A n CL B) u CC(Bn ce A).

n ~e B and suppose thg p e CC(An ce B) u CC(Bn c ~ A )Pick . H~p_~uchthatHn(AnclB)=0andHn(BncCA)=0.LetA'=HnAand 8' = H n B. Then A' and B' are a-compact subsets of @ Dand p E c l A' n c t B' so by Theorem 3.40 we may assume without lossof generality thal A'n cC B' f 0 so pick q E A' n ce B'. Then q E H n (A r l ce B), a contradiction. PIWJ k t p E ce-A

We isolate some specific instances of Theorem 3.40 that we shall often use.

Corollary 3.42. Let D be a discme space. (a) k t A and B be o-compact subsets of D*.

If A i l cC B = ct A n B = 0, then c t ~ n c e ~0.= (b)k t A and B be counto6k subsets of BD. If A r l cf B = CC A n B t 0, then cLAnctB=0. (c) Let A ond B be comtoble subsets of D*. I f A 17ct B = cf A 17B = 0, then clAnctB=O. PmoJ Countable sets are o-compact. By Theorem 3.18 (e) D* = BD\e[D] is aclosed subset of BD so a-compait subsets of D* are osompact in pD.

D

The following notion will be used in the exercises and later.

A

Definition3.43. Let X be a topological space and let E X. Then A is smngly discrete if and only if there is a family (U,),E~ of open subsets of X such that x E U, for each x E A and U, ilU, r 0 whenever x and y are distinct memben of A. E x d s e 3.4.1, Suppose thatJ : D + E is a mapping from a discrete space D to a discrete space E. h v e that f : OD + BE is injective i f f is injective, sujective if f is surjective and a homeomorphism if f is bijective. Exercise 3.43. Derive Theorem 3.40 h m Corollary 3.41. Exerdse 3.4.3. Prove that any infinite regular space has an infinite smngly discrete sum. Exercise 3.4.4. Let X be an extremally disconnected regular space. Rove that sequence can converge in X unless it is eventually constant.

~KI

Exercise 3.45. Let D be any set. Rove that no proper F,-subset of D*can be dense in D*. Exerdse 3.4.6. Let D be any set. Prove. that no zero set in D' can be a singleton. (A subset Z of a topological space X is said to be a zero set if Z = f - ' [ ( o ] ] for some continuous function f : X + [O. I].) (Hint: Use Theorem 3.36.)

Exereise 3.4.7. L= k) : k E N] G q . Then by Lemma 4.26 (yr. . x. : k , n E N and kcnJ~q.pand(x~.y.:k,n~Nandkcn)~p.q.

a

As a consequence of Theorem 4.27. we see that neither (BN, +) nor (BN. .) is commutative, and hence by Theorem 4.25 neither is a left topological semigroup. (As we shall see in Chapter 6, in fact the center of each is N.) Exercise 42.1. Rove that if S is a left zero semigroup, so is BS. In this case show that BS is not only a semitopological semigroup, but in fact a topological semigroup.

Ermtsc 432. 'Rove that if S is a right zero semigmup. so is flS. In this case show that flS is not only a semitopological semigroup, but in fact a topological semigroup. (Note that because of our lack of symmehy in the definition of the operation on flSthis is nor a "dual'of Exercise 4.2.1.)

For many reasons we are interested in the semigmup S* = flS\S. In the first place, it is the algebra of S* that is the "new" material to study. In the second place it turns out that the strucm of S' provides most of the combinatorial applications that are a large pan of o w motivation for studying this subject. One of the first things we want to know about the "semigroup S*" is whether it is a semigroup.

Theorem 428. Let S be a semigmup. Then S' is a subsemigroup of flS ifand only if for any A E Pj(S)andfor any infmifesubset B of S there exists F E P J ( B ) such that X-' A i s f i i f e .

nrEF

Proof: Necessity. Let a finite nonempty subset A and an infinite subset B of S be given. Suppose that for each F E P J ( B ) , x - ' A is infinite. Then [ x - ' A : x E B ) has the p r o m that all of its finite intersections are intinite so by Comllary 3.14 we may pick p E S* such that ( x - ' A : x E B J E p. Pick q E S* such that B E q. Then A E q p and A is finite so by Theorem 3.7, q . p E S, a contradiction. (Recall once again that we have identified the principal ultrafilters with the points of S.)' Sufficiency. Let p, q E St be given and suppose that q . p = y E S, (that is, precisely, that q . p is the principal ultratilter generated by y). Let A = ( y J and let B = (x E S : .?-'A E pJ. Then B E q while for each F E 4 ( B ) , one has x - ' A E p so that n E F x - ' is ~ infinite, a contradiction. 0

nXeF

.

nZeF

CoroU~ry4.29. Let S be a sernigmup. IfS is either right or left cmellative then S* is a subsemigmup of SS.

We can give simple conditions characterizing when S* is a left ideal of flS.

Definition 430. A semigroup S is weakly leftcancellntive if and only iffor all u, v E S. ( x E S : ux = vJis finite. Similarly, S is weakly right cancellafiveif a i d only if for all u, v E S, [ X E S : x u = u ] is finite. Of course a left cancellative semigroup is weakly left cancellative. On the other hand the semigmup (M.v) is wcakly left (and right) cancellative but is far from being canccllative.

When we say that a h c t i o n f isfinite-toane, we mean that for each x in the range o f f . f -'[[x)] is finite. Thus a semigmup S is weakly left canccllative if and only if for each x E S. Ax is finite-to-one.

lheortm 431. Let S be an infinite semigmup. Then S* is a left ideal of fJSifand oniy g S is weakly I@ cancellnrive.

pmof: Nsessity. Let 5 y E S be given, let A = ~ ~ - ' [ [ x )and l suppose that A is

infinite. Piik p E S* fl A. l h n y .p = x, a contradiction. Sufficiency. Since S is idinite, S* # 0. Let p E S*. let q E fJSand suppose that

q.p=x~S.Then[x)~q~pso[y~~:~-'[x)~~)~qandishencenonempty. So picky E S such that y-'[x) E p. But y-'[x) = ly-'[(x)] a conwadiction.

SO ly-'[(x)]

is infinite,

The characterization of S* as a right ideal is considerably more complicated.

Theorem 432. Let S be an infinitesemigroup. Thefollowing stntementsm equiwalent. (a) S* is a right ideal of 9, S. (b) Given any h i r e subset A of S, any sequence ( z n ) g 1in S, and any one-to-one sequence ( x , ) z l in S, there exist n -z m in N such tharx. - z, f A. (c) Given any a E S, any sequence ( z n ) z , in S, and any one-to-one sequence (x,)zl in S, there exisf n c m in N such t h t x. .,z # a.

. z,,, : n, m E W andn < m] G A. Pick p E fJS such that ([z, : m > n) : n E PI) G p and pick q E S' such that [x" : n E N] E q. which one can do, since {x. :n E N) is infinite. Then by Lemma 4.26, A E q .p so by Theorem 3.7. q . p E S. a contradiction. The fact that (b) implies (c) is trivial. (c) implies (a). Since S is infinite. S* # 0. Let p € fJS, let q E S,and suppose that q p = a E S. Then [s E S : s-'[a) E p ) E q so choose a one-to-one sequence (x,)zl such that [x,, : n E MI g [s E S : s-'[a) E p). Inductively choose a sequence in S such that for each m E N, zm E n:=l x.-'(01 (which one can do since m x. - I [a] E p). Then for each n < m in N, x, .zm = a. a contradiction. 0 PmoJ (a) implies (h). Suppose [x.

.

(z.)zl

n,=,

-

We see that cancellation on the appropriate side guarantees that S*is a left or right ideal of fJS. In particular, if S is cancellative, then S8 is a (two sided) ideal of fJS.

Corollary 433. Let S be an infrnite semigmup. (a) F S is left cancellative, then S* is a left ideal of fJS. (b) IfS is right cancellafive. then S* is a righr ideal of SS.


0

Comllarg 434. Let S be an infinite semigmup. If S* is a right ideal of fJS, then S is weakly right cancellatiwe.

84

4

Bs

pmof: Let a , y E S and soppose that p,-'[(a)] is infinite. Chocwe a one-to-one sequence ( x . ) z t in p y - l [ [ a ) ]andforeachm E N l a t.= y. 0

i

Combmy 435. Let S be an infinite semigmnp. US is weakly right cancellatiw and for all butfinitely many y E S. Ay irftrite-to-one, then S is a right ideal ofpS. Pmof: Suppose S is not a right ideal of @Sand pick by Theorem 4.32(c) some a E S, a one-to-one sequence ( x n ) z lin S,and a sequence (z.):, in S such that x. zm = a for all n -z m in N. Now if (t. : m E MI is infinite, then for all n E W, A ~ ~ - ' [ [ O ) ] is infinite, a contradiction. Thus [zn : m E N) is finite so we may pick b such that [ m E W :zm = b ) is infinite. Then, given n E N one may pick m > n such that zm = b and conclude that X . .b = a. Consequently, pb-'[[a)] is infinite, a contradiction.

.

a

Somewhat surprisingly, the situation with respect to's* as two sided ideal is considerably simpler than the situation with respect to S* as a right ideal.

Theorem 4.36. k t S be an infkite semigroup. Then S* is an ideal of @Syand only if S is both weakly lefi cancel&tive and weakly right cancellative. P m f Necessity. W r e m 4.3 1 and Corollary 4.34. Sufficiency. Theorem 4.3 1 and Corollary 4.35. m e following simple fact is of considerableimponance, since it is frequently easier to work with S* than with BS.

Theorem 437. Lct S be an infinitesernigmup. IfS" is an ideal of @S.then the minimal left ideals, the minimal right ideals, and the smallest ideal of S* andqfpS a n the same. Pmf

For this it suffices to establish the assntions about minimal left ideals and minimal right ideals, since the smallest ideal is the union of all minimal left ideals (and of all minimal right ideals). Further. the proofs are completely algebraic. so it suffices to establish the result for minimal left ideals. F i t assume L is a minimal left ideal of pS. Since S* is an ideal of @Swe have L E 'S and hence L is a left ideal of S*.To see that L is a minimal left ideal of S*,let L' be a left ideal of S with L' E L. Then by Lemma 1.43(b). L' = L. Now assume L is a minimal left ideal of S*. Then by Lemma 1.43(c). L is a left ideal of pS. If L' is a left ideal of pS with L' E L, then L' is a left ideal of S* and consequently L' = L. 0 Exercise 43.1. Prove Corollary 4.29. Exercise 43.2. Rove Corollary 4.33. Exercise 4 3 3 . Give an example of a semigroup S which is not left cancellative such that 'S is a left ideal of pS.

-

4.4 K ( p S ) and Its Closure

85

B e & 43.4. Give an example of a semipup S which is not right cancellative such .. that S' is a right ideal of 6s.

~ d s 43.5. e h v e that (W. +)and (-We, +) arc both left ideals of (pZ, +). (Here -FIB = (-p : p E N*) and -pis the ulmfiltcron 2! gengeaeratcd by (-A :A E p).)

nmeN

Exerdse 43.6. Let T = ctgz nZ. (a) Rove that T is asubsemigroup of (PZ. +) which contains all of the idempotents. (Hinl: Use the canonical homomorphisms from Z onto Z.) (b) Rove that T is an ideal of

(PZ..).

4.4 K(BS) and Its Closure In this section we detmaine pmisely which ultrafilters arc in the smallest ideal K (BS) of /3S and which are in its closure. We b m w some terminology from topological dynamics. The terns syndetic and piecewise syndctic originated in the context of (N,+). In (N, +), a set A is syndetic if and only if it has bounded gaps and a set is piecewise syndetic if and only if there exist a fixed bound b and arbitrarily long intervals in which the gaps of A an bounded by b.

La S be a semigroup. (a) A set A C S is synderic if and only if there exists some G E lPf (S) such that S = U,,c i-' A. (b) A set A G S is piecewise syndetic if a d only if there is some G E lPf(S) such that {a-'(UIGG r-'A) :a E S} has the finite intnsection ptopcrty. Definition438.

Equivalently, a set A

S is piecewise syndetic if and only if then is some G E

PI (S) such that for every F E IP/(S) there is some x E S with F .x E UtrGI-'A.

We should really call the notions defined above righi syndeiic and righr piecewise synderic. If we were taking ,9S to be left topological we would have replaced "S = U,,, CIA" in the definition of syndetic by "S = UrsGAt-I". We also would have replaced "a-l(U,,c r-'A)" in the definition of piecewise syndetic hy "(U,,G ~t-')a-I". We shall see in Lemma 13.39 that the notions of left and right piecewise syndetic arc different. The equivalence of statements (a) and (c) in the following theorem follows fmin Theorem 2.10. However, it requires no exm effort to provide a complete proof here, so we do.

Theorem 439. Lei S be a semigmup and lei p

E

PS. The fobwing memenis are

equivaleni. (a) p E K W ) . (b) F o r a l l ~ € ~ . { x € S : x - ' ~ € ~ ) i s s y n d e t i c .

(c) Forallq ~ f l S , p ~ p S . q . p .

4.4 K ( p ) and Its Closure

87

pmf (a) implies (b). Pick by Theorem 4.40 some p E K@S) with A E p. Now K(@S)is the union of all minimal left ideals of flSby Theorcm 2.8. So pick a minimal left ideal L of @Swith p E L and pick an idempotent e E L. 'Ihen p = p . c so pick E S such that y-'A E e. Now by Theorem 4.39 B = (z E S : z-'(y-'A) E e} is syndctic. so pick finite G S such that S = U,,c t-I B. Let D = (X E S : X-'A is cenaal). We claim that s = U,c,.o '-ID. Indeed,letx E Sbegivenandpickt € G s u c h t h a t t . ~E B . Then (I.%)-'(y-IA) E eso(t.~)-'(y-~~)iscenhal. B ~ t ( t - x ) - ' ( y - ~ A )= (y . z .x)-'A. ThuSy.t.~EDso~E(~.t)-~Dasnq~i~~d. (b) implies (c). This is hivial. (c) implies (a). Pick x E S such that x-' A is central and pick an idempotent p~ K ( ~ S ) s ~ c h t h a K x - ~fAp. ?henA E x.pandx.p E K(~S)sobyTheaem4.40, A is piecewise syndetic.

Recall from Theorem 2.15 and Example 2.16 that the closure of any right ideal in a right topological semigroup is again a right ideal but the closure of a left ideal need not be a left ideal.

Theorem 4.44. k t S be a semigroup. Then cC K(,TS) is an ideal of ,TS. Pmof: This is an immediate consequence of Theorems 2.15.2.17, and 4.1.

0

Exelrise 4.4.1. Let S be a discrete semigroup and let A E S. Show that A is piecewise syndetic if and only if there is a finite subset F of S for which cCgs (UtsFr-I A) contains a left ideal of BS. Exembe 4.42. Let A E N.Show that A is piecewise syndetic in (N, +) if and only if there exists k E N such that. for every n E N,there is a set J of n consecutive positive integers with the property that A intersects every subset of J containing k consecutive integers.

Exerdae 4.4.3. Rove that (&F

(N+).

2% : F E

+(a)l is not piecewise syndetic in

Exerdse 4.4.4. Let A g N. Show that K(,TN. +) g if and only if, for every k E N. there exists nt E N such that every subset of N which contains nk consecutive integers must contain k consecutive integers belonging to A. Exercise 4A.S. Let C be a group and let H be a subgroup of G. Rove that H is syndetic if and only if H is piecewise syndetic. (Kit: By Theorcm4.40. pick p 6 K(@G)such that H E p and consider {x E G :x - ' H E p).) Exercise 4.4.6. h t G be a group and let H be a subgroup of G. Show that the index of H is finite if and only if ceac(H) n K(0G) # 0. (By the index of H, wemean the number of left cosets of the form aH).

4 PS

88

Exerdsc 4.4.7. Let G be a group. Suppose that G can be expnxsed as the union of a finitcnuinberofsubgroups. H1,Hz ,Ha. Showthat,forsanei 6 [1,2 ,....n ) , H, has finite index.

,...

4.4.8. Let S be a discrete semigroup and In T be a subsemigroup of S. Show l # 0. (Apply Corollary 4.18 and T h m m 1.65.) that T is c e n d if (rebsT)fK(pS)

Exrrdre 4.4.9. Show that if S is commutative, then the closure of any right ideal of FS is a two sided ideal of BS. (Hint: See Theorems 2.19 and 4.23.)

Notes We have chosen to extend the operation in such a way as to make (pS, .) a right t o p logical semigrwp. One can equally wen extend the opcration so as to make (BS, .) a left topological semigroup and in fact this choice is often made in the literature. (It used to be the customary choice of the first author of this book.) It would seem then that one could find two situations in the literature. But no, one instead finds four! The reader will recall that in Section 2.1 we remarked that what we refer to as Wght topological" is called by some authors "left topological" and vice versa. In the following table we include one citation from our Bibliography. where the particular combination of choices is made. Obviously, when referring to the literature one must be careful to determine what the author means.

I Called Rinht I

-.

D. Coutinuous

A, Continuous

Called Left

I

I451 . . 11781

1

I ~opologi& I Topological 1 1

[ 1921 . 12151

I 1

'

Example 4.6 is due to J. Baker and R. Butcher [6]. The existence of a mnpactification with the propeniesof pS given by Theorem4.8 is [40. Theorem 4.5.31, where it is called the % A C-compactification of S. The fact that an operation on a discrete semigroup S can be extended to pS was fvst implicitly established by M. Day in 1751using methods of R. Arens [3]. The first explicit statement seems to have been made by P. Civin and B. Yoad [MI. These mathematicians tended to view p S as a subspace of the dual of the real or complex valued functions on S. The extension to BS as a space of ultrafilters is done by R. Ellis 186. Chapter 81 in the case that S is a group. Corollary 4.22 is due to P. Milnes [1841. The results in Section 4.3 are special cases of ruults from (1301 which are in hum special cases of results due to D. Davenpon in [72]. The notion of "central" has its origins in topological dynamics. See Chapter 19 for the dynamical definition ind a proof of the equivalence of the notions.

Exercise 4.4.7 was suggested by I. Protarov. This result is due to B. Neumann [186], rho proved something stronger: if the union of the subgroups Hiis imdundant, then every Hihas finite index.

.

.

Chapter 5

j3S and Ramsey Theory -Some Easy Applications

We describe in (his chapter some easy applications of the algebraic smchuc of flS to the branch of combinatotics known as "Ramsey Theory".,

5.1 Rarnsey Theory We begin our discussion of the area of mathematics known as R m e y meory by illushating the subject area by example. We cite here several of the classic theorems of thc field. They will all be proved in this book. The oldest is the 1892 result of Hilben It involves the notion offinire sum. which will be of continuing interest to us in this book. so we p a w to introduce some notation. Recall that in a noncommutative semigroup we have defined xi to be the product in increasing order of indices. Similarly, we take x. to be the product in increasing order of indices. So, for example, if F = (2,5,6,9),thm ~ . E F X . = X z .XS . X 6 .Xg. We introduce separate notation ("FP" for ''finite pmducts" and 'W for '%finite sums") depending on whether the operation of the semigroup is denoted by . or

nnEf

n:=l

+.

Dellnition 5.1. (a) Let (S. -) be a semigroup. Given an infinite sequence in S. FP((x,,)z,) = { n , , ~ x . : F E Pl(N)). Given a finite sequence (x.);=~ in S, m((x.):='_,) = (~..Fx. : F E P ~ ( l l . 2 ..., . ml)). in S. (b) Let (S, +) be a semigroup. Given an infinite sequence (x,):=~ FS((x,)gl) = (ZnEFx,, : F E Pf(W)}. Given a finite sequence (x.)a, in S, FS((x.):=:=,) = (Z.,FX,, : F E Pf(i1.2, ... ,ml)).

Theorem 5.2 (Hilbea [1161). L c r r E N &let N = UI=, At. For each m E N there exist i E (1,2. . r ) , a sequence (x,):=~ in N. andan intnire set B N such thatfor each a E B, a +FS((X.);=~) E Ai.

...

Pmof: l h i s is a consequence of Corollary 5.10.

0

The next classical result is the 1916 result of Schur, which allows one to omit the franslates on the finite sums when m = 2.

1

Tkcmrn 5 3 (Schw [221]). Let r (1.2

E

N and let

N =

....,r ) m d x d y i n N w i t h ( x , y . x + y ) E A i .

u=,Ai.

Them exist i E

pmof: This is a consequence of Corollary 5.10.

0

One of the most famous results of the field is the 1927 result of van dcr Waerden '~onochrome"arithmetic progressions. statement "let r E N and Ai" is often replaced by "let r E N and let N be r-colored", in which let N = case the conclusion ". . . E Ai" is replaced by ". ..is monochrome".)

me

E N and let N = U:=l Ai. For each 1 ~ N t h e r e e x i s t iE ( 1 . 2 ...., r J a n d a , d ~ N s u c l r t h a t ( a , a + d a+fd)zAi.

Theorem 5.4 (van der Waerden [2381). Let r

,...,

pmof This is Corollary 14.2.

a

Since the first of the classical results in this ~ I Ware due to Hilb+lt. Schw. and van der Waerden. one may wonder why it is called "Ramsey Theory". The reason lies in the kind of result proved by Ramsey in 1930. It is a more general smctural result not dependent on the arithmetic suucture of N.

Definition 5.5. Let A be a set and let K be a cardinal number. (a) [A]' = ( B & A : IAI = w ) . (b) [A] a f (a). Stnce c E Y, pick b E W such that c = b f (b). Let F = ( a ,b). I f one had b c a, then one would have c = b f (b) c a f (a),so max F = b. Then m ( F ) = a E X and M ( F ) = b f (b) = c E X . so F E M - ( [ x ] n m - ' [ x ] . tick p E B(Pf(N))such that A 2 p. By Lemma 3.30 p E T. 0

+

+

+

+

+

We can now prove the promised theorem yielding adjacent sequences whh non adjacent unions in o m cell.

Theorem 5.27. Lcr f : N + N be a nondecnasing function and let PI(N) = U : = I A ; . Thenrhenexisrsomei E (1.2. rJandasequence ( H . ) z l inP,(N) such that

....

+

Pmof: Let M and m be as defined in Lemma 5.25 and let Y = {a f (a) :a E N). tick B\N and by Lemma5.26 and Theorem 2.5 pick an idempotent p in (B(P,(N)), w) such that M ( p ) = B ( p ) = x. We show that for each P1 E p, there is a sequence (H,,)z, in Pl(N) such that

xE

.

Then choosing i E { 1.2, . .. r) such that Ai E p completes the proof. So let U E p be given. For each G E IP/(N) and any V G P f ( N )let G-'V = ~ ~ E ~ ~ ( ~ ) : G U F E V ) ~ ~ ~ ~ ~ ~ P ~ThenbyLenma ' = [ G E U : G - ~ U 4.14. U - E p and for each G E U * .G-IU* 6 p. We claim now that, given my H E Pl(N), ( I € P f O : m ( J ) > M(H)J E p. Indeed, otherwise we have some r j M(H) such that [ J E P f ( N ) :m ( J ) = t ) E p so that %(p) = I , a contradiction. Now, by Lemma 3.31, ( H E Pl(N) : M ( H ) E m[U*I)E p so pick HI E U' such that M(H1) E m[U*].Inductively, let n E PI and assume that we have chosen in Pf(W) such that for each k E (1.2,. . .,n J .

(3) if k w 1, then m(Hd = M(Hk-I). Hypothesis (1) holds at n = 1 and hypotheses (2) and (3) an vacuous there. If n = 1, pick H e U' such that m(H) = M(H1). Otherwise, by hypothesis (2). pick H E U* n {.G-IU. : G E na~~((~j);::)]

n

such that m(H)

t

M(H.). Let V be the family of sets J E J'f(N) satisfying

(i) m(J) > M(H), (ii) H U J E U*. and (iii) G U H U J E U* for dl G E naFlJ((~~)~:~), (iv) M(J) e m [ U ' n n { G - ' ~ ' :

G E ~~F~J((H~);,~)H.

We have seen that ( J E Pf(W : m ( J ) > M(H)J E p and H-lU* 6 p because H E U'. If n > I and G E n a ( ( ~ , ) f then H E G-'U' so G U H E U' so (G U H)-a e(* E p. Thus the family of sets satisfying (i). (ii), and (iii) is in p. Finally, if G E naFU((Hj);,l). then G E 'U' by hypothesis (I). so G-'PI' E p. T ~ U SU* n (G-I U* : G E ~~Fu((H~);,,))E p so

:),

n

by Lemma 3.3 1. Thus V e p and so is nonempty. Pick J E V andlet H.+I = H U J. Then M(H.+I) = M(J) andalsom(H.+l) = m(H) = M(H,,) so hypothesis (3) holds. Further, J E V so

wr hypothesis (2) holds.

To complete the p m f . we show that n a F U ( ( ~ j ) ~ ~g: )U'.. So let 0 # F E 11.2 n+1)suchthatforeachr~F,t+1 * F . I f n + 1 e F , t h e n U l e F H, E U * by hypothesis (1) at n. so assume that n I E F. If F = {n+1). thenwehave He+, = H U J E U.. Thus.assumethat F # (n+lJ, let K = F\(n + 1). and let G = U,,x H,. Then UrpFH, = G U H U J E U*.

.....

+

Exercise 5.4.1. Let f : N -t W be a nondcrreasing f u ~ t i o nand let Z be any infinite subset of [a + f (a): a E WJ. Show that the conclusion of Theorem 5.27 can be strengthened to require that min H, E Z for each n E PI. (Hint: In Lemma 5.26. choose x E Z\N.)

5.5 Compactness ~ o soft the combmataria1results that we shall pmve in this book arc infinite in nature. We deal h m with a general method used to derive finite analognes from the infinite versions. This version is commonly referred to as "compactness". (One will read in the literature. "by a standard compacmess argument one sees.. .".) We illustrate two forms of this method of proof in this section. first deriving a finite version of the Finite products Theorem (Corollary 5.9). For either of the common fonns of compacmes arguments it is more convenient to work with the "coloring" method of stating results in Ramsey h r y .

Definition 528. Lu X be a x t and let r E M. (a) An r-coloring of X is a function ~p : X + (1.2,. ,r). (b) Given an r-coloring (p of X, a subset 6 of X is said to be mnochmme (with

..

respect to v) provided (p is constant on 6. Theorem 5.29. Let S be a countable semigroup and enumemte S as ( s n ) z l . For each randrnhRthereuistsn ~Msuchtharwhenever(s,: t E (1.2,.... n)]isr-colored. there is a sequence (x,):!, in S such that F P ( ( X ~ ) ~is, ~monochitme. )

Pmof: Let r, rn E W be given and suppose thc conclusion fails. For each n E N choose an r-coloring (p, of {st : t E {I, 2.. . .,n)] such that for no sequence (xl)y=, in S is F P ( ( X ~ ) ~monochrome. !~) Choose by the pigeon hole principle an infinite subset 81 of N and an element . given L E W a ( l ) E (1.2,. .. r ) such that for all k E 61, (pk(sl) = ~ ( 1 ) Inductively, with L z 1 and an infinite subset &-I of N choose an infinite subset Bc of Bc-1 and an element a(() E (I. 2. . . . ,r ) such that min Bc 2 L and for all k E Bt. h ( s r ) = a ( 0 . Define an r-coloring r of S by r ( s d = a((). Then S = U:=l r-'[[i)] so pick ,r ) atxl an infinite sequence {x,)zl in S such by Corollary 5.9 some i E (1.2, ! ~ {sl.sz.. ) ,sk) that FF'((x,)E,) g T-'[(ill. Pick k E N such that F P ( ( X ~ ) ~ C and pick n E 6 k . We claim that FP((xl)~=,)is monochrome with respect to co., a contradiction. To see this. let a E FS((x,);",,) be given and pick L E (1.2. . ..,k) such thata = st. Then n E Bk E Lit sovn(a) = a(L). Sincea E F P ( ( X ~ )C~r-'[(ill. ~) i = r(a) = T(Q) = a(L). Thus p,, is constantly equal to i on FP((x,)y!.=,) aschimedo

.

...

..

The reader may wonder why the t m "compacmess" is applied to the pmof of Theorem 5.29. One answer is that such results can be proved using the compacmess theorem of logic. Another interpretation is provided by the pmof of the following theorem which utilizes topological compacmns.

Theorem 5.30. For each r and m in M there exists n E W such that whenever and (y~)y=~ in M such that (1.2, . .. n) is r-colored, there exist sequences

.

FS((xr)y!,) and FP((yr)y=l) a n contained in 11.2.. FP((yt)y=l)is monochrome. -

...nl

and F S ( ( X ~ ) ~U= ~ )

P w f : Let r. m E N be given end suppose the conclusion fails. For e a ~ hn E N choose an r-coloring (p, of (I, 2.. ,nl such that there are no sequences (x,)Y_, a d (yl)y=, in N for which FS({XI)Y=~)end FP((Y~):=~)are contained in (1,2.. .. n ) and FS((xt)Ll) U FP((Y&) is monochrome. m Let Y = X n = ,(1.2,. .. r), where (1.2.. .. r ) is viewed as a toplogical s m with the discrete topology. For each n E W M n e p,, E Y by

..

.

.

.

Exelriw 55.1. Rove. using compactness and Theorem 5.6. the following version of Ramsey's Theorem. Let k, r, m E W. There exists n E W such that whenever Y is a set with IYI = n and [ylk = Ui=,Ai, there exist i E (1.2,. . . r ) and B E [Y]" with [B]'

.

Ai.

Notes The basic reference for R m c y Theory is the book by that title [I I I]. CoroUary 5.10 was originally proved in [I 181. with a purely elementary (but very complicated) combinatorial proof. A simplified combinatorial pmof was given by J. Baumgartner (151. and a proof using tools from Topological Dynamics has been given by H. Furstenberg and B. Weiss [IOO]. The first proof of the Finite Sums Theorem given herc is due to E Galvia and S. Glazer. This pmof was never published by the originators, although it has appeared in several surveys. the first of which was [661. The idea for the consrmction occurred to Oalvin around 1970. At that time the Finite Sums Theorem was a conjecrure of R. Graham and B. Rothschild [I 101, as yet unproved. Calvin asked whether an "almost translation invariant ultrafilter" existed. That is. is there an ultrafilter p on W such that whenever A E p. one has { x E S : A - x E p ] E p? (In terms of the measure p corresponding to p which was introduced in Exercise 3.1.2. this is asking that any set of measure 1 should almost always translate to a set of measure 1.) Calvin had invented the

1 ,

&mction used in the tint proof of nKorcm 5.8. and kmw that an affiaffirmative. answer to his question would provide a proof of the conjecture of Graham and Rothschild. oneof the cumnt authors tried to answer this question and succeeded only in showing that, under the assumption of the continuum hypothesis, the validity of the conjecture of Graham and Rothxhild implied an affirmative answer to Galvin's question. With the subsequent elementary proof of che Finite Sums T h m m [118]. Galvin's almost mslation invariant ulbafiltm became a figment of the continuum hypothesis. Galvin was interested in establishing their existence in ZFC, and one day in 1975 he asked Glazer whether such ulbafilters existed. When Glazer quickly answered "yes". Galvin med to explain that he must be missing something because it couldn't be that easy. In fact it was that easy, because Glazer (1) knew that bh' could be made into a right topological semigroup with an operation extending ordinary addition and (2) Iww the charafteriZati0n of that operation in terms of ulnafiltm. (Most of the mathematicians did not think of ON as a space of who were a w m of the algebraic structure of ultralilters.) In terms of that characterization, it was immediate that Galvin's almost mslation invariant ultrafilters were simply idempotents. Theorem 5.12 is a result of F. Galvin (also not published by hi). An elementary proof of Comllaty 5.22 can be found in 1301. a m u l t of collaboration with V. Bergelson. Theorem 5.27 is due to A. Blass in [44] whete it was proved using Martin's Axiom, followed by an absoluteness argument showing that it is a theorem of ZFC.

Part 11 Algebra of jlS

Chapter 6

Ideals and Commutativity in BS

A very suikhg fact about BN is how far it is from being commutative. Although (BN, +) is a natural extension of the semigmup (N, +), which is the most familiar of all scmigroups, its algebraic smcture is amazingly complicated. For example, as we shall show in CoroUq 7.36, it contains many copies of the h e group on 2 m. The. I' @(m+n) = @(n)or@(m+n) = @(n)+l. Foreachm. (n E N : @(m+n) = @(n)) E p or [n E N :@(m + n ) = @(n) 1) E p. In the first case.

ScP

w.

+

+

In the s e c ~ n dcase. &I-+ p) = k p ) lrNow (m E _N :$(m-+ p) = & p ~ }E q or lm 6 N : @(m+ P) = @(P)+ 1) E 9.20 @(q+ P) = @(P)or@(q p) = @(@+ 1. Since there arc Zc different values of @(q),we can chm? q E A* satisfying @(q) $

+

I ~ P ) , ? ( P ) + ~ IThen9 . ~~so@(~+q)=6(q)#@(q+p)sop+q#q+~. and thus p cannot be in the center of (BN. +). Now forany m, n E W,@(m2") = @(m)+ n = @(m)+@(Zn). Now given p E W and q E A*, one has

And similarly??(q .p) = &q) +?(PI. NowJet p E W.We show that p is not in-the center of (BE,.). Since @ i sfiniteto-one. @(p) E M* so pick r E W* such that @(p) r # r @(p). Since maps A bi@tively_to BN. pick q E A* such that &q) = r . Then &p . q) = $(p) r# r+@(p)=@(q.p)sop.q#q.p.

-

+

+

+

Remark 6.11. ltfollowsfmm Theorem 4.24 that N is the topologicalcenter of (BN. and of (BN. .).

+)

We shall in fact see in Theorems 6.79 and 6.80 that there is a subset A of N* such that A and N'\A arc both left ideals of (BW. +) and of (BN, .).

Theorem 6.12. M contains M infinite d r c m i n g sequence ofidempotents. Pmof: Let (A"):, be an infiniteincreasing sequenceof subsetsof N such that An+, \A. is infinite for every n. Let S. = [m E W : supp(m) c A,, 1. We observe that. for evcry n. r E W, we havek+m E 2'NnS, wheneverk, m E 2' WnS, andsupp(k)nsupp(m) = 0. Thusit follows frornTheorcm4.20(with A = (S,, n2'N :r E W)) that T. = Z n W is a subscmigroup of BN. We shall inductively consmct a sequence (em)=, of distinct idempotents in N satisfying em+!5 e, and en E K(T.) for every n.

6.1 The Semigmup H

111

We fmI choose et to be any minimal idempotent in TI. We then assume that ei has been chosen for each i E (1.2.. . m). ~~Thtonm1.60,wecanchooseanidcmpotcntc~+1 E K(Tm+~)forwhichem+l 5 We shall show thatem+t # emby showing that em b K (Tm+l). To see this, we choose anyx E M* n (2" :n E Am+1\Am).Let

..

Since M n Sm = 0,em $ ;i?. However, it follows from Exercise 4.1.6, that y + x + z ~ ~ f o r ~ y , Sz oE~ c~~ n . tainstheidcalT~+~+x+~~+~ofT,+~ and therefore contains K(Tm+~).Hence em $ K(Tm+t). We have saen that it is remarkably easy to produce homomorphisms on 8. This is related to the fact Ihat H can be defined in t e r n of the concept of oids, and the algebraic shucture of anoid is very minimal. Theorem6.15 shows that the only algebraic information needed to define H is the knowledge of how to multiply by 1.

Definition 6.13. (a) A set A is called an id if A has a distinguished ekment 1 and a multiplicationmapping((I) x A)U(A x (I)) toA withthe property that l a = a1 = a for every a E A. (b) If for each i in some index set I. Ai is an id, then A; = (x E X i r l Ai : [i E I :xi is an oid. (c) If S =

# 1) is finite )

@,,Ai is an oid and x E S. then supp(x) = (i E I :xi # 1).

Notice that we have already defined the notation "supp(x)" when x E N. The comspondence betwoen the two versions will become apparent in the proof of Theorem 6.15 below.

Detlnltlon 6.14. Let S = Ai be an oid. If x, y E S and supp(x) nsupp(y) = 0. thenx.yisdefinedby(x.y)i = x i . y i f o r a l l i E I. Notice that in anoid. onedoes not rcquircx .y to be defined if supp(x)nsupp(y) # 0.

Theorem 6.15. Let A = [a, 1) be an id with fwo ekments and for each i E o,let A i = A. Let S = &, Ai and let H = n,,,ctps(x E S : min(supp(x)) 2 n). m e n d the opemrion . to all of S x S arbitrarily and then extend the operation . to pS aS in Theorem 4.1. Then H is topologicaIly and algebrnicaIly isomorphic to R

P m $ Define (p : N + S by

andnoticethat(pisone-to-one. p[Nl= S\{i).andforallx e. N, supp(p(x)) = snpp(x). = H. In particular, for all n E M. (p[ZnP1]= (x E S : min(supp(x)) 2 n) so that

(pm

112

6 IdealsdCanmutativityin~S

Since (p is one-to-one, so is 5 by Exercise 3.4.1. I h u s Gtnis a homeomorphism onto

H. To see that

1

Fm is a homomorphism, let p, q E 8. Then by Remark 4.2.

+

Now, givea any x E W, if n = max(supp(x)) 1. then for all y e 2"N. (p(x ~ ( x )(~(y).so that since q e M and A,(,) and Fare continuous.

.

+ y) =

Notice that in the statement of T h w m 6.15, we did not require the arbitrary extension of the operation to be associative on S. However, the thwrem says that it nonetheless induces an associative operation on H.

Exercip 6.1.1. Show h i H has 2' pairwise disjoint closed right ideals. (Hint: Consider (6Yp) : p E A') where A = (zn:n E N).) Exercise 6.12. Show that no two closed right ideals of BN can be disjoint (Hint: Use Theorem 2.19.)

Exercise 6.13. Show that no two closed right ideals of W* can be disjoint (Hint: Consider Theorem 4.37 and Exercise 6.1 2.) Exercise 6.1.4. Show that there are two minimal idempotents in (BN, +) whose sum

is not in r. the closure of the set of idempotents. (Hint: Let S denote the semigroup described in Example 2.13. By Corollary 6.5 t h m is a continuous homomorphism h mapping H onto S. Show that there are minimal idempotents u and v in W for which h(u) = e and h ( v ) = f . Observe that idempotents minimal in H are also minimal in BN.)

6.2 Intersecting Left Ideals In this section we consider left ideals of BS of the form BS . p. Definition 6.16. Let S be a semigroup and let s the semiprincipal left idea! of S generated by s.

6

1

S. The left ideal Ss of S is called

Note that the semiprincipal left ideal generated by s is equal to the principal left ideal gemrated by s if and only if s E Ss.

113

6.2 1nfersu:tingLeH Ideals

m i t i o n 6.17. Let S be a semigroup and let p E BS. Then C ( p ) = ( A x ~ ~ . ~ - l A ~ p J .

~beorrm6.18. Ler S be a semigroup and let p C ( P )c ql.

E BS. Then BS

.p

S :for all

= ( q E BS :

p m f Let r E BS. Then given A E C ( p ) one has S = ( x E S : x - ' A E p) so AE~.P. . For the other inclusion, let q E BS such that C ( p ) c q. For A E q. let B(A) = [x E S : X - ' A E p). Weclaim that [ B ( A ): A E q Jbasthe finiteintersationproperty. To see this observe that ( B ( A ) : A E q J is closed under finite intersstions and that i f B(A) = 0, then S\A E C ( p ) E q. Pick r E BS such that [ B ( A ): A E q } E r. Then 0 q=r.p. Theorem 6.19. Suppose that S is a countable discrete semigmup and that p. q E BS. I f ~ S ~ p n B S . q # 0 , t h m s p = x q f o r s o m eEs Sandsomex E p S , o r y p = t q f o r some I E S a n d s o m y E BS.

Pnmf Since BS .p = and 6.7 . q = 5 we can apply Corollary 3.42 and deduce that sp E for some s E S. or else tq E 5 for some t E S. In the first case, sp = xq for some x E pS. In the second cax,rq = yp for some y E BS. 0

5

Corollary 620. Let S be a countable gmup. Suppose t h a ~ p and q a n ekments of BS forwhichpS.pngS.q#0. T h e n p ~ p S . q o r q ~ p S . p .

Pmof By Theorem 6.19, s p = xq for some 5 E S and some x E BS. or tq = yp for some t E Sand some y E BS. 1n the first case, p = s-lxq. In the second = t-Iyp. 0

Corollary 6.21. Let p and q be distinct ekmnrs of BW. Ifthe I& ideals BN BW + q are not disjoint, p E BN + q o r q E pW p.

+

+ p and

Proof Suppose that ( B N + p) fl (BW + q ) # 0. It then follows by Theoran 6.19, that m + p = x + q f o r s o m e m ~ N a n d s o m e x~ B W , o r e l s e n + q = y + p f a s o m e n E N and some y E BN. Assume without loss of gemrality that m p = x + q for some m E W and some x E pN. I f x E No,then -m x E N*. because W* is a left ideal o f Z*. as shown in Exncise4.3.5. So p = -m + x q E BW + q. Suppose then that x E W. Then x # m , for otherwise we should have p = -m + x + q = q. I f

+ +

+

x>m.-m+x€Wandagainp=-m+x+qEB~+q.Ifx<m,-x+mEW

andsoq=-x+m+p~BN+p.

0

By Corollaries 6.20 and 6.21, we have the following.

Remark 6.22. V S is a counrable gmup or if S = W, any nvo semiprincipal lefr idea& of pS are either disjoint or comparablefor the relationship of inclusion.

6 Ideals end Commutativity in ,9S

114

Recall frpm Chapter 3 Ihat if T C S we have identified BT with the subset ce T of ,9s.

Corollnry 6.23. Let S be a countable smigmup which can be embedded in agmup G, andlere, f E E(S). I f B S . e f l B S . f # O , e f = e o r f s = f . PmoJ ByTheonrm6.19, wemay assumethatse = x f fasomes E Sandsomex E BS. This implies that e = s-'xf in BG. and hen- that ef = I-'xff = s-'xf = e. We can now show that, for many semigroups S. thc sNdy of commutativity f a idcmpotents in BS is equivalent to the study of the order relation 5.

CoroUary 6.24. Suppose that S is a c o d & dirfnte semigmup which can be embedded in a gmup, and that e , f E E(BS). Then e and f commute ifandonly if e 5 f orf je. P ~ JBy' definition of the relation 5 , if say e 5 f . then e = ef = fe. Conversely, suppose that e f = f e. Then BS .e nB S . f # 0. and so.by Corollary 6.23, we may suppose that ef = e. Since e f = f e , this implies that e 5 f .

Corollary 625. Suppore that S isa countablediscrete semigmup which can be embedded in agmup, andrhar C isasubsem'gmupof 6 s . v e . f E E ( C )satisfy CcflCf = 0, r h m p S . e n g S . f =O. PmoJ I f BS . e f l B S . f # 0 , we may suppose by Corollary 6.23. that ef = e. But then e E C e f l C f , a comradiction. 0 Exe6.2.1. and C ( P )=

Let S be a semigroup and let p

nvs.P).

BS. Prove that C ( p ) is a filter on S

Exercise 6.22. Let S be an infinite semigroup and let p E S*. Let C,(p) = (A c S : (x E S :%-'A O. p] is finite). h v e that C d p ) is a filter on S, C,(p) = n ( S * .p). andS.p=[q~BS:C,(p)Eq).

6 3 Numbers of Idempoteats and Ideals Copies of W

-

The semigroup EU arises in B M . We shall see. however, that copies of W can be found in S' for any infinite cancellative semigroup S. We introduced the notatiaFP((x.)~=,)in Definition 5.1. We extend it now to apply to sequences indexed by linearly ordered sets other than W.

Definition 626. Let S be a.s&group, let ( D . 4 ) be a linearly ordered set. and let ( x , ) , , ~be a D-sequence (i.e., a set indexed by D ) in S.

j

6.3 Numbers of Idempolenta and Ideals

115

n,,~

(a) F a F E PjPI(D), q is the product in increasing order of indices. (b) FP((~s)s,D)= I ~ ~ xsE :FF E Pj(D)). (c) The D-sequmce ( x ~ ) , ~has D dktinctfinife pmducts if and only if whenever F. G E P j ( D ) and ~ , , F x , = lT,,~x,. one has F = G. xo

If additive notation is being used we define finitesums" analogously.

&F

xs, FS((X,),,~); and "distinct

T h m m 6.27. Let S be a discrete semigmup and let ( x , ) z , be a sequence in S with distinctfiniteproducts. Let T = ct(FP((x.)~,,,)). Then T is a subsemigmup

nzc1

of 6.9 which is algebraically and topologically isomorphic to EE PmoJ It was shown in Lemma 5.11 that T is a subsemigroup of PS. We define a mapping f from FP((x.)Z1) to 2M by stating that f (n,,,,=x,,) = C n E 2"~for every F_ E Pf((W).S i f is a bijection (because of the distinct finite products assum@on), f : ct(FF'((x.)K,)) + B(2H) is a homeomorphism. We shall show that f i is ~a homomorphism from T onto W. Let T, = FF'((x.)&,,). Then f [T,] = 2"M and so

&

To see that is a homomorphism. it suffices by Thunem 4.21 to show that for each s E FP((x,)Kl) there is some m E N such that f (st) = f (s) f (r) for all r E FP((x,,),&,). Solets G F P ( ( ~ . ) ~ ~ ) a n d p i cEk FPj(N)suchthats = ll.,px.. n Letm = m a x F + l .

+

We now Nm our attention to the effect of certain cancellation assumptions. Much more will be said on this subject in Chapter 8.

Lemma 6.28. Let S bc a discrete semigmup and l e t s be a cmcehble element of S. ForeverytESandp~flS (i) ifS is right cmcel[ofive and sp = rp, then s = r. and (ii) ifS is left cancellative and p s = pt, then s = I. Pmof: (i) Suppose that S is right cancellative. Let r E S. let p E 6s. and suppose that s # t. We shall define a function f : S -+ S by stating that f (su) = ru forevery u E S and that f ( v ) = s2 for every v E S\sS. We note that f is well-defined because every element in s S has a u$que expression of the form su. and e a t f has no fixed points. We also note that since f o A, and A, agree on S. o m has that f (sp) = tp. Now by Lemma 3.33 we can partition S into three sets Ao, At. A 2 such that Ai n f [ A ; ] =-0 for each i E (0.1.2). Choose i such that sp G A,. Then by Lemma 3.30. f [Ail E f (sp) = tp. so sp # tp.

(ii) This is pmved in essentially the same way.

= 'I

Recall that a s e m i p u p S is weakly left cannllative if and only if f a all u , v E S. ( x E S : ux = V ] is finite. and that S is weakly right cancellative if and only if f a all u, v E S, (x E S : xu = v ) is Iinite.

r:

Lerrrm~6.29. Let S be an infinite weakly lep canceNarive semigroup and let I denote the set of righr idenrities ofS. Lct K be an infinite cardinal with K 5 IS],and let ( s ~ ) ~ , be a one-to-one x-sequence in S. If T is a subset of S with cardinality K , then them aim a one-to-one K-sequence( ~ A ) A , , in T such that

P&J For each u , v E S. k t A"., = (s E S : u = us]. 'since S is weakly Left cancellative, A",, is finite. We note that this implies that I is finite. because I E A,,. for every u E S. We consrmct ( t ~ ) ~inductively. _ n 1 + m . 0

+ a. xi=,

,

+ +

+

+

6 Ideals acd n d t u l i v i t y in BS

132

Lrmat.670. Ifii,6 E @z,Zx z l a i z i = and lbil 5 2'. then a' = 6.

xEl biu.andforeachi ..

E N , lail

sr

~ m o f Suppose : i # 6. Sinceii and6 each have only finitely many nonzero coordinats, we may pick the largest n such that a,, # b. and assume without loss of generality that a, r b. Then Cy=la;zi = X:=l biz; so z. 5 (4-

- bnkn = C:d(bi - a;)zi 5

"-1

i+l 2 zi < zn,

a conhadiction.

Definition 6.71. For each n E

N.B,, = n= :,

Lemma 6.72. For each n E N. B,,

c.! A .:

is a mnempty Ga subset of

N* such that

P m f By Lmuna 6.68. we have that B. # 0. Since for each m. min AT = 2," m have B. s N*. To verify (a), let p, q E B.+I. To see that p by Lemma 6.69. for each x E A:+,. A+:, s -x

.&+,

+ q € B,, let m E N bc given. Then + A:+' so

andconsequently A:+' E p+q. since^:" C A:,wehaveAT E p + q asrequid. To verify @), let p E B,, and suppose that we have some r E B.+t n cL(p BZ). T h c n ~ : + ~E r s o A 6 , 1 n ( p + & Z ) $ 0. Pickq E f l Z s u c h t h a t ~ : + ~E p + q a n d l e t ~ = ( x E Z : - X + A ; + , €4). ThcnC~psopickxeCnAf,andpick ii E 8; such that x = E ~ l a ; z i . Let m = msx(i E N : ai 0). Since A: E p, . x and y are in C, picky E A: n C and pick b E 8; such that y = CE, b ; ~Since pick w E (-x A:+,) f(-y l A:+,). Then x w E A,: and y w E A:+, such that x w = CE, cizi and y w = CEO=, diz;. Then so pick Z and d in w = Z z I(c; 4;)~;= C z l (d; b;)z;. By Lemma 6.68. for each i , Ic; -ail < 2' and Id; bil < 2' so by Lemma 6.70 c' - a' = 2 - 6. Now b E 13: sob; = 0 for 1 bi i < n m and thus 2' = 2"' Since a; = 0 for i r m we have that for i z n + m . d ; =bi+c;. Butthen

+

+

+

+

-

el+,

+

+ -

+

+

+

-

xE,,+, - -

a contradiction.

0

The following lemma is quite general. We state it for semipups written additively because we intend to use it with (875,

+).

i

6.7 Ideals and Deasity

133

6.73. Lct ( S , +) beacomprctright topologicalsmigmupmrdforeclehn E N, D, be a nonempry closed subset of S such that for each n, Dm+I+ Dncl s D,. ~ i w mn y sequence (q.)gl in S with each q. E D. there is a sequence (p.);=l with p E Dn. such that, for each n, p.+l+ q,+l = p,. -a

pmofi pick some r € No. For n and m in N with n 2 m. kt t., = qm. By downward inductiononn,for n c m. let t., = t.+~,, +qn+l and notice that each I., E D. For m h n E N.let p. = r- lim I., and notice that, since D. is closed. p. E D. Since meS

form > n. tn.m = fn+l,, +qn+l one has that

Theorem 6.74. mere i s o sequence ( p . ) z l in N. such thot ( p , +BZ)=, isa strictly increasing chain of principal right ideals of f3Z and (cC(p. B Z ) ) g I is a strictly inceasing chain ofprincipal closed ideals of f3Z.

+

Pmof: Pick some q, E B, for each n E N. By Lemma 6.72, we have the sequences (B&, and (q.)gl satisfy the hypotheses of Lemma 6.73 with S = f3Z so pick a sequence ( p . ) z l with each p, E B. and each p. = p.+~ q.+~. Then one has immediately that for each n , p. f3Z G p+ .1 BZ and SO. of course. ce(p. + G cC(p,+l PZ). Further. by Lemma 6.72(b). for each n,

+

+

+

+

pa

so both chains are strictly increasing.

6.7 Ideals and Density The conceptof density for a set of positive integen has intcnsting algebraic implications in BN. We shall show that the set of ultrafilters in BN all of whose members have positive upper density is a left ideal of (BW.+) as well as of (BN, .),and that the same statement holds for the complement in N ' of this set.

L m a 6.75. Let S be an arbitrary semigmup and let R be o partition regular set of subsets of S. Let A n denote the set of ultrofilters in BS all of whose members are supersets of sets in R. If R has the pmpeny that sA E R for every A E R and every s E S, then A a is a closed left ideal of BS. PmoJ It follows from Theorem 3.1 1that An is non-empty and it is immediate that Ax

is closed. To see that An is a left ideal, let p E An. It suffices to show that Sp g An. for then (f3S)p = c l ~ s ( S pE) As. So lets E Sand let B E sp. Then s - ' B E p so pick A € S s u c h t h a t ~E S - ' B . T h e n s ~ € RandsA g B.

Definition 6.76. Lct A

Z(m

N. We d e k e the udpcr density Z(A) of A by

+ A)

3 a(A)

and &A)

= LZ(A),

m

Dcenition 6.78. We define A E @Nby

~=(~€pN:a(A)>OforallA~~]. Theorem 6.79. A is a closed 1 4 ideal.of (pN. +) and of (pN. .).

Prwf: We apply Lemma 6.75 with R = (A N : Z(A) > 0). It is clear that R is partition regular. By Remark 6.77we have m A E R and mA E R for every A E R and every m E N. Thus our conclusion follows.

+

Theorem 6.80. No\ A is a lep i&al of (PN, +) and of (BN. .).

Prwfl Let p

E

N*\A and let q E PN. Ihm is a set B E p such that a(B) = 0.

N we put f (n) = 'Bn(1'2""'n)l. ~ o f ( n )+ 0 a s n + m. or n each m E N. we choose n, E N so that f (n) < whenever n > n,. We put B, = (n E B :n z n,) and observe that B, E p. It follows from Thmrcm 4.15 that B,) E q p and P = mB, E q p We shall show that these S = U,,,(m sets both have zero upperdensity. It will follow that p + q and pq are both in N*\A. Suppose then that m € N and that n E B, and that m n < r for some r E N. This So, for a given value implies that n, < r because n > n., and hence that f (r) < of r, i f f (r) # 0, the number of possible choices of m is at most Jr7fi;T.Since we also haven E B n 11.2.. ..,r), the number of possible choices of n is at most f (r)r. Thus the number of possible choices of m n is at most r m . It follows that For each n

E

3

+

+

+

+

Isn11,2.....r)I r The satne argument shows that

5.

im.

l ~ n ( i . 2 ... . r)I 5 r

m.

This establishes the claim that a(S) = a ( P ) = 0 and it follows that PNa\A is a leh ideal of (PW. +) and of (BN. .I. 0

+

Notice that it did not suffice to show in the pmof of Theorem 6.80 that n p and np are in N'\A for every p E N8\A because N*\A is not c l o d Notice also that Theorems 6.79and 6.80pmvide an alternative proof that the centers of (PW. +) and of (PN. .) are bath equal to W.

.

.

Notes

m e fact in Theorem 6.9 that BN has 2' minimal right ideals is due to J. Baker and p. Milnes [lo]. while the fact that BPl has 2' minimal left ideals is due to C. Chou [61]. The concept of an oid (" '@ id' being unpronounceable") is due to 1.(2041, who showed that the semigroup smcture H depended only on the oid struchxe of FS((Zn)Z,). Theorem 6.27 is due to A. Lisan 11781. Theorem 6.32 is due to T. Budak (net Papwan) Exercise 6.1.4 is fmm [38], a result of collaboration with 1.Berglund Theorem 6.36, one of three nonelementary results used in this book that we do not prove, is due to M. Rudin and S. Shelah. Theorem 6.36 and Lemma 6.37 deal (without ever explicitly defining them) with the Rudin-Keisler and Rudin Frolik orderings of ulaahlters. See Chapter 11 for more information about h e orderings as well as a dixussion of their origins. For a simpler pmof that there exist two points in Pi* that are not Rudin-isler comparable, see [182. Section 3.41. The pmof of Theorem 6.38 is from [MI. Theorem 6.46 is due to D. Parsons in [193]. Theorem 6.53 is, except for its weaker hypotheses, a special case of the following theorem which is due to E. van Douwen (in a letter to the first author). A p o f , obtained in collaboration with D. Davenport, can be found in [74].

Theorem Let S be an infinite cancellarive semigmup with Cardi~lilyr . Then?exists a decomposition 1 of Ux(S)with the following propenies: (1) (2) (3) (4)

111 = z2=. Each I E 1 is a Iej? ideal of BS. For each I f 1 andeach p E I , cC(pS) E I. Each I E 1 is nowhem dense in U. ( S ) .

Theorem 6.63 is from [1801, a result of collaborationwith A. Maleki. The results of Section 6.6 are from [149], a result of collaboration with J. van Mill and P.Simon. Sometime in the 1970's or 1980's M. Rudin was asked by some, now anonymous, analysts whether every point of Z* is a member of a maximal orbit closure under the continuous extension B of the shift function a , where a(n) = n I . This question was not initially recognized as a question about the algebra of PZ. However, if p E BZ, then Z(p) = 1 + p and so, for all n E Z, Zn(p) = n p. Thus the orbit closure of p is PZ + p and so the question can be rephrased as asking whether every point of ,925 lies in some maximal (proper) principal left ideal of PZ. One could answer this question in the aflinnativeby determining that there is no strictly increasing sequence of principal left ideals of PZ. The results of Section 6.7 are due to E. van Douwen in POI.

+

+

If S is a discrete semigroup, it is often quite easy to find large p u p s contained in BS. For example. the maximal groups in the smallest ideal of ,EN contain 2' elements. More generally, as we shall show in this chapter, if S is infinite an: cancellative with cardinality n , BS contains algebraic copies of the hw group on 2* generatas. This provides aremarkable illustration of how far BS is fmm being commutative. However. it can be tantalizingly difficult to find nontrivial small groups in BS. For many years, one of the difficult open questions about the algebra of ,EN was whether or not ,EN contained iny nontrivial finite groups. This question has now been answered by E. Zelenuk. and we give the proof of his theorem in Section 1 below. Whether or not BN contains any elements of finite order which are not idempotent still remains a challenging open question. Of course, there are many copies of Z in W, since Z p pmvides a copy of Z if p denotes any idempolent in N*. It is consistent with ZFC that there are maximal p u p s in N' isomorphic to %, since M&:s Axiom can be used to show that there are idempotents p in W* for which H ( p ) ,the largest p u p with p as identity, is just % p. (See Theorem 12.42). It is not known whether the existence of such an idempotent can be proved within ZFC. Because we shall be coRFtnrcn'ng several topologicalspaces in this chapreq some of which a n not necessarily ha us do^ we depart for this chapter onlyfmm our standing assumption that all hypothesized spaces are Hausdofl

+

+

7.1 Zelenuk's Theorem The pmof of Zelenuk's Theorem uses the notion of a lcfr invariant topoIogy on a group. Definition 7.1. Let G be a group. A topology 7 on G is lefl invariant if and only if for every U E 7 and evety a E G. aU E 7. Notice that a topology on G is left invariant if and only if for every a E G, A. is a homeomorphism. Notice also that to say that (G..) is a group with a left invariant

7 is the saw as saying chat (G. -,7 )is a left topological group, i.e., a p u p .. is a left topological &group. The next result Lemma7.4. is unfortunately rather lengthy and involves a good deal of notation. We shall see after the proof of this lemma how a topology on G and a set satisfying the hypotheses ofthis lemma arise naturally fmm the assumption that pG a nonhivial finite subgroup w l l e G does not.

x

Definition7.2 (a) F will denote the free semigroup on the letters 0 and 1 with identity 0.

@) I f m E o and i E (0.1.2, ...,ml, sp will denote the element of F consisting of i 0's followed by m i 1's. We also write u, e s: (so that uo = 0 ). (c) If s E F. I(s) will denote the length of s and supp s = [i E 11.2. ... I(s)) : si = 1) where si is the i" letter of s. ( d ) I f s . t E F , w e s h a I I w r i t e s ~ ~ t i f m a x s u p p s + <minsupp I t. (e) If s , t E F, we define s t to be the element of F f a which 1(s t ) = max[l(s),l ( t ) )and (s t)i = 1 if and only if si = 1 or ti = 1. .

-

+

.

+

+

ST

+ .+ST

Given any t E F. t has aunique representation in the form t = + s r l .. whereO5io<mo n if and only if t E u.F\Iu, : m E W]. Also, for each n E N,x[u. Fl = X(u.) by condition (12) of Lemma 7.4 so {x(t) : t E F and minsupp t > n] = X(u.)\[e) E yl. For each n E N.k t

:u:

n

foteacbie(1:2 ..... n). si E F, c(si) Ir p 1 (mod p2), and if i -z n, si cc si+l].

A. =[x(sr+sz+...+s.):

+

We show by induction on n that A. E yl". Since AI = B, f o r n = 1. So let n E Nand assume that A. E yln. We claim that A.

ly

~the , assenion is hue

E (a E G : ~ - I A ~ + EI YI]

sothatA.+~~y~"y~=~~"+'.SoIeta~A,andpicks~~cs~c~...ccs~inF such that each ~ ( s i = ) rp 1 (mod p2) and a = x ( s l + 52 ... 8"). Let

+

+ +

D = ( x ( t ) : r E Fandminsupp t >maxsupps.+l). T h e n B , n D e y l s o i t s u f f i c e s t o s h o w t b a t ~ , n D ~ n - ' ~ . +T~o.t h i s e n d l e t r e F such that min supp r > max supp h 1 and c(r) o r p 1 (mad pZ). By condition (LO) of Lemma 7.4.

+

X(S!

+

+ Q + . . . + h ) ~ ( f ) =X(SI + S 2 + . . - + S m

-kt)

+ + . .+

s o x ( s ~ s2 . s.)x(t) E Am+' asrequired. Now ylP+' = YI so Ap+t f y1 so pick some o E Ap+1 0 B,. Then a = x(sl

+ sz +. ..+ sp+l)

whmeachc(si) ~ r p 1+ (mod p2) andsl cc q cc .. . c c s p + lThen c(st+sz+...+sP+l)

= (rp+I).(p+l)

(modp2) ( r + l ) p + I (modp2)

and hence a E B,+1 so 8, f l B,+! # 0, a conwadiction.

0

146

7 GmupsinBS

CoroIlnry 7.18. W conln,'~no nontrivialfinite subgmups.

P m J Since Z*confains no nontrivial finite subgroup. neither d a s N*.

0

R e d l that apariial rlnptication a a a X is a h d o n mapping some subm 2 of X x X to X. Given a partial multiplicahon on X and poinls o and b of X, we say that a b is e d if and only if (a, b) E Z. We shall sometimes express the same fact by saying "ob E X". (The l a m terminology is converuent when the partial multiplication on X is induced by a multiplication on a larger structure.)

DeRnitim 7.19. Let X be a topological space with a distinguished element e and a panial multipticaiion. We shall say that X is alocalkflgmupif there is a left topological group G in which X can be topologically embedded so that the following conditions hold: (i) e is the identity of G. (ii) the partial multiplicationdefimd on X is that induced by the multiplication of G. (iii) for every a E X, t h m is a neighborhood V(o) of e in X for which aV(o) X, and (iv) for evay a E X. ax i l X is a neighborhood of a in X. We shall say that X is a regular 1 0 4 k~? gmup if X can be embedded in a Hausdofl zm, dimensional left topological group G so that these four conditions hold. Notice that if G is a left topological group with identity e, (hen every open neighborhood of e in G is a local left group. Notice also that in any local left group, one may presume that V(e) = X.

DeBnition7.20. Let X and Y be local left groups. We shall say that a mapping k : X -* Y is a local homomorphism if. for each a E X, there is a neighborhood V(o) of e in X such that b E V(o) implies that o b E X, k(o)k(b) E Y and k(ab) = k(o)k(b). We shall say that k is a loco1 isomorphism if it is a bijective local homomorphism and k-' is a local homomorphism. Lemma 7.21. Let X ond Y be h i lcft gmups with distinguished elements e ond f respectively and k t k : X -+ Y be o local homomorphism Then k(e) = f . I f k is continuous at e. then k is continuous on all of X. Pmof: Pick groups G and H containing X and Y respectively as guaranteed by the definition of local left group. For each a E X pick V(a) as guaranteed by the definition of local homomorphism. Since c E X and e E V(e), one has k(ej = k(eej = k(e)k(e) so k(e) is an idempotent in H and thus k(e) = f . Now assume that k is continuous a t e and let a E X. Let W be a neighborhood of k(a) in Y and pick open U E H such that k(a) E U f Yl W. Then Y fl k(a)-'U is a neighborhood of f in Y so pick a neighborhood T of e in X such that k[T] c Y ilk(a)-I U and pick open R G such that e E R ilX 5 V(a) ilT. Then a R ilX

c

f '

ir a neighborhood of o in X and, by the defi~tionof local left p u p , OX n X is a neighborhood of a in X. We claim that k[oR n a x n XI g W. To sec this let c ~ a R n o X n X a n d p i c k bE R n X s u c h t h a t c = a b . T h e n b E V ( a ) n T s o 0 k(c) = k(nb) = k(o)k(b) and k(b) E k(a)-'U. Thus k(c) E U n Y W.

Theorem 722. Let X and Y be countable regular lord lep zroups without isolated points. Then there is o local isomorphism k : X + Y. IfY isfusf countable, then k con be chosen to be continuous. IfX and Y ore bothfirst countable, then k con be chosen to be a homeomorphism pmof: We shall apply Lemma 7.4 with p = 1. so that the functions h and gp arc trivial. Let e and f denote the distinguished elements of X and Y respectively. We can define x(r) E X and X(t) X for every t E F. so that the conditions in the statement of Lemma 7.4 are satisfied. We can also define y (t) E Y and Y(t) G Y so that these conditions are satisfied with x replaced by y and X replaced by Y. Define k : X 4 Y by k(x(t)) = y(t) for each t E F. By condition (9) of Lemma 7.4, x ( 0 = x(s) if and only if y(t) = y(s) so k is well defined and one-to-one. By conclusion (12) of Lemma 7.4, k is defined on all of X and k[Xl = Y. Now let a E X, pick t E F such that a = x(t). and Let n = I(t) + 1. Pick (since X is a local left group) a neighborhood Vl (a) such that a b E X for all b E Vl (a) and let V(a) = Vl(o) n X ( u n ) By conditions (1) and (2) of Lunma 7.4 and the fact that x(un) = e, V@) isnmighborhmdof e i n X . Let b e V(o). S i c e b E Vl(o),ab e X. By condition (12) of Lemma 7.4 pick u E u.F such that b = x(v). Then by condition (10) of Lemma 7.4, x(t + v) = x(t)x(v) = a b and y(t v ) = y(t)y(v) = k(o)k(b) anduxrsequenrly k(nb) = k(a)k(b) as required. Thus k is a bijective local homomorphism. Since k-' : Y -+ X is charactaized by k(y (t)) = x(r) for each r € F. an identical argument establishes that k-I is a local homomorphism. Now assume that Y is first countable, and let {W. : n E N) be a neighborhood base at f. Then we may assume that for each n € N,Y(U.+I) 5 W,, by condition (13) of Lemma 7.4. Given any n 6 N,by condition (12) of Lemma 7.4. Y(u.+l) = y[u.+lF] = k[x[u.+' F]] = ~[X(U.+~)] so X(u.+l) isaneighborhoodof econtained in k-'[WJ Thus k is continuous a t e so, by Lemma 7.21, t is continuous on X. Similarly, if X is fist countable, we deduce that k-' is continuous. 0

+

Exerdse 7.1.1. Let G be a countable group with no nontrivial finite groups. Show that

G' contains no nontrivial compact groups. (Hint: An infinite compact subset of @G cannot be homogeneous by Theorem 6.38.)

Exercise 7-12. Let G be an infinite commutativegroup which does conlain anonvivial finite subgroup. Show that G* also contains a nontrivial finite subgroup.

+

Exerdse7.13. Let p E W. Let pl = p and for n E N,let p.+l = p. p. (We cannot use n p for the sum of p with itself n times because n p is the product of n with p in (BPI, .). As we shall see in Comllaiy 17.22, n p is never equal to the sum of p with

7 GmupsinpS

148

i w f n times.if n s 1 and p E N*.)Show that. if p. = p for some n > 1in N, p must be idempotent. 7

73 Semigroups Isomorphic to IRI

nz,

We remind the rrada that I4denotes d j ~ ( 2 " N ) . Recall from Chapter 6 that a gooddeal is hownof t h e s t n ~ ~W. ~ (Moreinformation of will be found in Section7.3.) We show in this section that copics of W arise in many contexts. In particular. G* contains copies of W whenever G is a countable abelian p u p (Corollary 7.30) or a countable h e group (Corollary 7.3 1).

De6nitioa 723. Let X be a sulwet of a semigroup. A hmction & : o -+ X will be called an W-mapif it is bijective and if &(m n) = )(m)&(n) whenever m. n E N satisfy max supp(m) 1 .c min'supp(n).

+

+

(We remind the reader that, if n E o , supp(n) E P f ( o ) is defined by the equation n = E(2' :i E supp(n)).) In the following theorem, and again in Theorem 7.28, we shall be dealing with two topologies at the same time. The spaces BG and BX arc constructed by taking G and X to be discrete. while the "neighborhoods" refer to the I& invariant topology on G and the topology it induces on X.

Thnrm 7.24. Let G be a gmup with a Ie@ invariant zem d i m e n s i o ~ Haw&@ l topology, a d k t X be a countable subspace of G which contains the identiry e of G and has M isolatedpoints. Suppose also that,for each a E X, a x nX isa neighborhoodof a in X andthat. foreach a E X, then isa neighborhood V(a) ofe in X, with V(e) = X. for which aV(a) E X. Then there is o countable set [V,, : n E N) ofneighborhoods of e in X for which cljx V,,\[eJ is a subsemigmup of BG. Funhermore, there is on H-map Y= :o -r X such that de$nes an isomorphismfmm W onto Y. In the care in which the filter of neighborhwdr of e in X has a coumable base, Y cmt be taken to be n ( c e g x W : W is a neighborhoodof e in X)\(e].

n2,

P m f We apply Lemma 7.4 with p = I so that IZpI= 1. We define h : X + ZIto be the constant map. We then observe that the hypotheses of Lemma 7.4 are satisfied. and hence x(t) and X(t) can be defined for every t E F so that the conditions stated in this lemma will hold. Weput V" = X(u.) andY = c t ~ X(u.)\[eJ. x Weshowthat Y isasemigroup by applying Theorem 4.20 with A = [X(u.)\(e] : n E N]. Now for each n E W, X(u.)\[e) = x[u.F]\[el = [x(t) : t E F and minsupp ( t ) w n] by condition (12) of Lemma 7.4. Given any m E Nand any u E u,F\(eJ. let n = maxsupp (u) 2. Then if w E u.F\[e) we have v w E u,F\(eJ and x(u w ) = x(u)x(w) by condition (10) of Lemma 7.4 so x(u)(x[u.Fl\~el) G x[u,Fl\[e) as required by Theorem 4.20.

nz,

+

+

+

7.2 Semigmaps Isomorphic to I4

149

We define 9 : F + a, by stating that O(t) = z{Zi: i E suppt). By condition (9) ofkmma7.4, fortl. t2 E F. we have 9(tl) = O(e) if and only if x(tl) = x(t2). n u s we define a bijective mapping : o -+ X for which B o O = x. The mapping @ :Do -+ @X is then also bijective (by Exwise 34.1). By condition (10) of Lemma 7.4, +(rn n) = @(m)S(n)whenever rn, n E o satisfy max supp(rn) + I c min supp(n), for we then have m = j ( s ) and n =-O(t)for some s, t E F with s < c r. s o @ is an_H-map. By Lemma 6.3 @(p q) = Jr(p)$(q) for every p. q E 8. Thus JIle is a homomorphism. Furthermore.

+

+

+

$m= nzlceax $[2w1=nz,C ~ ~ X X ~ =UY.. ~ I \ ~ ~ ~ This establishes that defines aeonthous isomorphism fmm W onto Y. Finally, if there is a countable base [ W. : n E PI) for the neigbborboodr of e in X. the sets X(u.) can bechosen so that X(U.+~)c W. forevery n E W (by condition (13) 0 of Lemma 7.4). So Y = n[cLgx W : W is a neighborhood of e in X)\(e).

Lemma 7.25. k t 7be a left invariant zemdimeiuioml Hrmsdo~topologyon (Z. +) with a countable base and assume that o has no isolotedpoints in this topology. Then the hypotheses of Theorem 7.24 hold for G = Z and X = LO.

+

Pmof: Given any a E X , a X is a cofinite subset of the Hausdorff space X and hence is open in X. For each a E X let V(a) = X. 0 Theorem 7.24 has several applications to subsemigroups of PN. The following is om example and others are given in the exercises.

Comllarg 7.26. Theset to Mi.

nzlc t 6 ~ ( n Nisalgebmicallyandropological~'isomorphic )

Proof: L e t 3 = ( a + n Z : a E Zandn E N ) . Givena.b,cEZandn.rn E N , i f

c~(a+nZ)n(b+rnZ),thenc~c+nrnZ~(a+nZ)n(b+mZ)soPIisabasis for a left invariant topology T on Z.To see that 7 is zero dimensional, notice that if

+

c E Z\(a + nZ), then c + nZ is a neighborhood of c missing (a nZ). To see that 7 is Hausdorff, let a and b be distinct members of Z and pick n 6 N with n > la - bl. T h e n ( a + n Z ) n ( b + n Z ) =O. Thus by Lemma 7.25. Theorem 7.24 appties. So ~ n[clam(W\(0)) : W is a neighborhood of 0 in o ) = n n E N ( c C 6nW is algebraically and topologically isomorphic to A.

Lennu 7.27. L a G be a countably injnite subgroup of a compact N t r i c topological group C. Then with the relative topology, G is a Haurdofl zero dimrmioml first countable topological group without isolatedpoints. Pmof: Let d be the metric of C. That G is a Hausdorff first countable topological group is immediate. To see that G is zero dimensional. k t x E G and let U be a

naghborhood of x in G. Since G is countable, for only countably many r E iR is

1,.

( y ~ C : d ( ~ , y ) = r ) ~ G # 0 . P ~ ~ k r E ~ ~ ~ ~ h t h ~ t [ y ~ C : d

and[y e G : d ( x , y ) < r ) c U . T h e n ( y ~ G : d ( x . y ) 0. The first possibility is what we wish to prove, and so it whl be sufficient to mle out the second. This is done by noting that, if j r 1, since b,,. # e, the equation ab,, b,, . . bmj = e implies that b, F which is a contradiction. If j = 1. it implies that a-' = b,, contradicting our assumption that a - I < b,, .

.

FZ,

.,...

.

.

.

,,...

,,_,.

Lemma 8.49. The expressionfor an element of T m a P-pmducf is unique,

Pmof: We apply Lemma 8.48 with a = e. We observe that we cannot express e as a P-pmduct b,, b,, ..bnj with i > 0. To see this. note that otherwise. if i = 1. we

.

180

8 CaneeIIation

.,-, . ..

should have b., = e. contradicting om assumption that e f P. I f i z 1, we sh&j have b., E F contradicting our definition of a P-pmduct. .bmk= bn,bnz...be,. where these are P-products, Lemma 8.48 Thus, if b,. b,. implies that k = I and that m, = n, for every i E (1.2. ... k).

.

Lemma 8.49 justilks the following definition.

+

Definition 8.50. Define : T + N by stating that $ ( x ) = k if x = b,,, b,,, where this is a P-product. As usual

v

...b,,,,

6 :T :+ ON is the continuous extension of ).

Theorem 8.51. La G be a countably infinite discrete gmup and let p E G*be right cancelable in _BG. Then T, is a compact subsemigm_up of G*which conrains Cp. Funhermom. @ is a hamontorphism on Tm satisfying *(p) = 1 and +[CPl = BN.

Pmof:L e t ~ ~ T ~ d ~ ~ p r e ~ s ~ d ~ a P - p r D d u c ... t : bx., = bE~n, sbm, ,t a n d if y E T., thm xy E T, and *(xy) = Jr(x) +(y_). It follows from 'Theorems 4.20

+

and 4.21 that T, is a subsemigroup of G* and that ) is a homomorphism on T., Foreveryn € NandeveryA E p,wecanchooseb E PJlA. S o ~ n ~ , , n @ - ' [ ( l ]#] 0. It follows that

This shows that p E Too and hence e a t C, c Tm. It also shows that $ ( p ) = 1 and hence that $[c,] = BN,because @[Cplis a compact subsemipup of Btal which contains I. 0 Theorem 8.51 allows us to see that C, has a rich algebraic structure. Comllarg 8.52. Lrr G be a countably infinite discrete group and Iet p e G*be right cancelnble in f3G. The stmignmp Cp has 2' minimal lefr ideals and 2' minimal right ideah. Each of these conrains 2' idempotents.

Pmok We apply Tkorem 6.41. The result then follows because, if L is a left ideal of f3N. @-'[L] n C, is a left ideal of Cp: and the corresponding remark holds for right ideals. We recall that every left ideal of Cp contains a minimal left ideal and every right ideal of C, contains a minimal right ideal, and the intersection of any left ideal and any right ideal w a i n s an idempotent (by Corollary 2.6 and Theorem 2.7). 0 We remind the reader that, if e and f are idempatents in a semigroup. we write f 5 e i f e f =fe= f. LcmmP 853. k t el and cz-be idempotents in BN with e2 < el, and let fl be an idempotent in-Cp for which Jl(f1) = el. There is an idempotent fi in C, for which f2 < f i a n d W f 2 ) = e2.

t 8.5 Compact Semigmups

z

p m j *tm+ t h a t 7 - t r c e ) ~ n c p f i @,because t 3 - i [ ~ c 2 ~ ~ n ciscwtaincd pvi bthis set. So $-'[[e2]]nCpfi is acompactsemigroup andthuscontainsanidempotent f (by Theopn 2.5). We put f i = fi f . It is easy to check that f2 is an idempotent ~tisfyingIlr(f2) = ez and fi c fi.

Corollary 8.54. Ln G be a countably infiite discrete gmup and let p E G* be right wlcelable in BG. semigroup C p contains infinite decreasing chains of idempotents. in BW by Theorem 6.12. p m j mere is ad-ing sequence of idempotents (e.):, BV Lemma 8.53 we can inductively choose a decreasing sequence ( f . ) E , of idempo-, tents in C p for which ?(f.) = en forevery n E W.

Corollary 855. Lcr G be a countabty infrnite discme gmup and let p E G' be right cancelable in BG. Eve7 maximal gmup in the smallest idrol of Cp contains a copy of the free gmup on 2' genemtors. Pmof: Let q be a minimal_idempo~t in Cp. men $(q) is aminimal idempotent in /3N by Lemma 8.53 and so (+(q))pW($(q)) contains agroup F which is acopy of the free group on 2' generators (by Corollary 7.36). For each generator x in F, we can choose

y E qCpq such that + ( y ) = x. These elements generate a group in qCpq which can be mapped homomorphically onto F and which is therefore free. 0

Comment 8.56. If we do not specify that p is right cancelable. the prsceding resulh may no longer hold. For example, if p is an idempotent, Cp is just a singleton. Less trivially, if p = 1 e , where e is a minimal idempotent in OZ, Cp is contained in the minimal left ideal BZ+e of BZ and contains no chains of idempotents of length greater

+

than 1.

Theomm 8.57. Let G be a countably inhire discnrr gmup and Iet p E G* be right cancelable in /3G.The semigmy Cp does nor meet K V G ) . Pmo$ We can choose x E P r l G* with x # p. We can then choose Q G P such that Q E x and Q $? p. We put P' = P\Q. Let ( b A ) z , enumerate P' in increasing order. Then define FL,PA. T'. Ti, and T& in terms of P' analogously to Definitions 8.45 and 8.46. Notice that for each n , F . G FL,and hence P; g P,. T; , :Tn. and T& d, TTm. We claim that T& n K(/3G) = 0. Suppose instead that we have some y E T& fl K(BG). Then by Theorem 1.67, y E (/3G)xy. So pick some u E /3G such that y = uxy. For each a E G , the set X. = (b. : b. E Q , b. > a. and b. > a-I) E x . For each b. E X ., we have T. E y. So (ab.v : a E G , b. E X,. and v E T.] E uxy. by Theorem 4.15. Since T' E y, there must be elements a E G , b. E X.. and u E T. for which 0b.u E T'. We note that b.u is a P-product and so. by Lemma 8.48. it follows that b. E P'. This contradicts our assumption that b. € Q and establishes that T; n K ( ~ G=) 0.

182

8 Cancellation

Now Cp

T& (by Theorem 8.51 with P' in place of P)and hence CpnK (PC) s.0,

Comllvg 858. If G is a couniably infinite discme group. there is a right canee&blc elemntp in flG forwhich Cp g C!(K(,~G))\K(~G).

-

-

Pmof We can c h m e a right cancelableelemmt p of ,9G in K (BG) (by Corollary 8.26). Since K(pG) isacornpact subscrnigroupof BG (by Theorem4.44). Cp g K(flG). By Theorem 8.57, Cp n K(BG) = 0. 0

Lemma 8.59. Let G be a countably infinite discreie group, let p

E G* be right came&ble in B G, andlet q be an idempotentin Cp. Then then is some idempotent r E (flG)p which is I ~ - ~ min aG' l such that q SR r.

Pmo$ We can choose a 5.q-manimal idempotent r in G* for which q 5~r by Thcorem 2.12. Suppose that r $ (BG)p. There is then a set E E r such that (pG)p n B = 0. F a every a E G, we have E 6 a p and heme B, = a-'(G\E) 6 p. We consider the set A of all P-pmducts b,,,b,, . ..b,,, with the property that, for each i E [I. 2,. .. k). bnj$ E and if i r 1, b., E Bu faevery u E F . We observe that, for a P-product of this f o n a we have b,, E A fa evev i E (2.3. . k). We also note that A n E = 0. For each n E N, we define A. to be the set of all P-products b%bU2.. b., in A n T, f a which b., E B, whenever u E F,. We shall show that A. is a compact semigroup which contains p. That it is acompact semigroup foUowsfmmTheorem4.20 and the observation that, if u E A, is expressed as a P-product b., b,, . . .b.,, them uA. G A, whenever n w nt. Now. for each n E Pi, we have A. E p, because n.e~m~b,:brEPnandbr~~wI€~.S~~€n,,NAn. Since A. is a compact semigroup which contains p. C, g A, and so qE An. F a each a E G. let Q. denote the set of P-products b., b., . . .b,, with b., =- a and b., z a-'. Then Q. E q and so (au :a E E and u E Q,J E rq @y Theorem 4.15). Since A E q = rq. there exists a E E and u E Q. such that au E A. By Lemma 8.48, it follows that a E A. This contradicts the assumption that a E E.

.

.,-,

nj;:

n,

-

nmGN nnGN

-

... .

n, -

The following theorcm may be surprising because all idempotents in K(BN) are 5.q-minimal (by Theorems 1.36 and 1.38). Theorem 8.60. Let G beacowrtably infinitediscretegroup. Foreveryq E Gg\K(pG), there is a c ~ - m i m aidempotent l of G' in K(j3G) 0 (pG)q.

-

Pmof By Theorem 6.56 we can choose an element x E G' f a which xq is right cancelable in BG. By Corollary 8.26, there is an element p E K(pG) which is righi cancelable in flG. Since pxq is also right cancelable, it follows from Lemma 8.59 that there is a 5~-maximalidempotent of G' in (flG)pxq. Now is an ideal of flG (by Theorem 4.44). and so (BG)pxq 5 K(BG). 0

nworem 8.61. Let G be a countobly infinite discrete gmup and let p E G* be right c ~ c e l a b l ein PC. There is an injective mapping 6 : T + N with the following properties:

...

Pmof Wedefine6 by stating that@(b.,b., b,,) = Cf=l2n1 wheneverb,, b., .. .b,, is a P-product Then $ is well defined by Lemma 8.49, and is hivially injective. We note that, if b,, b., ...b., E T. then b,, E P . and-so b., f {a E G : a 5 bat and hence n~ z n. It follows that $[TJ E 2"N and that $[Tml C W. Suppose that x E T is expressed as a P-product b.,b,, . . .b.,. Then, if n _z nk and y E T., we have +(xy) = y ( x ) $ 4 ~It) .follows from Th4.21 that 6 is a homomorphism on T,. Sime 4 j s injective (by Exercise 3.4.1). it follows that 5 defines an isomorphism from T, onto $[Tm]. Now $[PI E (2" :n E W ) . Since P E p. (2" :n E N) E

+

kp).

Corollary 8.62. Let G be a countat,Ey in@ite discrete gmup and let p E'G* be right cancelable in BG. There isanelementq E N*.for which (2" :n E N) E q andCp(pG) is algebraically and topologically isomorphic to C9(pN). P,mf We put q = &p), where 6 is the mapping defined in Theorem 8.61. Then $[Cp(pG)]is a compact subsemipoup of pN which contains q, and hence $ [ C p ( p ~ )2] C,(pW). Similarly, $ - 1 [ ~ 9 ( p Nn ) ]T, a compact subsemigroup l . $[Cp(f3G)I = Cq(flW)and of pN containing p and so CP(BG)g 5 - 1 [ ~ q ( p N ) So $defines an isomorphism from Cp(,9G)onto Cq(pN).

5

Theorem 8.63. Let G be a countably infinite discrete group and let p E G* be right cancelable in pG. Then T, is olgebmically and topologically isomorphic to 8. Proof By Theorem 7.24 (taking X = G and V ( a ) = G for all a E G), it is sufficient to show that we can define a left invariant zero dimensional topology on G which has the sets ( e )U T,, as a basis of neighborhwds of e. As we observed in the pmof of Theorem 8.51. if x E T,is expressed as a P-product b,,b,, . .b,,, then xT. S T, if n > m t . Let 33 = { ( x )U xT. :x E G and n E N ] . We claim that 9 ) is a basis for a topology on G such that for each x E G. [ ( x )U xT, : n E N ] is a neighborhood basis at x. To see this. letx, y E G , 1etn.k E W,and let a E ((XIUxT.) n ( ( y )U yTk).

.

(i) If a = x = y, let r = max(n,k). (ii) I f a = x = yb,, b,, ...bmi where b,, b,,

...b,, is a P-product and b,, E Pk, let r = max(n. mi I ) . . (iii) If a = xbr,br, ...br, = y where br,br, ...bi, is a P-pmducts and b,, E P. let r = max(lj I . kt.

+

+

184

8 Caacellation

...

...

bm, when br,br, ...bl, and b, bm, (iv) I f a = xbhb12 b~, = ybm, &, P-produch. 4, € P.. and 4, E Pk,1e1 r = max{i, + 1, mi + 1).

...b,

Then (a1 U a c E ((XI U xTd ft ([yl U yTd as required. The topology generated by is clearly left invariant. Suppose that a E G\Tm. If we choose n such that a < b,, arid a < b;!, it follows from Lemma 8.48 that aT, fl T, = 0. 'Ihus the sets (el U Tmam c l o p in our topology. We now claim that T, = 0, because a c b. implies that a f T. To see this, suppose that a = b,,, b., . .b., where this is a P-product and b., E P.. We note that b, > b.. since otherwise we should have b,, E F,,. Sob,,, > a and therefore I > 1. We then have b,. E F,,,_,,contradicting our definition of P-product

'

.

This shows that our topology is H a u s d d , and completes the proof.

0

Lemma 8.64. Lei G be a countably infinite discrete gmue and I d p be a right came/able element of Go. Suppose that x E BG and y E T., Then xy E T implies thm

XET. Pmof Let X E x and let Z den& the set of all products of the form ab., b., .. .b., where b., b,. b,, is a P-product a E X, a c b,,, and a-I < b., . By Theorem 4.15, Z E xy. Hence, if xy E T n Z # 0. It then follows from Lemma 8.48, lhat 0 TnX#0andhencethatT~x.

...

So far, in this section, we have reshicted our attention to PC. when G denotes a countably infinite discrete group. However, some of the results obtained have applications to (PW. +) and (BPI, .). the following theorem being an example.

-

Theorem 8.65. Lct S beacountably infinitediscretesemiimup which conbeembedded in a countable discrete gmup G. Then K(BS) contains 2' nonminjd idempotents, infinite chains of idemporents. and idempotents which are S R - m i m din G*. Pmf By Comllary 8.26 (with T and S replaced by S and G respectively), t h m is an element p e K@S) which is right cancelable in PC. Taking this p to be the element held fixed at the start of this section, we can choose the set P so that P g S. Consequently T c S so that T c BS. Now K(BS) is a compact subsemigroup of PS, by Theorem 4.44, and is thetefore a compact subsemigroup of BG. So Cp(BG) 5 K(BS). By Corollary 8.54, CP(pG) contains infinite chains of idempotents. and so the same statement is hue of K(BS). Let el be any non-minimal idempotent in @I. Since el is not minimal, we can choose an idempotent e2 in for which e2 c el. By Lemma 8.53. we --then choose idempotents f~and fi in Cp(BG) satisfying fz < fi. Wfi) = el. and $(fi) = ez. Since f ~ f~ , E K(pS). f i is anonminimal idempotent in K(BS). There are 2' possible

-

choices of el (by Corollary 6.33) and therefore 2' possible choices of f l . So K(BS) contains 2<nonminimal idempotents. By Lemma 8.59, if q is an idempotent in Cp(BG), there is an idempotent r (BG)p which is SR-maximalin C* and satisfiesq R r. Since q = rq andq E Tm. it follows from Lemma 8.64 that r E T g BS.

-

185

Notes

-

-

Weknowthatr=xpfasanex EBG. ByLemma8.64.x ~ F a n d s o Er (BS)p. since K(pS) is anideal ofBS, it foUowS that r E K(BS). 0

MOSIof the molts in this section also hold for the semigroup Cp(BN),where p denotes a right cancelable element of N*. The proofs have to be modified. since N is not a group, but they am essentially similar to the p m f s given above. We leave the details to the reader in the following exercise. Exercise 85.1. Let p be a right cancelable element in BN. By Theom 8.27, thm is a set P E p which can be arranged as an increasing sequence (b,)E, with the property that, for each k 6 N, Pk = (b,, :b, k < b,+, 1 E p. We define T to be the set of all sums of the form b., b,. b,,,, where, for each i E (2.3, ,k]. b.,+l bSi > I 2 3 ... b,,_, .We shall refer to a sum of this kind as a P-sum. WcdefineT.tobcthesetofsumsofdP-sumsfwwhichb,,+~-b,, z 1+2+...+n and we put T, = Tn Prove the following statements:

+

-

+ + -..+ + ++ +

...

nneN

The expression of an integer in T as a P-sumis unique. T, is a compact subsemigroup of pN which contains Cp. There is a homomorphism mapping C p onto BN. C pcontains 2' minimal let3 ideals and 2<minimal right ideals, and each of these contains 2' idempotents. (5) C,,contains infinite chains of idempotents. (6) There is an element q E (F :n E N] for which Cq is algebraically and topologically isomorphic to Cp. (7) T, is algebraically and topologically isomorphic to H.

(1) (2) (3) (4)

Notes MostofthensultsotSeetions8.1.8.2,8.3,and8.4arefrom[461 (amultofcollaboration with A. Blass ). [125], [l54], and [228]. T h m m 8.22 extends a result of H. Umoh in [235]. T h m m 8.30 is from [1291, where it had a much longer proof, which did however provide an explicit description of the elements of cC K @I). Theorem 8.63 is due to I. Rotasov. Most of the other theorems in Section 8.5 were proved for pN in [88], a result of collaboration with A. El-Mabhouh and J. Pym, [157], and [228]. and were pmved for countable groups by I. Rotasov.

.

Chapter 9

Idempotents

We remind the reader that, if p and q are idempotenw in a semigroup (S..), we write: if P . ~ = P ; p C ~ q if q . p = p : p s q if p . q = q . p = p .

L

The= relations are reflexive and transitive, and the lhird is anti-symmetric as well. Wemaywritep < q i f p S q a n d p # q . Wemayalsowritep k(b))), and G € J'j({a,, : n 2 m)) such that b . F = G. Then F G and b = n ( G \ F ) E m((a.),m=,).

n

n

2

9.1 Right Maximal ldemporena

We do not know whether the sum of two elements -in N'\K(BN)

M whether the sum of two elements in N*\K(,SN) can be in

can be in K(BN).

K(BM). We do know

+

that, if p is a right cancelable element of BN, then. for any q E BN, q p E K W N ) implies*€ K(BPd)by E e 8.2.2, and that, if p is a right cancelable clement of PN\K(PN), then q p E K(PW implies that q E K U N ) by Theorem 8.32. There are idempotents p in B A which also have this pmpaty. As a consequence of the next theorem, one sees that if p is a right maximal idempotent i n a n d if q E pN\K (PA), then q p $ K @ N . The comesponding statement for K(BW is also true: if p is a right maximal idempotent in BN\K(flW) and i f q E bW\K(BW). then q + p $ K(BN). We s h d not prove this hen. However. a proof can be found in [159].

+

-

+

Theorem 98. Let G be a countably infiitediscrete gmupand let p be a right maximal idempotent in G*. Then, i f q E BG\K(BG), qp (e K(BG). PmoJ Suppose that q p E KcPC). Then q p = eqp for some minimal idempotent e E BG by Theorem 1.64. Let C = [ x E G* :x p = p). By Theorem 9.4. C i s a finite right zero semigroup. Now q $ (BG)C. To see this we observe that, if q = ur for some u E PC and some r E C , then q = qr = q ( p r ) = (qp)r E K(BG), conhadicting our assumption that q $ K(BG). Since(pG)Ciscompact, wecanchooseQ E q satisfying~n(,TG)c= 0. Since q $ K(BG), q # eq and so we may also suppose that Q $ eq. We also claim that eq 6 (,9G)C. I f we assume the contrary, then eq = u r for some u E BG and some r E C. By Corollary 6.20, this implies that q E @G)r or r E (BG)q. We have ruled out the first possibility. and the second implies that p = rp E (BG)qp g K(BG). However. by Exercise 9.1.4, p $ K(BG). So wecan choose Y E eq satisfying Y r l (f3G)C = 0 and Y n Q = 0. Now qp E cL(Qp) and eqp E ct(Yp). It follows from Theorem 3.40 that ap = yp f a s o m e a ~Qandsomey ~ ~ , o r e l s e x ~ = b ~ f o r s o m e x ~ ~ a n d s o m e b ~ ~ . If we assume the Erst possibility. the fact that a # y implies that y E G*, by Lemma 6.28 and thus a w l y E G* by Theorem 4.31. The equation p = a-Iyp then implies that a-' y E C and hence that y E (PG)C.cannadieting our choice of Y . The second possibility results in a contradiction in a similar way. 0

Dellnition 99. Let S be a semigroup and let p be an idempotent in S. m e n p is a stmngly right maximal idernpotent of S if and only i f the equation x p = p has the unique solution x = p in S. Observe that uivially a strongly right maximal idempotent is right maximal. We now show that strongly right maximal idempotentsexist in N*. by giving a proof which illustrates the power of the methods introduced by E. Zelenuk. As we shall see inTheorem 12.39, strongly summableultrafilters are strongly right maximal. However,

192

9 Idempolenta

by Corollary 12.38, the existencc of smngly summable ulaafilten cannot be deduced in ZFC. It was an open question for several years whether the existence of strongly rinht maximal idcmvotcnts in W' wuId be deduced in ZFC.Tlis question has now been Awered by I. Rotasov. We do not know whether it can be shown in ZFC that them a e any right maximal idcmpotents in PI* which are not smngly right maximal. how eve^, E Zelmuk has shown that Martin's Axiom docs imply the existence of idempotents of this kind (247l.

Theorem 9.10. There is o stmngly right mmr'ml iohpotenr in W.Funhemon,for every right ~ i m a idempotent ! e in Z*, there is an B - m p +from o onto o subset of Zfor which Jr (e) is defined ond is a strongly right marimal idempotent in N*.

-'

P m $ We know that them are right maximal idcmpotents in Z' by T h m 2.12. Let e be an idempotent of this kind let C = [x E Z ' :x + e'= el. and'let C- = [x E j3Z : x + C 2 C]. By Theorem 9.4 C is a finite right zem subsemigroup of j3Z so in pacticular, x + C = C for every x E C. Then C- = (0)UC. (Certainly (0)UC g C-. Assume x E C-\(O]. Then by Lemma 6.28, x $$ Z. Let y = x + e. ?hen y E C w x+e=x+e+e=y+e=e.)Let(p-=nC-=(UEZ:C-E~).ByLemmas

7.6 and 7.7, there is a left invariant zero dimensional topology r on Z for which Q- is the filter of neighborhoods of 0. WechwseasetU ~ZsuchthatOandeareiniTbutf e n i f f EC-\(0,e]. We notethatforevery f ~ C " \ ( 0 , e ] , { n ~ Z : n +f ~ Z \ U ) ~ e b e c a u s e ef+= f E Z\U. We put v =u n In E z : n f E Z\UI andobservethat0~V a n d V E e . L e t X = V ' = ( n € V : - n + V ~eJandnote that 0 E X and X E e. Let n E X. We shall show that

-

nfEc-,,o,r, +

-

+

(i) there exists W(n) E Q- for which n (W(n) fl X) g X. (ii) (n X) r l X is a neighborhood of n in X for the relative topology induced by r.

+

and (iii) n is not isolated in the relative topology on X induced by r. Before verifying (i), (ii), and (iii), note that if xw 6 W for each W E Q-, then the net (xw)wsr- (directed by reverse inclusion) bas a clustc~point f in C-. To see this. suppose instead that for each f € C- one has some U(f ) 6 f and some W( f ) E cpsuchthatxw, $ U(f) forall W' ~ ( p - w i t hW' 2 W(f). Let W'= (Ufec- U(f))n W(f)). Then W' E Q-. Pick f E C- such that xw, E U( f). Then since W' 2 W(f). we have a conhadiction. To verify (i), suppose that for every W E Q-. thew. exists rw E W n X such that n rw B X. Pick a cluster point f E C- of the net (rw)w,,-. Since each rw E X one has f E jf, and thus f = 0 or f = e. Since each rw E Z\(-n + X) one has f €Z\(--n+X). ButOe - n + X a n d b y L e m m a 4 . 1 4 , - n + X = - n + V ' ~ e s o e E -n + X, a contradiction. Toverify(ii)weshowthatthereissome W E (p-suchthat(n+W)nX g (n+X)nX. Suppose instead that for each W E 9- there is some sw 6 W such that n sw E X

(nfEc+

+

,

. P'

+

9.1 Right Maximal Idcmpotcnts

193

but s w $ X. Pick aclusterpoint f E C- of the net (sw)weC-. Since each sw E Z\X, f $%andsince-hsw-n_+X.n+f.~x. Butsince f $x,onehas f $ (0.e) m,sincenEV,n+ f pU>X,acontradiction. To verify (iii), let W € q-. We show that there exists m E W\(O) such that n + m E X ( s o t h a t ( n + W ) n X # ( n ) ) . Since W E ( ~ " . ~ E C - C W S O W \ ( O ~)e . Also, n E V* so by Lemma4.14, -n V* E e. Pick m E (W\(O)) f(-n l + V'). Having established (i),(ii). and (iii). for each n E X pick W(n) E (p' as guaranteed by (0. choosing W(0) = X. Now we have shown that the hypnhesca of Theorem 7.24 are satisfied with G = Z and V(n) = W(n) fl X. Thus there. is a countable set (V, : n E Pi) of neighborhoods of 0 in X for which Y = cegx V.\(O) is a is an isomorphism subsemigroupof BZ and there is an W-map $ :o + X such that from W onto Y. For each n. there is some W. E (pa such that W. n X E V. and thus eachV,,EesoeEY. Now the equation 5 e = =_has the unique solution x = e in ~ J ( o ) . It follows has the unique solution x = $F1(e) in B and that the equ~tionx $-'(e) = hence that @-'(e) is a strongly gght maximal idempotent in W. Since by Lemma 6.8 all idempotents of W* are in H,*-'(e) is a strongly right maximal idempotent in No. 0

+

, : a

+

+

Theorem 9.11. There are 2' stmngly righr idempoients in H Consequently there are 2' srmngly right m'ml idempotenrs in W*.

PmoJ We know that there am 2' right maximal idempotents in Z ' by Theorem 9.1. For each idempotent c of this kind, there is an W map & from o onto a subset of Z for which g-' (e) is defined and is a strongly right maximal idempotent in W (by Theorem 9.10). For s y given e, there are at most c right maximaljdeppotents f in Z* for which ~ ; r ; l (e) = @I; (f ), because this equation implies that (+;'(e)) = f , so that @f is to f. and there. are at most c W-maps from o to subsets of Z. a B map taking $;'(e) Thus there are 2' distinct elements of the form q;'(c). 0 Corollary 9.12. Let G beacowrrablyinfiniredbcmtegmupwhichcan bcafgebmically embedded in a metrizable compact topological gmup. Then there a n 2' stmngly righr man'ml idempotenu in G*. P m J This follows immedia~elyh m Theorem 9.1 1 and Theorem 7.28. J3xelrlse 9.1.1. Which idanpotent in (BZ, and the unique 5~-maximalidempotent?

0

+) is the unique S~-maximalidempotent

Exercise 9.1.2. Let S be a semigroup and let p. q € E ( S ) . Show that p SR q if and only if pS q S and that p SL q if and only if Sp C Sq. Exercise 9.13. Let q be an element of ,9N which is not right cancelable in BN. Show that there is a right maximal idempotent p E BW for which p q = q. (Hint: Use

+

Theorems 8.18 and 2.12.)

The following exercise contrasts with Ihe fact that we do not know whether idempotents in K (fix)can be left maximal. Exercise 9.1.4. Lel G he a countably infinite discrete group. Show that no idempotent in K ( p G ) can be right maximal in G*. (Hint: Use Theorems 6.44 and 9.4.) Exercise 9.1.5. Let p and q be any two given elements of BZ. Show that the equation x + p = q either has 2' solutions in BZ or else has only a finite number. (Hint: you may wish to use Theorem 3.59.)

9.2 Topologies Defined by Idempotents In this section we investigate left invariant topologies induced on a group G in each of two natural ways by idempotents in G*. The first of these is the topology induced on G by the map a H ap from G to Gp. Theorem 9.13. Let G be an infinite discrete group and let p be an idempotent i G*. There is a left invariant zero dimensional Hausdoif topology on G such theflrer (o- of neighborhoods of the identity e consists of the subsets U of G fo ( x E pG : x p = pJ G cepe U. This topology is extremally disconnected and ' same as the topology on G induced by the mapping ( p , ) , ~ : G + G*. Furhe if(p,),~ and ( p , ) p induce the same topology on G , then pBG = qpG. Consequen there are at least 2' distinct topologies which arise in this way

-

Notice ~ that by Lemma 6.28, ep is injective. In parti Pmof Let 4 = ( 4 ) 1 (a E G : a p = pJ ( e )so b y Lemmas 7.6 and 7.7, with C = { p }(so that BG : x p = p)), we can define a zero dimensional Hausdorff left invariant topolo G, for which 9- is the filterof neighborhoods of e. We show first *at the topology we have defined on G is the one induced by 7 be the topology defined by q- and let V be the topology induced by Bp. In th of Lemma 7.6 we showed in statement (ii) that the sets of the form U- = (a E G : cepc Ul where U F p form a basis for the neighborhoods of e in 7 and in sta (iii) that each U- E 7. Hence {bU^ : b E G and U E p) fonns a basis for 7. b E G and U E p , b U - = ~ ~ - ~ [ h ~ -u~] ]-so~7[ ~ V. e ~T~o s e e t h a t v U E V and let b E U.Pick V c G such that b E B , , - ~ [ C ~ V~ ]~ c U.TheQb-' E U. and b E b(bc' Since G p is extremally disconnected as a subspace of BG (by Lemma 7, follows that G is extremally disconnected. NOWsuppose that p and q are idempotents in G* which define the same to on G. Then the flap r,b = b.= 0 Op-I is a homeomorphism fmm G p to G4 ' r,b(ap) = aq for d l a E G . Now q = + ( p ) = r,b(pp) = lim a-P pq s o 4 belongs to the principal right ideal of pG defined by p. ~ i d

c

Vr

'

op &, . is a homeomophism. y belongs to the principal light ideal of f?Gdefined by + Thus pa(; :- q B C Now f?Gcontains at least 2' disjoint minimal right ideals (by Corollary 6.41). Choosing an idempotent in each will produce at least 2' different topologies on G. The second method of inducing a topology on G by an idempotent p is more direct, { A U { e ) : a E p} as a neighborhood base fore.

Theorem9.14. Let G be an infinite discrete group with identity e and let p

E G*.Let x = [ A u { ~ ) : A~ p ) a n d l e t T =[ U c G : f o r a l l a € U , a - l ~e p ) . T h e n T i s theftlest lefi invariant topology on G such that each neighborhood of e i s a member of K. Thefilter of neighborhoods of e is equal to X ifand only i f p is idempotent. In this the topology 7 is Hausdo@

pmof It is immediate that 7 is a left invariant topology on G. (For this fact, one only p to be a filter on G.) Let V be a neighborhood of e with respect to 7 . Pick E U g V . Then U E p and thus V E p. Also V = V U {el so

[I E

7 such that e

V

N.

E

Now let V be a left invariant topology on G such that every neighborhood of e with espect to V is a member of X. To see that V E 7, let U E V and let a E U. Then ood of e with respect to V so a-'U = A U {el for some A E p f neighborhoods of e with respect to 7 .Assume first that M = X. see that pp = p, let A E p. Then A U (el is a neighborhood of e so pick

U

E

7

h that e E U 5 A U {e).Then U E p. We claim that U 5 {a E G : a - ' A E p). To - ' p) hisend leta E U . Thena-'U E panda-'^ E a - l ~ ~ [ a -S 'O) ~ - ~ A U { ~ E assume that pp = p. To see that M = X , let V E X . Then V E p. Let ' = ( x E V : x - ' V E p]. ThenB E p. W e s h o w t h a t B U [ e ) E 7 . (Then c V so V E M.)To see this, let x E B U (el. Now e-IB = B E p and, if f h e n b y Lemma4.14.x-'B E p. assume that pp = p and let a # e. Then a p # ep by Lemma 6.28, so pick \ a ~ .Let A = B\(a-'B U { a ,a-I)). Then a A U ( a } and A U { e ) are disjoint Orhoods of a and e respectively. o

:' :.: Thefollowing theorem tells us, among other things, that the topologies determined

.. , iaTheorems 9.13 and 9.14 agree if and only if p is strongly right maximal. &;.om,,, wemWin .Corollary 9.12that, if G is acountable group whichcan be embedded in a

'"?& ,,

G'.

etnzable topological group, there are 2<strongly right maximal idempotents 9-15. Let G be agmup with identiry e and let p be an idempotent in G*. Let invariant topology on G such that X = { A U { e ) : A E p } is the filter of of e. Then 7 is Hausdogand the following statements are equivalent.

Thc idempotent p is stmngly righr maximal in G', (c) The map (&)IG is a homeomorphisnrfmm (G. 7 ) onto Gp. (d) The topology 7 is regular and exrremally disconnected. (e) The topology 7 is zem dimensional.

(b)

Pmof The topology 7 is Hausdorffby Theorem 9.14.

(a) implies (b). Let q E G* such that qp = p and suppose that q # p. Pick B e q\p withe f B. Now G\B = (G\B) U ( e )is amighborhood of e so pick a neighborhowl U of e which is closed with respect to 7 such that U 5 G\B. Then U E p = qp so pick b E B such that b-'U E p. Since U is closed, G\U is a neighborhood of b so bK1(G\U) E p, a contradiction. (b) implies (c). By Theorem 9.13, the topology induced on G by the map (&)lc has q- = (U G : ( x E j3G : xp = p) s clpc U) as the filter of neighborhoods of e. Now. by assumption (x E BG : xp = p) = ( p ,e). (By Lemma 6.28, ap # p if a E G\(e).) Thus. given U 5 G, ( x E j3G :xp = p) E ceDc U if and only if U E p ande E U. Consequently p- = X so this topology is 7. (c) implies (d). By Theorem 9.13. the topology induced on G by the map (P,,)~~is regular and extremally disconnected. 0 Each of the implications (d) implies (e) and (e) implies (a) is trivial.

c

Theorem 9.16. Let G be a discnte group with identi0 e. and let p be an idempotent in G'. fet 7 be the lefr invariant topology defined on G by taking ( A U ( e ) : A E p] as a base for the neighborhoods of e. Let V be any topology on G for which G h a no isolatedpoints. Then V cannot be strictlyfiner than 7.

G G : for all a E U,a-'U E p). Suppose that V and pick V 6 V \ 7 . Since V $ 7 pick a E V such that a-'V f p. Then G\a-' V E p so (G\a-I V) U ( e )is aneighborhood of e with respect to the topology 7 and thus a ( ( ~ \ a - ' V )U ( e l ) = (G\V) U ( a ]is a neighborhood of a with respect to 7. and thus also with respect to Y . But then ((G\v) U ( a ) )n V = ( a ]is a neighborhood 0 of a with respect to V, a contradiction.

P m f : By Theorem 9.14, 7 = {U

7

Corollary 9.17. Let G be an infinite gmup. I f p is a strongly right m i m a l idempotent in G*, then the lefr invariant topology 7 on G defined by taking ( AU ( e ) : A G p) as the filter of neighborhoods of the identity e is homogeneous. zem dimensional. Hausdo& extremally disconnected and maximal among all topologies without isolated points. Distinct strongly right maximal idempotentsgive rise to distinct topologies. Pmof Sincethetopology isleftinvariant. itisvividly homogeneous. By Theomn9.15,

7 is Hausdorff, zero dimensional, and extremally disconnected. By Thmrem 9.16.7 is maximal among all topologies without isolated points. By Theorem 9.15, 7 is the topology inducedon G by the function (&JIG. Givendistinctright maximal idempotents p and q. one has P # qp so p 4 qSG. Thus. by Theorem 9.13. (pp)lc and (pq)lc induce distinct topologies on G. 0

9.2 Topologies Defined by Idempotents

197

Combined with CmUary 9.12, the following Corollary shows that them are 2c ~ i s t h lhomeomorphism ~t classes of topologies on %that are homogmeous, zero dimensional, Hausdortf, extremally disconnected, and maximal among all topologies without isolated points.

Corollary 9.18. Let G be an infinite gmup. Let K

= I ( p P G* : p b a stmngly righrmaximal idempotent in Ga)l.

1f21~1-z K , then then are m least I. distinct homeomorphism classes of ropologies on G that are homogeneous, zem dimensional, Hausdo6 exrremally disconnected, and maximal among all topologies withow isolated points.

P m J By Corollary 9.17 distinct strongly right maximal idempotents give rise to distinct topologies on G , each of which is homogeneous, zero dimensional, Hausdorff. examally disconnected, and maximal among all topologies without isolated points. Since any homeomorphism class has at most lGllCl = 2ICI members, the conclusion 0 follows. The use of ulnafilters leads to panition theorems for left topological gmups. The following theorem and Exercises 9.2.5 and 9.2.6 illustrate results of this kind. Let X be a topological space. Recall that an ultrafilter p on X is said to converge to a point x of X if and only if p contains the filter of neighborhoods of x.

Theorem 9.19. Let G be a counrably inJFnitediscrete gmup with identify e. Suppose a lej? invariant topology r on G for which there is a right cancelable

that there is

ulrmfilter p E G* converging to e. Then G can be ponirioned into w disjoint subsets which are all r-dense in G.

Prwf: Let@ : G x G + N bea bijection. Foreach b E G let Xb = (apd(".b) :a E G). Notice that if a , b. c , d E G and ap*@"'b'= c ~ " ' . ~ )then , @(a. b) = 4(c, d ) so that a = c and b = d . (Ifap" = cpm where. say, n w m, one has by the right cancelability of p that apn-'" = c E G , while G* is a right ideal of ,8G by Theorem 4.31.) We claim that for each b E G , cCgc Xb n cegc (UdEG,lbl Xd) = 0. Suppose instead that for some b E G , cegc Xb n cegc (UdsG\lbl Xd) # O. Then by Theorem

n cepe (UdGc,lb, Xd) # 0 or cepe Xb n (UdEC,lb, Xd) # 0. Since the latter implies a version of the f o m r , we may assume that we have some x E Xb n ceac (UdEG\lbl Xd). Then for some a E G. x = a ~ " ' ' ~ ' and x E : c E G and d E G\{b)). As we have already observed, apd(n.b) # c ~ ~d ' d# ( b ,~s o ~ ~ x Eceg~(cp : c E G and m > @(a,b)). Let n = $ ( a , b). Then apn E c l P ~ ( c p :m c E G and m > n ) E G'p" so pn E a-'G*pn E G'p". Thus. by Theorem 8.18 p" is not right cancelable. a contradiction. Since ctbc ~b n c t g c (UdeG\,b, Xd) = 0 forevery b E G , wecanchoose a family ( A b ) b eof ~ pairwise disjoint subsets of G such that Xb 5 CLBCAh for each b. We may Ab SO that (Ab : b E G ) is apartition of G. replace A, by G\ UbEC\,cl 3.40. either Xb

)

198

9a 1 t-

N o w I c t b ~ G .W e c l a i m t h a t A b i s r ~ i n G .Noticethatforepchn EN,^

w ~ ~ gtoei sbecause @y Exercise 9.2.3) the u l t d l t e in ~ ~G* that conmge to e f m a subsemigroupof p.7. Let V be anonempty r-open subaaof G and picka E V. nea a-' V is a r-neighborhood of e so a-' V E p*",b'. Since also Ab E ap*C.b). we have V n A b f 0. Corollary 920. Let G be a gmup with a ley? invariant topolo~yr with nspccr to wh&j, G hns no isolatedpoints. Ifthen is a comkable basisfor r then G can be purnrnIMlCd inro w disjoint subsets which a n all r-dense in G.

Prwf: Let [V. : n E N] be a basis far the neighborhoods of the identity e of G. By Theorem 3.36 the interior in G* of V* . is wnempty and thus by Theorem 8.10 there is some p E V,*which is right cauce@ble in PG ( w h m /3G is the ~ t o n e & c hwmpa~tificationof the discrete space G). Thus Theorem 9.19 applies. o

nzl

n'R,'

Exercise 9.2.1. Let S be any discrete semigroup and let p be any idempotent in S. Prove that the left ideal (0S)p is examally disconnected. (Hint: Consider Lemrm 7.41.) Exercise 933. Let G be an infinite discrete group and 1e1 p and q be idempotcnts in G'. Show that the foUowing statements arc equivalent

*

(a) q R P. (b) The function : Gp -t Gq defined by $(up) = a q is continuous. (c) The topology induced on G by (p&c is fincr than or equal to the one induced by (P&.

(Hint: Consider the proof of Theorem 9.13.)

Exercise 9.2.3. Let G be a p u p with identity e and let r be a non-discrete left invariant topology on G. Let X, = n{cf,4U\le)) : U is a r-neighborhood of e in G). (So X, is the set of nonprincipal ultrahltm on G which converge to e.) Show that X, is a compact subsemigroup of G*. (Hint Use Theorem 4.20.) Show that if r' is also a non-discrete leh invariant toplogy on G,then r = r' if and only if X, = X,,. The following exmise shows that topologies defined by idempotent8 can be characterized by a simple topological property.

Exercise 9.2.4. Let G be a group with identity e and let p be an idempotent in G'. L-4 7 be the leh invariant topology defined on G by choosing the x t s of the form U U (el, w h m U E p. as the neighborhoods of c. Show that for any A G and any x E G, x E CLT A if and only if either x E A or *-'A E p. Then show that, for any two disjoint subsets A and B of G, we have c t r A ncL7 B = (A n e t 7 B) U ( c e A ~ n B). Comcrscly, suppose that 7 is a left invariant Hausdorff topology on G without isolated points such that for any two disjoint subsets A and B of G, o m has cL7 A fl cC7 B = ( A n c e B ~ ) u ( c t ~ i \ n B ) .L e t p = (A E G : c E cCj-(A\(e))). Showthat

is an idempotent in G* such that the 7-neighborhoods of e are the sets of the form U U I C I . W ~ ~ UP. E

925. La G be a countably infinite discrete group with identity e and let p k an idempotent in G*. Let r be the left invariant topology defined on G by choosing the sets of the form U U (el, where U E p. as the neighborhoods of e. Show that G cannot be partitioned into two disjoint sets which are both r-dense in G.

~xerrbe93.6. Let G be a ~ n n t a b l yinfinite discrete group with identity e and let p be a right maximal idempotent in G*.Let D = [q E G* : q p = p). T h D is a finite right zero subsemigroup of G* by Theorem 9.4. Let r be the left invariant topology defined on G by choosing the subsets U of G for which D U ( e ) c regc U as the neighborhoods of the identity. Suppose that ID1 = n. Show that G can be ptitioned into n disjoint rdense subsets. but cannot be pdtioned into n 1 disjoint rdense subsets. (Hint: Use Corollary 6.20 to show that (pG)q n(pG)ql = 0 if q and q' arc distinct elements of C.)

+

The following exercise provides a conhast to Ellis' 'Ihconm(Corollary 239). Exerdsc 92.7. Let p be an idempotent in N* and let 7 = {U C 2 :for aU a E U. -a U E p). Then by Theorem 9.14. (Z. 7 ) is a semitopological semigroup which is algebraically a group. Rove that it is not a topological semigroup. (Hint: Either [Z2"(2k 1) : n. k E o ) E p or [22"f1(U 1) :n, k € o) € p.)

+

+,

+

+

9 3 Chains of Idempotents We saw in Corollary 6.34 that non-minimal idempotents exist in S* whenever S is right canceUativeand weakly left cancellative. In this section. we shall show that every nonminimal idempotent p in Z*Lies immediately above 2' non-minimal idempotents, in the sense that there arc ZC non-minimal idempotents q E Z* satisfying q p, which are maximal subject to this condition. Shis will allow us to construct ol-sequences of with the property that each idempotent inthcsequencccomsponding idempotents in Z*, to a non-limit ordinal is maximal subject to being less than its predecessor. Whether any infinite increasing chains of idempotents exist in F is a difficult open question.

Theorem 9.21. k t G be a cowuably infttite discrete group, let p be any non-minimal idempotent in G*. and let A g G with Tin K(BG) # 0. Then then is a set Q E (E(G8) n Ti)\K@G) such that

IQI = 2'. (2) each q E Q satisfies q c p, and (3) each q E Q is mazimol subject to the condition that q < p. (1)

200

9 Idempo~

Pmof. By Tfieonms 6.56 and 658. there is an infinite subset B of

A such that&

following statements hold for every x E 8': ( 9 P $ (BG)xp. (ii) xp is right camelable in BG, (iii) (BG)xp is maximal subject tobeing aprincipal left ideal of BG strictly contained . in (BG)p, and (iv) for all distinct x and x' in B*, (BG)xp and (BG)x'p are disjoint Let x be a given element of B* and let C denote the smallest compact subsemigmup of BG wbich contains xp. Pick an idempotent u E C. By Theorem 8.57, u 1s nM minimal in G'. We can choose an idempotent q E (BG)xp which is S~-maxirnalin G' and satisfies qu = u (by Lemma 8.59). Let v = pq. Using the fact that qp = q, it is easy c check that v is idempotent and that v 5 p. Now a # p, because v E (6G)xp and p p (,SG)xp, and so v c p. Let D = (W E E(G*) : v c w -z p). Then D G (w E BG : wq = p i ) since. if u c w -z p, one has p q = v = wv = wpq = wq. Thus. by Theorem 9.6. D is finite. If D # 0. choose w maximal in D. If D = fl, let w = v . We then have an idempotent w E G* which satisfies w c p and is maximal subject to this condition. We shall show that w E (,SG)xp. Since u E (,SG)w fl (,SG)xp, it follows from Comllary 6.20, that w E (BG)xp or xp E (BG)w. The lint possibility is what we wish to prove. and so we may assume the second. We also have w E (,SG)p and so (BG)xp 5 (pG)w (,SG)p. Now (BG)xp is maximal subject to being a principal left idealofBG sbictly containedin (BG)p. Thus (BG)w = (,SG)xpor (BG)w = (,SG)p. The fvst possibility implies that w E (BG)xp, as claimed. The second can be ruled out because it implies that p E (BG)w and hence that p w = p, contradicting our assumption that pw = w. We now claim that w $ K(/?G). To see this, since v 5 w it suffices lo show that v $ K(pG). We note that xuu = xpqu = xpu E C and that C n K(,SG) = 0 (by Theorem 8.57). So xvu p K(BG) and therefore v $ K(BG). Thus we have found a non-minimal idempotent w E (pG)xp which satisfies w c p, and is maximal subject to this condition. Our theorem now follows from the fact that there are 2' possible choices of x. because IB'I = 2' (by Theorem 3.59). and that (pG)xp and (,SG)xrp are disjoint if x and x' are distinct elements in B'. 0 Lemma 9.22. Let G be a countably infinite discrete gmup. Let (q= );,, be a sequence of idempotents in G* such that, for every n E N,q.+l c~ q., Ifq is any limit poinl of the sequence ,:q )(, then q E (BG)q, for every n E N and q is right cancelable in /?G. Proof For every n E N, we have q, E (BG)q. whenever r r n. Since (pG)q. is closed in BG. it follows that q E (BG)q.. Suppose that q is not right cancelable in BG. Then q = uq for some idempotent u E G' (by Theorem 8.18). So q E c~((G\(el)q)and q E ce (q, : n E N),where e denotes the identity of G. It follows from Theorem 3.40 that aq = q' for some

1 1

a E G\[e) and some q' E c l [q. : n E NJ.or else q. = xq for some n E PI and some x E BG. Assume first that q,, = xq for some n E N and some x E pG. Then q E pGq.+l so q, E /3Gq.+1 and thus. q. SL q.+l, a contradiction. Thus we have aq = q' for some a E G\[e) and some q' E c t (q. :n E M). Smce aq E ct[aqn : n E NJand q' E ct(q. : n E W). another application of Thwrem 3.40, allows us to deduce that aq. = q" or else aq" = q. for some n E N and some q" E cL(q. : n E W). Since the equation aq" = q. implies that a-'q,, = q N ,we need only refute the first of these equations. Assume first that q" = q, for some m E W. Then qm = aq. = aq.9. = qmqnso qm SL q. Also a q d m = qmqm= qm = aq. so by Lemma 8.1, q.qm = q,, so that q, 5~ q,. Thus m = n and consequentlyaq, = eq. so by Lemma 6.28. a = e, a contradiction. Thus q" is a limit point of the sequence (q&. So, as already established, q" E (pG)q,,+l and thus q. = n-'q" E (pG)qn+l so that q. 51. q.+l, a contradiction.

Theorem 923. Lcr G be a countably infinite discrete group and let p be m y nonminimal idempotent in G*. There isan w l - sequence (pa),,,, idempotents in G* with thefollowing pmperries:

of distinct non-minimal

(1) W = P. (2) for every a, p E wl, a c p implies that pp 0.

Observe that U. and D, an the set of points of P S for which value from above and below respectively.

5 approaches its

Ddlaitlon 105. Let Sbe a subsemigroup of @. +) and l a a E R Then a is irrational with respect to S if and only if

208

10 Homomorphisms

Saying that a is irational with respect to S means that h. is one-to-one on S and that 0 p ha[S\M1. Notice that "inational with respect to W is simply "irrational". Notice also that if IS1 c c, then the set of numbers irrational with respect to S is dense in R. Remark 10.6. Suppose that S is a subsemigmup of 09, +)and fhai a > 0 is irmtioml with rrspect to S. k t U. D and Z be the sets defined in Lemma 10.1 with h = ha. Then U = Ua. D = Da and Z = Z,\(O). Lcmma 10.7. Let S be a subsemigmup of respect to S. Then Ud U Dm = S*.

(R,+) and let a

7

0 be irmtioml with

P m f :This is immediate from Lemma 10.1 and Remark 10.6.

0

The following rrsult is of interest because, given the cohtinuity of pp, it is usually easier to describe left ideals of BS. (See for example Theorem 5.20.) Theorem 10.8. Let S be a'subsemigmup of (R.+) and let a z 0 be irrational with respect 10 S. Then We and Dm are right ideals of BS.

Pmof:f i s is immediate fmm Lemma 10.1 and Remark 10.6.

0

In the following lemma, all we care about is that cCflS(Na)n U, # 0 and that d p s ( N a ) n D, # 0. We get the smngerconclusion for free, however. Lemma 10.9. Let S be a subsemigroup of (W,+) and let a > 0 be irrational with respect to S. Thenfor each a E S\(01, cCgs(Wo) ilXu # 0 and ctps(Na)n Y, # 0.

P m f : We apply Lemma 10.1 with h replaced by h, and S replaced by No. Then c f p ~ ( N arl) X, = U rl Z # 0 and cCps(Na) n Y, = D n Z # 0. For the remainder of this section we resnict our attention to PI. Notice that, while h. is a homomorphism on (flW. it is not a homomorphismon (BN. .). However. close to0 it is better behaved.

+),

Theorem 10.10. Lcr S = N and let a be a positive irrational number: Then Xu and Ymare left ideals of (BPI, .).

PmoJ We establish the statement for X,, the proof for Y, being nearly identical.

&,

Letp E Xa andletq E BN. We€irstobservethat,foranym. n E N.if 1 f,(n)l < then f.(mn) = mf,(n). To see that qp E X . let 6 > 0 be given (with 4 5 For each m E N, let B, = ( n E W : 0 < f,(n) < ): and note that 9, E p. Thus by Theorem 4.15. (mn : rn E N a n d n ~ B , ) ~ q p . s i n c e ( m n : m ~ N a n d n ~ B , ) ~ { k ~f ,W( k:) O < s 0 with 8 < E a. Let C = ( n E N : O < fa@) m)). So a second application of Theorem 3.40 shows that there exists n z m in N and u ~cC({t,: n s m ) ) s a t i s f y m g ) ( b + t . + q ) = a + ) ( c + u + q ) a @ ( b + u + q ) = a @(c I,, q). By condition (i) of Lemma 10.24, neither of these equations can hold if v E Y*. So u = r, for some r > m. However, by condition (ii) of Lemma 10.24, neither of these equations can hold if r # n. Thus r = n. E In. -a). Byadding@(y+q)onthe So@(b+t.+q) = a'+)(c+r.+q),wherea' rightof thisequation, wesee that @(b+q)+@(b+y+q) = a'+)(c+q)+@(t,+y+q). By Lemma 10.25. Q(r, +y +q) is right cancelablein PC. So@(b+q) = al+@(c+q) and)(c+q) E G + b ( b + q ) .

+

+ +

+

Lemma 10.27. For every c E T. @(c q) E G

+ p.

Pmof:Let Y be the set guaranteed by Lemma 10.24 and let y E Y*. F a every b E T. wehave@(c+q)+@(b+y +q) = @(b+q)+@(c+y+q). ItfollowsfromComllary 620that@(b+y+q) E B G + @ ( c + y + q ) o r @ ( c + y + q ) ~ P G + @ ( b + y + q ) . In either case, Lemma 10.26 implies that @(c q ) E G @ (b q). By choosing b to be the ~dentityof T,we deduce that @(c q) E G p. 0

+

+

+

+

+

In the statement of the following t h e o m , we remind the reader of the standing hypotheses that have been apply~ngthroughout this section.

Theorem 10.28.

b t S and T be countably infinite. comrmrrative, and cancellative semigmupsmdasswne that T has an identify. Let G be the gmup generated by S and assume that ) : T ' -+ S* is a continuous injecfive homomorphism Then there is an injective homomorphim f : T + G such that f (x) = )(x) for every x E T*.

Pmof: By Lemma 10.27, foreacht E T, themexistsa E G forwhich@(t+q) = a + p . This element of G is unique. by Lemma 6.28. We define f : T + G by stating that @t q) = f (r) p. It 1s easy to check that f is an injective homomnphism, and so f is also an injective homomorphism (by Exercise 3.4.1 and Corollary 4.22). Since )(t q) = f (1) + p forevery r E T, it follows by continuity that@(x+q) = 7 ~ x 0 p forevery x E BT. Let 'I be the set guaranteed by Lemma 10.2%and let y E Y*.-Fw e v y x E T*, w _ e h a v e ~ ( x ) + @ ( y + q ) = 4 ( x + y + q ) = f ( x + ~ ) + ~ = f ( x ) f+( y ) + p = t ( x ) )(y q). Now )(y q ) is right cancelable in BG, by Lemma 10.25. So f (XI= @(XI. 0

+

+

+

+

+

+

+

10.3 Homomorphisms from T ' into S

217

C o d q 10.29. Let f denote the mapping d @ d in Theorem 10.28 and let T'

=

f-'[S]. Then T' is a subsemigmup of T forwhich T\T1 isfnite. PRX$ It is immediate that T' is a subsemigroup of T . _By T h m m 10.28, T ( x ) = $ ( x ) E S* for every x E T'. Thus, if x E T'. then S E f ( x ) so T' = f - ' [ S ] E x by Lemma 3.30. Consequently T\Tr is finite. 0

Thmrem 10.30. Let S and T A countably infinite, cornmufafive, and cancellative semigmups andassume that T has an idemMfyand that $ : T* -+ S' is a continuous injective homomorphism. There is a subsemigmup T' of T for which T\Tf isfinite, and an injective homomorphism h : T' + S for which h ( x ) = $ ( x )for every x E T'. Proof. Let f denote the mapping defined in Theorem 10.28. We put T' = f -'IS] and let h = fir.Our claim then follows from Corollary 10.29 and Theorem 10.28. o The necessity of introducing T' inTheorem 10.30. is illushated by choosing T = o and S = N. Then T*and S* are isomorphic, but there are no injective homomorphisms from T to S. In this example. T' = T\[O). In the case in which S is a group we have S = G and h e k e T' = fm'[S] .= f e l [ G ]= T .

Theorem 1031. Let S and T be countably infinite, cornmutafive, and camellarive semigmups and assume thar T has an identify. Suppose that 0 : pT + pS is a continuous injective ho~ornotphism Then there is an injective homomorphism f : T -r Sforwhich0 = f . Pmof Let $ = O I p . We observe that $ [ T W g ] S,because x E T * implies that $ ( x ) is not isolated in pS and hence that $ ( x ) 6 S. B_yTheorem 10.28 there is an injective homomorphism f : T + G such that @ ( x )= f ( x )for evety x E T*. Let Y denote the set guaranteed by Lemma 10.24 and let y E Y*. For each t E T, we have@@ y q ) = 0-W 0 ( y + q ) = @(r) $ ( y q ) and also 0 0 y + q ) = 70 Y + = f 0 )+ f (Y + q ) = f (0 + $(Y + q ) . BYLemma 10.25. $(Y q ) is right cancelable in pG. So f ( t ) = @(t).Now f ( t ) E G and 6 0 ) E pS so f ( t ) e S.

+

+ +

+

+

+

+

+

Theorem 1032. Lct S and T be countably infinite, commutative, and cancellative semigroups andassume that T has an identity. Then S* doesnot contain any topological and algebraic copies of pT.

Pmof This is an immediate consequence of Theorem 10.31.

0

We donot know theanswer tothefoIlowingquestion. (Although, ofcourse, algebraic copies of Z are plentiful in N*. as are topological ones.)

Question 1033. Does N* confain an algebraic and ropological copy of Z? E x e ~ . l O 3 . 1 .Show that the only topological and algebraic copies of N* in N* are the sets of the form (kN)*,where k E W.

I

10 Ho-

218

Exerdse 103.2. Show that the only topological and algebraic copies of Z* in Z ' the set9 of the fonn (kZ)*, where k E Z\(O).

a

Exercise 1033. Show that the only topological and algebraic copies of (N*. +) ia (N*, .)arc those induced by mappings of the fonn n H k" from N to itself, whae k E N\{lJ.

Exercise 103.4. Let S be an a t e d i m t e canccllative semigroup. Show that S* contains an algebraic and topological copy of N. (flmt: By Theormi 8.10. there is a. element p E S* which is right cancelable in BS. Define p, for each n E N by stating that pl = p and p . + ~= p. p for every n E W. Then (p. : n 6 NJ is an algebraic and topological copy of N.)

+

Exerrise 10.35. Show that N* contains an algebraic and topological copy of o. (Hint: Let p be an idempotent in N* for which FS((3")zl)E p and let q E ~ ( ( ( 2 . 3 ": n E N)) n N*. Definex. for every n E o by stating that y, = p, X I = p q p and &+I = x. + X I for every n E N. Then (x. : n E w ) is an algebraic and topological copy of o . To see that (x, :n E o ) is discrete. consider the alterations of 1's and 2's in the ternary expansion of integen. Forexample, (&F, 3' C,,F~2 . 3 ' XrEF33[ : max Fl < min F2 and max F2 < min F,) E X I . )

+ +

+

+

10.4 Isomorphisms Defined on F'rincipal Left and Right Ideals In this seftion, we shall discuss continuous isomorphisms between principal left ideals or principal right ideals defined by idempotents. Throughout section, S and T will denote countably inhnite discrete semigroups. which are commutative and canceUative. G will denote the group generated by S and H will denote the group generated by T.

Lemma 1034. Suppose that p and q an nanminimal idempotem in S and T* nspective3.. and that f :q B T q + p BS p is a continuous isomorphism Ir!

+ +

+ +

S ' = ( ~ E G : ~ + ~ E ~ ~ S ) ~ M ~ T ' = ( ~ E 7ken@(q)=pand H : ~ + ~ E B T ) . then is an isomorphism f : T' + S' m h that +(I q) = f (I) p for every I E T'.

+

+ +

+

+ +

PmoJ Since 9 is in the center of q BT q. @(q) is in the center of p BS p. So @(q) = a p for some a E C (by Theorem 6.63). Since @(q) is idempotenk it follows from Lemma 6.28 that a is the identity of G and hence that p = f (9). Foreveryt E T1.t+q = q + ( t + q ) + q isinq+pT+qandisinthecenterofthis semigroup. So g ( r q) is in the center of p ,l?S p. It follows from Theorem 6.63 that *(r q) = s p for some s E G. We note that s E S' and that the element s is unique (by Lemma 6.28). So we can define a mapping f : T' -+ S' such that $0 + q ) = f ( I ) + p for every r E T'.

+

+

+ +

+ +

It is easy to see that f is injective and a homomorphism. To see that f is smjectivc, 1 e t s ~ S ' S. m ~ s + p i s h t h e c m t a o f p + B S f p.s+p=@(x)forsomexinthe center of q pT q. We must have x = r q for some r E T' by Th6.63, and a this implies that f ( t ) = s.

+ +

+

em^ 1035. L a pond q be nmininaal identpofentc in S and T* rrspcaivcly. Lrt Sf = (a E G :a+p E BS)andlerT' = ( b E H : b+q E BT). I f @ :pT+q + pS+p is a continuous isomorphism, then $(q) is o ~nminimalidempotent, BS + p = BS + @(q),and there is an isomorphism f : T' -+ S'for which @(t q ) = f (t) + $(q) for every t E T'.

+

+

+

+

p = BS $(q). On the one hand. $ ( q ) E BS p and so BS + $ ( q ) E j3S + p. Oa the other hand @-'(PI= q and so p = $($-'(p)+q) = p+@(q) andthenforeBS+p E BS+ $(q). It follows that @(q)is a nonminimal idempotent (byTheorem 1.59). Wealsoobse~ethat(aE G : a + p E BS) = (a E G :a+$r(q) E BS). Tosee.tor example,that[a E G : a+p E 0s) g (a E G : a+$(q) E BSl,leta E Gandassume

p m J We first show that pS

+

thata+p~BS.Thena+@(q)=a+$(q)+p=$(q)+a+pEpS+BSEBS. The result ~ n follows v from Lemma 10.34 and the obsuvation that $ defines a

+ +

wnrinuous isomorphism from q + pT + q onto $(q) BS $(q). (To see that $(q)+BS+$(q) E $(q+BT+q),letr E $(q)+BS+@(q). T h e w = r+$(q)so r = $(v)forsornev E pT+q. T h e n q f v E q+BT+qand$(q+v) = $(q)+r = r.)

a We omit the proof o f the following lemma, since it is essentially similar to the preceding proof.

Lemma 10.36 Let p andq be ~ M l i n i m aidcmpotents l in S* and T* respectively. Let St = (a E G : a+p E pS)andletT' = ( b E H :b+q E BT). I f @ :q+BT -+ p+BS

+

is a continuous isomorphism then @(q)is a nonminimal idempotent, p pS = $(q) + gS, and them is an isomorphism f :T' -r S' such that $0 q ) = f ( t ) @(q)for every t E T'.

+

+

Lemma 1037. Suppose that y E G9\K(/3G). Lcr f : H -+ G be an isomorphism Then them is an element r E T' for which j ( x ) + y is right cancelable in BG. :BH -+ BG is an isomorphism and so & K _ c ~ H ) ]= K(pG). By Lemma 6.65. T n K(BH) # 0. and so f[Tln K(f3G) = f [T flK(BH)] f 0. It follows from 'lheonm 6.56 (with S = G ) that there is an element v E-G* such that f [TI E u and v y is right cancelable in BG. Since v E TIT].v = f ( x ) for some x E T*.

Prwf: We note that

+

Theorem 1038. Suppose that G and H are counrabk discrete commutative gmups and that L and M am principal lefi ideals in BG and pH respectively, defined by nonminimal idempotents. I f @ : M -t 4 is a continwus isomorphism there is an isomorphism f : H + G for which $ = f i ~ .

no

10 ~om~morphism~

PPZR$ LetL = pG+pandM = BH+q,whempandq arenonminimalidcmpolm*lin

G* d H * respectively. By applyingLemma 10.35 with G = S = S a n d H = T = TI, we see that $(q) is nonminimal and t h m is an isomorphism f : H + G such tha~$(t + q) = f(t) $(q) for every t E H. By continuity, this implies that $(x+q) = T(x) +$(q)forweryx E BE. Forx E BH,wehave$(q+x+q) = f(q+_x)+!b(q) = f"(a)+T~)+$(q)and also$(q+x+q) = $(q)+*(x+q) = $(q)+f (x)+$(q). Sof (q)+f(x)+$(q) = $(q) + f i x ) + $(q) for every x E OH. By Lemma 10.37. we can choose x E BH such that f ( x p $(q) is right cancelable in BG. Tbus $(q) = f (q). This implies that $(x q) = f (X q) for every x E pH. 0

-

+

+

+

Theorem 1039. Suppose that L and M a n principal IejY ideals in ON defined by nunminimal idempotents. If$ : M + L is a continuous Lromorphisrn then M = L and $ is the identie map. Pmof Let p and q be nonminimal idempotents in BN for which L = BN + p and M = p N + q . TbenL=pZ+pandM=pZ+q.since,givenr E B Z , r + p = r + p p E BW p because N* is a left ideal in BZ (by Exercise 4.3.5). So L and M are principal left ideals in BZ defined by nonminimal idempotents in Z*. It follows fium Theorem 10.38 that there is an isomorphism f : Z + Z for which $ = f i ~ Now there are precisely two isomorphisms from Z to itself: the identity map r H t and the map r H -I. We can ~ l e _ o uthe t possibility that f is the second of these maps. because this would imply that f (q) $ BN. Hence f is the identity map and so is Jr.

+

+

Theorem 10.40. Let G and H be countable commutativc groups and let L and M be principal right idealr in G* and H* respectively, dqined by nonminimal idempotents. Suppose that there is a continuous Gomorphism $ : M + L. Then there is an isomorphism f : H -+ G for which f [MI = L. Pmof Let p and q be nonminimal idempotents in G* and H* respectively for which L s p + BG and M = q + p H . By applying Lemma 10.36 with G = S = S' and H = T = T'. we see that $(q) is nonminimal and them is an isomorphism : H + G suchthat$(t+q) = f (t)+$(q). Itfollows. bycontin$y,that$(x+q) = f (x)+$(q) for every x E M. Putting x = q shows that M q ) = f (q) $(q). On the other haid, we also have, for every x E H, that $(q x qj = T(q) .h~+$(q)_and$(q+x+q) = $(qJ+W_+x+q) = $(q)+f(q)+f(x)+$(q). for wery * E BH. We can So f (4) f (x) + $(q) = y ( q ) f (4) f ( x )) 4 $ choose x E H* for which f (x) $(q) is right cancelable in BG (by Lemma 10.37) and so f"(q) = S ( q ) R q ) . Itfollowseasily that $(q)+B_G = f"(q)+flG. We alsoknow that $(q)+BG = L 0 (by Lemma 10.36). So = f (q) BG = *(q) BG = L.

t

+

+

+

+

+

+

T[M]

+

+ +

+

+

+

Tbe preceding results in this section tell us nothing about principal left or right ideals defined by minimal idempotents.

.

If S is any discrete semigroup, we know that any Wo minimal left ideals in BS are both homeomorphic and isomorphic by Theorems 1.64 and 2.11. However. the maps which usually define homeomorphisms between minimal let3 ideals are different from those which define isomorphisms. It may be the case that one minimal left ideal cannot be mapped onto another by a continuous isomorphism. It is tantalizing that we do not know the answer to either of the following questions.

Question 10.41. Are there two minimal le~?ideals in BRI with the property that one cannot be mapped onto the other by a continuous homomorphism? Question 10.42. Are there w o distinct minimal Iej3 ideals in PRI with the property that one can be mapped onto the other by a continuous homomorphism?

Notes Most oftheresult of Section 10.1 arc from [331, resultsofcollaboration withV. Bergelson and B. Kra Theorem 10.8 is due to J. Baker and P.Milnes in [lo]. Theorem 10.18 is from [228] and answers a question of E. van Douwen in [XO]. Mosl of the results in Section 10.3 were proved in collaboration with A. Maleki in [180].

'Chapter 1I

The Rudin-Keisler Order

Recall that if S is a discrete space. then the Rudin-Keislcr order SRKon BS is dehned bystatingeafforanyp,q ~ B S . p i ~ ~ q i f a n d ~ n l y i f t h e r e i s a f u n cft i:oSn+ S for which f (q) = p. Alw dthat if p. q E BS, we write p C R K q if p SRK9 and ~ F R ~K ~ ~ n d ~ ~ ~ t e p a ~ ~ 9 i P.f ~ 5 ~ ~ 4 a n d q 5 As a consequence of Thkrem 8.17 one notes that while the relation SRKis clearly reflexive and transitive. it is not anti-symmetric. The relation SRK does induce an antisymmetric partial order on the set of equivalence classes of the equivalence relation ~ R K .

The RudiiKeisler order has proved to be an important tool in analyzing spaas of ulhahlters. It shows how one ultrafilter can be essentially differcat from another. There are deep and difficult theorems about the Rudin-Keisler order which we shall not attempt to prove in this chapter. Our aim is to indicate some of the connections between the Rudin-Keisler order and the algebraic structure of BS. If S is a discrete semigroup. one would expect that the Rudin-Keisler order on BS would have little relation to the algebra of BS,because any bijective mapping from one subset of S to another induces a Rudin-Keisler equivalence between ulaafilters, and a bijstion may have no respect for algebraic structure. Nevertheless, there are algebraic properties which are related to the RudiiKeisler order. For example, if S is countable and cancellative. an element q E S' is right cancelable in @S if and only if p < R K pq and q C R K pq for every p E S*as we shall see in Theorem 11.8.

Ex-

11.0.1. Let S be a discrete space and let p q E BS if and only if p E S.

€

BS. Show that p 5 q for every

Exercise 11.02. Let S be a countable discrete space and let p E BS. show that (q E BS : q SRKp} has cardinality at most c. Exercise 11.03. Let S be a discrete semigroup and let a E Sand p a p SRKp and pa I R p. KShow that a p pa Z R Kp if S is right cancellative.

%RK

E BS. Show that p if S is left cancellative and that

Exercise 11.0.4. Let S be a countable discrete space and let p be a P-point in S*. If q E S* satisfies q ~ R p. K show that q is also a P-point in S*.

11.1 Conndons with Right Cancelability

11.1 Connections with Right Cancelabiity In this section we shall explore the connections between the relation R K and right cancelability in BS. T h w arise from the fact that both concepts arc related to the topological property of smmg discreteness.

Definition 11.1. Let S be a discretespace and let p, q E BS. The temorpmducf p Q q of p and q is defined by ~Q~=(A~SXS:(S:(~:(S,~)EA)E~]E~].

The reader is asked to show in Exercise 11.1.1 that p- Q q- is an ulaafilter on S x S. I f A S SxS.thenA e p@qifandonlyifAcontainsasetofthefom( ( s ,t ) : s E P and t E Q,), where P E p and Q, E q for each s E P. Note that. if s. r E S, we have s 8 t = (s. t ) where. as usual, we identify the point (s,1) with the principal ulaPfilter it generates. If we choose any bijectio56 : S x S -+ S. can rrgard pOq as being an ultra6lur on S by identifying it with # ( p @ q), where # : p(S x S ) + BS is the continuous extension of 6. In panicylar. if s. t E S, then s Q t can be identified with the element #(s, t ) of S. In this section, we shall sometimes think of Q as being a binary operation on S, assuming that a bijection 6 has been chosen. Of course, different bijections will result in p @ q being identified with different elements of pS. These will, however, all be Rudin-Keisler equivalent. So it does not matter which one we choose for sNdying properties of the relation S R K .

we

Lemma 113. Let S be a discrete space and let p, q E S*. Then p 4 4 Zy,l

XI. Define yp : FS((xn)zl)+ Pf(W by ( p ( z t E ~ x O = [XI : t E F). Let 91 = [A E Pf(N) :yp-'[Al E p). 7h-n 91 is o union ulnafitez

Notice first that by Lemma 12.32, the function (p is well defined. One has then immediately that 91 is an oltratiltcr. To see that 1L is n union ultnrfilter, let A € 1L. 'Let A = (p-l[Al. Then A E p so pick a sequence (y,)gl such thatFS(CV,):,) GA and FS((y,Jz,) E p. For n E N, let F,, = yp(y,,). By Lemma 12.34. one has for each H E Pf(N) that yo(&^ y,,) = UIEHFn Consequently, one has that

p,&

o

FU((F.)Z,) 2 A andFU((Fdz,) E 91.

Theorem 12.36. Lcr U be a mion &mfiIfer and let q = m%(U), whrm m = : p(pf (N))-t ON is the confi~ousatmsion of max : 3'f(W) -r N. Then q is a P-point of W. Plwf: Notice that q is a noaprincipal u l ~ l t cbecause r 91 is and the function max is finiteto one on ,F"(N). To see that q is a P-poiat, let (A,)E, be a sequence of membess of q. We need to show that them is some B E q such that B* G A.'. We may presume that Al = N and that A.+I G A. and n # A.+I for all n. @I particular A,, E B.) Dcfim f :N + M by f (I) = maxln E N :x E A"). Let

n?=,

,n:

and and pick i E 10.1) such that 3% E U. Pick a pairwise disjoint sequence in such that W((F,,)Z1) G Bi and W((F,)Z,) E 91. We may pmume that min F,, min F,,+l for all n. Since FU((F,)?=l) E 91. we have that (maxG : G E FU((F.)zl)) 6 q. Since

e(W)

12 UIwfilters Gcrrnted by Finite Sums

248

We show first that i # 0. so suppose instcad that i = 0. Pick k E N suchthat fa n g k,maxF, s m a x F 1 . Sinceqisnonprincipal[maxF. : n E N a n d n z k ) e q . Let f = min FI. Now Ac+l E q so pick n 2 k such that max F. E A ~ + IThen .

a conwdiction. Thusi = 1. LetB=[maxF, : n EN). N o w , i f n , k ~ N a n d m a x F . E B\At, t h a n 5 min F. c f (max F,,)< kso IB\AkI < k. ThusB* A.*asrequimj.

nz,

0

We have need of the following fact which it would take us too far afield to prove.

Theorem 12.37. The existence of P-points in M' c m m t be established in ZFC. P&

r222.

VI, 941.

0

Corollary 1238. The existence ofshongly sunvnable ulnafilterscan notbe established

in ZFC. n it is consistent relative to ZFC that there are no P-points Pmof: By l l ~ e ~ r n12.37 in N*. Theorem 12.35 shows that if strongly summable ultrafilters exist, so do union ultrafilters, while Theorem 12.36 shows that if union ultmilters exist so do P-points in

W*.

0

F.xercise 12.3.1. As in T h e o m 12.31. let U be a union ulaafilter and let p = : F E A ) : A E 91). Define f : P f ( N T) N by f ( ~ = ) ZIEF21-I. Rove that f (U) = p.

SF^'-:

Exercise 12.33. As in Theomm 12.35. let p be a strongly summable ultra6ltcr and pick a sequence (x.)zl in N such that FS((x.)zl) E p and for each n E N, x.+l r 4x:,,xt. Define (p : FS((x.)z,) -t P,(N) by (P(C~FX,)= [x, : I E F ) and extend yp arbih;lrily lo the rest of N. Let U = [A S Pf(W) :cp-'[A] E p). Prove that G(P) = U.

12.4 Algebraic Properties of Strongly Summable Ultrafilters Recall that for any discrete semigroup (S, +), an idempotent p in BS is strongly right m i m l in f3S if and only if (q E BS : q p = pl = (p]. We saw in Theorem 9.10 that strongly right maximal idempotents in f3Pd exist.

+

Theorem 1239. Let p E BN be a strongly surnnrable ultrajlte,: Then p is a srmngly right ll~~n'mal idempotent.

249

12.4 Algebraic Roperties

Proof. By Theorem 12.19, p is an idempotent. Let q f p be given. We show that p # p. Pick A e p\q and pick by Lemma 12.20 a sequence (X,);=~_N such that FS((xJK,) G A, for each m E N. FS((x,)~,,,) E p. and X.+I 2 4 xl for all n. It suffices to show that

4

+

(In fact equality holds, but we do not care about that,) Indeed, one has then that p andthus p # q p. ( Y EN: -Y +FS((xn)El) E PI q s o F s ( ( + ) z I ) To this end, let a E N such that -a + FS((x.)zl) E p. Then

e +

e

+

Pick F andG in 9if(N) such that yl = & ~ x ~ - a n d y = z ~ I , G X , . By L e m 12.34. F n G = 0. Also, a yl and a n are in FS((x,)zl). So pick H and K in P f ( W XI and a + A = EIGK XI. Then such that a yl =

+

+

+

+

+

%X%Vf XI = &PAK XI Z ~ E P2x1 ~K Thus, by h I l l l M 12.32 SO & G A H X ~ GAH = FAK and G flH = F n K. Since F n G = 0. one concludes from these equations that F E H. S o a = &H\FXI and hencea € FS((x,,)zI) as required.

We saw in Corollary 7.36 that maximal group6 in K(BN) are as large as possible (and highly noncommutative). That is, they all contain a copy of the free group on 2' generators. We set out to show now that if p is strongly summable, then H(p) is as small (and wmmutative) as possible, namely a wpy of %.

Lemma 12.40. Let (x,)zl be a sequence in N such thatfor each n, x.+l 4 Z:, x,. Ler F, GI, andG2 befinire subsets ofN such that G I n G2 = 0 and F U GI u GZ# 0. Lcra=mio(FUG~UGz)andlet

Pmof: Assume that t # 0. Let F' = F\(GI U G2). let GI' = ( G I \ F ) U (G2 n F), and let GZ' = G2\F. Then F', GI'. and G2' are pairwise disjoint and

' = E ~ S F '-X ~

-EnEG2'k.

& E ~ , l ~ n

12 Ultratiltus GmcmAby Fite Sums

250

S i n e e t f O , F ' U G ~ ' u G ~ ' 0. f Letb=max(FiUGI'U-G2')and.notiamotbza. Then if b s F', we have

Otherwise b E (GI' U G 2 9 so that

The foUowing lemma is exceedinglytechnical and we apologize in advance for that fact It will be invoked twicc in the p m f of the main result of this section. b4 seqllCnCeS in N a n d a s s m that Lemma 12.41. Ln (x,b:,. (y.):,. and for each n, x.+l > 4 x,. Let L E N and assume that whenever k, j > L, one has SOPIW H ~ E ,g ~ ( N such thor zk yj = Z n f ~ t .xn. J L C ~b = max Ht,t curd (USthat wheneverminlk, jl r L one has min Hk.j > b. L e t t = ( & H ~ . ~ \ H ~ , ~x.) + , yt

x,,I

+

-

P 4 Wehtshowthatforanyk > f.Hk,t\Hk,e+~ = Hc,c\He,t+~ andHt,k\Hc+~.k = Ht.c\Hf+l+c. Indeed, consider the eqoatiws

and Zk

+ Yt+I = CneHt,c+l X n ,

From the first two equations we have

while the last two equations yield

Thus by Lemma 12.33, one has that Ht,c\Hk,t+l = He,c\He,e+l. Likewise. working with z t + ~ zt. one gets that He.k\Hf+l.k = He.t\He+l.c.

-

124 Algebraic Ropnties

Xb+l

Xb+l

xb+l

sndsoltl < -.Similarly. 1st c -andthusIs+rl < -. 4 4 2 Now let k > f be given. Then zk yc = &H,,, x,, so

+

Tocompletethe proof, it thmfmsuffices toshow thats = -1. Fmm thederivatioos above we have that z k yk = C n € ~xn k & E ~ k xn r s and we also know that Zk Yk = & ~ t , Xn. ~ Thus L E H ~ ,Xn - &A Xn %€GI Xn = S 1. k t Kt = (Fk\Gk) U (Gk\Fk) and let K2 = Fk n Gk. Thm

+

+

+

++ -

+

As amnsequence of the next cheorrm (andTheorern 12.29) it is consistent with ZFC that Ume are maximal groups of (BN, +) that are just copies of Z. We do not know whether the existence of such small maximal groups can be established in ZFC.

Theorem 12.42. k t p be a stmngly sumnuable ultrafilter: Ifq E W and them exists r ~ N * s u c h t h o t q + r = r + q = p , t h e n q ~ Z + p. InpaniculazH(p)=Z+p.

Prwf: Let us notice first that since, by Theorem 4.23, Z is contained in the center of (075, +), Z p is a subgroup of (BZ. An4 by Exercise 4.3.5, Z p 5 BN. Consequently Z p H(p).The "in particular" conclusion then follows from the

+

+

+).

+

12 Ultrafitera Oemtcd by Fimte Sum

252

k t ssscrtian immediately, because any member of H(p) is in W and is imrntible wim respect top. So assume we have q and r in N* such that q r = r q = p and s u p p that (l+q,q.-l+q),pickAl i epsuch q p Z + p . 'Ihmlikewise,p$! Z+q. Sincep $ Al q, A1 q, and 1 A1 # q. By Lemma 12.20 wemay presume (by that -1 passing to a subset) that AI = F S ( ( X I , ~ where ) ~ ~ )for each n, XI,,,+I > 4 qj. and for each m, FS((xl,,):-) E p. Let BI = (x E N : -x AI E q ] . Since p = r +q. Bl E r. tick yl E BI. Let

+

e

+
1.letFk = J ~ . t , ~ n J ~ . ~ . ~ a n d 1 Gk = h , ~ . t ~ J ~ , zByLemma12.41 ,k. wehavcthat Irl < -xl.bl+l andforeachk > I, 4 Zk = XncFb X1.n f yC 3 &ck X1.n f. Next let L = 111 (or t = 2 if It1 5 1). Lett' = ( ~ n e ~ ~ xt.") , ~ -, yt. ~ For \ ~ ~ , ~ ~ ~ ~ k 7 l , k t Fk' = Jt.r,c n Jt.r.t+l and Gk' = Jt.t.k n Jt.c+l,t. Then again by Lemma t' and yk = EnEGkv xt." 1'. 12.41 we have that for each k r C, zk = CnEFt,xt.., To complete the +of we show that t = r'. Indeed, once we have done this we will At)), a ha* It+] = ~ I E ~ t + I , +x I~ .Enf At while zt+l E Ct+l g Ct g (W\(t contradiction. Now, given n, xt, E At G FS((XI,.)~=~)so pick T . such that x e , = EmETm XI.^ Also, if n # k, then XI.. E FS((XI,,)~=~) so by Lemma 12.34, T. fl fi = 0. Let k = L 1. We show that ifn E Je,k,t, then min T . 2 a t 2. Indeed, zt yk E A t 5 W(XI.,),&+~) so pick H with zt yk = E ~ ~ Hand x min I , H ~ 3 at 2. On the other hand. usingat the second equality the fact that the T.s' an pairwise disjoint ~ ,= ~C , ~n E ~ ~where ~ . Ln = UmEJ 1 forwhichnbtc korkb3 6 n. SoI#b(n)-#b(k)l ) 3. By-mnma13.4. E 6b(k)+ @b(p)+(O.1). &b(m.p++n.p) E 4b(n)+@b(p)+(-1.0,1.2)md@b(k.1) It follows from Lmuna 6.28, that I#&) #b(n)l 5 2. This contradiction shows that k=n. Suppose now that rn_f 0. Let r E N-k a prime sati-g r r Iml n. For any a E W, we have h+(m)hp(p) +hp(n)h,-(p) = h+(n)h,a(p). Sincer is not afactor of Iml, it followsthat h,n(p) = 0. That is, P N E p for each a e N. Forie[l,2 r-l],let

-

+

....,

m u s Ai is the set of positive integers whose rightmast nonzero digit jn the base r expansion is i.) Pick i E (1.2. .. . r 1) for which Ai E p. For each x E N,let f (x) denote the largest integu in o for which rf(') is a factor ofx. Wehave[mx+s : I EAiands ~rf"'+'N)€m.p+n.~(b~Thcaem4.15) and nAi E n . p. Thus there exist x, y E Ai and s E rf(')+'W for which mx s = ny. We note that f (rnx s) = f (x) and f (ny) = f (y). So f (x) = f ( y ) = t . say. NOW x t irc (mod r"') and y E ir' (mod rf+l). So mx s = mire (mod rt+') and ny Inir' (mod #+I). It follows that m I n (mod r ) and hence that m = n. (b)implies(a). Assumethat k = n. Ifm = O.thenrn.p+n.p = k.pforall p E @Z.

.-

+

+

+

Km=nmdp+p=p,thenbyLemma13.1,n.p+n.p=n.(p+p)=n.p.o Corollary W.19. Let p E N ' .

Then p # (-p)

+ p.

Pmof This is immediate from Theorem 13.18. Corollary 1320. Let p E Z* and kt n, m E Z\{O]. Then (n ondp.(n+m)#p.n+p.m.

0

+ m) .p # n .p + rn .p

P m j Since, by Theorem 4.23. Z is contained in the center of (pZ,.), it suffices to s h o w t h a t ( n + m ) . p # n . p + m . p . Supposeinsteadthat(n+m)-p = n . p + r n . p . 0 Then by Theorem 13.18.11+ m r rn. so n = 0. acontradiction. Theorem 13.18 has special consequencesfor the semigroup 8. Theorem 1331. Let q, r E H a n d let p E 2'. Then p . (q

+

.

+ r ) # p .q + p . r and

. ( p q) .r # p r + q .r. In particular there are no instances of the validity of either distributive law in W.

13 Multiple Stmclmcs in pS

266

+

. +

+

r E A. so by Theacm 13.14. p (q r ) g Z' Z*. Since (Z\(O). .)is cancellative. we have by CoroUaty 4.33 &at Z*is a subaemigroup

P m f By Lemma 6.8, q

o f ( ~ Z . . ) a n d s o p p q + p per E * + Z W . S i m i l a d y ( p + q ) . r f Z * + Z * a n d p . r + q . r ~ Z * f 2'.

0

We now set out b establish the topological rarity of instances of a distributive law in W, if inany such instancesexist at all.

Delhritlon 13.22. Let A s N. A is s h u b $ thin set if ind only if A is infinite and whenever(m. n) and(&,1) andistinctelementsofh'xo,onehas I(n+mA)n(l+kA)I c W.

Lemma 13.23. Let B be rm infiite subset of N. 7'hem is o doubly thin sef A

E B.

Pmqf Enumerate N x o as ((m(k), n(k)))zl and pick sl e B. IDductively k t k E N and assume that q.Q, . ,s k have been chosen. tick st+, z st such that

..

st .m(l)+n(l) -n(i) : i , I , t ~ ( 1 , ,..., 2 k)). Let A = {sk :k E h'). Tben A is infinite. Suppose that one has some I # i such thal

+

k x ( m O , n(1)) f (m(i), n(i)), thwc is at most one x E A such that n(l) m(l)x = n(i) +m(i)x. If there issuch an x = s., let b = max(u, i. I), and otherwise let b = maxfi, I). Pick z E (n(1) m(l)A) n (n(i) m(i)A) with z r n(l) m(l)sb and z > n(i) +m(i)sb and pick t, k E R such that z r n(1) +m(l)s, = n(i) +m(i)sk and notice that t > b and k > b. Then st # s t so assume without loss of generality that t < k. But then n ( 0 mWsr n(i) m(i) o a contradiction.

Now, s

+

+

*= +

+

-

Lnmnn 1324. Let A be a doubly thin subset of N. let p E A*. and let B 5 N x o. ~nce{n+m.p:(m,n)EB)nct(n+m-p:(m,n)E(h'xo)\B]=0. Pmof Supposethatq e cC[n+m.p : (m, n) E B)ncefn+m.p : (m. n) E (Nxo)\B). Order N x w by < in order type o.m a t is. < linearly orders El x o so that each element has only finitely many predecessors.) Let

-

Notice thac C f Dl = 0. Thus witbout loss of gemdity we have N\C E q. Pick (m,n) ~ B s u c h t h a t n + m . p e N \ C . Nown+mA € n + m . p . n + m . p ~ N * , and forevery (n', m') < (n,m). I(n' m'A) fl( n mA)1 c a,since A is doubly thin.

+ (n + mA) \ U(n,,n7 0, (x r, x) E p. (b) Lcr p, q a B(R), l e t x = a(p), and let y = a(q). I f p a x+. then p q e (X y)+. lfp E X - , then p + q E ( x y)-. In (c) B(W)\R = U U D and U and D a m disjohf right ideals of (Em), particular; B(W) is not commutative. (dl The set 0+ is a compact subsemigroup of ( B e ) ,+). (e)Ifx € W a n d p a B i l & , t h e n r + p = p + x . (0The set 0' is a nvo sided ideal ofthe semigmup (B(0, l)d. .).

+

+

-

+

+

+).

Pmof: The pmof of (a) is a routineexercise. (e) is a consequence of Theorem 4.23, and (c) and (d) follow from (b). We establish (b) and (0. To verify (b) assume first that p E x+. To see that p + q E (x + y)+. let r > 0 be given and let A = (x + y, x y 6). To see that A E p q. we show that

+ +

+

.-. ?

.:

13.3 Ultnfltcn on 1Near 0

269

(x,x+c) G IZ E R : -z+A €9). Soletz E (x,x+E). btts =min{z-X,X+E-z). eq Sincea(q) =y.wehave(y-6,y+6) ~ q a n d ( y - 6 . y + 6 ) S -z+Aso-z+A as required. The proof that p q E (x y)- if p E x- is nearly identical. Toveri&(0,letpEO+andletq €/3(o,l)d.To~eeLbtp.q€Of andq.pEOC, let c 7 0 be given. Since (0. c) (0.1) = (0, E). we have (0, c) t (x E (0, 1) : x-'(0, E) E 4) (SOthat ( 0 . ~ 1E p - q) and (0. I) G (x E (0.1) : x-'(O, 0 E p ) (SO that (0, E) E q . p). 0

+

+

.

We shall restrict our attention to 0+, and it is natural to ask why. On the one hand, it is a subsemigroup of (BRd. +) and for any otherx E R,x+ and x- are not semigroups. On the other hand we see in the following theorem that O+ holds all of the algebraic structure of (B(W). +) not already revealed by B.

Theorem 1330. (a) (O+, +) and (0-. +) a n isomorphic. (b) Thefunction ~p : IIb x ((0) U O+ U 0-) + B(B) defined by (p(x, p) = x + p is a continuous isomo~hismonto (B(R). +). Pmofl (a) Define r : O+ -t 0- by r(p) = -p, where -p = (-A : A E p). It is mutine to verify that r takes 0+ one-to-one onto 0-. Let p, q G O+. To see that r ( p + q ) = r(p) +r(q) itsuffices, since r(p+q) and r(p) r(q) are bothulkafilters, to show that r ( p + q ) 2 r(p) r(q). So let A E r ( p q). Then -A E p q so

+

+

+

+

B=(xEIR:-x+-AE~}E~

+

+

and hence - B E r(p). Then -B c {x G B : -x A E r(q)) so A E r(p) r(q) as required. (b) To see that (p is a homomorphism. let (x, p) and (y, q) be in & x ((0)uO+UO-). Since p + Y = Y p. we have (p(x, P) V(Y,4) = (P(X Y,P 4). To see that (p is one-to-one, assume we have (p(x. p) = ~ ( yq). , By Lemma 13.29b)

+

+

+

+

wehavex=rr(x+p)=u(y+q)=y.Thenx+p=x+qsop=-x+x+q=q. To see that ~p is onto B(W), let q E B(W) and let x = ~ ( 4 )Let . p = -x + q . Then

q=x+~=W,p). To see that p is continuous, let (x, p) E 4 x ( ( 0 )U OC U 0-) and let A E x p. Then -x A E p so (x) x (cCpb(-x + A ) ) is a neighborhood of (x, p) contained in

+

+

~p-'lctBWd A]. As we have remarked, OC has an interesting and useful multiplicative structure. But much is known of this shucture because. by Lemma 13.29(0. 0+ is a two sided ideal of (B(0, l)d. .),so K(p(0, I),+..) G 0+ and hence by Theorem 1.65. K(O+, .) = KWO, l)d, .I. On the other hand, OC is far from being an ideal of (B(S). +),so no general results apply to (O+, +) beyond those that apply to any compact right topological semigroup. We first give an easy characterization of idempotents in (O+. +).whose pmof we leave as an exercise.

Theorem 1331. Then exists p = p + p h O+ with A E p ifand only ifthere is some sequence ( x n ) z lin (0, 1) such thor x. c o n v c g e s a n d F S ( ( ~ , ) ~2~A. )

xEl

13 Multiple Smmna in BS

TI0

PI&

n

Thisis Exercise 13.3.1.

As a compact rigbt topo1ogical semigroup, 0+ has a smallest two sided ideal by

Theorcm 2.8. We now twn our attention to characterizing the smallest ideal of 0+ and its closure. If (S. is a disnete semigroup we know From Thcornn 4.39 tha~a point p E BS isinthesdlesti&alofBSifandonlyKforeachA E p.(x E S : - x + A E p ) is syndctic (and also if and only if for all q e BS, p E BS g p). We obtain in ' k o ~ m 13.33 a nearly identical charactcrizahon of K(O+, +).

+)

+ +

Delinition 1332. A subset B of (0.1) is syndetic -0 if and only if forevery c > 0 thcn exist some F E J',((O, c)) and some 6 > 0 such that (0.8) G UIeF(-r 8).

+

Tbconm 1333. La p E Of. Tlvfollowing starmrents om equiwht.

+

Pmt$ (a) implies (b). Let A E p, let B = {x E (0.1) : -x A E p). and suppose that B is not syndetic near 0. Pick c z- 0 such that for sll F E P,((O, a)) and all 8 > 0, (0, 8)\ U,,F(-r + B) 0. Let 9 = ((0. 8)\UCEF(-r B) : F E Pf((O, a)) and 8 > O\. Then g has the finite intersectionproperty so pick r € B(0. l)d with $ G r. S i ((0.8) :8 > 01 G r we have r E O+. Pick a minimal left idcal L of O+ with L E 0+ r, by Corollary 2.6. Since K(O+) is the union of all of the minimal right ideals of O+ by Theorem 2.8, pick a minimal right ideal R of Of with p E R. Then L nR is a groupby Thwrem 2.7, so let q be the identity ofLilR. T h e n R = q + O + , s o p ~ q + O + s o p = q + p s o B ~ q . Alsoq e O + + r s o pickw€O+suchthatq = w + r . Then(0,c)E wand(r E ( 0 . 1 ) : - r + B ~ r ) E w so pick t E (0, a) such that -1 B E r. But (0, I)\(-[ B) E $ S r, a contradiction. (b) implies (c). Let r E Of. For each A E p, let B(A) = {x E (0,l) :-x A E p) and let C(A) = (r E (0.1) : -1 B(A) E r). Observe that for any A], Az E p. one has B(AI n Az) = B(A1) fl B(A2) and C(A1 fl At) = C(Al) fl C(A2). We claim that for every A E p and every c z 0. C(A) fl (0, c) # 0. TO see this. let A E p and c > 0 be given and pick F E Pf((O. a)) and 8 > 0 such that (0.8) C UreF(-f B(A)). Since (0.8) E r we have UleF(-I B(A)) E r and B(A) E r. Then r E C(A) fl(0.r). hencethere is some r E F with -t Thus ((0. r) n C(A) : a > 0 and A E p ) has the finiteintersection pmpeny so pick q E B(O, l)d with ((0.6) flC(A) : s 0 and A E p ) G q. Then q E OC. We claim that p = q r p for which it suffices (since both are ulhdlten) to show that

+

+

+

+

+

+

+

+

+

+

+ +

p~q+r+p.~AEpbegiven.Then(tE(O,1):-t+B(A)Er)=C(A)Eq

soB(A)Eq+r,soA € q + r + p a s r e q u i d (c) implies (a). Pick r E K(O+).

If (S. +) is any d k e t c semipup, we lmow from Cnollary 4.41 +hat,given any discrete semigroup Sand any p E BS. p is in thec~OSM of the smallest ideal of @ Sif and ifeach A E p is piecewise syndetic. We obtain a similar result in O+. Modifying the definition of "piecewise syndetic" to apply to O+ is somewhat less stmightfonuard than was the case with "syndetic".

Dcenitiw 13.34. A subset A of (0.1) is piecmise syndctic near 0 if and only if there exist sequrnws

(F.)zland (6.)z1 such that

(1) for each n E N, F. E Pf (@. l/n)) and 6. E (0. lln), and (2) f a all G E Pf((O.l)) and all p > 0 there is some x E (0, p) such that for all x S Ute,(-r A). n E PI, (G n (0.6.))

+

+

Theorem 13.35. Lcf A E (0.1). 7 7 ~ K(O+) 1 n ctpp~.~),A # 0 ifand only if A is piecewise &etic near 0. Proof. Necessity. Pickp E ~(O+)flcC~(~,l), AandletB = (x E (0. 1) : -x+A E p]. By Theorem 13.33, B issyndeticnearo. Inductively forn E N pick F. E Pf((0.1 In)) and 8. E (O..l/n) (with 6. 5 &,-I if n > 1) such that (0.6.) G Ute,(-t B): Let G E Pf ((0, 1)) be given. IfGfl(O.61) = 0, the conclusion isaiviaL so assume G n ( 0 , 6 1 ) p 0 a n d l a H =Gn(0,6,). Fareachy E H,let

+

m(y) = max{n E N :y

.

4

6").

..

+ ,....

E (1.2,. m(y)}, we haw. y E UrgFa(-r B) so pick r ( y , n ) ~ F ~ s u c h t h M y ~ - t ~ , n ) + B . ' I h c n p i v r n y ~ H a n d n ~ { 1m(y)). .2 onehas-(r(y,n)+y)+A~p. Now let P > 0 be given. Then (0, p ) E p so pick

For each y E H and each n

x E (0. P)

nn ,

n:3)c-ctlv9

+ Y)+ A).

Thengivean ~ N a n d yE Gn(O.&).onehasy E Handn m(y)sorCy,n)+y+x E A so' -NY, n) + A G U,,F.(-' A). Y+x Sufficiency. Pick and (6.)zl satisfying (I) and (2) of Definition 13.34. Given G E Pf ((0.1)) and p z 0. let

(F.)zl

C(G, p) = lx E (0. a) :for all n E M. (G n (0.6,))

+

+ x c U,,,

(-1

+ A)).

By assumption each C(G. p) # 0. Further. given G I and Gz in q((0.1)) and

C ~ I ~2 . > 0. one has C(G1 U Gz. min(p1. p21) G C(GI. P I ) n C(G2, ~ 2 so)

W G . p ) :G E Pf((O.1)) and p > 01

hasthefiniteintmectionpropmysopickpE p(O,l)dwith{C(G. s): G E dp/((0.1)) and p > 0) g p. Note that since each C(G, p) G (0. p), one has p E 0+.

13 Multiple Strucnms in BS

272

Now we claim that for each n E N. O+ n E Nand let q E O+. To show that U,,F"(-t

+ p G clp n, then exists r E Bk such that Bx\ supp(2' 2;) # 0 for each i E (1.2, .. n). Indeed, if r E Bk and Bk G supp(Zr z) then supp(z) Tr Bk = Bk\(r). Consequently I(r E Bt : there is some i E (1.2,. .. .n) with Bk E supp(2' zi))l 5 n. For i E (1.2.. ... n). let z0.i = &H yi.,. Pick the least L such that 2' s n. Now given i we have 2"+Ilzo.i and 2'-' c n 1 so 2'-' E Be-'\ supp(z0.i). Pick ro E Be such that Bt\supp(Z'Q z0.i) # 0 for each i E (1.2. . , n ) and let 21.; = zo,i 2Q. Inductively choose r j E such that supp(ZrJ 2j.i) f 0 for each i E (1.2.. . . n) and let z,+l.i = z,,; 2'j. Continue the induction until t + j = k Z r eyi., ~ foreach i E (1.2,. n) andleta = 2" +Zrl +. .+Zrt-'. where z2'

-

.

+

+

.

+

+

+

+

.

..

+

.

..

.. +

.

Theorem 14.18. There is aset A g N such that A satisfiesthe conclusion of the Centml Sets Zkorem (Theorem 14.1 1)for (N. +) bur K(j3M. +) n cC A = 0. Pmof Let A be as in Lemma 14.17. By conclusion (a) of Lemma 14.17 and Theorem 4.40 we have that K(j3N. +) n cl A = 0. For each e E N, let ( ~ t . , , ) ~ ,be ~ a sequence in N. Choose by condition (c) of Lemma 14.17 some a1 and H I with a [ & r ~ y1.1 I E A Inductively, given a,, and H., let L = max H. and pick m > n in N such that for all i E (1.2,. ,n). a. &H. yi.~< z2-. Pick by condition @)of Lemma 14.17 (applied to the sequences m ( ~ ~ . e + t )(~2,t+r),,~. z~. ... ( ~ ~ + ~ . t + ~ )sot mthat , ~ min . H.+I z e = max H.) some i (1.2 .....n + 1 ) , a . + 1 + Z ~ ~ ~ .y+i ~ . ~E A ~ N P . a.+l and^,+^ s u c h t h a t f o r a ~ E

+

+

.

..

We n m proacd toderive a combhatorial&mc&imtion of central sets. This characterization involvr~the following generalization of the notion of a pieawise syndetic set.

DeRnitlon 1419. Let (S,-) be a semigroup and let A E P(S). Thm A is c o l k c ~ . wisepiecewise syndetic if and only if there exist functions G : Pf ( A )+ P f (S)and x : Pf ( A )x Pf ( S ) -+ S such that for all F G 9fPf.S) and all 3 and X in Pf (A)with 3 E 3one has F .X ( X ,F ) E U I G C ( F , t - l ( n3). Note that a subxt A is piecewise syndetic if and only if ( A ) is collationwise p i m i x syndetic. An alternate characterization is: A is coUcctionwise piecewise syndetic if and only if there exists a function G : P f ( A )-+ P f ( S ) such that

has the finite intRsection property.

ere

Y - ~ ( G ( P ) ) - ' (3) ~ = u , , c , , , y - l t - l ( n ~ ) , sa Exercise 14.5.1.) j/ In the event that the family and the semigroup are both countable. we have a considerably simpler characterization of collectionwise piecewise syndetic.

Lemma 14.20. Assume the semigmup S = (rr, : n

E N) and (A. : n E N ) g P(S). Thefmnily [A. :n E N ) is colleciionwisepiecewisesyndetic ifand only i f t h &a sequence ( g ( n ) ) z lin Nand a sequence ( y n ) z lin S sueh thatfor dl n , m E N with m p n. I(") -I hy., azym,. .. am~mlE Uj=1 aj Ai).

.

(n:*'

Pmof: For the necessity, pick functions G and x as guaranteed by Definition 14.19. Given n E N,let

mdy, = ~ ( ( A Az.. I,

..,A"), (a~.az.....an)). Tben if m 2 n, letting

For the sufficiency, let ( g ( n ) ) z land ( y . ) z , be as in the statemem of the lemma. Given 3 E * ( [ A n : n E N)). let rn = max(k E N : At E 3)and let G ( 3 ) =

T b e o ~ m14.21. Let (S. .) be an inhire srmigmup and kt A C P(S). Them aisu p E K (PS) with A p ifand only ifA is coUectionwiscpiecewisesyndefic.

Sufficiency. PickG :Pf(A) + Pf(S)andx : P f ( A ) x g ( S ) by the definition. For each 3 E 9f((A and each Y E S, let

-+

Sa~gmlced

Then (D(3. y) : 3 E +(A) and y E S) has the finite intasection propmy so pick u E 8.9 such that (D(3. y) : 3 E Pf (A) and y E S) G u. We claim that

To!hisend,let3

E

Pf(A)andkty

€

S. ThenD(3.y)

U,Eccr,f-'(nO so U t e o ( ~ , t - ' ( n 3 )E Y .

G

(x : y . x E

=roqni=d.

By (*) we have that for each 3 E Pf(A). 0s. u G u,,o(p,t-'(nF). Since PS - u is a left ideal of PS, pick a minimal left ideal L of PS such that L c /3S .u and pick q E L. Then for each JC € Pf(A) we have UrEG(mt-'( X) E q so pick r(X) E G(X) such that ( I ( x ) ) - ' ( ~ x ) E 4. .

n

292

14 The Central Sets Theorem

For each 3 E Pf(A). let E ( 3 ) = M X ) : X E PfW and 9 G XI.nK. ( E ( 3 ) : F E Pf (A)] has the finite intersection propmy. So pick w E pS such tha ( E ( F ) : F E P ~ ( A ) } C W . L ~ ( ~ = ~ . ~ . T ~ ~ ~ € L G K1 ( A g p, let A E A. Since E((A)) E w , it suffices to show that E({A)) {r E S : r - ' ~ E q]. so let 3f E P f ( A ) with (A] G 3f. Then (t(X))-'(nd() E q and Ourcombinatorialcharacterization of central is basedon an analysis of the proofs of Theore.mS.8. The imponant thing to notice about these proofs is that when one chooses x, one in fact has a large number of choices. That is. one can draw a tree, branching infinitely often at each node, sothat any path through that trec yields a sequence ( x , ) z t withFP((x.)zl) G A. (Recall that in FP((x,,)El), theproducts are takenin increasing order of indices.) We formalize the notion of 'Wbelow. We recall that each ordinal is the set of its prcderrssors. (So 3 = (0,1,2) and 0 = 0 and, iff is the function ((0,3), ( I , 5 ) , (2,9), ~ I(0,3). (1.5). (2.9)t.) (3,7), (4,5)t, then f i =

DeAnition 14.22. T is a tree in A if and only if T is a set of functions and for each f E T, domain(f ) E o and range(f ) G A and if domain(f) = n > 0, then E T. Tisatreeifandonlyifforsome A. Tisatreein A. The last requirement in the definition is not essential. Any set of functions with domains in w can be converted to a tree by adding in all reshictions to initial segments. We include the requirement in the definition for aesthetic reasons - it is not nice for branches at some late level to appear from nowhere.

Ddinitlon 14.23. (a) Let f be a function with domain(f) = n E w and let x be given. Then f-x = f U ((n, x)). @)GivenaaeeTandf € T , B f = B f ( T ) = ( x : f-XET]. (c) Let (S, .) be a semigroup and let A E S. Then T is a *-me in A if and only if T is a tree in A and for f E and n E B/. B/-, x-I B/. (d) Let (S, .) be a semigroup and let A 5 S. Then T is an FP-tree in A if and only if T is a tree in A and for all f E T, BJ = (17,,Fg(t) : g E T and f g and 0 # F G dom(g)\dom(f)).

c

The idea of the terminology is that an FP-tree is a tree of finite products. It is this notion which provides the most fundamental combinatorial characterization of the notion of '*central". A *-tree arises more directly from the proof outlined above.

Lemma 1424. k t (S, .) be an injnite sernigmup and let A G S. Let p be an idemporent in BS with A E p. There is an FP-tree T in A ~ u c hthat for each f E T. Bf E P. Pmof: We shall define the initial segments T" = (f E T : dom(f ) = n) inductively. Let To = (0) (of course) and let Cn = A'. Then Cn E p by Theorem 4.12. Let TI = (((0.x)) : x E Cn).

I I

b

293

14.5 Combmatorial Characterization

Inductively assume that we have n E N and have defined T. so that for each f E T. one has FP((f (t))y,-d) E A'. Given f E T,. write Pf = FP((f (t))y;d),let Cf = ~'n~,,,,x-~~*,andnorethatby~emma4.14,C~ E p. LetT.+l = I f - y : f E T, and y E C f )T h m given g E T.+I. onehas FP((g(t))Y4) E A'. T. Then T is a tree in A. Om The induction being complete. let T = sees immediately from the construction that for each f E T . Bf = C f . We need to show that for.each f E T one has Bf = ( n , , ~ g ( t ): g E T and f P g and 0 f F E dom(g)\dom(f)). Given f E T and x E B f , let g = f-x and let F = dom(g)\ dom(f ) (which is a singleton). Then x = ll,,F g(t). For the other inclusion we first observe that i f f ,h E T with f E h thm Pf E Ph so Bh E B f . Let f E T,, and let x E g(t) : g E T and f $ g and 0 # F c dom(g)\dom(f)). Pickg E T with f gandpickFwith0 # F dom(g)\dom(f) such that x = & g g ( t ) . First assume F = ( m ) . Then m 2 n. Let h = g,,. Then f E h and h-x = glm+l E T . Hence x E Bh E Bf as required. Now assume . g(t), and let IF1 1 I , let m = max F , andlet G = F\(m). Let h = gl,. let w = y = g(m). Then y E Bh. Let Pf = FP((f (1)):;) and P . = FP((h(t))ysl).Weneed toshowthatw.y ~ B f . T h a t i s , w e n e e d w . y ~ A ' a n d f o r a l l zP~f , w . y E Z - I A * . N o w w ~ ~ a n d ~ ~ B h s o y ~ w - ~ A ' s o w ~ y ~ A * . L e t z ~ P ~ . T h e n z . w and y E Bh soy E ( Z . w)-'A' and so w . y E z-~A*.

Us

(n,,~

n,,G

Theorem 1425. Let (S, .) be an injinite semigmup and let A c S. Statements (a). (b). (c),and ( d )are equivalent and are implied by statement (e). I f S is countable, then aN jive statements are equivalent. (a) A is centmL (b) There is a FP-tree T in A such that ( B f : f E TI is collectionwise piecewise

syndetic. (c) There is a *-tree T in A such that ( B f : f E TI is collectionwise piecewise syndetic. (d) There is a downwarddirecredfamily ( C F ) F of ~ Isubsets of A such that (i)foreach F E I andeachx E C F thereexists G E I with Cc E x - I c F and (ii) {CF: F E I ) is collectionwise piecewise syndetic. (e) There is a decreasing sequence (C. )Elof subsets of A such that (i)for each n E Wand each x E C. there exists m E N with C, E x-I C,, and (ii) (C. :n E N)is collectionwisepiecewise syndetic. Pmof: ( a ) implies (b). Pick an idempotent p E K ( B S ) with A E p. By Lemma 14.24 pick an FP-tree with { B f : f E TI E p. By Theorem 14.21 ( B f : f E T ) is collectionwise piecewise syndetic. (b) implies (c). Let T be an FP-me. Then given f E T and x E B f . we claim that Bf-x E x - ' B f . Tothisendlety E Bf-,andpickg E TandF Sdom(g)\dom(f-X) such that f - x g and y = g(t). Let n = dom(f ) and let G = F U In). Then g(t) and G E dom(g)\dom( f ). so x . y E Bf as required. x .y =

n,,c

nrE~

(c) implies (d). Lct T be the given *-Ira%S i (Bf : f E T)is cOUe*i~nwi~ piefnvise sjmdetic, so ia Bf : F E Pf(T)). Obis can be seen directly or by hmking Theorem 14.21.) Let I =T (O /$' and for each F E I . let CF = Bf. Then [CF : F E I ) is coUeCtio~wisepiecewise syndehc, so (u) holds. Let F E I and l e t x ~ C pL . e t G = [ f - x : f E F ] . T h e n G ~ I . N o w f o r e a ~ b ~f F w e b a v e Bf-= C x - ' ~ f so Co E x-~CF. cL CF. By Themem 4.20 M is a subsemigroup of (d) implies (a). Let M = @S. Since [CF : F E I) is collectionwise piecewise sylldetic, we haw by ThtOnm 14.21 that MnK(pS) jA 0 so wemay pickaminimal Ieftidcal L of pS with LflM # 0. Tbm L fl M is a compact subsemipup of pS which thus contains an idempotent, and this idempotent is necessarily minimal. That (e) implies (d) is trivial. Finally assume that S is countable. We show thaI (c) implies (el. So let T be the given *-tree in A. Then T is countable so enurnexate T as (f.)zl. For each n E N. Bfk. Then [C, : n E M) is collcaionwise piecewise syndetic. Let let C, = n E N a n d l e t x ~ C , , . Pickm €Nsuchthat(fk-x:kE(1.2 ....,n)] E [ f , : t E (1.2 .....m)]. Then C, = Bfi E Bfi-, E f l , l x - l ~ f i =x-'c.. 0

tnfEF

nfEF

n,,,

n;=,

n:!,

n;=,

We close this section by pointing out a Ramscy-theontic consequence of the charsacrization. Ai. Them CoroIhry 14.26. Let Sbem infinite semigroup, let r E N, and k t S = mit i E [I. 2,. I ) mufan FP-tree T in Ai such that (Bf :f E T)is collectionwise

.

..

piecewise syndctic.

Pmofi Pick an idempotent p E K(pS) and pick i E (1.2, Apply Theorem 14.25.

.

... r ) such that Ai

E p. 0

Ex& 145.1. Let Sbeasemigroup. Rovethatafamily A g Z(S)is colleztionwia piecewise syndetic if and only if there exists a function G :Pf (A) + Pf ( S ) such that [y-' (G(P))-'(n F) : y E S and F E Pf (A)) bas the finite intersection property. (Where Y - ~ ( G ( F ) ) - ' ( ~ r)= u,,,,,,Y-lt-'(n ~ 1 . )

Notes The original Central Sets Theorem is due to Furstenberg [98, Proposition 8.211 and

applied to the semigroup (N.+). It used a different but equivalent definition of c d . SeeThmrem 19.27 fora proof of the equivalenccof the notions of central. This original to take values in Z.Sina version also allowed the (finitely many)sequences (yc,,,)~=l any idempotent minimal in (PW. +) is also minimal in (BZ. +). and hence any c e n d set in (W. +) is central in (Z, +),the original version foliows from Theorem 14.1 1.

'Iheidea forthe @of theCenaal Sc& Tbmrcm(aswell as the proofs in this chapter of vsn der WaerdetfsThaonm and the Hales-JewenThearem) isdue to Furstenberg and Katmelson in [W]. where it was developtd in the context of enveloping semigrmps. The idea to convat this proof into a proof in @ Sis due to V.Bergelson, and the constmction in this context first appeared in [27]. Corollary 14.13 can in fact be derived from the Hales-Jcwen Theorem (CoroUary 14.8). See faexample [28, p. 4341. The material in Section 14.5 is from [148]. a d t of coUaboration with A. Malelti. except for Theorem 14.21 which is from [147l, a result of collaboratioa with A. Lisan.

Partition Regularity of Matrices

In this chapter we present several applications of the Cenaal Sets Theorem

waxem

14.11)andofitsproof.

15.1 Image Partition Regular Matrices Many of the classical results of Ramsey nKory arr natutally stated as instances of the following problem. Given u, v E N and a u x v matrix A with non-negative integer entries, is it hue that whenever N is finitely colored there must exist some .? E Nusuch that the entries of A.i are monochrome? Consider for example van der Waerden's Theorem (Corollary 14.2). The arithmetic progression [a, a + d , a + 2 4 a + 3d) is precisely the set of entries of

Also Schur's Theorem (Theorem 5.3) and the case m = 3 of HiIben's Theorem (Theorem 5.2) guarantee an affirmative answer in the case of the following two matrices:

This suggests the following natural definition. We remind the reader that for n E N and x in a semigroup (S. +),nx meansthesumofx withitself n times, i t . the additive version of x". We use additive notation here because it is most convenient for the matrix manipulations. Note that the requirement that Shave an identity is not substantive since

F

15.1 Image Partition Regular Matrices

297

one may be added to any semigroup. We add the requirement so tbat Ox will make sense. (See also Exercise 15.1.1, where the reader is asked to show that the central sets in S are not affected by the adjoining of a 0.) Deenition 15.1. Let (S, +) be a semigroup with identity 0, let u, v E N, andlet A be a u x u matrix with entries from o.Then A is image panition regular over S if and only if whenever r E N and S = U:=, Ei, there exist i E (1,2,. ,r ] and i E (S\(O))" such that A? E EjU.

..

It is obvious that one must =quire in Definition 15.1 that the vector x' not be constantly 0. We make the stronger requirement because in the classical applications one wants all of the entries to be non-zero. (Consider van der Waerden's Theorem with increment 0.) We have restricted our matrix A to have nonnegative enlries, since in an arbitrary semigroup -x may not mean anything. In Section 15.4 where we shall deal with image partition regularity over W we shall extend the definition of image partition regularity to allow enaies from Q. Deanition 152. Let u, v E N and let A be a u x v matrix with *tries from Q. Then A satisfies thefirsf entries condition if and only if no row of A is 0 and whenever i , j E (1.2,. . . , u ) and k = min{t E (1.2 v) : a i , ~# 0) = min{f E (1.2, .... v) : aj., # 0). then ai,k = aj.k > 0. An element b of Q is afirst entry of A if and only if # 0). thereissomemwi of A such that b = aLkwhere k = m h { t E [I, 2. . .. u ) :

,...,

.

' of all sets that Given any family R of subsets of a set S, one can define the set 2 meet every member of R. We shall investigate some of these in Chapter 16. For now we need tbc notion of central* sets. IMMtion 153. Let S be a xmigroup and let A G S. Then A is a centraP set of S if and only if A n C # 0 for every central set C of S.

Lemma 15.4. Let S be a semigmup and kt A G S. Then thefollowing statements a n equivalent. (a) A is a cennar set. (b) A is a member of every minimal idempotem in BS.

(c) A n C is a cenml set for every central set C of S.

Pmoj: (a) implies (b). Let p be a minimal idempotent in BS. If A $ p then S\A is a cenual set of S which misses A. (b) implies (c). Let C be a central set of S and pick a minimal idempotent p in flS suchthatCEp. ' I h c n a l s o A e p s o A n C e p. That (c) implies (a) is trivial. 0 In the following theorem we get a conclusion far stronger than the assertion that mahices satisfying the first entries condition are image partition regular. The stmnger conclusionisof some interest in itsown right. Moreimportantly, thestrongerconclusion is needed as an induction hypothesis in the proof.

298

I5 Partition Regularity of Maaices

TbcMcm 155. k t (S. +) be an infinite c o m smigmup ~ with identity 0, let u, v E N, and &t A be n u x v matrix with nfriesfrom co which sutigksthefirst m e condition Let C be cenfml in S. for everyfirst entry c of A, cS is a cmtml* set. then there exisfsequmces ( X I , ~ ) E(x~.d,,~. ~ , d(l .... (xvr)z1 in S such that for eve^ F E pf ((N, ?F E (S\(O))" and A i F E Cu. whem

Pmof: S\{O) is an ideal of S so by Corollary 4.18.0 is not a minimal idempotent. and thus we may prcsume that 0 p C. We proaed by indnction on u. Assume f a t u = 1. We can assume A has no repeated w s . so in this case we have A = (c) for some c E N such that cS is central*. Then C n c S is a central set so pick by Theorem 14.1 1 (with the sequence y l , = 0 for each n and the function f constantly equal to 1) some sequence (k,,)E1 with FS((k,)=,) C C rl cS. (Infact here we could get by with an appeal to Theorem 5.8.) For each n € N pick some XI, E S such that .k = cxl,,,. The sequence XI..)^^ is asrequired. (Given F € 4 ( W . &F .k f 0 so % € F X1.n f 0.) Now let u E W and assume the theorem is m e for v. Let A be a u x ( v 1) mmix with entriesfrom o which satisfies the first enhies condition, and assume that whenem c is a first entry of A, cS is a central* set By m g i n g the w s of A and adding additional w s to A if need be. we may assume that we have sow r E {I, 2, ..,u - 1) and some d E N such that

+ .

0 i f i € (1.2. ..., r ] ai.1 = d i f i E { r + l , r + 2

I

.....u).

Let B be the r x u matrix with entries bi,, = ai.j+!. Pick sequences (ZI.~):~, ( z ~ . ) ~...~, (. z ~ , ~ )inE S , as guaranteed by the inducuon hypothesis for the matrix B . F o r e a c h i e { r + l , r + Z ...., u J a n d e a c h n ~ W . l e t

and let y , . = 0 for all n E N. (For other values of i, me may let y i , take on any value in Sat all.) Now CfldSiscentral, sopickbyTheorem 14.11 sequences h S a c d (H.)zl in Pf(N) such that m a H. c min H,+I for each n and for each i E (r, r + 1. u).

.

...

(If (kL)zl and (H;)zl are as given by Theorem 14.1 1, let k,, = V,+=and H, = HL, foreveryn.andfori E ( r , r + I . ...,u l , l e t f i ( n ) = n i f n < i a n d l e t f i ( n ) = i i f n 2 i.)

15.1 Image Parrition Regollr M d a s

299

+

Note in parcicularthat each S, = k,, E,.H, y,., e C C dS, so pick X I , S such bat k. = d x ~ , " .For f .E 12.3,. ,v 11. let xj, = E ~ E Hzj-I.,. . We claim that the scquCWXS (x~..):=~ are as reqoired To see this, let F E Pf(N)be given. We need to ~ E (1.2 u). s h a v t h a t f o r e a c h j ~11.2 ..... ~ + l ) . & ~ x j fOmdforeachi

.. +

....,

For the lira &on m e that if j > 1. then = xrEczj-1.1 where G = U I E ~ H m I. f j = 1 . h d X n E ~ x = ~ ,Cns.=(km E t E ~yr,d . E C. To establish the second assenion, let i E ( 1 . 2 . . u) be given. . Case 1. i 5 r. Then

...

+

where G = UnEFH.. C a s e 2 . i > r . Thm

We now preseat the obvious partition regularity m U q . But note that with a little

effort a stmnger result (Theorem 15.10) follows. Corollary 15.6. k t (S. +) be an in@ite c o m m u ~ v semigmup e and let A be afinite matrix with entriesfmm o which sah'sfies the first rnm.es condition. Iffor each first erury c of A, cS is a central* set, then A is image pam'tion regular over S.

u,,

P m J Let S = Ei. Then some Ei is cenbal in S. I f S has a two sided identity. thenTheomn 15.5 applies directly. Otherwise,adjoin an identity 0. Then, by Exmise 15.1.1. Ei is central in SU (0). Fmrher, given any first entry cof A, c(SU (0)) iscentral* in S U ( 0 ) so again Themm 15.5 applies. Some common, well behaved. semigroups fail to satisfy the hypothesis that cS is a central* set for each c E N. For example, in the semigroup (N..), [x2 : x E W) (the multiplicative analogue of 2 s ) is not even a central set. (See Exercise 15.1.2.) Consequently B = W\{x2 : x f N) is a central* set and A = ( 2 ) is a first entries matrix. But thm does not exist a E N with aZ E B. To derive the stronger partition result, we need the following immediate corollary. For it, one needs only to recall that for any n E N, nN is central*. (In fact it is an IP* set; that is nN is a member of evcry idernpotent in (BN, +) by Lemma6.6.) Corollary 15.7. Any finite manix with eniriesfrnn o which sofisjies thefirst enrries condition is image panition regular over N. 0

15 Wltition R q u h t y of Mabins

300

Corollary 15.8. La A be a h i r e marrix with entries from. w which sorisfirs the fim entries condrtion and let r E N. There exl'st9 k E N such that whenever (1.2. ...,k ) & r-colon4 the= exiits .? E (1.2, ...,k)" such char the entries of A? a n rnonochmmc

Pmof: This is a standard compactness argument using CoroUary 15.7. (See Section 5.5 or the proof that (b) implies (a) in Theorem 15.30.) See also Exercise 15.1.3. 0

'

As we nmrnked aftex Definition 15.1, in the definition of image panition ngularity we demand that the entries of .? me all non-zero because that is the natural vmio. for the classical applications. However, one may reasonably ask what happens if one weakem the conclusion. (As we shall see. the answer is "nothing".) Likewise, one may strengthenthe definition by requiring that all entries of A? be non-zero. Again, we shall see that we get the same answer.

+)

Definition 15.9. Let (S. be a semigroup with identity 0, Ie4 u, v E N,and let A be a u x v matrix with entries from o. (a) The matrix A is weakly image pam'tion regular over S if and only-if whmever r E N and S = U;=l E,,there exist i E (1,2. .. . ,r ) and i E (Su\(0)) such that A? E EiU. (b) The matrix A is strongly imagcpammtionregular over S if and only if whemvcr r E N and S\(O) = U{=l El,there exist i E (1,2, . r ] and .? E (S\(O))" such that

..

Ax' E

EiU.

.

Theorem 15.10. Let (S, +)be a commutativesemigroup with identify 0. 7 hfollowing statements a n equivalent. (a) WheneverA is a h i r e matrix with entriesfmnr o which

satisfies thefirst entries condition. A is stmngly image partition regular over S. (b) WheneverA is afinite matrix with entriesfmm o which sotisfis thefirst enhies condition A is image partition ngular over S. (c) WheneverA is afvrite ma& with enniesfmm o which sariipes thefirst entries condition, A is weakly image partition regular over S. (d) For each n E N,nS # (0).

P m t That (a) implies (b) and (b) implies (c) is trivial. (c) implies (d). Let n E Nand assume that nS = (0).Let

Then A satisfies the first enmes condition. To see that A is not weakly image partition regular over S, let Et = (0) and let E2 = S\{O). Suppose we have some.? =

'1 .

153 Kcmcl Partition Regular Matrices

301

~ ~ \ {and 6 ] some i < (I. 2) such that A?

E (Ei)3. Now O = n q so i = 1. Thus +nxz =OsoLhatxz = X I + X Z =Oandhence.?=O,acontradietim. (d) implies (a). Let A be a finite matrix with entries from o which satisfies the first entries condition. To see that A is strongly image partition regular over S,let r E N and let S\[O)= Ei. Pick by Corollary 15.8 some k E W such that whenever (1.2,. . , k) is r-colored, there exists .? E ((1.2, ... ,k))" such that the entries of A.? are m~ochrome. There exists z E S such that [z, 22.32,. .,kzl n (0)= 0. Indeed. otherwise one would have k!S = (0).So pick such z and for i E (1.2,. . . r ) . let Bi = (t E (1.2.. .. ,k) : rz E E i } . Pick i E (1.2,. ...r ] and 9 E ((1,2, ...,k))" such that A$ E (Bi)". Let.? = jz. Then A? E (Ei)'. 0 XI = X I

UT=,

.

.

.

Exercise 15.1.1. Let (S,+) be a semigroup without a two sided identity, and let C be central in S. Adjoin an identity 0 to S and pmve that C is central in S U (0). Exerdse 15.1.2. Prove that (x2 : x E central) in (N..).

N) is not piecewise syndetic (and hence'not

Exercise 15.13. Prove Corollary 15.8. Exerdse 15.1.4. Let u, v E N and let A be a u x v matrix with entries from o which is image partition regular over N. Let T denote the set of u l M t e r s p E @N with the property that. for every E E p. there exists .? E PT for which all the entries of A.? are in E. Prove that T is a closed subsemigroup of BN.

15.2 Kernel Partition Regular Matrices If one has a group (G.+) and a matrix C with integer entries, one can define C to be over G if and only if whenevqr G\(O)is finitely colored there will exist i with monochrome entries such that C.? = 0.Thus in such a situation, one is saying that monochrome solutions to a given system of homogeneous linear equations can always be found. On the other hand, in an arbitrary semigroup we know that -x may not mean anything, and so we generalize the definition. kernelplrt'fion regular

Definition 15.11. Ln u , u E N and let C be a u x v matrix with entries from Z. Then C+ and C- are the u x v matrices with entries from o defined by czj = (IciVjI + q j ) / 2 and c; = (lci,jI - q j ) / 2 .

Thus, for example, if

c=(.

1 -3 -2

0 2)g

1 0 0 then

Note that C = C+ - C-

c+q0

2)

0 3 0 and C-=( 0 2 0

)

'

m

I5 PartitimRgolarityof~0~1

+)

IkMUm 15.U Let (S. be a scmigmup with identity 4 l a u. v E Nand let C be a u x' v matrix with enhies fmm Z.Then C is kemlpam'tion regular o w S if and d g if whenever r E N and S\{0) = U:=, Di, then exist i E (1.2,. ,r ] and x' E (Di)u

..

such that C+x' = C-x'.

The condition which guamntees kernel partition regularity owr most semigroup & known as the columnr condition. It says that the columns of the mahix can be g a t h a into groups so that the sum of each group is a linear combii$on of columns horn prcecding groups (and in particular the sum of the first group is 0).

-- .

Dcgnitlon 15.13. Let u , v E N. l a C be a u x v matrix with enhies horn Q rmd l a cl. a,. . . E; be the columns of C. Let R = Z or R = Q. The matrix C satisfies the columns condition over R if and only if there exist m E Nand 11.Iz. . I,,,such that

..

.

.....Im)isapartitionof(1.2.....v].

(1) {11,12

(2) &I, 6 = 6. (3) If m > 1 and r E ( 2 . 3 . . .. m]. let JI = R such that XI,,, 6 = Xi,,, 61.r.cj.

.

u!-'

1-1

Ij. Then there exist (61.i)i,,, in

Observe that one can effectively check whether a given maaix satisfies the cohmms condition. me problem is, however. NP complete because it implies the ability to determine whether a subset of a given set sums to 0.)

Lemma 15.14. Supposethat C isa u x m matrix with entriesfmm Z which sati$es the firnenbiescondition. L . e t k = m a x { ~ c i ~ ~j :) (Ei {. 1 , 2,...,u)x{1,2 ,...,m ) ] + l , and let E be the m x m matrir whose entry h mw t Md column j is

Thm C E is a ntolrix with enhiesfmn, o which also satisfiesthefirst entries condition Pmef Let D = CE. To see that D satisfies the first enhies condition and has entries horn o,let i E (1.2,. ,u] and lets = minlt E (1.2,. . m ] : ci., # 0 ) . Thm for j < s. d;:.j= 0 and di,, = ci,,. If j s s, then

..

..

A connection between mahices satisfying the columnscondition and those satisfying

the first enhies condition is provided by fhe following lemma

a-

Lem~15.15.Lnu.v E N a n d I r t C b e a u x vmafrixwith&sfmnrQwhich satisfiesrhrcol~condirionowrQ.Themadnm e (1,2. vJmdavxmmatrix B with enrriesfmma,thatsari.$es thefirst enfries conditiomsuch that C B = 0,where 0 is the u x m matrix with all zero mnies. IfC ssolirfus the columns condition over Z then the morriX B can be chosen so that its onlyfirst entry is 1.

...,

Pmof: tick m E N, (It):=,. (J&. md ((&,i)isr,)E2as gu~fanreedby the columns condition for C. Let B' be the v x m matrix whose enby in row i and column t is given by - 6 if i E J: 0

ifi

CUj,, 4.

We observe that 8' satisfies the first eohies condition, with the frrst non-zem enby ineachmbeingl. WealsoobsmethatCB'= 0. Ondeed,letj E (1.2. ..., u)and I E (1.2 ml. I f f = 1,then &cj.i .bi,, = &r, cj.i = Oandift w 1,then Er-1 Cj,i ' b;,, = &J, -8t.i ' cj.i Cis,, cj,i = 0.) We can choose a positive integer d for which dB' has entries in 2. ?hen dB' also satisfie8the first entries condition and the equation C(dBf) = 0. If all of the numbers 6i. are in 2,let d = 1. L c t k = m a ~ ( l d b ~ , ~ l : ( i , j ) ~...., [ 1 , 2v ) x ( 1 . 2 ,...,m)]+l.aadletEbethe m x m matrix whose enby in IUW i and column j is

.....

+

By Lemma 15.14. B = dB'E has entries in o and satisties the lirst enhies condition. Clearly. CB = 0. Now,giveni ~ ( 1 . 2 v),choosese(1.2 ,...,m)suchthati E I,. I f t < s , then b;, = 0 while if t > j, then e:,j = 0. 'Ihus if j c s, then binj= 0 while if j 2 s. then bi, 3 E/=, dbi,Ikj-f. In particulrn the first nonzero entty of row i is d.

....,

We see that, not only is the columns condition over Q sufficient for kernel partition regularity over most semipups. but also that in many cases we can guarantee that solutions to the equations C + i = C-x' can be found in any central set.

+)

Theorem 15.16. Ln (S. be an injinire commutative semigroup wirh idem.ty 0, let u, v E R and let C be a u x v nrofrir with enm'esfivm Z. (a) If C sarisfis the columns condition over Z thenfor any cenmrl subset D of S, them aists ? i D" such that C+i = C-L (b)If C sofisfies the columns condition over Q andfor each d E N dS is a cmtml*set in S. thenfor any central subset D of S. there &IS 2 E DVsuch that C+x' = C-2. (c) IfC satisfies the columnr condition over Q and for each d E N, dS # (01, then C is kernelpm'rion regular over S.

P n $ (a) Pick a u x m mamix B as guaranteed for C by Lequna 15.15. Then the only firstentry of B is 1 (and IS = S is a central* set) so by Theomn 15.5 we may pick some 2 E Sm such that B i E DU. Let 2 = Bi. Now CB = 0 so C+B = C-B (and all entries of C+B and of C-B are non-negative) so C f i = C+Bi = C-BZ = C-i. (b) This pmaf is nearly identical to the proof of (a). (c) F'ickamauix Bas g u m t e e d for C by Lemma 15.15. Let r E N andlet S\(O) = D,. By Theorem 15.10chwseavecta~E (S\(ODm and i E (1.2.. .. , r ] such that BE E (D;)", and let i = LIZ. As above, we wnclude that C + i = C+Bi = C-B? = C-i. 0

Exercise 15.21. Note that the matrix (2 -2 1) satisfies the columns condition. Show that there is a set which is cenhal in (N,.) but contains no solution to the quation x12- x3 = ~ 2 ' . (Hint: Consider Exercise 15.1.2.)

15.3 Kernel Partition Regularity Over N Rado's Theorem In this seetion we show that matrices satisfying the columns condition over Q arc precisely those wbicharekemel partition regularover (N, +)(which is Rado's Theorem) and are also precisely those which are kernel partition regular over (N, .). Given a u x v ma& C with integer entries and given 2 E NVand 9 E NUwe write iC = j: tomean that for i E (1.2,. ..,u], xjC,-; = yi.

ny=,

Theorem 15.17. La u. v E R and let C be a u x v ma;& following sratements are equivalent.

with entriesfmm 2 The

(a) The mnrrix C is kernel partition regular over (N, +). (b) The mar& C is kemelpa&ion regular over (N..). That is, whenever r E N a@ w\(l] = Ui=,D;, thereexist i E (I. 2,. . ., r ] a n d l E (Di)"such thnriC = I.

us,

P m f (a)irnplies(b). Letr E WandIetR\{I] = D;. Foreachi E (1,2, .. ., r!, letA;= (n E W : T e Di). Picki E (1.2.....r J c $ ~ dEj (Ai)"suchthatC~=O. Foreach j e (1,2,..., u}.letxj = 2 " J . T h e n i C = 1. (b)implies (a). Let ( p ; ) z , denote the sequenceof prime numbers. Each q E Q\(O)

can be expressed uniquely as q = nE1p?, where e; E Z for every i. We define 9 : Q\{O] -+ Z by putting +(q) = C z l ei. (Thus if q E R. O(q) is the length of the prime factorization of q.) We extend 9 to a function I) : (Q\(O])" + Zuby putting *(f); =#(xi) foreach i E (Q\(OI)" and each i E (1.2.. . . v]. Since 4 is a homomorphism from (Q\{O], .) to (Z, +),it follows easily that @ ( i') = C g ( l ) for evety l E (Q\(0]IV. Let {Ei)l=lbe a finite partition of N. Then (@-'fEi] OR):=, is a finite partition of N\(IJ, because 9[N\(I]] G W. So pick i E (1.2.. ,k) and

.

..

2'

15.3 Kaml Partition Regularity Over N

2 E_ (#-f[Eil n Nusuch that; *(I) = 0.

= i. Then $(?)

E

305

(EiY and C@(.Z)= $(zC) = 0

Definition 15.18. Let u, v € N, let C be a u x v matrix with entries from Z.and let J and I be disjoint nonempty subsets of (1.2,. . . v ) . Denote the columns of C as

.

- - ..,

+

Cl, C2,. C. (a) If there exist (zj)jcj in Q such that t, = EjEJzj+ then E(C, J , I ) = 0. (b) If there do not exist (zj)jtj in Q such that 3 = EjeJz,Zj, then

xjCi,, xjE1

E ( C , J. I ) = {q : q is a prime and there exist (zj)j.j in o, aE(1.2 q-L),d~o.a n d j ~ P such that EjEjzjZj aqd Cjer 3 = q d + l j ).

.....

+

Lunma 15.19. Let u . v E R. let C be a u x u mufrix wirh entriesfrom Z, and let J and I be disjoint mnempfysubsetsof (1.2, ... v). Then E(C, J , I ) isfinite.

.

Pmof: If (a) of Definition 15.18 applies, the result is trivial so we assume that (a) does not upply. Let b = Ej.r i,. Then b is not in the vector subspace of Q" spanned by (Z,),EJ SO pick some 3 E Qu such that 3 Z, = 0 for each j E J and 3 i # 0. By multiplying ky a suitahre m e m e of Z we may assume that all entries of 3 are integers and that lir . b s 0.Lets = ti! . 6. We show now that if q E E(C. J, I), then q divides

.

.

5.

Letq E E(C,J. I)andpick(zj)j.jino.a ~ ( 1 . ..... 2 q-1),d c o , a n d y ' ~ Z " such that Ej,j z j 3 aqd EjE,Zj = qd+I?. Then

+

.

Since 3 - Zj = 0 for each j E J we then have that aqds = qdt'(G j) and hence a s = q ( 3 7). Since a E (1.2, . . .,q 1). it follows that q divides s as claimed. ir

-

The equivalence of (a) and Ic) in the following theorem is Rado's Theorem. Note that Rado's Theorem pmvides an effective computation to determine whether a given system of linear homogeneous equations is kernel partition regular. Theorem 1530. Let u. v E R and Iet C be a u x v morrir with entriesfmnt Z. The following statements are equivalent. (a) The matrix C is kernel panition regular over (N. +). @) The marrir C is kenelpartition regular over (R. .). That is, whenever r E N a* N\{I] =U:=, Di,thereerisri E (1,2, .... r l a n d l ~(Di)"suchthariC = 1. (c) The matrix C satisfies rhe columnr condition over Q. Pmofi The equivalence of (a) and (b) is Theorem 15.17,

(a) implies (c). By Lrmma 15.19, whenevez J and I are disjoint wnempty s u b of 11, 2, v), E(C, J, I ) is finite. Consequently, we may pick a prime q such that q>-(lZjGr ~ , j I : i E ( 1 . 2 ,.... ~ ) a n d B #J E ( 1 , 2 v))andwhcmmJ andI;1redisjointnomrnpcywWof(1.2.....v ) , q f E(C.J.1). Given x E N, pick a(x) € 11.2.. ,q 1) and ((x) and b(x) in a, such that x = a(x) .qtCX) b(x) .qf(x)+ That is, in the base q expansion of x, L(x) is the number of rightmost zeros and a(x) is the rightmost noavm digit. Foreacha E (1.2. ....q - 1 ) l e t A. = (x E W\{lJ : a(x) = a ) . Picka E (1.2.. ..,q 1) and XI, XZ, ,xa E A. such that C i = 0. Phtion(1.2 ,....vlaccordingtoL(x,). T h a t i s , p i c k m ~ N . s e t s I 1 . 1...., ~ I, aodoumbmll

0) such rhar t k matrix

/

-1

0

...

0 \

is kernel pam'tim rc$ujar over N. (c) Then exist m E N and a u x m matrix B wirh enm'esfmm Q which satisfies tkfirsz eniries condition such that given any 7 e W' rhem is some 2 e Nu with

A i 5 89. (d) Then exisr m E tiand a u x m mawh C with entries+ Z which satisfies the first entries condition such that given any 3 E Mm there is some 3 E Nu wirh Ax' = Cj7 (e) ntnrrxirtm~NMdauxmnronirDwirhmtriesfmnrowhichsan'sfiestk first mtries condition such thor given any 3 E Nmthere is some f E NYwith A? = 07. (0There exist m G N. a u x m matrix E wirh mtricsfmm o,and c E PI such that E satisfiesrhejirst enfries condition. c is the onlyjirst entry of E, and given m y

y'~N*t(lcmiSsomei€N"xti~hA?=E~.

Pm# (a) implies (b). Given any p E N\(I), we let the stan h e p coloring of N be the function

.. -

x:=o

defined as follows: given y E N, write y = a,pJ, where each at 6 (0.1. . ,p 1) and a. # 0: i f n > 0, up(y) = (a.,an-1.i) WheZT i E n (mod 2); if n = 0, up(y) = (q.0.0). (For example, given p z 8. if x = 8320100, y = 503011, z = 834. and w = 834012. all written in base p. then op(x) = a&) = (8.3.0). J+(W)= (8.3.1). and up(?)= (5.0.1). L e t 6 . i ,.... 6 bcthecolumnsof~nradletd;.d;, d;,dmotetbccocolmnnsof the u x u identity matrix. Let B be the matrix

....

..;

whefe SJ, 52. s, aze as yet unspecified positive ~~tionak. Dmotc the columns of B by bi, b2. b.+,. Then

....

.

/ ,

15.4 Image Pmiticn Regularity Over N

311

.

.

..

( ~ l ( p )&(PI). . .. ,~ ~ ( pof) (I, ) 2,. u) as f011ow.s. M i E w such that *i is monochrome with respect to the start base p coloring and let y' = A.?. Now divide up (1.2,. ..,u) according to which of the yi's stan furthest to the left in their base p representation. That is, weget Dl(p), &(p), ...,Dm(p)so that

.

Thereareonly Gnitely many ordnedpmtitionsof (1, 2.. .. u). Therefore wemay pick an infinite subset P of N\(l), m E El, and an ordmd partition (Dl. &. , Dm)of (1.2, ...,u)sothatforallp€ P,m(p)=mand

...

...

..

..

We shall utilize (Dl, 9 , ,Dm)to find q.q,. . s and a padtion of (1.2.. , u 1))85 required for the columns condition. We proceed by induction. F i t we shall find ZI G (1,2, ... v), spcify 4 E-Q+ (s E Q : s z 0) for each i E El, let 11= E LU ( v Dl), and show that &,I, bi = 0. That is, we shall show that &E, 4 .6 = &o, &. In order to do this, we show that Ci,nl d is in the (1.2,. . .,v)-restricted span of (-5.6,. ..,-5).For then by Lemma 15.23(b) one has 4 = C:=, ai 6 , where each ai E Q and each cri 2 0. Let ,V ) :ai z 0) and for i E Ei. lets; = at. El = (i E (1.2. Let S be the (I, 2,. ,v)-restricted span of (6.6, &). In orderjo show that &b, 2. is in S it sufii&s, by Lemma 15.23(a). to show that &D, di is in d S. To this end. let c z 0 be given and pick p E P with p > u/c. Pick the i E Nu and y' E Nu that we used to get (Dl (p), &(p). . ,Dm(p)). That is, A? = y'. y' is monochrome with respect to the start base p wloring, and (Dl, &, ,Dm) = (Dl (p), Dz(p), . .,D,(p)) is the ordered @tion of (1.2,. ,u ) induced by the starting positions of the yi's. Pick y so that for all i E Dl. y(p, yi) = y. Pick (a. b, C) E (1.2.. .. p 1) x (0.1.. . . p - 1) x (0.1) such that up(yi) = (a, b. C) foralliE(1,2 ~).LecC=a+b/pandobsmethatl~L~p.ForifD~. y; = a ~ P Y + b . p y - 1 + ~ i . p ~ - 2 w h m O j z c ip,andhenceyi/p~=L+zi/p2;let Ai = zi/p2 and note that 0 5 Ai c l/p. For i 6 U;L=2D,. we have y (p, yi) 5 y 2; let Ai = yi/pY and note that 0 c Ai c Ilp. Now A i = y' so

+

.

+

xiED, ... ..

.

.

...

..

.

..

.

.....

-

...

.

-

S i c e xy=l (xi/(CpY)) .6 is in S,we have that &D,

4 E ct S as required.

15 Partition Regularity of Matrices

312

.

Now let k r (2.3.. .. rn) and =same we have. chosen El, Ez, s (1.2.. .,v), si E Q+ for i E E ~ and , Ij = El U (v Dj) as quired for the columns condition. Let Lt = Ej and let Mk = Dj and mumerate Mk in order as q ( l ) ,q(2). ...,q(r). We claim that it suffices to show that CieD, is in the (1.2.. .. v)-restricted span of (?I.&. ,G,Liq(1). &2), ....&). which we will again denote by S. Indeed, assume we have done this and pick by L e n t 15.23(b) a1.q.. . .,a, in ( x E Q : x 2 0 ) and e ( l ) , 6,(z), ,Sq(,) in Q such that C i + o k 4= X ~ = I a i . G + C & , 8 , ( i ) . ~ qk( ti )E. * = (i E ( 1 . 2 ,..., v)\Lk :ai s 01 and for i E Ek, k t si = ai. Let I k = Ei U (v DI). Then

.

u:;

uz+:

uzj

.

.?

4

..

...

+

and

as required for the columns condition. In order to show that C i e ~di, is in S, it suffices. by Lemma 15.23(a) to show that 4 is in EL S. TOthis end, let w 0 be given and pick p E P with p > u/e. Pick the iE Nu and E Nu that we used lo get (Dl (p). Dz(p), ..., D,(p)). Pick y so p-11 x { O , l , ...,p t h a t f o r d l i e D t , y ( p . y i ) = y . Pick(a,b,c) E (1,Z 1) x ( 0 , l )such that a,(yi) = (a. b, c ) for all i E (1,2,. .. ut. Let C E a blp. For i E Dk, yi = a . pY + b pr-l zi pY-Z where 0 + zi < p, and hence Dj. y;/pY = L zi/pZ; let Ai = zilpZ and note that 0 5 A; < Ilp. For i E we have y (p, yi) 5 y 2: let Ai = yi/pY and note that 0 iAi -z I l p . (Of course we have no control on the size of yilpv for i E Mk.) Now A i = 9 so

,...,

+

.

-

+ .

.

+

u=k+l

I 15.4 Image Partition Regularity Over N

Thus ELl (xi/pY). G = & ~ ~ b / p Y .& ) Consequently.

313

+ Xiah e . 4 +

&,D~ Ai

.&

11 XidIk 4 - (2;-1 (xi/(lpv)) .G + Xie&f, (-yi/[lpY)) .&)I! = II xy=t Xis~~(Ai/!) 5 Xy=k XiGD,

k/tl

< U/P C

c.

.. .... +

which suffices, since xi/(tpy) > 0 for i E (1.2.. ,v). u v) = Having chosen I I , Iz, .,I,, if (1.2, Ij, we me done. So Now assume(l.2 ,..., u + u ) f L-&Ij. LetIm+1 = ( 1 . 2 ....,u+v)\l$=lIj. (1.2 U) = Uy=l Dj.so[-dl. - d ~ . c (& : i E U;(sl Ij)andhencewe can write XiEL+, as a linear combination of (6 : i E Uy=, I,). (b) implies (c). By Theorem 15.20 M satisfies the columns condition over Q. Thus by Lemma 15.15. t h m exist some m E (1.2, ...,u v) and a (u v) x m matrix F with entries from o which satisfies the first entries condition such that M F = 0, the u x m matrix whose entries are all zero. Let S denote the diagonal v x v matrix whose diagonal entries are sl,4.. . .,s,. Then M can be written in block formas ( A S - I )

..

,....

6

- - ....-4)

+

and F can be written in block form as

+

( ),where I denotes the u

x r identity matrix

and G and H denote v x m and u x rn matrices respectively. We observe that G and H are first entries matrices and that ASG = H, because M F = 0.We can choose d E N such that all the entries of dSG are in o.Let B = d H . Then B is a first entries matrix. Let y' E Nm be given and let I = dSGy'. Then Ax' = By' as required. (c) implies (d). Given B. let d E N be a common multiple of the denominators in . such that Ax' = B i = Cy'. entries of B and let C = Bd. Given y', let ? = d y and pick ? (d) implies (e). Let C be given as guaranteed by (d). By Lemma 15.14, there is an m x rn matrix E for which D = C E has entries from o and also satisfies the fint entries condition. Given y' E Nm,we define i E Nm by i = Ey'. Pick x' E Nu such that A? . = Ci. Then Ci = CEy' = D j . (e) implies (9. Let D be given as guaranteed by (e). and for each j E [I, 2, . .. m), pickw, E Nsuchthatforanyi E (1,2. .... ulif j =min(r E (1.2. ....m) : di,, $ 0 ) . then di,j = wj. (That is, w, is the first entry associated with column j. if there are any ,w,). Define first entries in column j.) Let c be a common multiple of (WI,wz, the u x m matrix E by, for (i. j ) E (1.2, ..., u) x (1.2, .. .,m), ci,j = (c/wj)di,j. Now, given 7 E Nm,w e d e h e ? E N m by, for j E (1, 2.. ..,m). zj = (c/wj)yj. Then Di = E j . ( 0 implies (a). For each c E N, cN is a member of every idempotent in pN by Lemma 6.6. and is in particular a central* sel. Thus by Corollary 15.6 E is image partition regular over W. To see that A is image partition regular over M, let r E N and let M = U:=! Fi. Pick i E (1, 2,. . .,r ) and y' E Am such that Ey' E Flu. Then pick i E MU such that A? = Ey'.

.

...

314

.

15 Partitism Regulariv of Matrim

Statement (b)of T l m 15.24 is a computable condition. We illustrate i b use by determining whether the mahica

.

an image partition regular over N. Consider the matrix

0 3s1 IS2 -1 2.~1 3 s 2 0 -1 where sl and ~2 arc positive ration!^. One quickly sees that the only possible choice for a set I1 of columns summing to 0 is 11 = ( 1 . 2 . 3 . 4 ) and then, solving the equations

3s1 2.q

+ 32

=1

+ 352 = 1

om g a sl = 2/7 and 32 = 117 and one has established that

is image partition regular. Now consider the matrix

48,

0

-I

Is1 -1q

0

6S2

0

-1

where again sl and Q arc positive rationals. By a laborious consideration of cases one sees that the only non zero choices for st and sz which make this matrix satisfy the columns condition are sl = 315 and q = -215. Consequently,

(i i) is not image partition regular.

Exercise 154.1. Rmn that the matrix 2

0

2 -2

3

is image partition regular.

Exe& 15.4.2. Let A be a u x v matrix with entries from o. Rovc that A is image partition regular over (N,+) if and only if A is image partition regularover (M..). (Hint: Consider the proof of Theorem 15.17.)

I55 Maaices with Pmr*s from Fklds .

.

315

15.5 Matrices with Entries from Fields

In the g c n d situation whae one is dealing with arbitrary commutative semigroups, we restricted our coefficients to have enhies from o.In the case of (N. +),we allowed the entries of mahices to C o w from 0.Them is another natural setting in which the maies of a efficient matrix can be allowed to come from other sets. is the case in which the s e m i p u p is a vector space over a field. We show here that the appropriate analogue of the k t entries condition is sufficientfor image partition regularity in this case, and that the appropriate analogue of the columns condition is sufficient for kernel partition regularity. We also show that in the case of a vectorspace over afinire field, the columns condition is also necessary for kernel partition regularity. We begin by generalizing the notion of the first entries condition from &finition 15.2. Definition 15.25. k t F be a field let u, v E N.and let A be a u x v matrix with entries from F. Then A satisfies thefirst entries condition over F if and only if no mw of A is 6 and whenever i, j E (1.2.. ..,u) and k = m i n { t ~ { I , 2..... v ) : a i , , # O ) = m i n ( t ~ { 1 , 2,.... v):a,,,#O). then ai.k = aj.k. An element b of F is afirn enhy of A if and only if them is some row i of A such that b = ai,k where k = min{t E 11.2,. . . v) : ai., 0). in which case b is thefirst enrry of Afkm column k.

.

+

Notice that the notion of 'Hrst enhiescondition"fromDefinition 15.2 and the special case of Definition 15.25 in which F = Q not exactly the same since there is no requirement in Delinition 15.25 that the first enhies be positive (as this has no meaning in many fields). We now consider vector spaces V over arbitrary fields. We shall be dealing with vectors (meaning ordered v-tuples) each of whose enhies is avector(meaning amember of V). We shall use bold face for the members of V and continue to represent v-tuples by an arrow above the letter. Tbe following theorem is very similar to Theorem 15.5 and so is its proof. (The main difference is that given a vector space V over a field F and d E F\(O), one has dV = V so that dV is automatically central* in (V, f).) Accordingly. we leave the proof as an exercise. Theom 1536. Lct F be afeld and let V be an infinite vecror space over F . Lct u, v E Nand let A be a u x v matrix with enm~esfiumF which smisfies thcfirst entries condition over F. Le C be central in (V. +). Then there exist sequences ( x ~ . ~ ) ~ = , , (xz..)zl. ....(xu..)z, in V such rhor for every G E Pf(N), .?G E (V\{O))" and A i G E CU, where

/ L E G XI.. \

316

15 Partition Regularity of Mattices

Pm$ This is Exercise 15.5.1. The assertion that V is an infinite vector space over F reduces, of course, to the a d o n that either F is inlinite and V is nonh*ial or V is infinite dimensional over F.

Comhry 1527. Let F be afield and Id V be an infinite vector space over F. Lcr u. v E Nand Iet A be a u x u matrix with entriesfrom F which thefirst entries conditionover F. Then A is~lron#Iyimage purrition regular over V . T h t is, whenever r E Wand V\(O) = U;-, Ei. there uisr i E (1.2.. .. r ) and? E (V\(O])Y such that A? E EiY

.

PI&$ We lint observe that ( 0 ) is not central in (V. +). To see this, note that by Corollary 4.33. V* is an ideal of (PV, +) so that all minimal idempotents are in V*. Consequently om may choose i E (1.2. ... r ) such that Ei is central in (V. +). Pick. by Theorem 15.26 some i: E (V\(O))' with A? E EiU. 0

.

Now we turn our attention to kcmel partition regularity. We extend the definition of the columns condition to apply to matrices with entries from an arbitrary field. Delinition 1528. Let F be a field let u , u E N. let C be a u x u matrix with entries from F, and let &. &, ... be the columns of C. The matrix C satides the columns condition over F if and only if there exist m E ti and 11,12.. ... I , such that

.c;

(3) If m > 1 and t E (2.3,. ..,m).let JI = F such that xi,,, 6 = &I, 61.i . 4

Ij.

Then there exist (8,,i)i,l, in

Note that Definitions 15.13 and 15.28 agree in the case that F = Q. Theorem 15.29. Lct F be afield let V be an infmilc vector space over F, Iet u , v E N. ond Ier C be a u x u matrix with entriesfmm F that satiqiies the columns condition over F. Then C is keml panition rcgular over V. That is, whenewer r -E ti rmd V\(O) = U:=, Ei, then exist i E ( 1 . 2, ... r ) and? E E,! such that C i = 0.

.

...

P m f : Pick m E M, 11.12, ..., I,, and for r E (2.3,. m ) , J, and (8,,i)i,J, as guaranteed by the definition of the columns condition. Define the u x m matrix B by, for (i. r ) E (1,2.. . . v ) x (1.2,. .. ,m].

.

We observe that B satisfiesthe fint entriescondition. We also observe that CB = 0. (Indeed, let j E (1.2.. . .,u ) and t E (1.2. ...,m ) . If t = 1, then cj.i . bi., = &I, cj.i=Oandift> 1 t h e n ~ i V , , ~ j . i ~ b i , ~-8,,i.cj.i+& =~i~j, cj.i=O.)

'

15.5 Matrices with Entries fnnn Fields

3 17

Now let r E N and let V\{O) = U;=[ Ei. Pick by CmIIary 15;27 some i E (1.2, r ) an$ j E (V\(0))m such that B j E EiY. Let f = B j . Then Cr' = CBf = O j = 0. 0

....

We see that in the case that kemd partition regularity.

F is a finite field, we in fact have a characterizationof

Theorem 1530. Let F be ajinitefirld, let u, u E N,and let C be a u x v matrix with entriesfmm F . Thefollowing storements a n equivalent. (a) For each r E N then is some n E N such that whenever V is a vector space of dimension ar least n over F and V\(O) = U:=, Ei then exist some i E (1.2 ,.... r J a n d s o m e f e ~ ~ ' s u c h t h a t ~ f = 6 . @) The matrix C satisfies the columns condition over F.

-

Proof: (a) implies @). Let r = IF1 1 and pick n E N such &at whenever V is a vector space of dimension at least n over F and V\(O) = U:=, Ei there exist some i E (1.2. ... r) and same Z E EiYsuch that C f = 6. Let V = X:=I F . Since in this case we will be working with v-tuples of n-tuples. let us establish our notation. Given f E VY.we write

.

...

...,u), xi = (xi (1). xi (2), ,xi (n)). We color V accnding to the value of its first nonzero d n a t e . For each x E V\(O). let y(x) = min{i E (I. 2. ..n) : x(i) # 0). For each u E F\{O). let Em= (Ir E V\{O) :~ ( ~ ( x=)a)) . Picksomea E F\(O) andsomef E EnYsuchrhat C f = 0. Lcl D = xi) : i E (I, 2,. .. u)) and let rn = IDI. Enumerate D in increasing order as (dl. 4....,dm). For each r E (1,2.. .. m). let I, = (i E (1.2,. . . u) : y(x;) = dl). For t E (2.3,. ..,m), let JI = Uj;; Ij and for i E I,, let = -xi(d~).a-I. To see $at these are as required for the columns condition. we first show that Xi,/, ? = 0.Tothisend let j E (1.2, . . .,u) begiven. We show that Xi.l, cj.i = 0. NOW ~f = 6 so X;='=,cj,i .xi = o so in particular, X;=t cj.; . xi(dl) = O. NOW if i E (1.2..... u ) \ I ~ . t h e n d < ~ y(xi)soxi(dl) = O . ThusO= Xiell cj,i.xi(dl)= o.xi,/, cj.isozi,1, cj.i=o. 6 = &J, 4.; .?,we Nowassumem s 1 andt E (2.3, ,mJ. Toseethat &, again let j E 11.2, .. . .ul be given and show that &I, c1.i = XiEjra,.; . cj.; Then ~ ~ = l c j . i . =O~)inparticular.C,~=~cj.i.xi(d,)=O.Ifi xi E (1.2 ....,v)\(f,UJ,). thenxi(d0 = OsoO = Xie/, ~ ~ . ~ . x ~ ( d cj.i.xi(dr) ~ ) + C ~=~ Xi,,, ~ , cj.i.a+Ci,~, cj.i. and given i E {I, 2,

..

.

.

...

.

IS F'atitioa Regularity of Matrices

(b) implies (a). We use a standard compaemess apment. Let r E Nand suppose theconclusionfails. Forcachn E Nlet V, = (x E Xi,, F :foreachi > n,x(i) = 0). Then for each n. V. is an n d i n s i o n n l veclor space over F and V,, V,,+l. Now given any n E N, there is a vector space V of dimension at least n om F for which there exist El. E2.. . E, with V\(O) = Ei swh that for each i E (1.2.. ..,r ) and each 2 E EiY. C.? # 6. The same asmliou holds for any n-dimensional subspace of V as weH. Thus we can assume that we have some cp. : V,\(O)_+ (1.2, ...,r ) such that for any i E (1.2, ...,r ) and any 2 E (9,,-1[(i)])Y, cz 0. ChooseaninfinitcsubsetA~ofNsothatforn.mE A l ~ n e h a s ( p , ~= ~ ~( ~~not , , , , \ , ~ ~ . Inductively, given A,-!, choose an infinite subset A, of At-I with min A, ? t such that forn,m E A, onehas("nlK\l~l =qmlv,\,ol.Foreachrpickn(t) E A,. Let V = V. Then V is an infinite vector space over F. Forx E V\(O) pick the first t such that x E V,. Then x E V, E V,(,) because n(r) E A, and min A, 2 r. Define(p(x) = cp.(,)(x). By Theorem 15.29picki € (1.2,. .-..r) and2 E ((p-'[(i)])" V,. such that c.? = 6. Pick t E N such that (XI.x2, ... ,x.] We claim that for each j E (1.2, ,v ) one has w&j) = i. To set this, let j E (1.2....,~)begivenandpickthelesstssuchthatx~ E &.Thenn(s),n(r) EA, so q.(,,(xj) = %(,,(xi) = = i . Thus ?. E (p,,ct)-l[(ill)Y and C2 = 5, a conhadiction. 0

..

+

UF=l

...

c

Of course any field is a vector space over itself. Thus Corollary I f .27 implies thah if F is an infinite field. any matrix with entries from F which satisfies the first entries condition over F is strongly image partition regular o m F. Exrrcige 155.1. Row Theorem 15.26 by suitably modifying the proof of Theo-

rem 15.5.

Notes The tcfminology"image @tion regular" and "kernel partition regular" was suggested by W. Deuber. Matrices satisfying . . the first entries condition are based on Deuber's (m. p. c) sets (761. The columns condition was introduced by Rado [206] and he s h e d there that a matrix is kernel partition regular over (N. +) if and only if it satisfies the columns condition over Q. Other gmeralizations of this result w m obtained in [206] and [207]. The pmof that (a) and (c) are equivalent in Theomn 15.20 is based on Rado's original arguments [206]. It is shown in [162], a result of collaboration with W. Woan, that there

solutions to the systan of equations iCr 7 in any cenhal set in (N. .) if and only if C satisfies the columns condition o v a Z. The proof given here of the sufficiency of the columns mdition using the image partition regularity of matrices satisfying the first entries condition is based on Deuber's proof that the set of subsets of W containing solutions to all kernel panition regular manices is partition regular [76]. The material from Section 15.4 is taken from [142], a result of collaboration with I. Leader, where several other characterizationsof image panition regular matrices an given, including one which seems to be easier to verify. This characterization of the image partition regular matrices is quite recent Whereas Rado's Theorem was proved in 1933 [206] and (m. p. c ) sets (on which the first entries condition was based) were introduced in 1973 [76]. the characterization of image pattition regular matrices was not obtained until I993 [1421. Theorems 15.26. 15.29. and 1530 arc from [21], written in collaboration with V. Bergelson and W.Deuber, except that t k the field F was required to be countable and the vector space V was required to beof countable dimension. (At the time [21] was mitten. the Central Sets Theorem was only known to hold for countable commutative semigroups.)

-. Chapter 16

IP, IP*,Central, and Central* Sets

We saw in Chapter 14 that in any semigroup S, central sets have rich combinatonal content. Andourinwduction tothecombinatonalapplicationsof the algebraic shuctwe of p.7 came through the Finite Products Theorem (Corollary 5.9). We shall see in this chapter that sets which intersen FP((x.)El) forevery sequence (x.)z, (that is, the IP* sets) have very rich combinatorial structure, especially in the semigroups (W. +) and (N,.). Further, by means of the old and often studied combinatorial m i o n of spectra of numbm, we exhibit a large class of examples of these special sets in (Pi. +).

16.1 IP, IP*,Central, and Central* Sets in Arbitrary Semigroups Recall that we have defined a cenfml set in a semigroup S as one which is a member of a minimal idempotent in BS (Definition 4.42). As one of our main concerns in this chapter is the presentation of examples, we begin by describing a class of sets that arc central in an arbitrary semigroup.

Theorem 16.1. k t S be a semigroup andfor each F UFEP,(S)(F .XF) is cenhul in S.

E

Pf(S),let X F

E S. l k n

Then (BF : F E Pf(S)] is a set of subsets of S with the finite intersection pmpeny so choose (by Theorem 3.8) some p E BS such that {BF : F E Pf(S)) G p. We claim that PS. p g 7i for which it suffices, since pp is continuous, to show that ~ . ~ ~ 7 i . ~ o t h i s e n d ~, lSe. tTs h e n t J ~~ , ~ s - ~ ~ s o ~s~-s ol b~~ T h e o r c m 4 . 1 2 A E sp. By Corollary 2.6, there is a minimal idempotent q E p S . p and so A E q.

In the event that S is countable, the description given by Theorem 16.1 can be simplified.

CoroUarg 16.2. Let S = (a. :n E M ) be a cowttable semigmup and let (x.)z, be a sequence in S. Thcn (a. .x. : n. m E M and n j m ) is central in S.


0

We now i n d u c e some terminology which is due to Furstenberg [98] and is commonly used in Topological Dynamics circles. lk6nitim163. Let S be a semigroup. A subset A of S is an IP set if and only if there is a sequence (x.)El in S such that FP((x,)gl) E A.

Actually. as defined by Furstenberg. an IP set is a set which can be written as FP((x.)z,) for some sequence {x.)El. We have modified the definition be~auscwe already have a notation for F P ( ( x , ) ~ , )and because of the nice characterizationof IP set obtained in Thwrern 16.4 below. The terminology may be remembered because of the intimate relationship between IP sets and idempotents. However, the origin as described in 198) is as an "infinite dimensional parallelepiped". To see the idea behid that term. consider the elements of FP((x,)!=,) which we have placed at seven of the vertices of a cube (adding an identity e at the origin).

Theorem 16A. Let S be a semigmup and let A be a subset of S. Then A is an IP set if and only ifthem is some idempotent p E pS such that A E p.

P m f : This is a reformulation of Theorem 5.12.

0

In general. given aclass R of subsets ofa set S, one may define the class R* of sets that meet ever member of R. We have already done so (in Definition 15.3) for central sets. DeRnitl01116.5. Let S be a semigroup and let A ifforevery IPsetB S . A n B f 0 .

s S. Then A is an IP set if and only

Recall from Lemma 15.4 that a set is a central* set if and only if it is a member of every minimal idempotent. A similar characterization is valid for IP* sets.

Theorem 16.6. k t S be a sentigmup and let A 5 S. Tlu following statements are equivalent.

16 IP. IP.,Cenml. and Cenml* Sus

(a) A is an IF" set. (b) A is a member of every idempotenf of BS.

(c) A f l B is an IP set for every IP set B of S. Pmof: (a) implies (b). Let p be an idempotent of BS and suppose that A # p. 'Ihm S\A E p soby Theorem 5.8 thereis a sequence ( x , , ) ~ in ~Swith FP((x.)zl) G S\A. Tbat iq S\A is an IF' set which misses A, a contradiction. (b) implies (c). Let B be an IP set and pick by Lemma 5.1 1 an idempotent p of BS sucbthatB E p. Then A r l B e p s o A n BisanIPsetbyTheomn5.8. That (c) implies (a) is trivial. 0 As a trivial coosequence of Lemma 15.4. Theorems 16.4 and 16.6. and the dehition of central, m e has that

IF'*

* cmtral* * central * IF'.

One also sea immediately the following:

Remark 16.7. Lrt S be a sernigmup and let A and B be subsets of S. (a) IfA and B on I P sets, then A n B is an IP* ser. (b) IfA ond B am cennol* sets, then A n B is a centml* set. We s a now that in most reasonable semigmups, it is easy to produce a specific a n d * set which is not an IF'* set.

Tbeorrm 16.8. Let S be a semigroup and a s m e that (x.)z, is a sequence in S such thatFP((x.)~,) isnotpiecnvisesyndetic. 7'hen S\ F P ( ( x i ) z l )isa central* setwhich is nor an IF set. Pmof Trivially S\ FP((x.)zl) is not an IP* set Since FP((x&) is not piacwisc syndetic. we have by Theorem 4.40 FP((x.)E,) n K(pS) = 0 so for every minimal idcm~otmtp, S\ FP((x.)z,) E p.

Of course. since the notions of central aad 1P arc c b ~ z c byd membership in an idempotent, they arc partition regular notions. Ia hivial situations (see Exercises 16.1.2 and 16.1.3) the notions of central* and IP* may also be partition regular. Lemma 16.9. Let S be o sernigmup. (a) The norion o f I P is pnition regular in S i f and only ifpS has a unique idempotent. (b) 77renotionofcennal* ispartition ~ g u h r i nS ifandonly ifK(BS) has a unique idemporenr. Pmof We establish (a) only. the other proof being very similar. Necessity. Suppose that pS has two idempocents p and q and pick A A U (S\A) is an IP* set while neithct A nor S\A is an IP* set.

p\q.

Then

16.1 Sets in Arbitmy Semipups

323

Sufficiency. Let p be the unique idempotent of PS. Then a subset A of S is an IP* set if and only if A E p. 0 As a m n s e q of~ Lemma ~ ~ ~ 16.9 we have the following.

R m u L 16.10. Lrr S be a semigroup.

If the notion of lP. is partWon

rrgular in S.

then so is the notion of cmtml*.

Exercise 16.1.3 shows that one may have the notion of central* pattition regular when the notion of IP* is not. We see now that in more civilized semigroup. the notions of IPO and central* arc not partition regular. ('heomn 16.1 1 isin fact acomllary to Corollary 6.43 and Lemma 16.9, but it has a simple selfcontained proof, so we present it.) Theorem 16.11. Lrr S be an i n i i t e weakly lcp canceIlative semigmup. Then exin disjoint centml subsets of S. Consequently, neither the notions of c e n m P nor I P am panition regular in S. Pmofl Let K = IS1 and mumerate Pf(S)as (Fa).,,. Obsem that wheBES andIBI < x a n d F E q(S)thereissomex ESsuchthat(F.x)1lB=0. (Foreach u E F and u E B, (x E S :ux = v ) is finite by thedefinition of weakly leficaacellative so I U y e ~ U y . ~E lS~: ux = u)I -= K.) Choose xo E S and choose yo E S such that (Fo . yo) n (Fo xo) = 0. Inductively, let a < K and assume we have chosen (x,),, and (yo).,. in S so that

.

LetB = (Uo,,F,.xo)U(U,,,Fo.y,)andchoosex. E Ssuchthat (F..x,)nB =0 and choosey, E S such that (F, y.) n ( B U F, x.) = 0. By 'Ihcorem 16.1 we have that (U,,,F, . xu) and ((U,,F, .y,) sre disjoint central subsets of S.

.

.

In any semigmup, we see that IP9 sets satisfy a significantly stronger combinaknial conclusion than that given by their definition.

T h e o m 16.12. Let S be a semigroup. let A be an IP* set in S. and let (x):,. be a sequence in S. There is a pmducr subsyam (y,)g, of (x.)E1 such that m u ~ . ) z , ) c A.

Proof: Pick by Lmuna 5.11 an idrmpotent p

E

flS such that FP((x,)r-)

every m E Pd. Then A € p by Theorem 16.6, so Theorem 5.14 applies.

E p for 0

Exercbe 16.1.1. Rove Corollary 16.2.

Exercise 16.13. Let S beasetandleta E S. Definexy = a foranx and y in S. Show that a subset A of S is an IP set if and only if a e A and c o n ~ u e n t l ythat the notions of IP* and central* arc partition regular for this semigroup.

16 IP. IP*, Cenhll. and Central* Sets

Ex&

16.13. Recall the semigroup (N, A) where n Am = min(n, m). (See Exercise

4.1.11.) Verify eacb of the following assertions. (a) In (BN, A) every element is idempotent and K@M, A) = (1). (b) The P s e t s in (N. A) are the nonempty subsets of N and the central sets in (N, A) are the subsets A of N with 1 E A. (c) The only IP* set in (W, A) is N, while the central* sets in (W, A) are the same as the central sets. Consequently, the notion of IP* is not partition regular in (N, A) while the notion of central* is partition regular in (N, A). Exercise 16.1.4. Recallthesemigroup (N, v ) wherenvm = max[n, m]. (SeeExmise 4.1.1 I.) Verify each of the following assertions. (a) In (BN,v ) every element is idempotent and K(BM. v) = W. (b)The IP sets in (N, v ) are the nonempty subsets of N and the central sets in (N,v ) are the infinite subsets of N. (c) The only IP* set in (N, v ) is N. while the central* sets in (N. V) are the cofinite subsets of N. Consequently, neither the notions of IP* nor the notions of central* arc partition regular in (N, v).

16.2 IP* and Central Sets in W In this section we compare the additive and multiplicative structures of N. We begin with a trivial o b ~ e ~ a t i o n .

Lemma 16.13. k t n E N. Then Nn is an IP* set in W.+).

P m f This is an immediate consquence.of Theorem 16.6 and Lemma 6.6.

0

Now we establish that central sets in (N. +) have a richer shucturr than that guaranteed to an arbitrary commutative semigroup (and in fact richer than that passessed by central sets in (N, .) ).

Theonm 16.14. Let B be centml in (N. +). let A be a u x v matrix with enm'es

Imm Q.

(a) IfA is image panition regular, then there exists j E NUsuch that all entries of Ay' are in B. (b) IfA is kernelparfition regular, then there exists j E BV such that Ay' = 6.

P m f (a) Pick by Theorem 15.24 some rn E Nand a u x m matrix D satisfying the first entries condition with entries from o such that for any i E Nm there is some 3 E NU with Ay' = Di. By Lemma 16.13. for eachn E N. Nn is an IP* set and hence acmtral* set. Consequently, by Theorem 15.5, pick Z E Nm such that all entries of D t are in B and pick y' E Nu such that A j = Di. (b) Let d E N be acommon multiple of all of the denominatorsof enaies of A. Then dA is a matrix with entries from Z which is kernel partition regular. (If N = U ,,; Ci.

4

.

16.2 IP*and Cenbal Sets in N

325

..

pick i E (1.2,. .rl and i E CiYsuch that A.? = 6. Then (dA)i = 6.) 'Ibus by Rado's Theorem Clheorem 15.20) dA satisfies the columns condition over Q so, by Lemma 15.15. pick m E N and a v x m matrix C with entries from w which satisfies such th_atall the first entries condition such that dAC = 0. As in pan (a), pick; E

ym

enmesofCiareinBandlet~=Ci.'IhendA~=dACi=Oi=OsoA~=O. In fact, c e n d sets in (a, +) not only contain images of all image partition regular matrices, but all finite sums choosing at most one term from each such image as well.

(Yn)zl

+)

Definition 16.15. Let (S. be a semigroup and let be a sequence of subsets ofS. ThenFS((Y.)zl)= [&,,,=a. : F E Pf(M)andforalln E F. a. E Y,,).

Theorem 16.16. Let B be centml in (N, +). Let (A(n))zl entuneme the imngc partition mgular matrices with entriesfmrn Q andfor each n. let m(n) be the number of columns of A(n). There cristsfor each n E W a choice of i ( n ) E Wm(") such that, if Y,, is the set of entries of A(n)?(n), then FS((Y,)zl) G B. Pmof: F'ick aminimal idempotent in (BW, +) such that B E p. F'ickby Theorem 16.14 somei(1) E Wm(" such that all entriesof A(])?(]) arein B* = (a E B : -a+B E p). Let YI be the set of entries of A(l)i(l). Inductively. let n E PI and assume that we have chosen ?(k) E for each k E [I, 2,. . . ,n) so that, with Yk as the set of entries of A(k)?(k). one has FS((Yk);_,) c B'.ByLemma4.14,foreacha E B ' , - a + B ' ~ p s o

so pick ;(n

+ 1) E Nm"+') such that all entries of A(n + I)i(n + 1) are in ~ *nc-~ n + B*

Letting Y.+I be the set of entries of A(n

: a E FS((Y~);=~)I.

+ I)i(n + 1) one bas FS((yk);:f)

g B'.

o

A similar result applies to kernel partition regular matrices. Theorem 16.17. Let B be central in (W, +). Let ( A ( n ) ) z l enumerate the kernel pariition regular matrices with entriesfmm Q and for each n, let m(n) be the number of columns $A(n). %re existsfor each n E W a choice of i ( n ) E Nm(")such that A(n)i(n) = 0 and, if Y, is the set of entries of i(n), then FS((Y.)Zl) E B. Pmf. This can be copied nearly verbatim from the proof of T h m m 16.16.

0

The contrast with Theorems 16.14, 16.16, and 16.17 in the case of sets c e p l . in fact central* in (W, .) is striking. Recall from Section 15.3 th_e notation iA = 1 which represents the multiplicative analogue of the equation A? = 0.

Thrn C is c e n M in (N, .). A is m e mrion n g u h in (N. .), ond B is kemd portirion ngulor in (N. .). Howevec for no x E N is x2 E C &for no x' E C ' is is = 1.

The reader was asked to show in Exmise 15.1.2 that (x2 :x E N) is ncd cabal in (N. .). C q t l y , its complementmust be anailla.Trivially A is image panition regular in (N..) and B satisfies the colutuns condition over Q. so is kernel partition regular in (N. .) by Theorem 15.20. 0

P&

'Ibe following lemma establishes that dtiplication p n s e m s IP* sets in (N.

Lcanaa 16.19. k t A E N ond let n

E N.

+).

Thefillowin8 srcunnmrs o n equivdm

(a) A is M I P set in (N, +). (b) n-'A ison I P set in (N, +). (c) nA is an IP* set in (N. +).

A.

(a)implics(b). ~ s s u m i t h aAt is an IP*set and let Bbean IP Picka~cqucnw fl A # 0 so n-'A n B # 0. (XI)PO,~ such that FS((x,)E,) G 8. T h m FS((~X,)PO,~) (b) implies (a). Assume that n-' A is an IP* set, and let B be an IP set. Pick a sequence (xt)El such that FS((x,)E1) G B. By Lemma 16.13, Nn is an IP* set in (N. +) so pick by Theorem 16.12 a sum subsystem (Y,)PO_~ of (X,)PO,~such that FS((Y,)PO,~) E Nn. In particular n divides y, for each t. Consequently

P&

We see now that IF" sets in (N, +) are guaranteed to have substantial multiplicative shuchue.

Tbwrmn 16.20. k t d bc the set offinite sequencesin N (inctuding the empty sequence) ond let f : d + N. Let (y.)zl be o sequence in M ond let A be on IP* set in (N. +). Then then is o sum subsystem ( x n ) z l of (y.)zl such that whenever F P,'/O. mmF-I ),ondt E (1.2. ..., L), onehart. Z n e ~ xEn A. C = f((xdk_,

Pmof: Pick by Lunma 5.11 some p = p + p in /3Nsuch that FS((ym)zO=,) E p for each m E N. Then by Lemma 16.19, we have for each t E W that t-'A is an IP* set in

nfiy'

t-lA and note ihat BI E p. (N. +) and hence is in p. Let 81= FS((y.)zl) r l Pickxl E B1' and pick HI E P,(W such that XI = &H, y,. Inductively, letn E Nandassumethat wehavechosen (xi)bl. (Hi);=', and (Bi);=' k h t h a t f o r e a c h i ~ ( 1 , 2 , n):

...,

'

Only hypotheses (I), (3). and (4) apply at n = 1 and they bold hivially. Let k = max H. + 1. By assumption FS((y,)rd) E p. Again we have by Lemma 16.19 that for each t E N, t-'A is an IF*set in (N, +) and is thus in p. Foreachm E (1.2.

...,nllet

.

Em = ~ ~ j e r :x0j# F C {1,2.. .. n) andm = min F).

.

By hypothesis (4) we have for each m E (1.2,. .. nl and each a E Emthat a E Bm' and hence. by Lemma 4.14, -a B*. E p. Let

+

B"+I= FS((YI)%) n

f((x1)y-J

rxI

1-'A

n n",=,

n.~,(-a

+ B,').

Since X.+I E FS((y,)z). choose Then B,+I E p. Choose X.+I E B.+I'. %+I E PfC9 such that min H.+I 2 k andx,+~ = &H.+, Y,. Then hypotheses (1). (2). (3). and (5) arc satisfied directly. To verify hypothesis (4). let 0 # F E (1.2.. ..,n + 1) and let m = min F. If n + l $f F,the conclusion holds by hypothesis, so assumethat n + l E F. If F = (n+ I). = xn+l E B,,+I'. so assume F # (n + 1) andlet G = F\{n + 1). Let then XjCjp~xj a = Ejecxj. Then a E Em sox,+l E -a + Bm'so X j e ~ x = j a +x.+l E B*. as

m.

We thushave that (x.)Z, is asurnsubsystem of &)El.Tocompletethe proof, let F E Pf(M). l e t t = f ( ( ~ ~ ) ~ ~ ~ - ' ) , a n Ed l{1,2, e t t ...,C]. Letm = minF. Then by hypotheses (4) and (5). X j e xj ~ E Bm E t-'A so t X.,FX. E A.

.

Corollary 16.21. Let A be an IP* set in (N. +) and let (y,)zl be a sequence in N. Then is a swn subyam ( x B ) z lof (y.)zl such char FS((xn):,)

' J F W ( ~ n ) ~Cl )A-

Pmof: Let d be the set of finite sequences in N. Define f (@ = 1 and given ( ~ j ) & ~ define f ((xi)&l) = xj. Choose (x,)El as guaranteed by Theorem 16.20. Letting t = 1, one sees that FS({xn)zl) C A. To see that FP((x,)zl) C A. let F E Pf(N)andletn = max F. IfIFI = 1,thennjIjcFxj=x,, E A.soassumeIFI > 1 ~ j . ~ E~ A.o j andletG = F\{n). Lett = nj,cxj. Thmr 5 f ( ( x j ) ~ ~ ~ ) s o =t.x,,

nJmI

Theorem 16.20 and Corollary 16.21 establish that an IP* set in @. +) must have substantial multiplicative structure. One may naturally ask whether similar results apply to central* or central sets. On the one hand, we shall see in Corollary 16.26 that sets

which a n central* in (N, +)have significant mult@cative structure, in fact are cenhal in (N. .). On the 0th- hand, central sets in (N. +) need have no multiplicative strucave at all (7korem 16.27) while central sets in (N. .) must have a significant amount of additive strucntre (Theorem 16.28).

Deflnltion 1622. M = ( p E pN : for each A E p, A is central in (N, +)). As an immediate consequence of the definition of central we have the fouowing.

~cmark1 6 2 . M = ce E (K (BN, +)).

Theorem 1624. M u a leji ideal of (BN, .).

P m J Let p E M and let q

E BN. To see that q p E M,let B E q p and pick a E N such that a-'B E p. Then a-'B is central in (N. +) so pick, by Theorem 14.25 a decreasing sequence (C.)z, such that

(i) for each n E N and each x E C. there is some m E N such that C, & -x and (ii) (C. :n E N] is collectionwisepiecewise syndetic.

+ C.

Then (aC,,)z, is a decreasing sequence of subsets of B and immediately one has thatforeachn E Nandeachx E aC..thereissomem E NsuchthataC, g -x+aC,,. It thus suffices by Theorem 14.25 to show that lac,, : n E N) is collectionwise piecewise syndetic. That is, using Lemma 14.20, that t h m exist sequences (g(n));" and (x.)z, in N such that for all n, m E N with m 2 n,

If a = 1, the conclusion is immediate from the fact that {Cn : n E N) is collectionwise piecewise syndetic so assume that a s 1. Since (C. : n E N) is collectionwisepiecewisesyndetic,choosesequences (h(n))zl and (y.)zl in N such that for all n.m E N with m 2 n.

For each n E N, let x. = ay. and let g(n) = ah(n) + a . Given n, m E N with m 2 n and given k E (I, 2, ....in), pick r E (0, I, ....a - I] such that k r = a1 for some I E (1.2 ,....m). Thenforsome j E (1.2 ,....h(n))wehavey,,,+l E -j+n:=ICi a1 E -aj aCi so x, k E -(aj r) aCi. Since so that ay, a j + r < g(n) we are done.

+

+

+ ny=,

+

+ + n;=,

We saw in Corollary 13.15 that K(pN, .) n K(pN. +) = 0. We pause to observe now that these objects are nonetheless close.

Comllary 16.25. K@N. .) flcC K@N. +) # 0.

1

16.2 P*and Central Sets in N

329

P m 4 By Theorem 16.24. M is a left ideal of (BN, .) and consequently has nonempty intersection with K(BN, .) while by Remark 16.23, M E cC K ( p N , +). Corollary 16.26. Let A be central* in (N. +). l l u n A is central in (M..). Since M is a left ideal of P m $ 'BY Lemma 15.4. E(K (BN, +)) g 7i so M c (PA, .). M r l E ( K ( p N , .)) # 0 by Corollary 2.6 so A is cenwl in (N. .). 0

Theorem 16.27. T h e n is a set A E N which is central in (W, +) such that for no y a n d z i n N i s ( y , z , y z ] S A . Inpar~icularfornoyE N ~ S { ~ , ~ ~ ) ~ A .

+

Proof: Let X I = 2 and inductively for n E N choose x.+l (x,, n ) Z . Notice in particular that for each n , x. s n. Let A = [x. k : n , k E Nand k 5 n]. By Corollary 16.2 A is central in (N. +). Suppose now we have y 5 z in N with [ y , z , y z ) G Aandpickn E Nsuchthatz G [x.+l,x,+2 x.+n]. Then y s 2 so yz s 22 s 2x. s x. n so yz 2 x.+l 1 w (x. n)' 2 z2 2 yz, a conwadiction. 0

+

+

,....

+

+

We see.now thatsets central in (N;.)must contain large finite additive structure.

Theorem 16.28. Lcr A E N be central in (N,.). For each m sequence (x,)y=' such t h a t F S ( ( ~ , ) y =)~ )A.

E

N there exists ajinite

Proof: Let

By Theorem 5.8. all idcmpotenls in (pN. +) are in T so T # 0. We claim that T is a two sided ideal of (BN, .)so that K(pN. -) G T. Let p E T and let q E pN. To see that q .p E T, let B E q . p and let m E N. Pick a E N such that a-'B E p and pick (x,):!~ such that FS((X,):=~) E a-'B. Then F s ( ( a ~ t ) ; "C _ ~B. ) Toseethatp-q E T,letB ~ p - q a n d l e t mE N . L ~ ~ C = ( ~ E N : ~ - ' B E ~ ) . Then C E p so pick (x,):=~ such that F S ( ( X , ) ~ =E~C. ) Let E = FS((x,):=,). Thrn E is finite, so n O e E a - ' B E q so pick b E a e E a - ' B . T l e n F S ( ( b ~ , )E~ B. ~~) Since A is central in (N,.), pick a minimal idempotent p in (pN. -)such that A E p. Then p E K ( p N , .) G T so for each m E N there exists a finite sequence XI)^=^ such thatFS((x~):=~)C A. It is natural, in view of the above theorem, to ask whether one can extend the conclusion to infinite sequences. We see now that one cannot.

Theorem 16.29. There is a set A N which is central in (A. .) such that for no sequence (y,,)zl does one have F S ( ( y , , ) z l ) C A.

Pm$ Let XI = 1and fmn E N pick some X.+I

z nx.,.

I

La

By Corollary 16.2 A is central in (N. :). Suppose we have a sequence ( Y . ) ~with ~ FS((y.)zl) G A. We may presmoe that the sequence (y,):. is increasing since in any m n t an incrrasing sum subsystem of (y,)g, can bechosen. Pick m, n, and r in N such that n < r, y,,, E Ix,, 2s". 3x.. ,nx.1, and ym+l E lx,, 2r,. 3xr.. rxrl. Pick k E (1.2, . r ) such that ym+l = kx,. 'Ihcn

...

.

..

...

Given any central set A in (N, +), om has by definition that there is a minirml idempotent p in (BN. +) such that A E p. Since p = p p, one has that

+

+

A is centml) is central. That is, evey central set and so. in particular, [x E N : -x often translates down to a central set. Recall that for any p E BN, -p = (-1) . p E BZ. S i by Exercise 4.3.5. -W i a left ideal of (PZ, +).one cannot have p = p (-p) for any p E N*. If ooc had p = (-p) p for some minimal idempotent p in (BN, +)one would have as above that foranyA E p.[x e N : x + A ~ p Epandhence(x ] EN:.x+Aiscentml]iscentral. Howem, according to Corollary 13.19 such an equation cannot hold. Nonetheless. we are able to establish that for any central set A in (PI, +), [x E El : x A is central] is central.

+

+

+

the om^^^ 1630. Let p E K(BN. +) Md Ier L be a aminid I@ ideal Then is an idempotenz q E L such :hot p = (-q) p.

+

of (BN. +).

P m $ LetT = {q E L : (-q)+p = p). Givenq. r E PN, -(q+r) = (-q)+(-r) by Lenuna13.1,soifq.r E T , t h a q + r E T. Dehael: p Z + BZbyL(p) = -p. (That is, in thesemigmup (gZ, .). l = A-I.) Then.! iscontinuous and T = L n ( p , ~ t ) - ' [ [ ~ ] ] so T is compact Therefore, it suffices to show that T # 0. F a then. T is a compact subsemigroup of PN. hence contains an idempotent by Theorem 2.5. Since p E K@N, +), pick a minimal left ideal L' of BW such that p E L'. Now W is a left ideal of BZ by Exercise 4.35 ro by Lemma 1.43(c), L and L' are Iefi ideals of BZ. In panicular -L p L'. We claim that -L p is a left ideal of BN so that -L+p = L'. Toseethis,letq E -Landletr E BN. Weshowthatr+q+p E -L+p. -q = -(r q ) by Lemma 13.1 we have Now -q E L so -r + -q E L. Since -r r + q E - L s o r + q + p ~ - L + p . S i n c e - L + p = L 1 a n d p € L1wehaveT#O asquirrd. 0

+

+

+ +

proof: Kck a a idempotent p in (BN.+) such that A G p and pick, by Theonm 16.30. a minimal idempotent q in (BN. +) such that p = (-q) + p. Then [ x € N : x + A E p ) ~ q a n d [ x ~ N : x + A ~ p ) ~ ( x E N : x + A i s c e n t r aol ) . We do not know wh& every centrpl* s a often translates eitha up or down to anothn central* set. However we do have following strong contrast with Corollary 16.31 for P scts.

Theorem 1632 nirm is an IP set A of (N. +) such thatfor atl n 6 N. neithur n + A nor -n +A is an IP* set. Proof: Let (D,,).,z be a sequence of pairwise disjoint infinite sets of positive even integers such that for each n, min D. > 12nl. For each n E Z enmerate' D, in increasing order as (a(n. k))& and for each k E N,let y..t = 2"(n.U)+ P(".=-l). Foreachn E Z\(O], let B. = FS((y..k)&). Let C = [n +z :n € Z\(OJ and z E 8.1. Notice that if n E Z\(O) and z E B. then In1 < .i so C g N. Let A = W\C.-Then givenany n E Z\(O).(A+n)rlB-. =O,soA+nisnotanIP*seL We now claim that A is an IP"L set, so suppose instead we have a sequence (x),: with FS((x,,)g,) n A = 0. That is. FS((x,)=,) E C. By passing to a suitable sum subsystem we may presume that the sequence ,x(:). is increasing. Wefirstobsewethat ifn E Z\(O), u E n+B,,.andu = &F 2'.thenmaxF E D,. Indeed u = n + z where z E B, and z = &.q 2' where H 5 D. and H has at least two members (becauseeach y..t has two b i i digits). Thus, if n c 0, borrowing will not reach man H. Since then min H > 12nl one has max F = max H. We now show that one cannot have n E Z\(O} and i < j with [xi. xjJ G n B., so 2' ~ where man F E Dm.Let k = max F. suppose instead that we do. Now XI = z t a Since xi < xj we have that xj +xi = C t e 2: ~ where either max H = k or max H = k + 1. But as we have just sen. given any member of C, the largest element of its biiw suppott is even, so the latter case is impossible. The former case tells us that B..soifL=minD.we xj+xi E n + & . ButnowIxi-n,xj-n,xi+xj-n]c have that 2' divides each of xi - n, xj n, and xi + x j - n, and hence 2' divides n, a conmdiction. Consequently, we may choose for each i E FI some n(i) E Z\[O] such that xi E n(i) + B,,(i) and n(i) p n(j) for i j4 j. Choose i such that In(i)l r xi. Then xi = Z t E F 2' and xi + XI = z,ac2' where max F = maxG and consequently, xi + X I E n(i) + B,,(i). But now11 = (xi + X I )-xi is a difference of two members of B,,(i) and is hencedivisible by 2'. where[ = min Dng). This is aconhadiction because L > (2n(i)l > X I .

+

-

Exerdse 16.2.1. In the proof of Lemma 16.19 we used the fact that in the semigmup (N. +), one must have n-'(nA) = A. Show that in (N. +) one need not have n(nW'A)= A. Show infactthatn(n-'A) = A if andonly if A E Nn.

16 IP. IP*. Central, and Central* Seln

332

Exercise 1622. Rove that if r € A and A = Ci, and for only one i E (1.2,. .. r ) is there a sequence ( y m ) z lin N with F S ( ( y , ) g l ) g Ci, then in fact for this i , Cj is an IP*set (so that there is a sequence ( x , ) g l in A such that FS((x,)Z,) u ~ ( ( x x , ) c c,).

.

: -

163 IP*Sets in Weak Rings In this section we extend Corollary 16.21 to apply to a much wider class, called "weak rings". obtaining a sequence and its finite sums and, depending on the precise hypotheses. either all products or almost all products in a given IP* set

+,

Definition 1633. (a) A left weak ring is a triple ( S , .) such that (S. +) and (S. .) are semigroups and the left distributive Law holds. That is, for all x. y, z € S one has x.(y+z)=x.y+x.z. (b) A right weak ring is a triple ( S , .) such that (S. +) and ( S , .) are semigroups and the right distributive law holds. That is. for all x , y, z € S one has ( x y ) . z = x.z+y.z. (c) A weak ring is a triple ( S , .) which is both a left weak ring and a right weak ring.

+,

+

+,

In the above definition, we have followed the usual custom regardig order of operations. That is x y x z = ( x y ) ( x .z). Notice that neitherof the semigroups (S. 9 )nor ( S , .)isassumed to becommutative. Of course all rings are weak rings. Other examples of weak rings include all subsets of C that are closed under both addition and multiplication.

. + .

. +

Lemma 16.34. Let ( S , +) be any semigroup and let . be the operation making ( S , .) a

+, +,

+)

has a? least one elemenf righr zem semigroup. Then ( S , .)is lefr weak ring. I f ( & which is not idempotent. then (S. .) is not a right weak ring.

P m J This is Exerci~e16.3.2. Analogously to Lemma 16.19, we have the following. Notice that it does not matter whether we define a-'Ab-' to be a - ' ( ~ b - ' ) or (a-'A)b-I. In either case, y € a-' Ab-' if and only if ayb E A.

Lermnn1635. LetSbeaset.IetA

S,andleta,b~S. (a) I f ( S . .) is a left weak ring and A isan IPS set in ( S , +), then a - ] A isan IF+ set in (S. +). (b)v ( S , .) is a right weak ring and A is an IP* set in (S. +), then ~ b - 'is an IP* set in (S. +). ( c ) I f ( S , + , . ) i s a w e a k r i n g a n d A isanIP*setin(S,+), t h e n a - ' ~ b - ' i s a n I P set in (S. +).

+, +,

.

16.3

J P Sets in Weak Ringa

333

P m f It suffices to establish (a) s i n e then (b) follows from a left-right switch and (c) follows from (a) and (b). So, let ( x , , ) z t be a sequence in S. Then

so pick F E P j ( W such that C , , ~aF .x,, E A. Then, using the left distributive law we have that &F xn E K I A . 0 Recall that in FP((X,)F=~).the products me taken in inmasing order of indices.

Definition 1636. Let (S, .) be a semigroup. let ( x . ) z l be a sequence in S, and let k E N . Then AP((x&) is the set of all products of terms of (x.):=~ in any order with norepetitions. Similarly A P ( ( x , ) E l ) is the set of all pmducts of terms of (x,,)El in any order with no repetitions. For example, with k = 3, we obtain the following:

Theorem 16.37. Let (S, +, .) be a lep weak ring. let A be an IP* ser in (S, +), and let m be any sequence in S. Then there exists a sum subsystem ( x n ) E I of ( y . ) E l (Y.),=~ such that f f m ? 2 F E P j ( N with minF 2 m, and b E A P ( ( x . ) ~ ~ ~then ), b . & F X , E A. Inpmricujan

Proof:Pick by Lemma 5.11 some idempotent p of (BS, +) with

Then by Lemma 16.35. we have for each a E S that a - l ~is an IP* set in (S, +) and hence is in p. Let B l = F S ( ( y , , ) E l )and note that BI E p. Pick X I E BI' and pick HI E P j ( N ) such that xl = C~EH, y,. Inductively. let n E Nand assume that we have chosen (xi):=, (Hi);=, and (Bi ):=, such that for each i E (1,2,. . .,n ) :

.

= & e ~ ,Yr. ( 2 ) if i > 1, then min Hi > max Hi-,

(1) xi

(3) Bi E P. (4) if 0 f F & (1.2..

(5) if i > 1. then Bi

.

.

...i) and m = min F, then X j G Fxj

E n ( a - ' ~: a E A P ( ( x , ) ~ I ~ ) ) .

E Bm.. and

16 1P.IP.Central. awl Central* Sar

334

Only hypotheses (1). (3). and (4) apply at n = 1 and they hold trivially. Let k = max H. 1. By assumption FS((y,)z) € p. Again we have by Lemma 16.35 thatfmeacha E S . ~ - ' ~ i s a n ~ P * s e t i n ( ~ , + ) a n d i s r h u s i n ~ . Foreachm~{1.2. njlet

+

...,

.

By hypothesis (4) we have for each m E 11.2,. .. m ) and each a B,' E p. Let and hence. by Lemma 4.14, -a

+

E

Em that a E Em*

m e n &+I E p. &msc &+I G B~+I'. S k *.+I E FS((y,)%), choose H.+I E Pf(N) such that min &+I 2 k and X,+I = &H.+, yf. Then hypotheses (1). (2). (3). and (5) are satisfied directly. To verify hypothesis (4). let 0 # F S {I. 2,. ,n 1) and let m = min F. If n+l $ F. theconclusion holds by hypothesis, so assume that n+ I E F. I f F = (n+l). then zj;jEF~ =jX"+I E B~+I.. SO assume F # {n 1) and let G = F\{n + 1). a = z j r ~ x j .Thena E Em Wx.+l E - a + EmgW E j G i x j =a+x,,+l E Bm*as required. We thus have that (xn)Elis a sum subsystem of (y,,)zl. To complete the proof, let m 1 2, let F E Pf(N)with min F 2 rn and let a E AF('x.);):'. Then by hypotheses (4) and (5). xj,rxj E 8. G a-' A so that a &F xt E A. o

.. + +

.

Observe that {b . x, : in E 2 and b E AF'((xn);i1)) is the set of all products (without *petition) horn (x& that have their largest index occurring on the right. Notice that the proof of Theorem 16.37 is nearly identical to that of Theorem 16.20. Essentially the same proof establishes a much shunga conclusion in the event that one has a weak ring rather than just a left weak ring.

+.

Theorem 1638. Let (3, .) be a weak ring. let A be an ZP* set in (S. +), and la (y.)zl be any sequence in X. Then then exists a sum subsysrem (x.)zl of (y,,)zl in Ssuch thntFS((x,)zl) UAP((x.)zl) E A.

Pmof Modify the proof of Theorem 16.37 by replacing induction hypothesis (5) with: (5) if i > 1, then

Thm replace the definition of

with

J

16.3 IPI Sda m Weat Rings

335

Theomus 16.37 and 16.38 raise the natural qucstion of whethex the stronger amelusion in fact holds in any left mak ring. The example of Lemma 16.34 is not a countmxampk to this question be~ause,in this left weak ring, given any sequence (x.)Z"=, one has AP((x,)zl) = (x, :n E N).

Theorrm 1639. Let S be Ihefree semigmup on the two distinct lettersa and b and kr hom(S. S)be the ser of homomorphisrns/mrn S to S. Let o be the usual composirion of fwtctions anddfie an operation 8on horn(& S)asfollows. Given f. g E h o d s , S) andu1,u2, ut E (a,bl.

....

( f 88)(ulu2.. .ut) = f (ul)"8(ul)"f

(u2)-8(~2)-.

.,-f ( ~ , ) - g ( ~ t ) .

Then@om(S. S),8,o) i s a ~ w e a k r i n g M d t h e n u i s t m I P * s e r A in @om(S. S). e) and a sequence (fm)Kl in hom(S. S) such rkor no swn subsystem (g,)gl of ( f n ) z l has AP(k")K,) G A. Proof: The wification that (hom(S. S). 8 , o) is a left weak ring is Exercise 16.3.3. Notice that in order to define a member of hom(S, S)it is enough to define its values at a and b. Definef~6 hom(S, S) by fl(a) = a b and fl(b) = b and define inductively forn E N, &+I = f ,8fi. Notice that foreachn E N. fn(a) = (ab)" and f.(b) = b". Let A = hom(S. S)\[ f, o f, :r, s E N and r > a]. Notice that, given r, s E N. I

, I

fr(f,(a)) = fr((aW)) = (f,(a)f,(b))* = ((06) 6 )

.

(Wehaveused the Fact that f, is a homomorphism.) In panicular, notice that if f, o f, = fm 0 fn, then (r, s) = (m. n). Weclaimthatifm,n,r,s.C,r~N,h=f o f n . k = f r o f r . a n d h @ k = ftof,, thene=rn=randr=n+s.Indeed

((ab)'bf)' = (h 8k)(a) = h(a)-k(a) = ((ab)"bm)"((ab)'b')'.

Suppose now mat (h),:

is a sequence in hom(S. S) with

W ( h . ) z ~ ) G [f, 0 f, :r,s

E

Nand r > sl.

Then. using the factjust established,there is some r and for each n some s(n) such that h, = f. 0 f,(.). Then h18h28--.8h,=f,of, where t = x i = I s(n) 2 r, a coneadiction. Thus A is an IPS set in (hom(S, S). +). Rnally. suppose that (g& isasum subsystemof ( withAP((g.)z,) A. Then gl = Cn,n f. for some H E .P,-(N) so g1 = fk where k = C H. Then gk+l = ft for some t w k and thus gk+l o 81= ft o fi $ A;i a conhadiction.

f,)zl

c

+.

Exmlse 163.1. Show that if (S. .) is a weak ring in which (S, +) is commutative. n E Wand the operations on the n x n mahices with entries from S an defined as usual (SO,for example, the cnhy in row i and column j of A . B is E;=I ai.k. bk j ) , then these mahices form a weak ring. Give an example of a left weak ring (S. .) for which (S, +) is commutative, but the 2 x 2 mahices over S do not form a left weak ring.

+.

16 IP, P. Central,and Central* Sar

336

Exercise 16.33. Prove Lemma 16.34.

.

Exerdse 16.35. Rove that (hom(S. S). @. o) as dcsaibed in Theorem 16.39 is a left weak ring. Errrdse 16.3.4. We know that the left weak ring (hom(S, S).@. o) ofTheorem 16.39 is not a right weak ~g because it does not satisfy the conclusion of 'Ihcorem 16.38. Establish this fact directly by producing f, g. h E hom(S, S) such that (f fB g) o h # f 0heJgoh.

16.4 Spectra and Iterated Spectra

+

Spectra of numbers are sets of the form (LnaJ : n E N1 or (Lna y J : n E N] where a and y are positive reals. Sets of this form or of the form {LnaJ : n E A ) a (Lna yJ : n E A] for some specified sets A have been extensively studied in number theory. (See the notes to this chapter for some references.) We are interested in these sets because they provide us with a valuable collection of rather explicit examples of IP* sets and central* sets in (N.+). By the very nature of their definition, it is easy to give examples of IP sets. And anytime one explicitly describes a finite partition of N at least one cell must be a central set and it is often easy to identify which cells are central. The situation with respect to IP* sets and cenhal* sets is considerably different however. We know from Themem 16.8 that whenever (x.)zI is a sequence in N such that FS((x,)=,) is not piecewise syndetic, one has N\FS((x.)E,) is a central* set which is not IPa. So, for example. (CtCF 2' : F E P,(N) and some t E F is even) is central* but not IP* because it is N\Fs((~"-')~,). We also know from Lemma 16.13 that f a cacb n E M. Nn is an IP* set. But at this point. we would be nearly at a loss to come up with an IP* set which doesn't almost contain Nn for some n. (The set of Theorem 16.32 is one such example.) Inthis section we shallbeutilizingsomeinfomationobtained in Section 10.1. Recall that we defined there for any suhsemigroup S of (R. +) and any positive real number a. functions g, : S + Z.f, : S + and h. : S -+ T by g d x ) = LXL ;J. f&) = xu g&). and h. (x) = n (f,(x)) where the circle group T = R/Zand n is the projection of R onto T.

+

+

[-4. i),

-

Mnitfon 16.40. Let a > 0 and let 0 < y < 1. The function g,.~ :N + a,is defined by g,,,(n) = Lna yJ.

+

We denote by. ;g

the continuous extension of g,,y h m pN to pw. Notice that

8e = g,,:.

Recall that for a > 0 we have defined Z = [ p E pS : ?,(p) = 0).

Lemma 16.41. Let a > 0. let 0 < y < 1. Md let p

E

Z,. Then g;(p)

= &(p).

16.4 Spsmand ltcratcd S p m

337

pmof:Lete=min(y.l-y)andletA=(n~N:-c< fdn) < ~ ] . T h e n A e p s o it suffices to show that g., and g, a p e on A. So let n E A and let rn = g,(n). Then m-s 0).

Definition 17.1. A combirutorially rich ultmjilter is any p G ha r l A n K (BN,.) such thatp.p=p. We shall see the reason for the name "combinatorially rich" in Theorem 17.3. F i t we observe that thcy exist.

Le-

17.2 Then exists a combinatoriolly rich ultrafilter.

+)

P m f By Theorem 6.79 we have that A is a left ideal of (BPI. and so, by ComUary 2.6 contains an additive idempotent r which is minimal in (PA. +). By Remark 16.23 we have r E ha and consequently M n A # 0. Thus by Theorems6.79 and 16.24. Mn A is a left ideal of (PN. .) andhence contains a multiplicative idempotent which is minimal in (BPI. .).

17 Sums and Roducta

Recall the notionof FP-meintroduced in Definition 14.23. We cmll the corresponding additive notion an FS-tree. Since any member of a c o r n b ' i y rich ultrafilter is additivelycentral, it mustmntainby Theorem 14.25 anFS-tree T such that [B, : f E T ] is collectionwise piecewise syndetic, and in particular each Bf is piecearise syndetic. (Recall that Bf is the set of successas to the node f of T.) Notice, however, that in an arbitrary central set none of the Bf's need have positive upper density. (In fact, mall from Theorem 6.80 that W\A is a left ideal of (BN, +) and hence there are central sets in (W, +) with zero density.)

The0~llll7.3. Let p be a combinororially rich ulhofijter and let C E p. (a) C ir cenml in (W, +). (b) C is central in (N, .). (c) Let ( A ( n ) ) g lenumemfe the imagepanition regularmatrices with entrirsfrom Q andfor each n, let m(n) be the number of columns of A@). %re existsfor each n E W a choice of i ( n ) E Nm") such that, if Y. is the set of enfries ofA(n)i(n), then FS((Y")Z,) c c. (d) Let (A(n))zl enumerate the kernelpartition regularmamh?~with entriesfmm Q andfor each n, let m(n) be the number of columns g A ( n ) . There &ts for each n E N a choice of i ( n ) E Wm(") such thaf A(n).?(n) = 0 and, if Y,, is the set ofentries ofi(n), then FS((Y.)El) 5 C. (e) There is an FS-tree T in C such t hfor each f E T, a(Bf) z 0. (0There is an FP-tree T in C such t h for each f E T, Z(B,) s 0.

.

Prof. Conclusion (a) holds because p E M while conclusion (b)holds baause p p = P E K(BN. .). Conclusions (c) and (d) foUow from Thmms 16.16 and 16.17 respectively. To verify conclusion (e), notice that by Lemma 14.24 them is anFS-tree T in A such that for each f E T. Bf E p. S i c e each member of p has positive upper density the conclusion follows. 0 Conclusion (0 follows in the same way. As a consequence of conclusions (a) and @) of Theorem 17.3 we have in particular that any member of a combinatorially rich ultrafilrcr satisfies the conclusions of the Cenhal Sets Theorem (Theorem 14.11). phrased both additively and multiplicatively. There is naturally acomponding partitionresult. Notice that Corollary 17.4applies in particular when C = W. Corollary 17.4. Let C be a central* set in (W,+), let r E N, and let C = Uj=l Ci. There is some i f [I. 2, . ..,r ) such t h each of the conclusions of Theorem 17.3 hold with Ci replacing C.

Prof. By Lemma 15.4 we have ( p E K@N, +) : p = p + p) a combinatonally rich ultrafilter p. Then C E p so some Ci E p.

F so M F. Pick

-

17.2 Pairwix Sums nod Roducts

17.2 Pairwise Sums and Prpducts We know from Corollary 16.21 that any IP* set in (N, +) conlains FS((x.)E!P_!! u F P ( ( X . ) ~ ~for ) same sequence in N. It is natural to ask whether t h m is some pambon analogue of this result We give a strong negative answer to this question in this smtion. That is we produce a finite partition of N such that no cell contains the pairwise sums and products of any injective sequence. As a consequence, we see that the equation p p = p . p has no solutions in N*.

+

Detinition 17.5. Let {xn)Elbe a sequence in N. (a)PS({xn)zl)= {x. +x, : n, m E Nandn # m]. (b) PP((x.)zI) = (x. x, :n. rn E N and n # rn).

.

The partition we use is based on the binary repsentation of an integer. Recall that for any x e N, x = E,,,w(x) 2'. Definition 17.6. Let x E N. Then (a) a(x) = max supp(x). (b) E x $! (2' : t E w), then b(x) = max(supp(x)\(a(x))).

..

,a(x)l\ supp(x1). (c) C(X)= max(1-1.0.1,. (d) d(x) = min supp(x). (el If x $! (2' : t G w], then e(x) = min(supp(x)\(d(x)]). When x is written (without leading 0's) in binary, a(x). b(x), c(x), d(x), and c(x) are respectively the positions of the leftmost 1. the next to leftmost 1, the leftmost 0. the rightmost 1. and the next to rightmost 1. Remark 17.7. Let x E N\(2' : t E N] and let k E o. (a) b(x) 2 k ifandonly ifx 2 2"") 2k. (b) b(x) 5 k ifand only ifx < 2"(') + 2'"'. (c) C(X)2 k ifand only ifx c 2°(X)+' - 2'. (d) c(x) 5 k ifand only ifx 2 2"")+' 21r+'. (e) e(x) = k ifand only if k > d(x) and there is some m E o such thnt x = 2'+'m 2' 2d(r).

+

-

+ +

We now introduce some sets that will be used to define the partition that we are seeking. Definition 17.8. (a) Ao

= (x E N : a(*) is even and 20") c x c rb)+f 1.

A = (X E N : a (x) is even and 2a(x)+: c x c 24(X)+'1. A2 = (x E N : a(x) is odd and 2a(X)c x c 2a(x1+4I. A3 = (x E N : a(x) is odd and 2'(")+f c x < 2a(X)+'1. . Aq

= (2' : f E w).

(c) (CO.CI) is any partition of Nsuchthatforall k E W\(l). k + 1 E Coif and only i f 2 E CI. Notice that a partition as specified in Definition 17.8(c) is easy to come by. Odd numbers may be assigned at will, and if the numbers less than 2k have been assigned. assign 2k to the cell which docs not contain k 1.

+

Remark 17.9. (a) Ifx, y E A0 U Az, then a(xy) = a(x) + a(y). (b) Ifx, Y E AI U As. hen a(xy) = a(x) +a(y) 1.

+

We are now nady to define a partition of N. When we write x ss y (mod R)we mean. of course, that x and y are elements of the same member of 3.

+

Dellnitinn 17.11. Define a partition 92 of N by specifying that A4 and 2N 1 are cells of R and for any x, y E N\((2W -C 1) U A4). x a y (mod 3)if and only if each of the following statements holds. (1) Fori E ( O , 1 , 2 ) . x ~~ ~ i f a n d o n lE~Bi. if~ (2) For i E (0. 11, d(x) E Ci if and only if d(y) E Ci. (3) a(x) b(x) 5 d(x) if and only if a(y) b(y) 5 d(y). (4) a(x) c(x) 5 d(x) if and only if a(y) cCy) 5 dCy). (5) a(x) b(x) ia@) b(y) (mod 3). (6) a(x) 0 a ( ) (mod 2). (7) e(x) r e(y) (mod 2). (8) x = y (mod 1)6.

-

-

-

-

Lerrmu 17.12. Let ( x n ) z l be a one-to-one sequence in N. If PSI(X,,)~~) U PP((x.)zl) is contained in o m cell of thejmrtition d,then (d(x.) : n E N) is unbounded. ProOK Suppose instead that (d(xm).:n E N) is bounded and pick k € W such that for infinitely many n, d(x.) = k. If k = 0. then we would have PP((X,,)~=~)G 2N 1 so PS((x,,)zl) E 2W 1, which is impossible. Thus we may assume that k 2 1. If k > 1. then pick n < r such that d(x.) = d(x,) = k and either

+

+

Then d(x. d(xn

+ x,)

=

k

+ 1 and d(x,,x,)

+x,), d(xnxr) E Ci.

= 2k so one can't have i E (0.1) with

Thus we must have that k = 1. Suppose first that for hlinitely many n one bas

d(x,,) = 1 and e(x,,) = 2. Pick n < r and u. u E w such that u = v (mod 2) and x,, =2+4+8uandx, =2+4+8v.Thenxm+x, = 1 2 + 1 6 . ( Y ) i r 12 (mod 1)6 while x.x, = 36 48(u v) 64uv 4 (mod 116. Consequently one has infinitely many n with d(x.) = 1and e(xJ w 2. Rckn ir and u, v E o such that d(x,) = d(x,) = 1.3 5 dx,) Ie(x,), u = v (mod 2). and

+

+ +

Consequently we have some e > 2 such that e(x.) = e(x,) = C. Then x.

and x.x, = 4 so that e(xn

+ x,)

+x, = 4 +2'+' + 2"'

u +v (T)

u +v + (u + ~ ) 2 ~ - -+' ~ + 2'+' + 2'+3(2t-3 + 2

~

= L + 1 while e(x.x,) = e + 2, again a contradiction.

2

~

4 0

As aconsequenceofLemma 17.12. we know that if (x.)zl is a one-to-one sequence with P S ( ( x . ) ~ , )U PP((x,,)zl) contained in one cell of the panition 3.then we can assume that for each n, a(x.) < d(x.+l) and consequently. the= is no mixing of the bits of x. and x,,, when they are added in binary.

)

344

17 S u m and Products

Lemma 17.13. k t (x& be a on#-to-one sequence in N. If PS((xn)zl) u PP((xn)El) is cON&lined in one cell of the partition R,then (n E A : x. E Ao] infinite o r (n E N : x. E Aj] is infhite

Prm$ One cannot have PS((x,,)z1) E A4 and one c a n n O t h a ~ e P S ( { x ~ ) g ~ =2N+ ~) 1

+

so one has a(x. x,) ia(x.x,) (mod 2) whenevcr n and r are distinct m e m h of A. By the pigeon hole principle we may presume that we have some i E (0.1,2,3.4) such that (x. : n E N] E Ai. If o m had i = 4, then one wouldhave PP((xn)zl) c 4 and hence that PS((x,)zl) g 4,which we have already noted is impossible. By Lemma 17.12 we have that [d(x.) : n E Nl is unbounded so pick n E A such thatd(x.) > a(x1). Thena(x. + X I ) = a(x.). Suppose i = 1, that is {x, : n E N) g AI. Then by Remark 17.9, a(x.xl) t a(x.) a(x1) 1 so ~(x,,xI)is odd while a(x. + X I ) is wen. 0 Similarly, if i = 2, then a(x,xl) is wen while a(%, + X I ) is odd.

+

+

Lemma 17.14. Lct Cr.)El be a one-to-one sequence in N. If d(x.+l) z a(x.) for each n. PS((X.)F=~) U PP((x,)zl) is contained in one cell of the partition W and (x,, :n E N] E Ao, then (a(x.) b(x.) : n E N] is bounded

-

-

Pmof: Suppose instead that (a(x.) b(x.) : n E N) is unbounded. Pickn E N such that a(x,,) b(xJ > d(x1). Then, using the fact that d(x,,) > o(x1) we have that

-

a(x. + X I ) - b ( x . + x ~ ) =a(x.)-b(x.)

> d ( x ~ )=d(x.+xl)

while by Lemma 17.10

Lemma 17.15. Lct (x.)gl be a one-to-one sequence in H Ifd(x.+~) > a(x.) for each n, PS((x,,)zl) U PP((x.)z,) is contained in one cell of the pam'tion R,and (x. : n E N) C A,. then {a(xn) c(xJ : n E N) is bounded

-

Proof This is nearly identical to the proof of Lemma 17.14.

0

Theorem 17.16. There i s m one-to-one sequence (x.)z, in Nsuch that PS((xn)E,)U PP((x,)zl) is contained in one cellof theparfition R. Pmofl Supposeinstead we have such a sequence. By Lemmas 17.12,17.13,17.14, and 17.15 we may presume that for each n E N. d(x.+l) r a(xJ I and either

+

- b(x,) :n E N) is bounded, a - c(x.) : n E N) is bounded. Assume lint that {x. : n E Nt G Ao and [a(x,) - b(x.) : n E A) is bounded. Pick some k G N and n < r in N such that a(x.) - b(x,,) = a(x,) - b(x,) = k. Then a(x, + x.) - b(x, + x,) = a(+) - b(x,) = k. (i) (x. :n E N] E Ao and {a(x.) (ii) [x. : n E N] E A3 and {a(x.)

.;

17.2 Pairwise Sums and Rcdwts

- k +2. Thus a(xrxn)- b(xrx.) E (k- 1,k - 2 ) so a(x,x,,) + - b(x, +x.) (mod 3). a wnhadiction. Finally assume that k : n € lal) S An and [o(xd - c(x.) : n E N) is bounded.

so b(x,x.) 5 a(x,xn) b(x,x.) $ a(x, x.)

We may presume that we have some k E W such that a(x,J - c(x.) = k for all n By the pigeon hole principle, pick n c r in N such that e i k

- +

E

W.

k 1. and consequently c(x,x.) 5 a(x,x.) Assume that x,, < 2°(x.)+l(1 2-k): and i, < y @ - ) + l ( l 2 4 ) t . we have a(xr x.) - c(xr x,,) = a(x,) - c(x,) = k. Since x,x, c 2a(X,x0)+1(1- 2 - 9 = T"x")+l 2a(xrx")+1-kand hence c(x,x.) > a(x,x.) - k 1, one has c(x.xn) = a(x,x,) - k 1. But then a(x,x.) - c(x,x.) f a(x, +x,) c(x, +x.) (mod 2), a conhadiction. Thuswemusthavethatx, 2 ~ ~ ( ' ) + 1 ( 1 - 2 - ~ )andx, i ) T(Xr)+l(l-2-k)f.

+

+

+

-

-

+ -

-2-')4 anda(x,+x,) - c ( ~ , + ~ , , ) = a(xr+x.) = a h ) sox, +x. w 2"(xr+X")+1(1 a(x,) c(x,) = k. so picking i E (0. 1.2) such that i 1 I k (mod 3 ) we have that x, +x, E Bi and consequently x,x. E Bi. ~ ( k " ~ l + l-( l2-k) = 2a(x.&)+l 2&sJ+l-k SO ~ ( x r x d5 x,xm a(x,x.) k. By Lemma 17.10 a(x,x.) - c(x,x.) 5 k so a(x,x,) - c(x,x.) = k. Notice that (1 2-'-' - 2-2k-2) < (1 - 2-')f, a fact that may be verified by

-

-

+ -

-

17 Sums and Rodueta

346

Thus since x,x.

E B,

we have that k = a(x,x.)

- c(x,x,)

t

i za k

- I (mod 3). a

Recall from Theorem 13.14 that if p E BN and Nn E p for infinitely many n. then thm do not exist q, r. and s in N* such that q .p = r +S. The following corollary bas a much weaker conclusion. but applis to any p E No.

+

.

PmoJ Suppose instead that p p = p .p and pick some A E R such that A E p p. L a B = ( x ~ N : - x + A € ~ ) n ( ~ ~ N : x - ~ A ~ ~ ) a nEdBp. Inductively, ickx~ let n E N and assume we have chosen (xt);,, Pick

.

Then PS((x.)z,) U P P ( ( x , ) ~ , ) E A, contradicting Theorem 17.16. Theorem 17.16establishesthat one c?.nnotexpect any mt of combined additive and multiplicative results from an infinite sequence in an arbitrary finite partition of M. On the other b&d. the fouowing question remains wide open. Question 17.18. Ut r, n E M. IfN = U:=, Ai, must them exist i E (1.2,. a one-to-one sequence (x~):,~ such rhar FS((xt):_,) U FP((X,):,~) c Ai?

.

.. r ) and

We would conjhcturr sfmngly that the answa to Quation 17.18 is 'yes". However. the only nontrivial case for which it is known to be true is n = r = 2. (See the notes to this chapter).

17.3 Sums of Products Shortly after the original (combinatorial) proof of the F ~ t Sums e T h m m and its comllary, the Finite Products Theorem. Erd6s asked [891 whether. given any finite partition of N. there must exist one cell containing a sequence and all of its "multilinear combinations". We know, of course. that whatever the precise meaning of '4nultilinear combinations", the answer is "no". (Theorem 17.16.) We see in this section. however. that a cemin regularity can be impoxd on sums of products of a sequence. Recall that, if n E N and p E BW, then n . p is the product of n and p in the semigroup (BW. .) which need not be the same as the sum of p with itself n times. (We already know hom Theorem 13.18 that if p E W'. then p p # 2 .p.) Consequently we introduce some notation for the sum of p with itself n times.

+

Definition 17.19. Let p E pN. Then ot(p) = p and. given n E N, o,+l(p) = %(P) P.

+

-:

.

Of come, if p € N, tben for all n, n p = u,,(p). The question naturally arises ar to whether it is possible to haven .p = o.(p) for some n E N\(1) and p E N'. We shall see in CornUary 17.22 that it is not possible.

Lemma 1720. k t n E N, let p E W. let A E p and Irr B E a,,(p). Them is a om-to-one sequence ( x l ) z l in A such that, for each F E [Nln, &P xr E B. prwf: This is Exercise 17.3.1.

Tbeorem 17.21. Lcr n E N\(1). There is afrnite panition R of W such that there do not exist A E 3! anda one-to-one sequeke ( x l ) z l such that

.

(1) for each t E N, n x, E A, and (2) whenever F E [Nu. X t Ext~ E A.

and let R = (Ao, Al. Az, A,]. Suppose that one has a one-t0;One sequence (x~)PO_~ and i E 10, 1.2.3) such that

.

(1) for each t E El. n x, E Ai, and (2) whmeva F E [N", &F X, E Ai.

BY the d m n hole vrinciple. we may presume that we have some j

so that i f j

E

(0.1,2,3) such

+ 2 (mod 4). a contradiction.

0

P m J Supposethatn.p=un(p)andpickA

E RsuchthatA € 8 8 . ~ . '~henn-'A € p so. by Lemma 17.20. choose a sequence (x,)El inn-' A such that for each F E [Nln, ErcFx, E A. This contradicts Theorem 17.21. 0

Recall that given F. G E P,((N) we write F < G if and only if max F < min G. Delinition 17.23. Let ( x t ) z I be a sequence in N and let m E N. Then

17 Sums md Produns

348

P m f I f m = 1,this is just the hitepmducts theorem (Themem5.8). In theproof, we a!~dealing with two semigmups. so the notation B* is ambiguous. We shall use it to refer to the semigroup (ON..), so that B* = (x E B :x-lB E p). We do the m = 2 case separately because it lacks some of the complexity of the general thmrem so assume A E p p. Let BI = (x E N : -x A E p). m e n BI E p so BI* E p. Pick XI E BI'. Inductively, let n E Nand assume that we have chosen (~t):=~in N and BI 2 B2 2 . 2 B,, in p such that. if 0 # F 2 (1.2,. ,n). k=minF,andl=maxF.then

+

+

..

..

+

...,

n) and a E Ek. a E Bt. E BI SO -a A E p and Now. given k E (1.2. E p so B.+I E p. Choosex,,+l E B.+I'. n 1). k = min F , and To verify the induction hypotheses, let 0 # F G (1.2,. C = max F. If I < n, then both hypotheses hold by assumption. Assume that C = n. 'Ihen (1) holds by assumption and n,,,vxt E Ek so (2) holds. Finally assume that L = n + l . Then(2)isvacuous..Ifk = n+I,then &FX, = x,+l E Ifk 5 n. xt E Ek sox.+l E &+I G (&G xr)-lBke and thus then let G = F\(k). Then (1)holds. The wnshuction being wmplete, let FI.F2 E Pf(N with FI < Fz. Let k = min F2 and let L = max FI.Then

... +

n,,~

This completes the proof in the case m = 2. Now assume that m s_ 3. Let BI = (x E N : -x + A E a m - ~ ( p ) )and choose X I E El'. Inductively, let n E Nand assume that we have chosen (xk);=, in Nand (&);=I in p so that for each r E (1.2, ,n) each of the following statements holds.

...

,...,m - 2). F1,Fz ....,Ft E g ( ( l . 2 , ...,r)) with Fl < F2 < ... < Fl. and r < n, then B.+I c (XE N : -x + (- ZIn,,, xr + A) E U~-L-I(P)).

(V) If L E (1,2

At n = 1, hypothesis 0 says that XI E BI*. Hypotheses (11). 0, and (V) are vacuous. and hypothesis (m) says that -XI A E am-1(p). P o r e ~ ( 1 . ,..., 2 m-i).let

+

55 = ((FI,Fz.

...,Fe) : FI, F2,. ...FLE Pf((l.2. ....n)) and FI < F2 < ... < Fe)

Given a E Ek, we have that a E Bk* by hypothesis (I and ) so a-'BkR E p by then by (m)we have

Lemma 4.14. If (FI, Fz. ...,Fmml)E

- Xyi' nf,Rx, + A If e E (I, 2. . . . ,m

E p.

- 2) and (FI, Fz, . . .,Ff) E 35 we have by hypothesis (m)that

So we let

and notice that B.+I E p. For simplicity of notation we are taking 0 = N in the above. Thus, for example. if n = m - 3. then Fm-z = F m - l = 0 and so

n

Choose *,+I E B.+I'. To verify hypothesis 0,let 0 # F g (1, 2,. . .,n + 1) and let k = min F. If n 1 @ F, then (I) holds by assumption, so assume that n + I E F. If k = n 1. we have x.+l E B.+l' d i i t l y , so assume that k < n + 1 and let G = F\{n 11. Then n , , ~ x ,E Ek soxn+l E (nrP~xr)-lBk'. Hypothesis @) holds trivially and hypotheses (IV) and (V) hold d i i y . To verify hypothesis (In). let t E 11.2, ...,m 1) and let FI,F2, Fc E Pf((1.2.. . n + 1)) with FI < Fz < ... < Fc. If e = I, then by hypothesis

+

+

+

.

.

-

....

350

17 S u m a d Products

-

O and (n).%F, XI E BI* G BI so &F, XI + A E um-t(p). So assume that C s 1. Let k = min Fe and let j = max FM. '&a by hypotheses O and 0, nteF, XI E Bk' E B,+t and by hypothesis (V)at r = j .

..

The induction being complete. we have thu wbeneva FI, F2,. ,F,,, E Pf(N) with FI < Fz < ... -z F, if k = min F, and r = max F,-I. then by (I O. ), and

(N). n r e ~ , E ~ t E*&+I

E - Cyi;;'nlEfi XI + A

.

Cornnary 17.25. Letr, m E NandletN = U;=I Aj. Thenther~exiaj E (1.2.. .. r ) anda sequence (X~)PO,~ such that SP,,,((x,)z,) Aj.

PIV@ Pick any idernpotent p in (BN, .) and pick j E (1.2.. MP).

...r ] such that At

e 0

There is a partial converse to Theorem 17.24. In this convme, the meaning of SP.((xI)E2) should be obvious. Notice that one does not require that p .p = p.

Theorem 17.26. Lct ( x l ) z l be a sequence in N. l f p E n and k in N SP,,((x,)zk) E u,,(p).

nzIF P ( ( x I ) s ) ,thenforall

PmaJ We p e e d by induction on n, the case n = 1 holding by assumption. So let n E Nand assume that facach k E N. SP.((X,)PO,~)E a.(p). Let k E N. We claim that

so that -a

+ SP.+1((xf)%)

E

p.

Weseenowthatonecannot necessarily expect t o f i n d S P , ( ( ~ ~ )a~n~d)S P ~ ( ( x , ) z ~ ) (= FP((xI)zI))in one cell of apartition, indeed not even SP,,,((xI)z,) USPI ( ( y , ) z l )

with possibly different sequences (x,)zl and ( y l ) z l .In fact muchsmngerconclusions are known -see the notes to this chapter. Theorem 17.27. Let m E N\(1). Them is afmite pam'tion R of N such that. given any A E R. Ihem do not &st one-to-one sequences (x,)El and ( y , ) z l with SPm((xr)El)E A andSP~((yr)PO,~) C A.

35 1

17.3 Sums ofRadu*s

4 ~ 1 n-aOll)+d(yl) )

> a ( y ~ y z ) - a ( y r ) r a ( y ~ y z ) - d ( y d > a(yryz)-d(ylyz).

Also, yl t rna(Y1)+rnd(yl) and yz > ma(n)so

. .

a canhadiction. Next we claim that 4 does not contain any S P , , , ( ( X ~ ) ~so~suppose ). that we have a one-to-one sequence (X,)PO,~with S P , ( { X , ) ~ ~C) A4. If for infinitely many 1's we have d(xJ = 0. we may choose F E [NJmsuch that for any r, s c F one has xf

le x,

q4 m (mod mZ).

Then ~ ( Z ~ F X=I1 )SO & F X ~ € S P ~ ( ( X ~ ) P O , ~ ) \ A ~ . Thus we may assume that each d(xf) > 0 and hence, by passing to a product subsystem as we did whcn discussing A3, we may presume that for each t, a(x,) < d(x,+l), so that there is no canying whcn x, and x,+l are added in base rn arithmetic. (Notice that by Exercise 17.3.2,if ( y , ) z , is a product subsystem of ( x , ) E l , then S P m ( ( ~ t ) gE I ) SPm((xt)zl).)

+

+

We may further assume that f a each t 2 m. a(x,) ? a(xm-1) d(x1) 2. We claim that there is some t 2 m such that XI < ma(x"-d(xl'-l. Suppose instead

+

17 Sums and Pmductn

k

t

+

and c ~ l l ~ e q u c nacn,, tl~ XI) = Z r = I . a ( ~ , ) I. Since. for each I, d(x,+l) w XI). we have that for each s E (m, m a h + X~+...+X,-I +x,)=a(x,)an&since

+ 1,. ...k),

k k k a(x,)iseven. Alsoa(x~+x2+. ..+%,-I +n,=,x,) = U(~,=~X,),SOU(I'I,,X~) is even, a contradiction. Thus we have some t 2 m such that x, < + m"(x~)-d(xl)-'. k t = XI + X 2 + . . . + X m - ~ + X r Then.a(y) =a(xt)andd(y) =d(xl). Andhecausethen is no carrying when these numbers an added in base m, we have

contradicting the fact that y E Aq. Now we claim that As d w s not contain any FP((X,)PO_~).So suppose instead that we have a one-to-one sequence ( x , ) z l such that F P ( ( x , ) ~ , ) A s As before, we rnd(") as can may presume that for each 1, d(x,+l) > a(x,). Now XI 5 ma(")+' be seen by considering the base m expansion of X I . Also, x2 < m'(Xz)+l so that XIX2 < ,,,~(xI)+Nzz)+~ - , , , ~ ( ~ Z ) + ~ ( since X I ) +also ~ . xlxZ > , , , ~ T I ) + ~and (XZ) a h ). a(xz), and a(x1x2)arc all odd. we have a(x~xz)= a(x1) a(x2) 1. Also

-

+

+

co~nradicting the fact thatxlx2 e As.

Finally we show that A6 docs nm contain any S P m ( ( x 1 ) ~ , )so , suppose instead that we have a one-to-one sequence ( x l ) E l such that SPm((xr)PO,I)C A6. ASin the considemtion of A4, we see that we cannot have d(xJ = 0 for infi~telymany t's and hence we can assume that for all t, d(x,+l) w a(xl). We claim that there is some t 2 m such that xr s m"(xt)tL ma(x*)-d(xl). For suppose instead that for each t 2 m. x, m"(X1)+l(l m-d(xl)-'). Then for some e, (1 - m-d(X!)-l)'+l < 11," so

-

-

Pick the iirst k such that

Then

,,

nk-1

- I xr > m ~ kr-n'(*)+k-l-"'

and xk

,,,a(xd,

and hmn

.... k ) , a ( x ~+ x ~ + . . . + x . - l +x,) =a(x,)soeach c:=~a(x,) + k - 1 - m arc odd, which is impossible.

Alsoforeacht E l m , m + l , a(xd as well as

-

Thus we have some t 2 rn such that x1 > ma(*)+' m ' ( X ~ ) - d ( x ~ ) . k t Y = +xr. Thena(y) = a(xl) andd(y) = d(x1) and y w ma'rr)+'

xt +xz +...+x,-I mds,)-d(x~)

=m a ( ~ ) t l ,,,n(y)-d(y),

acon~c~on,

-

0

We see now that we can in fact assume that the partition of Theorem 17.27 has only two cells. Corollary 17.28. Let m E N\(l). There ir a set B

g N such tho1

( I ) whenever p . p = p E pN B E p, (2) whenever p . p = p E pN. N\B E um(p), (3) them is M one-ta-one sequence ( x r ) Z I in N with SPm((xr)PO,l)g B. and (4) there is M one-to-one sequence (x,)PO_,in N with SPl((x,)ZI) g W\B.

Pmof Let R be the partition guaranteed by Theorem 17.27 and let B = U(A E R : there exists ( x r ) Z l with FP((x,)EI)2 A).

To verify conclusion ( I ) , let p .p = p E pN and pick A E R such that A E p. Then by Theorem 5.8 there is a sequence ( x , ) z , such that F P ( ( x r ) z I )G A, so A G B

SO BE^.

354

17 S m aad Rodncb

To verify conclusion (2). Imp. p = p E BN and pick A E R such that A E d p ) . Thenby Theorem 17.24pickasequence (y,)zl with SPm((yt)Zl)C A. By Theorem 17.27 there does not exist a sequence (xl)El with FF'((xf)zl) E A, so A n B = 8. To verify conclusion (3). suppose that o m had a one-to-om sequence (x,)PO,, with S P m ( ( x , ) ~ ,c ) B. By Lemma 5.11, pick p p = p E ngl FP((xr)z). Then by Theorem 17.26 SP,((xr)z,) E um(p) so that B E um(p), contradicting conclusion (2). To verify conclusion (4). suppose that one had a one-to-one sequence (x,)EI with FP((x,)zI) C N\B. Again using Lemma 5.11, pick p p = p E FP((x,)zk). Then FP((xf)Po,,)E p so N\B E p. contradicting ccmclusion (I). 0

.

.

nzl

Exerdre 173.1. Pmve Lemma 17.20. (Hint: See the proofs of Corollary 17.17 and Theorem 17.24.)

EXedse 1733. Let (y,)El. Modify the proof of Theonm 17.24 to show that if p p =p E FP((yl)L) and A E am(p), then them is a product subsystem m of (x~),,, such that SP,((X~)PO,~) E A. (Hinc: Seetheproof ofTheomn 17.31)

.

ngl

bf)zk

.

Exe* 173A. Let n, r E N and let N = U:=I Ai. Prove that t h m is a function f : (I.?, ..n] -t (1.2,. r ] and a sequence ( x l ) z l such that for each m E (1,2.. ..,n]. SPm((x,)zl) C A,@). (Hint: Use the results of Exercises 17.3.2 and

..

..

17.3.3.)

-

17.4 Linear Combinations of Sums Infinite Partition Regular Matrices

In this section we show that, given a finite sequence of coefficients, one can always find one cell of a partition containing the linear combinations of sums of a sequence which have the specified coefficients. We show fimher that the cell can depend on the choice of coefficients. In many respects, the results of this section arc similar to those of Section 173. However. the motivation is significantly different. In Section 15.4, we characterizedthe (finite) image panition rrgular matrices with entries from Q One may naturally extend the notion of image partition regulariry to infinite dimensional matrices. Definition 17.29. Let A be an o x o matrix with enhies from Q such that each mw of A has only finitely many nonzero enhies. Then A is i m g e partition q u l a r over N if Ci, thereexisti E (1.2,. ,r ) and? E Nm andonly if wheneverr e Nand P1= such that all entries of A? are in Ci.

..

"

17.4 Linear Combinatim of Sum

A simple example of an in6nite padtion regular rnahix is

Hcn A is a finite sums matrix. That is. the assertion that all entries of A: are in Ci is the same as the assertion that FS((x,,)gl)C Ci. We establish here the image partition regularity of certain infinite matrices and provide a smng contrast to the situation with respect to finite image partition regular matrices. For example, any central set C in (W,+) has the pmpmty that. givmany finiteimage partition regular matrix A. there must exist x' with all entries of Ax' in C . (Theorems 15.5.15.24 and Lemma 16.13.) Consequently, given any finite partitionof N,some o m cell must conah the entries of & for every finite image partition rrgular matrix A. By way of contrast. we shall establish here that a certain class of infinite matrices an image partition regular, and then show that there arc two of these matrices, A and B, and a two cell partition of W, neither cell of whichcontains all entries of A i and B3 foranyiandj. We call the systcms we are studying "Milliken-Taylor" systems because of their nlation to the MillibTaylor Theorem, which we shall pmve in Chapter 18.

De6~itfoa1730.( a ) A = [ ( a i ) ~ = l ~ N: m m ~ N a n d f o r a l l i ~ { l..... , 2 m-I), ai # a i + ~ ) . '3)Girm o' E A with length m and a sequence ( x t ) E ,in N,

'Ihc reason for requiring that oi # ai+l in the definition of A is of course that a E,,F x, + a Etstx, = a E,,NO X, when F < G. As a consequence of Corollary 17.33 below, whenmr o' E A and N is partitioned ( x r ) z I for ) some sequence into finitely many cells, one cell must contain m(o'. ( X , ) ~ O , ~ .This is the same as the assertion that a particular infinite matrix is image partition reguiar.

356

17 S m and Rcducls

For a m k . if

and.? = (xt)zl.thenMT((I. 2). (x,)PO,,) isthesetofentriesof A.?. (Thermvsof A are all mws with entries from (0, 1.2) such that (I) only finitely many entries arc non-zero, (2) at least one entry is 1, (3) at least one entry is 2, and (4) all occurrences of l come before any occurrences of 2.)

Pmof Assume first that m = 1. Then 41-'A

E p so by Theorem 5.14 there is a sum subsystem (x1)pO,, of (yl)EI such that FS((X,)PO,~) a 1 - I ~ .This says precisely that MTG. (xr)Fl) G A. Assume now that m ? 2 and notice that

c

so that {x E N : - a 1 x + A € a ~ p + a 3 p + . . . + a , p ) Ep.

Pick X I E BI* and pick H I E Pf (N)such that xl = &.H, y,, (Here we are using the BI E p).) additive version of BI'. That is, 81'= {x E BI : -x Inductively. let n E N and assume that we have chosen ( ~ r ) ; , ~in N, (B& in p. and (HI)",=, in Pf(W) so that for each r E (1.2,. ..,n ) :

+

(D I f O # F S ( 1 , 2 .....r ) a n d k = m i n F , t h e n E l E ~ x t ~ B i . . S B, and H, < H,+I. {1,2...., m - I ) , FI.&..... FEE PfP1({1.2

(II)I f r < n. then B,+I

(Ill) I f e E

... r~ Fc.then

...., r ) ) . a n d F ~ < FZ

4

17.4 Line= Combhtiom of Sums

(IV) f f F 1 . F..... ~ Fm-I

E

357

P ~ ( ( 1 .,..., 2 11). FI < FZ < ::. < Fm-l.Bndr < n.

thm Br+t E am-'(-

~ 2ai CtGW ' X I + A).

At n = 1, hypotheses 0 and (Vl)hold dictly, hypotheses 0. (IV), and (V) are A E a l p a3p ...+ a m p which is

vacuous, and hypothesis (111) says that -alxl true because xl E BI

.

For!

E

.

{I, 2, ... m

and fork E ( 1 . 2 .

+

+

+

- 11. let

....n], let +

and so -b Bt' E p. If Givm b E Ek. we have b E B I . by hypothesis 0, ( F I .F2,. .. & - I ) E F + - I then , by (m)we have (- EL;' ai XrsFi xr A) E a m p M that a m - l ( - ~ ? = y 1 a i + A ) E p. If ! E (1.2, .... m - 2 ) and ( F I .F2. . .; F d E Ft we have by (111) that

.

+

.

Lets = max H.

+ 1. Then we have that B.+I

Hen again we use the convenrion that

E p, where

n0 = N so.for example, if m = 2. then

Cbooae *.+I

S (s,s

and pick H.+I

E B.+I*

+ 1,s + 2.. ..) such that x.+l

=

& I . + YI. ,

Hypothes'i 0can be verified as in the sswd pmof of lborcm 5.8 and hypothcs'i 0holds trivially. Hypotheses 0, (V) and (VI) hold dinctly. To verify hypothesis (W, let C E (1.2.. .m 1) and l a Fl < F2 < . < Ft in Pf((I.2 n+1)).ff~=l,wehavebyhypocheses(9and@)that~~~,x, EBI so that -a1 &IF, xt + A E a2p a3p amp as required. So assume that L > 1 andletk=RlinFcandj=manFr-1.l'hen

,....

+

X t t ~X,I E

.. + ...

4* E &+I E ( X E N

..

+

+

: -atx (- X f z : ai X I c h X I A) E at+lp ae+2p +amp)

+ ...

+

@y hypothesis (V)at r = j ) so

+

ai XISF,X I ) A E at+lp+ar+2p

-&I

asrequired

The induction being complete, let FI < F2 < min F, and j = max F,-I. Then %F.

so

XL1ai

XI

E Bk*

c Bj+l

+... +amp

-. . < F,,, in P f ( N ) and let k =

C am-'(- X:slai E r C X~I i+ A )

o

xI E A.

Just as Theorem 17.26 was a partial cnmne to Theonm 17.24, we now obtain a partial converse m Theorem 17.31. Notice that p is not rcquind to be an idempotent.

= (a!,02,. ...am) E A let (X:)PO,~ beasequmcc in N,d l n F S ( ( x , ) s ) . Then MT(& (x&) E a l p + a z p +. . .+amp.

lheomrn 1732. Let;

PE

Pmol: We show, by downward induction on C E (1.2,

.

... m ) that for each k E Q

{ ~ : = t a i ~ r e f i x t : F t . F t +..., ~ , F m ~ P f ( ( k . k +,... l DandFc < Fe+l 0. Pick an open set U such that^ ~ U a n d p ( [ l )< We first claim that

e.

(t) if (I.),,,F is a (finite a countably infinite) indexed family of pinvise disjoint intewals contained in U and for each n E F , p(A t l I.) c (I - c)p(L), then p*(B\ UnEF In) 0. To see this, since in general p*(C U D ) 5 p'(C) p * (n ~ U,, I.) < p * ( ~ ) ~. n indeed d

+ p*(D), il suffices to show that

364

17 Sums pad Rodvcrr

...

Inductively, let n E N and assume we have chogea XI,xz, ,x, in B and Wtive hl, h2.. .. hn such that each [xi hi.xi +hi] g U,each p(A fl (xi -hi. xi +hi)) < (1 6)2hi.and[xi - h i , & +hi] Il[xi - hj,xj + hl] = Ofor i # j. Let d. = sup(h : there cxistx E B such that [x - h, x hl E U,

-

.

-

+

[ ~ - h , x + h l n U ~ ~ ~ [ ~ i - h i , x i + h i l = 0 . a n d p ( A ~ ( x - h . x< + h(1-~)2h}. )) Notice that d,, > 0. Indeed, by (t)one may pick x E B\Uf,i[xi hi, xi hi] 8) flU:,,[xi hi,xi hi] = 0 and and then pick 8 > 0 such that (x - 8 , x (x 8, x 8) E U. Then, since x E B, one may pick positive h < 8 such that @ ( A n(X h, x h)) < (1 c)2h. Pickx.+l E B and h.+l > such that [x,+l h.+~. *,,+I h.+ll g U ,

-

+ -

+

+

-

4

-

+

-

+

+

-

[ X ~ +-hm+~,xn+~ I + h , + ~ l n U : ~ ~-hi,xi t ~ i +hi1 =0.

+

and P(A n &+I -hn+l. xn+l hn+l)) < (1 -€)%+I. The inductive construction being complete, let C = B\ h.]. [x. - h., x. h,l)T=l is a painvise disjoint Then by (t) p8(C) > 0. Also. since ([x" - h.. x. = , so pick m E W-such that collection contained in (0, 1). ~ ~h. converges,

+

+

zZ4+l hn

-

< P*(C)/~.

+

men P( U&,+l tx. 36,. x. 3hnl) < P*(C) so pick x E C\( U~-+l[xn 3hn,xn+3h& Thenx E B \ ~ = l [ x . - h . . x n + h . l s o p i c k h >Osuchthat[x-h.x+hlE U, [x-h,x+hlnU~=,[x,-h..x,+h,l=8.andp(An(x-h,x+h)) m. Thenhkz + ? so that {x x t l s h hk 5 3ht and hence x E [xt 3hr.xk 3ht1, conhdictingthe - 3h., x,, 3hJ. fact thatx t U;?+,[x.

+ CEl

-

+

+

+

-

4

+

We also need another basic result about measurable sch.

Lemma 17.43. Let A be a meosumble subset of (0.1) such thuta(A) > 0. There exists B

E A such that B U (0) is compact a n d Z ( ~ \ B = ) 0.

Pmqf Foreachn E N,letA. = Afl(l/Zn, l / ~ ~ - ' ) d l eTt = (n E N : @(A") > 0). As is well known given any bounded measurable set C and any c > 0 there is acornpact subset D of C with p ( D ) > p(C) 6 . Thus for each n E T,pick compact B, 2 An with w(Bd > @(An)For n E N\T. if any. let B. = 0. Let B = B,,. Then B U (0) is compact. < a. Pick Suppose now that &A\B) = a > 0. Pick m E N such that x < 112" s u c h t h a t p ( ( ~ \ ~ ) n ( ~ . x ) ) />x F'ickn E Wwith 1/2" 5 x i1/2"-' and note that n > m. Then

A.

-

Uzl

A.

&

2'

-

Definition 17.44. Lolnma 17-45.

P = { p E 0+ : for all A E p, a(A) > 0).

L is a I@

ideal of (O+.

-).

Pmof: It is an easy exercise to show that if a(A U B) > 0 then either &A) z 0 or Z(B) s 0. C-uendy, by lheorem 3.11. it fol~owsthat [ p e ~ ( 0 , : for all A E p, &A) > 0) # 0. On the other hand, if p E B(O. l)d\O+ one has some E > 0 such that (6.1) E p and ;i((c. 1)) = 0. Thus 6 # 0. Let p G P and let q E O+. TO see that q .p E L. let A E q .p and pick x such that x-' A E p. Another easy exercise 0 establishes that &A) = a(x-'A) > 0. Theorem 17.46. Let p be a rnultiplicarive idempotmr in 6 and let A be a mearurable member of p. Then [x E A : x-' A G p and -x +A E p) E p.

Pr00f:LecB=(x~~:x-~A~p).ThmB~psinccp=p~p.LetC=(y~A:y is not a density point of A]. By Theorem 17.42, p(C) = 0. Consequently since p E L, C $ p s o B \ C ~ p .W e c l a i m t h a t ~ \ ~ ~ [ x € ~ : x - ' ~ ~ ~ a n d - x + ~ € ~ ) . Indeed, given x 6 C one has 0 is a density point of -x A so by an easy computation, d((o. I)\(-x A)) = o so (0. I)\(-x + A ) e p SO -X A G p.

+

+

+

Now let us define the kind of combined additive and multiplicative stnrctures we obtain. Definition 17.47. Let ( x " ) z l be a sequence in (0. 1). We define FSP((xn)zl) and a : FSP((x.)zl) + 9(Pf((W) inductively to consist of only those objectsobtainable by iteration of the following:

+

+

For example, if z = x3 xs .x7 . (xu xlo - xll). then z E FSP((x.)Zl) and [3,5,7,8, 10, 11) E a(z). (Of course, it is also possiblethat z = x4+x1z.x13, in which case also (4, 12. 13) E ~ ( z ) . )Note also that (x3 + x d .x7 is noL on its face, a member of FSP((x.)El). Notice that trivially FS((xfl)zl) U FP((X.)F=~) 5 FSP((X.)T=~). The proof of the following theorem is reminiscent of the first proof of Theorem 5.8. Theorem 17.48. Let p be a ntuhiplicative idempotent in B and ler A be a Baire set which is a member of p. Then there is a sequence (x.)zl in (0.1) such that FSP((xn)zl) G A

'Iltcn Az E p and, since multiplication by XI-'

and addition of -XI are howomorphisms. A2 is a Baire set. Inductively, let n E m ( 1 )and assume chat we have chosen (x,)::: and (A,);,, so that A. is a Baire set which is a member of p. Again invoking Themem 17.40, one has that Lx E Am : x-IA. E p and -x A. E p ) E p so pick x. E A, such that x.-'A. E pand-x.+A. E p. LetA.+l =A.~x.-'~.n(-x,,+A.). lbenA.+~ is a Baire set which is a member of p. The induction W i g complete, we show that FSP((x,,)El) E A by establishing the following stronger assertion: If

+

thq z

E

A,.

.

Suppose instead that this conclusion fails, and choose

wch that z (t A, and IF(is as smaU as possible among all such counterexamples. Now if F = [m), then z = x, E A,, so we must have IF1 z 1. Let G = F\(m) and picky E FSP((x,,)z,) such that G E u(y) and either z = xm y or z = x, y. Let r = min G. Since IGI < IF],we have that

+

and hence x,

+ y E A,

and x,

.y E A,,

contradicting the choice of z.

.

0

u=,

ComUary 17A9. k t r E Nand let ( 0 , l ) = A;. Ifeach Ai is a Baire set, then thereexisti E (1.2. r ) a n d a s e q u c n c e ( ~i n~ () O ~ ~, 1 ) ~ h t h a t F S P ( ( ~GJ ~ ~ )

....

A 0. By Lemma 17.43, pick B compact and Z(A\B) = 0. Thm A\B $ p so B E p.

A such that B u (0) is

Now, proceeding exactly as in i h pmof of lbborrm 17.48, invoking Thcorrm 17.46 instead of lbborrm 17.40. one obtains a sequence (x.)zl such that FSP((x,,)zl) 5 B N&etbatifctFSP((x,,)L1) 5 AU{O).theninparticUlmctFS((~~)~~) 5 AU(0) d hence, if F is any subset of N. 6nite or infinite, one has &F xn E A. Corollary 1751. Icr r E N and lct(O.1) = UA1 Ai. If each A, is a mcanrmble r ) and a sequence ( x n ) z l in (0. I) such that set, then t h e &.sf i E (1.2, c ~ F S P ( ( X ~G) Ai ~ ~U)(0).

.

...

P m f By Lemma 17.45. +is a l e f t i d d of (w,.),which is acanpract right topological semigmup by Lemma 13.29. Thus, by Corollary 2.6 we may pick an idempotent p Pick i E (1.2,. ,r ] such char A, E p and apply Theorem 17.50.

..

G

L. 0

Exercise 175.1. Let X be a topological space and let 8 = (UA M : U is open in X and M is meager in X). Show that 8 is the x t of Bake sets in X.

Notes Thematerialinsection 17.1 is tiurn [25], [27L and[31] (nsultsobtained in collaboration with V. Bergelson). The material in Section 17.2 is from [127l. (In [1271. some pains were taken to reduce the number of cells of the partition so that, in lieu of the 4610 cells of the partition we used, only 7 were needed.) , It is aresult of R. Graham(prescntedin [120. Theorem4.31) that wbenevc~(1.2, 252) =A~_lUA~.thmmustbesomex # y a n d s ~ n c i€ (1.2)with(x,y,x+y,x.y) c At, and consequently the answer to Question 17.18 is "yes" for n t r = 2. Thconm 17.21 isfrom[ll9]inthecasen = 2. Itisshownin[119]thatthepartitim R can in fact have 3 cells. It is a still unsolved problem of J. Owings [I881 as to whether there is a two cell partition of W such that neither cell conlains (x, +x, : n, m E N) for any one-to-one sequence (x,)z, (where, of course, one specifically does allow n = m). Theorem 17.24 is due to G. Smith [226] and Theorem 17.27 is a special case of a much more general result from [226], namely that given any rn # n in N,there is a nvo cell papartirion of N neither cell ofwhich contains both SP.((xl)zI) and SP.((Y~)PO,~) forany sequences ( x f ) z 1mtd (y,)zl. The results of Section 17.4 are from 1791. which is a result of collaboration with W. Deuber. H. Lefmann. and I. Leader. Infact a result much stronger than Theorem 17.35 is proved in [79]. That is, $2 and b are elements of A, neither of which is a m t i o ~multiple l of the other, then there is a subse~ A of N such that for no sequence (%,)El in W,is either MT(ii. ( x f ) z I )c A or MT(b, (x,)zl) S N\A. Most of the material in Section 17.5 is from [35], results obtained in collaboration with V. Bergelson and I. Leader. The pmof of the Lebcsgue Density T h e m (Theorem

...

17.42) is horn [l891. llle idea of considuing Baire or mC8SUCBbIe partitions arises horn research of Plewk Riimel and Voigl: Given a sequence ( t n ) z l in (0.11, such that 1, converges, define the set of all s m of the sequence by AS((r.)z,) = (&p h : 0 # F E N). In 11971, Riirnel and Voigt considered the question: If (0,ll = U!=, Ai, must there exist i E (1.2.. ..,r) and a sequence (t,)zl in (0, I] with AS((t,,)=,) E Ai? As they pointed out, one easily sees (using the Axiom of Choice) that the answer is "no" by a standard diagonalization argument. They showed, however that if one adds the requirement that each Ai has the property of Baire, then the answer becomes "yes". In [I951 Plewik and Voigl reached,tbe same conclusion in the event that each Ai is assumed to be Lebesgue measurable. A unified and simplified proof of both results was prexnted in [361, a result of collaboration with V. Bergelson and B. Weiss.

.*j$ .-,~

Chapter 18

Multidimensional Ramsey Theory

Several resub in Ramsey Theory, including Ramsey's Theorem itself, nanvally apply to more than one "dimension", suitably interpreted. That is, while van der Waerden's TheoremandtheFinite Sums Theorem, for example, deal with coloringsof the elements of W. Ramsey's Theorem deals with colorings of finite subsets of W, which can be identified with points in Cartesian products.

18.1 Ramsey's Theorem and Generaliiations In this section we present a proof of Ramsey's Theorem which utilizes an arbitrary nonp~cipalultrafilter. Then we adapt that proof to obtain some generalizations, including the Milliken-Taylor Iheorem. The main feanue of the adaptation is that we utilize ultrafilters with special properties. (For example, to obtain the Milliken-Taylor Theorem, we use an idempotent in place of the arbiaacy nonprincipal ultrafilter.) While many of the applications are algebraic, the basic tools are purely set theoretic.

Lemma 18.1. Let S be a set, let p E S*, let k, r E N,and let [flk = Ai. For each i E {I, 2.. . r), each t E (1.2.. ..,k ) , &each E E [Vf-I.de/ne B , ( E , i) by downward induction on t:

..

(1) Far E E 1s)'-'. Bk(E, i) = (y E S\E : E U (y) E Ai). (2) FortE{1,2 ,..., k - l ) a n d E ~ [ S J ' - ' ,

Thenforeach t E (1.2,.

..,k ) and each E E [s]'-'.

S\E =

B d E . i)

P d We proceed by downward induction on t. If t = A, then for each y

E S\E. E U (y] E Ai fm some i. So let t E (1.2.. . . k 1) and let E E [s]'-'.Then given y E S\E, one has by the induction hypothesis that S\(E U (y)) = U:,] B,+,(E U (yt, i) so for some i. B , + I ( E U {Y). 0 E P.

.-

Theorem 18.2 (Ramscy'sThmm). Lcr Sbcan in&teMMd[nk.

r E

N. YISlk =

Ubl Ai. rhmthcncxicti E (1.2. ...,r ) M d a n i n f u r i t e s ~ s e t ~ o f ~ w iEf hA,.[ ~ ] ~

Pmof: If k = 1, this isjust the p i g m hole principle. so arsumc that k z_ 2. . Let p be any nonpiincipal ulMter on S. Deiine M E , i) as in the statement of Lcmms18.1. ThenS = U:,l Bt(0.i) sopicki E (1.2, ..., r)suchthatB1(0, i) E p. Pickxt E Br(0.i) (sothat Bz((xI), i ) E p. Inductive1y. let n E W and assume that (x~):=~ has been chosen so that whencvu t E ( 1 . 2 ...., k-1)andmt < m z < . . . < m , 5 n , O M h a s

.

.

... -

t E (1,2. k I] and To see that B,+l((xm,,,.x,,. . ..xm,), i) E p whenml < m z < . . . < m t ~ n + l , l e t s u c h r a n d m ~ . r n z,..., m , b e g i v e n . I f m , ~ n . then the induction hypothesis applies, so assume that m, = n 1. If r = 1, the conctusion holds because x.+t E Bt(0. i). If t r 1, the conclusion holds because X.+I E Br((xml.xm2. xm,-, 1. i). The sequence (x.)zl having ban chosen. let m~ s mz < ... c mr. Then 0 x,. E Bk((x,, ,xm2,. ,xm,-, 1, i) so (xml,.x,,. ... xm,1 E Ai.

+

.... ..

.

Inorda toestablish some #ations special notation.

of Ramsey's 'Ihcorem.we introduce some

DeBaitim183. LetSbeaset,letkeN.andlet(p:S+ N. (a) If (D&

is a squence of subsets of S, then

[ ( D . ) ~ ~ . - ?= I ~{ (x1.x~....,x d : x i #xi fori # j in (1.2, .. ..k) andthereexistmt e m 2 c . . . <mkinNsuchthat foreach j E (1.2. .... k), x j E Dm, andforeach j E 12.3. k). mj > -?(xi-I)].

....

(b) If t E N and (D,):=I

is a sequence of subsets of S. then

..

..

[(D&=,. PI\ = ( (XI.XZ.. ,XI} :xi # x j f m i # j in (1.2.. . k) andthereexistm~< m z c - . .c m k i n ( 1 . 2 . ...,t) such that for each j E (1.2, k), x j E D, and for each j E (2.3, . ,k), m, s p(xj-~)].

..

.

...

Forexample, if S = N, ~ ( n=) n , and Da = [m E N : rn 2 n) for each n E N, then I ( D ~ ) ~ ~ .=( ~ { (Ix\~ , x r ,.. . .xtl

G N :XI

c xz
0, then ?. = (sl ,sz. . .. s.) and Zlo = 0. (b) If3 8 , then tail.(?) = ( S ~ + I , S . +.. Z.). . (c) If 3 e W(Q; u)", then lil = n . (d) I t 3 is a finite sequence and i i s a finite or infinite sequence. then (3, i ) will denote the sequence in which ? is followed by i. (Thus, if 3 = ( $ 1 . $2, . . . s,,,) and i = ( t ~12,. . ..),then (s,t ) = ( s l , s2,. .. s, rl, t ~..,.).) (e) A subset X of d is said to be dmosrdense in En(?) if X r l En(?) # 0 whenever i E En(?).

--

.

.

Notice that the notation indiffers slightly from our formal viewpoint since formally n = (0,1, ,n - 1) and (0.1. ... n - 1) is not a subset of the domain of the tixlction 3.

.

...

DeRnftion 1834. We define a condition (*)on a sequence (Tn).,, of subsets of d as follows: (*) For every n

w and every 3 E 8

(a) B.(3)f7Tnf Oand T".then Em(?)E 7".

@) if 3 E

We make some simple observations which will be useful in the proof of Carlson's Theorem. R-k

1835. Let 3 E 8. ( a ) I f X isalmostdensein En(?),then X isalmusrdense in En(;) forevery? En(& @) I f X n B . ( a = 0,then X n B.(i) = 0for every i E En(;). (c) F o r ~ iy E a. i E Em(?)fondonly ifii. = 3,. and tail.(i) E Bo(lail.(3)).

Lemma 1836. Let (Tn).,, be a sequence of subsets of 8 which satiq5es (r) and let 3 E 8. Thenfor each k E o,then exists i E B& such that,for every n r k in N and e v e v variable reducria i we have (7, rail.(?)) E.!'IT P m f : We first show that

(t) forevery 7 E 8 and evcry n E N.there exists G E Be(?) such that (7. tail.(G)) . Tlrl forevery variable nduaion ? of $.

E

..

We enumerate the finite set of variable reductions of iW as (7 I . iZ,.,im). We put ? O = tail. (?) and then inductively choose i I . ? . .., f E 8 so that the foUowing properties hold for each i E [I. 2. .. . ,m):

',

(i) iiE &(? i-l) and (ii) (;i,ii) T~i'l,

..

Suppose that i E (0.1.2, . ,m - 1) and that we have chosen ?'. Let 1 = [?'+'I. By condition (a) of (*). we can choose ri E 6,((ii+'.i i ) ) n T r . Let ?'+I = tailr(;).

men?'+' E Bo(?')and ( i i + ' , i ' + l ) = I E T ~ i i + ' ~

(6.. i "). Xiice i " E Bo(tail.(i)). we have Now let i be a variable reduction of 6.. Then i = i' for some i E ( 1 . 2 . . .. .m). Having chosen im. we p u t d =

a E 6"(i).

E Bo(ii),and hencethat ( i i , i m ) E ~ ~ ~ i i~i ) )(. So ( i (ii,?") ~ , = Wenotethat i m

( i i , taiI.(l)) E TF'I, by condition (b) of (*). Thus (t)is established. Now let k E o be given. By (7) we can inductively define elements To, i ' , of 6 with the following properties:

i2....

-

(i) ... = t ' = s . (ii) inE B.(in-') forevery n > k, and (iii) forevery n > k and every variable reduction 7 of

(7, tail.(;")) E Tlil.

Let i = (1'1, t22. 1'3,. ..). Then ? E B&. Let n z k be given and let ? be a variahle reduction of 4., Now = in-' 1, so i is a variable reduction of in-',1 and thus (i, tail,(?")) E rlil. Also (i, tail.(?)) E Blil((i.tail(in))) and so by condition @)of (*). (i, tailn(i)) E Tli1. o

4,

Corollary 1837. Let ? E 8. k E o. and X G 6. Then there exists i E B&) such that,for eveiy n s k and every variable reduction i of ii., either X is olmost dense in Blil((i,tail.(?))) orehe X n B1;1((7,tail.(?))) = 0. Pmof: For each n E o w e define Tn G 8 by stating that i E T nif and only if either X is almost dense in B.(i) or X n B,(i) = 0. We shall show that (Tn)., satisfies (*). and our claim will then follow from Lemma 18.36. Letn E oand i E 6. If X isahostdensein B.(?).theni E B,,(i)nTn. Otherwise there exists 2 E 6. ( i ) for which B, (I) n X = O and so I E B.(i) r l T". Thus satisfies condition (a) of (*).

To see that (Tn)., satisfies condition (b)of (*), let n E o.let ? E 1"a d let i E 4 ( i ) . Then B d i ) E B.(i). Consequently:if X is almost dense in B&), 0 X is almost dense in En(;) and if X n Em(?)= 0. then X f l B&) = 0.

Ddiattioa 18.38. (a) Let ;E d. then ~ ( =3{w E W(B: v ) : w is a variable redaced

a. ...,

word of ( b ) L e t ; ~ 6andletw E R(3. 'ibe.N(w.i)isihatI E Nforwhichthnruist al.az. a! E O U [v) suchthat w = Il(a~)-rz(az)-. ..-R(al).

Lemma 18.40. Lrr 3 E d. k E o and X E d. Thrn there exists 7 E B& such that, for every n E N sarirfying n > k and every variable reduction i of either

a,

Pmof:Wedetke a seq(T").,, of subsets of d by stating that i e T" ifandonly if either X n Bn+~((il..w , tailN(w,&(o)+n(i))) = 0 for every w E R(tail.(i)), or else X is almost dense in B.+l ((il,,. w , t a i l ~ ( , , ~ ] ~ ( a )(5)) + . forew E ~(tai~., (5)). We shall show that (Tn)., satisfies (*I. The claim then follows from Lemma 18.36. Let n E o and r' E d. We shall show that T" and i satisfy conditions (a) and (b) of (*). By Corollary 18.37, there exists 3 E Em(?)such that, for every rn r n in W and every variable reduction i of Zl,. X n Bl;l((i, fail,(;))) = 0 or else X is almost dense in Blil((i.tail,(;))).

Let

vo = (w E R(um(u')) : X n Bn+l((u'lm W, tail~(rn.a(ri))+n(Z))) = 0). VI = R(tail.(i))\Vo. and V2 = W(W v)\~(tail,(i)). We note that, if w E R(tail.(i)), then w E VI if and only if X is almost dense in B ~ + I ( @ w, I ~taih(w.&~ 0. Thenforeach k E Nthere exists H E Pf(N) with min H s k such that p(A n T ~ - ' [ A ] )s 0.

.

E N such that p(A) > p(X)/m. F a n E [ I ,2.. . ,m), let F. = [k n , k n 1.. ,k m]. Then each ~ ( T F , - ' [ A ] )= p(A) s p ( X ) / m so pickn.! E [1,2, ....m ) such that n < e a n d p ( ~ ~ m - ' [ A l n T ~ t - ' [ r A I0.) Let H = {k+n.k+n+l, ....k+t-1). ThenTnoTFt = T F ~ s o T F ~ - I [ =AT] ~ ~ - ' [ T H - ' [ A ] ]

P m J Pick m

+

+ + .. +

so ~ ( T H - ' [ An ] A) = ~ ( T F ~ - ' [ T H - ' [ An] A ] )

= IL(TF~-'[TH-'[AI]n T ~ - ' [ A I ) = ~ ( T F ~ - ' [n AT ] ~ - ' [ A ]> ) 0.

0

Lemma 1932. Let (X,8 ,p, (T,),,s) be a measure presrmMn8 system ~d let A E 91 satisfip(A) > 0. Forevery sequence (s,,)gl andeveryk E El. thereexists F E .Pf (N) such that Ir < min F and p(A n T;'[Al) > 0,where s = n m € Fs,,. P m f For F E Pf(IV), let RF = ThEFs,,.Then (RF),~&,'I(Pois a monotone action of Pf(N) on X. So we can choose a set F with the required properties by Lemma 19.31. 0

c

Theorem 1933. Let(X, 8,p, (Tr)rt~)bCame~represe~~ingsyslemandIet C S. Ifthere is some A E S with @(A) s 0 such that 1s E S : p ( A n T ; I [ A ] ) > 0 ) C C. then C is an I P set. Pmof: This follows immediately from Lemma 19.32.

0

Deanition 1934. Let S be a semigroup. A subset C of S is a d y m i c a l I P set if and only if there exist a measure preserving system (X. 21, p, (T,),,s) and an A E 21 with p ( ~ >) 0 such that [s E s : p ( n~ T;'[A]) > 0)E C.

+)

Recall that by Theorem 16.32. there is an IP* set B in (N, such that for each n E N, neither n B nor -n B is an IP* set. Consequently. the following simple result shows that not every P set is a dynamical IP* set.

+

+

Theorem 1935. k t B be a dynamicat I P set in (N, +). T h e n is a dynamical I P set C E B such thatfor each n E C. -n + C is a dynamic01 IP* set (and hence -n B is a dynamical IP* set).

+

19.4 Dynamically GwratedI P . Sets

409

Proof Pick a measure space (X, S , r ) , a measure preserving action (Tn)nEN of N on X,andasetA E Ssuchthatp(A) =-Oandln €N:p(AnTn-'[AI) s o ] s B . I . e t C = (n € N : p(A r l T,-'[A]) > 0). To see that C is as rrquircd, let n E C and let D =A~T.-'[A]. Weclaimthat [m E N : p(DilT,-'[DI) > 01 E -n+C. Tothis end, let m E N such that p(D i l T,-'[D]) r 0. Then DnT;'[D]

=A~T~'[A]~T~'[AI~T~~[A]]

r A r l T;'[T;'[A]]

= A n (T, o T,)-'

[A]

= A n T,&,[A] son+m€C.

.

0

R e d from Theorem 18.15 that an IP* set in M x N need not contain [w] x FS((x,)E,) for any sequenw (x.)zl. In the final result of this section we show that a dynamical IP* rat in the product of two semigroups with identities must contain sets of the form FF'((x,JEl) x FF'((y,,)zl). Indeed, it has a much stronger property: given any sequences (w,):=~ and (z.)El one can choose infinite product subsystems m (xn)n=l ~f(w.):=~ and ( y n ) E l of (z.)El with FP((x.)=,) x lWy.)El.) contained in the given dynamical IP* set. More than this, they can be chosen in a parallel fashion. That is if x,, = w,, then y. = zr. We only discuss the product of two semigroups for simplicity and because the generalization to arbitrary finite products is straightforward. In the proof of Theorem 19.36 we utilize product measure spaces. In doing so we shall use the customary notation. If (X. 8 ,p ) is a measure space and m E N, then (Xm,P, p m )denotes the measure space defined as follows: is the usual Cartesian denotes the a-algebra generated by the sets Xi='Ai where Ai E S for product, Sm each i E (1.2.. . . m], and pmdenotes the countably additive measure on 41'" such that p m ( ~ ~ = = l ~n:=.=, i ) p(Ai) for every sequence (Ai)rCl in d8. Any transformation T : X -+ X defines a transformation T : Xm + Xm for which T(xl,x2, .. ..x,) = ( T ( x l ) , T(xl), . . . , T(x,)). If T is measure preserving, so is T, because T)-'[X~=~AA =~X] ~ = ~ ( T - ' [ A ~for ] ) every Al.Az A, G X .

n,,~,

n,,~,

.

n,

.....

n,

n.

(n,

Theorem 1936. Lct SI and S2 be semigmups with identities and lei C be a dynamicd IF" set in S1 x S2. Lcr (w.):=~ be a sequence in SI and let (z.)?=~ be a sequence in S2. There exists a sequence ( H , ) z l in P,(W) such that (a) for each n, max H. < min H.+I and @) iffor eachn, x. = nreH, w, and y. = &H,

FP((x"):,)

zr, then

r c.

x FF'((Y")~')

Proof: Let S = SI x S2. Pick a measure space (X. 41.p), a measure prcsming action (T(I.l))(I.~)~~,~~Z of S m X, and B E a) with p(B) > 0 such that,

D = ((s. t) E s : P(B

n ~,,,)-I[BI) > 0) 5 C.

410

19 Relations Wlth Toplogid Dynamics

I

!

411

Notu

Notes

.

-

The n d o n of 'dyllamical system'' is often defined only for canpact metric spsas. 'Ihe greater generality that we have chosen (which is essential if one is going to lake flS as the phase space of a dynamical system) is also wmmon in the literature of dynamical systems. General references for topological dynamics include the books by R. Ellis [86] and 1. Auslander [S]. Theorem 19.8 and Lemma 19.9 are due to B. Balcar and E Franek in [I21 as is the proof that we give of Theorem 19.10. Theorem 19.10 is proved by R Enis [86] in the case that S is a p u p . Theorem 19.15 is due to S. Glasner in [IDS], w h m it is stated in the case that S is a countable abelian group. Theorems 19.17 and 19.18 are from [141], a result of collaboration with I. Lawson and A. Lisan. Theorem 19.19 is due to W. Ruppert in [219]. S. Glasner has recently published [I061 a p m f thal the envelophg semigroup of a minimal left ideal in pZ is not topologically and algebraically isomorphic to flZ via a map taking n to A:. As a consequence, the condition of Theorem 19.18 docs not hold in BZ. The equivalence of the notions of "ccnhal" and "'dynamically central" was established in the case in which S is countable and the phase space X is metric in [Dl. aresult of collaboration with V. Bergelson with the assistance of B. Weiss. The equivalence of these notions in the general case is a result of S. Hong-ting and Y. Hong-wei in [164]. Theorem 19.24 is due to 1. Auslander [4] and R. Ellis [851. Most of the results of Section 19.4 an from an early draft of [321, results obtained in collaboration with V.Bergelson. Lemma 19.31,is a modification of standard results about Poinc& recurrence. Theorem 19.33 for the case S = Z is due to H. Furstenberg

WI.

Chapter 20

Density

-Connections with Ergodic Theory

As we have seen, many multr in Ramsey Theory assat that, given a finite partition of some set, one cell of the panitin, must contain a specified kind of shucture. One may ask instead that such a smcture be found in any "large" set. For example. van der Waenlen's Theorem (Corollary 14.3) says that whenever N is divided into Gnitely many classes, one of these contains arbitrarily long arilhmetic progressions. S z e m d i ' s Theorem says that whenever A is a subset of N with positive upper density. A contains arbitrarily long progressions. Szemen%i's original proof of this theorem 12321 was elementary, but very long and complicated. Subsequently. using ergodic theory, H. Furstenberg [97] provided a shorter proof of this result This proof used his "correspondence principle" which can be viewed as a device for translating some problems involving sets of positive density in W into problems involving measure preserving systems, the primary object of study in ergodic theory. We present in Section 20.2 a proof of Furstenberg's CorrespondenaPrinciple using the notion of p-limit and in Section 20.3 a strong density version of the Finite Sums Theorem obtained using the algebraic structure of (BW,+).

20.1 Upper Density and Banach Density We have already dealt with the notion of ordinary upper density of a subset A of N, which was defined by Z(A) = lim sup n-m

IA n(1.2. .... n)l n

Another notion of density that is more useful in the context of ergodic themy is that of B a ~ c densiry h which we define here in a quite general context. Ddinilion 20.1. Let (S,.) be a countable semigroup which h s been ~ enumerafcd as ( s , , ) ~and , let A E S. The right Banoch densify of A is d:(A) = sup(@E B : for all n E

N there exist m 2 n and x € S

20.1 Uppa Density and Banach Dtnsity

If S is commutative, one has d: = d; and we simply write d9(A) for the Banach density of A. Notice that the Banach density of a subset A of S depends on the fixed enumeration of S. For example. in the semigroup (N,+),if N is enumerated as

i.

then d*(2N) = 1, while with respect to the usual enumeration of M, d*(2N) = When working with the semigroup (N, +) we shall always.assume that it has its usual enumeration so that

such that

IAn[x+l,x+2, m

.... x+mH

r a).

Lemma 20.2. Let (S, .) be a countable semigmup which has been enumerated as

+

m (sn),,=, and let A, B G S. Then d:(A U B ) 5 d:(A) d:(B) and d:(A U B) 5 d:(A) d;(B). B S is right cancellative, then d:(S) = 1 and if S is lefr cancellative. then d;(S) = I

+

Pmof: This is Exexcise 20.1.1.

Recall that we have defined A = ( p E fJN : for all A E p. Z(A) r 0). We introduce in a more general context the similar sets defimd in terms of d: and d;. Definition 203. Let (S, .) be a countable semigmup which has been enumerated as (s,,)z,. Then A:(S, -)= ( p E fJS: for all A € p. d:(A) s 0) and A;@, .) = ( p E fJS : for all A E p, d;(A) s 0).

Lemma ZOA. Let (S, .) be a countable semigmup which has been enumerated as (s.)z, andletA g S. 0, then n A;(S. .) # 0. (a) Ifd;(A) (b) .'fd;(A) r 0, then Ti r l A;(S, .) # 0.

x

Proof: We establish (a) only. By Theorem 3.1 1 it suffices to show that i f 3 is a finite nonempty subset of P ( S ) and d : ( U 9)s 0. then for some B E 3.e ( B ) r 0. This follows fmm Lemma 20.2.

414

20 Density

-Conoectims with Ergdc Theory

.

Recall fromlleorems 6.79 and 6.80 that both A andN.\A an leftidealsof (BN. +) and of (BN, .). By way of contrast, we me in thefollowing theorems that S*\A: is far from being a left ideal of 0.9 and S\A; is far fmm being a right ideal of /IS.

Theorem 20.5. Lct (S. .) be a countable right crnueIlativc semigmp which kas been enumerated as ( s n ) g l . Thcn A:(S. .) is a right ideal of BS.

P M One has that A:(S. .) # 0 by Lemma 20.4 and the fact from Lnnma 20.2 that d:(S) s 0. La p E .I. let 9 € PS, and let A E p . q. We need to show that d:(A)~O.LaB=(y€S:y-'AEq).ThenB~pso~(B)~O.Picka~Osuch thatforalln ~ N t h e r e c x i s t mz n a n d x ~ S s u c h t h a It B n

@I.* ....,s

~ .IX I

m To see that d:(A) 2 a , let n E N and pick m > n and x E S such that I B ~ I S I , S ~S .~ I . X I 2 a. LetC = B f l ( s 1 . q s,}.xandpickz E ,r. Y-lA. Then, since S is right cancellative. p,l~ is a one-to-one h c t i o n from C to A f l (st,ST. ... sm) xz so ~ A n ' ' ' ~ s"r ~ - ~ ' ~ ' ' x 2z' a. 0

n,

......

.....

.

.

T h e o m 20.6. Let (S. .) be a countable kfl cancellorive semigmup which has been enumemted as ( s , ) z l . Then A;(& .) i s a I& ideal of BS.

Pmof L* p E A;(S, .), let q E BS, and let A E q . p. We need to show that d;(A) z 0. Now ( y E S : y - l ~E p ) E q so picky E S such that y-IA E p. Then d;@-lA) > 0 so pick a > 0 such that for all n

€

N then exist m s_ n and x E S such

nx " S ~ ' s 2 ' " ' r s m l l z a. Now givcnm andx. sinceSisleftcancellative. m A, is aone-tmne function from Y - I A fix. ($1. sz. . . .,s, ] to A fl yx .(sl,s2, .. s,). and consequently, d; ( A ) 5 a. 0

that ''-IA

..

As a consequence of Thwrcms 20.5 and 20.6, if S is commutative and cancellative. then A*@) is an ideal of BS.

Theorem 20.7. Let (S. .) be a countable right cancellorive semigmup which has been enumerated as ( s n ) g l .For any A S S,i f A is piecewise syndetic, then then is some G E P J ( S ) such thmd:(UlpG 1-'A) = 1. I f ( S . .) = (N. +), then the converse holds. Consequently.

c

Pmof Let A S , assume that A is piecewise syndetic, and pick G E P f ( S ) such that for each F E $',(S) there exists x e S with F x E U,,G t-' A. To see that d:(U,,G ' - ' A ) = 1,letn E Nbegivenandpickx E Ssuch that

.

I

20.1 Uppx Lknsity and B a n d Density

Now assume that (S, .) = (N. +) and we have G E 3f' (N) such that dg(U,,, -t A) = 1. Let F E Pf(M) be given and pick k E Pi such thar F (1.2 ....,k). Weclaimthatthereissomex E Nsuchthat{x+I.x+2 x+k) c (Ute, -1 +A). To see this. pick some m > 2k2 and some y E N such that

+

....,

+

and pick v E N such that vk < m 5 (u 1% noting that v ? 2k. If for some i E (0, I ,...,y-l)onehas(y+ik+l,y+ik+2 .....y+(i+l)kI -t+A. then we arc done, so suppose instead that far each i E (0. I, . ,u 1).

.. -

-

so, sincem > uk,omhasthat (1 &)vk < v(k- l)+k sothat 21.1 v. acontradiction. 0 The final conclusion now follows f m Corollary 4.41.

The following result reflects the interaction of addition and multiplication in ON which was m e d in Chapter 13.

+)

E A*(N, and let q E /3N. To see that q . p E A*(N. +), let A E q p. Then [x E N : x-'A E p ) E q so pick x such that x-'A E p. Then d * ( x - ' ~ ) > 0 (where here, of course. we arc referriog to the additive version of d*) so pick u > 0 such that for all n € N there exist m ? n and y € R such that

Prooj Let p

.

'x-1An(y+1'y+2"..'y+m)' m IAnIxy

?a.Theng*ensuchn,m,andy.ooeharthat

+ I,xy+2. ....xy+xm]I m

so that dW(A)?

?U

-.a X

Recall that, given a subset A of a s e m i p u p (S. .) that we have defined a me T in A whose nodes are functions. Recall further that given a node f of T. the set of successors to f is denoted by 9, and that T is an FP-tree provided that for each node

416

20 Density

-Connectionswith Ergodic 'lheny

f of T , Bf consists of all finite products of'entries on paths extending f which occur after f. (By a path in T we mean a function g : o + A such that for each n E o,the restriction of g to (0.1, ,n - I) is in T.)

...

Theorem 20.9. Let ( S , .) be a countabli c m l l a t i v e $mrigmup which has been enumerated as (s.)zl. Let r E A and ler S = Ai. 77ure &st i E {I, 2,. . ,r ] and a tree T in Ai such thatforeachpath g in T , EF'((g(n))>) S Ai, andforeach node f ofT.d:(Bf) > Oandd,'(Bf) > 0.

.

Pmof: By Theorems 205 and 20.6, -) is a right ideal of 5 S and A;(S, -) is a left ideal of 5s. By Corollary 2.6 and Theorem 2.7 pick an idempotent p E A:(& .)n A;(S. .)and pick i E {I. 2. . ..,r ) such that Ai E p. Then by Lemma 14.24 thereisan P - k c T in Ai such that for each f € T. Bf E p. Thus, in particular, for each f E T , dF(B,) > 0 and d i ( B f ) > 0. 0 The following theorem is not a corollary to Theorem 20.9 because d * ( A ) > 0 does not imply that Z(A) > 0. The proof is essentially the same, however, so we leave that proof as an exercise. Theorem 20.10 is an interesting example of the strength of combinatorid results obtainable usingidempotentsin~N.Althoughmany of theRamscy-theoretic results which we have presented were first obtained by elementiuy methods or have subsequentlyk e n given elemenmy proofs. we doubt that an elementary proof of Theorem 20.10 will he found in the near future.

Theonm 20.10. Let r E A and let A = Ai. The= exist i E (1.2,. ..,r ) and a tree T in Ai such thatfor each path g in T.F S ( ( g ( n ) ) z o )S A;;and for each node f of T , Z ( B / ) > 0. Pmof This is Exemse 20.1.2. The following lemma will he needed in the next section.

Lemma 20.11. Let A G W such that k ( A ) = w r 0. Then there exists some sequence ( I k ) K 1 of intervals in W such that lim I l k 1 = m and k+ca

-

lim

IA n I k I = a.

141

Pmof This is Exercise 20.1.3. Exercise 20.1.1. Rove Lemma 20.2. Exercise 20.13. Using the fact that A is aleft ideal of (BPI. +) (Theorem 6.79),prove Theorem 20.10. Exercise 20.13. Rove Lemma 20. It.

20.2 'Ibe Comspondence Principle

~.

20.2 The Correspondence Principle

Recall that a meanrre preserving system is a quadruple (X. 8,p, (T,),,s) w h m (X, 21, p ) is a measure space and (T,),,s is a measure preserving action of a semig r w p S o n X. If the semigroup is (N. +). the action (T.).en is generated by a single function TI and we put T = TI and refer to the measurepnserving system (X. S ,p. T).

Theorem 20.12 (Furstenberg's Compondence Rinciple). Let A g N with d'(A) > 0. There ex& a measure preserving system (X, B.p , T)(in which X is a compact metric space and T is a homeomorphismfmm X onto X) anda set A' E 8 such rhat (1) p(A1) = d*(A) and (2) forevery F E d*(n ~

nnSF(-n

+A))

z p ( ~ ' nnnEF. T-~[A'I).

Proof: Let R = '(0, I1 with the product topology and let T : S2 -, S2 be the shift 1). (By Exercise 20.2.1 one has in fact that S2 is a defined by T(x)(n) = x(n metric space.) L e t t be the charafteristic function of A (viewed as a subset of Z). Let X = cC(Tn(f) :n E Z), the orbit closure of 6. Then X is a compact mehic space and (the restriction 00T is a homeomorphism from X onto X. For each n E Z, let D. = X n 1r.-~[(1]] = ( 8 E X : b(n) = 1) and notice that Dmis clopen in X. Let d be the Boolean algebra of sets generated by {D,, : n E Z] and let B be the a-algebra generated by {D. : n € Z). Notice that for each n E Z. TCD.1 = D.-1 and T-'[D.] = D.+I. Consequently, if B E 8,then T - ~ [ B ]E 8 (and T[Bl E 8 ) . Define rp : P ( X ) -, P ( N ) by q(B) = {n E N : Tn(S) E B). Let a = d*(A) and pick by Lemma 20.1 1 a sequence ( I t ) & of intervals in N such that lim Ilk I = m and

+

k-co

Pick any p E N* and define v : 9 ( X ) -+ [O, 11 by

Notice that v(X) = 1. Further, given any B, C S X, one has cp(Bfl C) = p(B) nrp(C) (sothat,inparticular,if BnC = O.thenrp(B)nrp(C) = O)andrp(BUC) = rp(B)Urp(C). Thus, if B and C are disjoint subsets of X. then

= p- lim I M B )n iki+ IMC)n keN lid

20 Density

418

(-I

-Connections with Ergodic Thmry

Next we claim that for any B V ( B ) )n so

+

w

G X , v(T-'[El) = v ( B ) . Indeed, ~ ( T - I [ B ]=)

+

because for any k E N,I(-1 q ( B ) ) n It1 and Ip(B) n It1 differ by at most I. Thus v is finitely additive and T-invariant on P ( X ) and hence in particular on A. Funher all members of A arc clopen in X,so if one has a sequence in A such thar each &+I E B. and B. = 0, then since X is compact there is sow k such that for all n 2 k. B. = 0 and hence lim v(B.) = 0 . n-m Therefore, we are in a position to apply Hopf's Extension Theorem. (This is a standard result in first year analysis cowses. See for example [115, Exercise 10.371 or [14. Satz 3.21.) The finitely additive measure v on A can be extended to a countably additive measure p on 8 where for each B E 8.

nLl

A, v ( T - ' M ) = v(C).ooeseca immediately that for all B E S. p ( ~ - ' [ B l = ) p ( B ) . Thus we have that (X.8.p . T ) is a measure p m e ~ n system. g Let A' = Do = {6 E X : 8(0) = 1 ) and observe that p(A1) = A and for each n E N.T-"[A'] = Dn and (p(D.) = (-n A) r l N . Thus

- Usingthisdescriptimofp(B)andthefactthatfod E

+

and given any

F

E

Pf(N).

= p-lim

k€N

IcWO)n rXEF d D n ) n hl lltl

When proving Szemerkdi's Theorem. one wants to show that. given a set A with P ( A ) s 0 and e E N,there is some d E N such that

( a ~ N : ( a , a + d , a + 2 ..... d a+td)SA)#O. As is typical of ergcdic theoretic proofs of combinatorial facts, one establishes this fact by showing that the set is in fact large. The proof of the following theorem require. extensive background development, so we da not present it here.

C o m h q 20.14 ( S z e W s Theorem). Ln A

E N m'rh d g ( A ) r

0. Then A con-

rnim arbitrarily long anthmeric prugnssimrr.

Pmof: PieL by Theorem 20.12 a measure pnsaving system ( X . 93, p , T) and a set A' E

8 such that

(1) p(A1)= dg(A)and (2)for emy F E Pf(FI). d * (n ~

Let C E

N be

n,

c-n

+ A)) r M A ' n a,,T-"~A'I).

given and for each i E (1.2,.

...CJ,let Ti

= T i . Notice that

f i . Tz.. ..,Tc are measure presming transformationsof ( X , 8,p ) that commute with each other. Pick by Theorem 20.13 some d E N such that p(A' f i ~ ; . - ~ [ Ar' ]0) and notice that for each i. Ti-d = T - ~ L.~ et .F = {d. 2d, 3 4 ...,Cdt. Then

&

Exercise 20Zl.Let Q = '(0, 1 ) and for 6 # 9 in Q. d e k e p(#, q ) = what k = minllrl : 1 E Z and C(r) # ~ ( 0 (and 1 of p(6.F) = 0). Rave that p a a mehic on R and the metric and product topologies on Q agree.

203 A Density Version of the Finite Sums Theorem The straightforward density version of the Finite Sums Theom (which would assen or perhaps d * ( A ) r 0 - would contain that any set A E N such that &A) s 0 F S ( ( x , , ) ~ , for ) some sequence { x , , ) ~is~ obviously ) false. Consider A = 2N + 1. However. a consideration of the proof of S z m d d i ' s Theorem via ergodic theory reminds us that the pmof was obtained by showing that a set which was only requiredto be nonempty was in fact large. Viewed in this way, the Finite Sums Theorem says that whenever r E N and N = I& A,. there exist i E (1.2.. .. r) and a sequence ( x J ~ such , that for each n, x.+l E Ai n C,,FX, Ai : 0 # F E (1.2, n ] ) . In this section we show that not only this set, but indeed many of its subsets. can be made to have pasitive upper density and specify how big these sem can be made to be.

-

nf-

+

.

...,

B

20 Density -Connections with Egodie Thmry

420

Lunma 20.15. Let A E N such t h n r z ( ~> ) 0 and let B be an i y k i t c subset of N. For mery 6 w 0 then uisr x c y in B such that $(A n (-(y x ) A)) > 6.

- +

Notice that for every t E N,;Imm

l ( - t + ~ ) n ( l , 2 , ....x.11

-

=a.

Xn

Enumerate B in increasing order as ( y i ) E l . Let c > O be given. If e 2 a2. the conclusion is hivial, so assume that c c aZ and let b = J-. Pick k E N such 40 thatk w -andpickC~Wsuchthatforalli E (1.2, kJandalln >C,

....

It sfices to show that far some pair ( i , j ) with 1 5 i c j 5 k, one has

-

+

since d((-yi + A ) r l (-yj + A ) ) = $(A n (-(yj - yi) A)). Suppose instead that this conclusion fails. Then in particular for each pair (i,j ) with 1 5 i j 5 k, there is some vi,j E N such that for all n 2 vi,j

Let n = max((L) U (v1.j : 1 5 i c j 5 k ] ) and let m = x.. For each i E (1.2, ...,k).

. -. -.

let Ai = (-yi + A ) n (1.2.. .. .m)and forany C E (1.2.. .. ,m).let p(C) = ILI m Thenforeachi~(1.2. ..., k ) , b c p ( A i ) c h , a n d f o r l s i < j i k , p ( A i n A j ) c a2 - c. Notice also that for C E {I. 2. .. . , k].

We claim that it suffices to show that

Indeed, once we have done so. we have k2b2 c 2ka

a conhadiction.

+k(k - 1)(a2 - c) so that

To establish the desired inequality, observe first that

'

so that

Next one establishes by a routine induction on m that for any function f : (1.2 ,..., rn]+R,

and thus by (I) we have

Now, for any t,

so that

Now, using (2) the mquLed inequality is established.

0

Notice that in Lemma 20.15, one cannot do better than Indeed,choose a set A 5 W by randomly assigning (with probability f ) eachn E W to A or its complement. Then for any r E W,Z(A fl (-r + A)) = f = Z ( A ) ~ .

Lemma 20.16. Let

p E A such that p

+ p = p, let A € p, and let e > 0.

+

-

P~w$ La B = (x E N : ; i (n~(-x A)) 2 Z(A)* 6/21. Since by Thenem 4.12, A. = (x E A : (-x A) E p) E p, it Suffices to show that B E p. Suppose instead &at N\B E p and pick by 'Iheorcm 5.8 a sequence (x.);, in N such that FS((x.)Z,) E N\B. Let C = {C:,, xf : n E W) and pick by Lemma 20.15 some n irn such that C;",, x, C:, xf E B. Since

+

-

this is a muadidon.

0

The proof of the following theorem is reminiscent of the proof of Lemma 14.24.

Notice that, following our usual convention that concerned with at the time. one has D(0, g) = A. Let To = (0) and inductively for n E w, let

n0 is whatever semigroup we sre

La T = TT,.Then trivially T is a aee in A. (Given x E B(g). x E C(0. g) D(0, g) = A.) Notice that forn E W andg E Tn.Bs = (x E B(g) : x > g(n - 1)).

Weshowfintbyindudonoanthatforallg E T.andall F C (0.1. .... n - 1). C(F, g) E P and Z(D(F, g)) > 62'F1. O ~ S C Wthat ~ (*) ifk = max Fandhisthe~snidonofgto(0,I, ....k).thenD(F,g) = D(F, h) and C(F, g) = C(F, h). To ground the induction assume that n = 0. Then g = F = 0 and D(F. g) = A so that a(D(F. g)) s 8 = @.Let c = Z(A)' - 6'. Thm

20.3 A Density Version of tbe Fbite Slnnr l?muun

423

so that by Lemma 20.16. C(F. g ) E p. N w l e t n ~ o a n d a s s u m t h a t f o t a l l g ~ T ~ a n d a 10.1 I l F,..., ~ n-1). C(F, g) E panda(^(^, g)) r 6''". Let g E Tm+land F E (0. I.. ...n) be given. By the observation (*). we may assume that n E F. Let h be the rrJtriction of g to (0. I.. ,n 1). let H = F\[n), and let x = g(n). Then h E T. and x E Bfh). We now claim that

.. -

+ D(H. h)). To see that D(H, h) r l (-x + D(H, h)) E D(F, g), let z E D(H, h) n (-x + D(H, h)). Then z E D(H, h) S A. Let Y E m((g(t))tsF) and pick G E F such that y = &c g(0. We show that z E -y + A. If n G. then y E FS((h(t))tEH)and h e m . since z E D(H, h). z E -y + A. We thus assume that n E G. If G = (n), then z E -x + D(H. h) E -x + A = -y + A as required We therefon assume (**I

D(F, g) = D(H, h) n (-x

that K = G\(n) # 0. Then K E H. Let v = & ~ g ( r ) = X,," h(t). Then ~EFS((h(t))~c~)~~x+z~D(H,h)~-v+Aadthusu+x+z=y+z~Aas required. To see that D(F, g) E D(H. h) r l (-x D(H, h)). let z E D(F. g). Since m ( ( h ( t ) ) t s ~-E ) m ( ( g ( f ) ) t ~ ~z ) E , D(H. h). Since x = g(n) E FS((g(t)),e~), x z E A. To see that x z 'E D(H, h). let y E FS((~(~)),,H).Then y x E FS((g(f))re~) SO Z E -(y x) + A SO x z E -y A. Thus ('*)is established. D(H, h) E p. Thus Since x E B(h) E C(H, h), we have -x

+

+

+

+

+

D(H, h) n (-x

+

+

+

+ D(H, h)) E p. +

That is D(F, g) E p. Now since x E C(H. h) and D(F, g) = D(H, h) n (-x D(H. h)), ~ ( D ( F 8, ) ) r 6Z'H'f'= 8"". Let y = a(D(P. g)) s~"' and let c = ~ ~ 8 2 " ' y2. Let

-

+

E={zED(F,g):-z+D(F.g)~pmd -~ ( D ( F .g) n (-2 D(F,~))) > I(D(F.~))' - CI.

+

-

+

.

BY emm ma 20.16. E E p. since Z(D(F. g)12 = (82IF' y)2 - c = 62'm+1 we have that E = C(F, g) so that C(F, g) E p. The induction is complete. Now let f be any path of T. We show by induction on IFJ,using essentially the first proof of Theorem 5.8. that if F E Pf(cu), n = min F. and h is the restriction of f to 10. I , . n - 1). then X r E ~ f ( f )E D([O, 1. .n 1). h). first assume that F = ln1,letgbetherestrictionof f to(O.1, ....n),andlethbetherestrictionof f to(0, 1. .... n - I). Theng= hU((n, f(n)))with

... -

...

f (n)

E

.

B(h) G CUO, I,. .. n - I), h)

as required.

E DUO, I , . ..,n - 1). h)

.

Now assumethat IF1 > I. let G = F\(n),andlet m = min G. Let h, g, and k bethe restrictions off to (0. I , . .. n I), (0.1. .. . n). and (0. I , . ... m 1) respectively. men & z f ( f ) E DUO, 1, ..., m - 11.k) G D(I0, 1 n1.g)

.-

-

.....

424

20 Iknsity

...,

-Connectioar with Ergodic lbcory

andD(IO,l, n I , g ) = D ( l O , l , ... , n - l I , h ) n ( - f ( n ) t ~ ( ( o , l , . . . . n-11.h)) by (**). Thus &G f (1) f (n) E D((0. I. . ,n I), h) as required. In particular. conclusion (1Xa) holds. To establish conclusion (I)@), let F s Pf((o), pick n w max F. and let h be the restriction off to ( 0 . 1 , ... n 1). The conclusion of ( I ) @ )b precisely the statemen\ proved above, that ~ ( D ( Fh)) , s 621F'. As we observed above, for n € N and g E T,,.Bg = [x E B(g) : x r g(n 1 ) ) and thus B, E p so. since p E A. J ( B ~s) 0. 0

+

.. -

.-

-

Notes The notion which we have called "Banach density" should perhaps be called "F'olya density" since it appears (with refmnce to N) explicitly in [1961. To avoid proliferation of terminology, we have gone along with Furstenberg [98]who says the notion is of the kind appearing in early works of Banach. The definition of Banach density in the generality that we use in Section 20.1 is based on the definition of "maximal density" by H. Umoh in [237]. It is easy to consttuct subsets A of N such that for all n € N,a(U:=, -1 A) = d ( A ) > 0 . On the other hand, it was shown in 11261 that if d * ( A ) > 0, then for each E > 0 there is some n E N such that dl ( U,: -t A ) s 1 - e. Consequently, in view of Theorem 20.7, it would seem that A*(N, is not much larger than c! K @ N , +). On the other hand it was shown in [73],a result of collaboration with D. Davenport, that ct K(@N,+) is the intersection of closed ideals of the form GUN' p ) lying strictly between ce K(@W,+) and A*(N, +). Theorem 20.13, one of three nonelementaq results results used in this book that we do not prove, is due to H. Furstenberg. Lemma 20.15 is due to V. Bergelson in [18]. Theorem 20.17 is from 1241. a result of collaboration with V. Bergelson.

+

-

+)

+

+

Chapter 21

Other Semigroup Compactifications

Throughout this book we have investigated the s t r u m of pS for a discrete semigroup S. According to Theorem 4.8, OS is a maximal right topological semigroup containing S within its topological center. In this chapter we consider semigroup compactifications that am maximal right topological, semitopological, or topological semigroups, or topological groups. defined not only for discrete semigmups,but in fact for any semigroupwhich is also a topological space.

21.1 The ZAnie, WAS', As), and a A 9 Compactifications Let S be a semigmup which is also a topological space. We shall describe a method of associatingS with acompact right topological semigroupdefinedby aunivmalproperty. The names for the compactifications that we intmduce in this section are taken fmm [39]and [40]. These names come from those of the complex valued functions on S that extend continuously to the specified compactification. The four classes of spaces in which we are interested are defined by the following statements. DeRnitton 21.1.

(a) 8 1 ( C ) is the statement "C is a compact right topological semi-

groupP*. (b) rlr*(C) is the statement "C is a compact semitopological semigmup". (c) rlr3(C) is the statement "C is a compact topological semigroup". (d) 84(C) is the statement "C is a compact topological group".

Lemma 21.2. Let S be a set with IS1 = K r o. let X be a compacr space, and let f : S -t X. Iff [Sl is dense in X, then 1x1 j 22'. Pmofl Give S the discrete topology. Then the continuous extension j o f f to PS takes PS onto X M that 1x1 IpSI. By Theorem 3.58. IpSl = 2.' In the following lemma we nced to be concerned with the technicalities of set theory more than is our custom. One would like to define Fi = [(f, C) : Oi(C). f is a

continuous homomorphism from S to C . and f [S] E A(C)). However,there is no such s& (Ie existence would lead quickly to Rusell's Pafadox.) Notice that the requiremenu in ( 1 ) and (2) concerning the topological center are dundant if i # 1.

Lcmm;.21.3. k t S be an infbtite semi~mupwhich is also a topological space and k t i E ( 1 , 2 , 3 . 4 ] . There is a set Fi ofordcnd pairs ( f . C ) such that (1) jC(f, C) E Fi, thcn WC),f is a cominuour honwmorphisrn~5~1 S to C, and f ~ s rl ~ ( c )rmd . (2) given any D such that rVi(D) and given any contimow homomorphism g :S + D with gtS1 S A(D), thcn u i s r ( f . C ) E Fi and a continuour one-to-one honmorphim (p : C -r D such thm (p o f = g.

$ = [ ( f , ( C , 7. .)) : C c X , 7 is a topology on C . .is an associative binary operation on C . and f : S -r C ) .

so 9 is a set (Moreformally, the axiom schema of separation applied to the set

.

and the statement T C X,7 is a topology on C. is an associative b i m q operation on C , and f : S + C" guarantees the existence of a wt 9 as we have defined it. In defining 9 we have, out of necessity, &paned from the custom of not specifically mentioning the topology or the operation when talking about a semigroup with a topology. In the definition of the set F, we renxn to that custom writing C instead of (C. 3". 9. Let

Fi = { ( f ,C ) E $ : \Yi(C). f is a continuous homomorphism from StoC,andf[Sl c A(C)). (Notioe again that the requirement that f [S] c A(C) is redundant if i # 1.) Trivially F; satisfies conclusion (I). Now let g and D be given such that Uri(D)and g is a continuous homomorphism from S to D. Let B = ce g[S]. By Exercise 2.3.2, since g[Sl G A(D). B is a subsemigmup of D. Further, if D is a topological p u p (as it is if i = 4). then by Exercise 2.2.3 B is a topological group. Since the continuity requirements of Wi are hereditary, we have Y i ( B ) . Now by Lemma 21.2, IBI 5 '2 = 1x1 so pick a one-twne function r : B -+ X and let C = r [ B ] . Give C the topology and operation making r an isomorphism and a

21.1 The LAC?,WAS', AS', and 8APComprctibcmions

homeomorphism fromB onto C. Let f = r o g and let p = r-I. Then ( f , C ) E is a continuous one-to-one homomorphism from C to D, and p o f = g.

427

E , (p

Theo~m 21.4. Lrr S be an i n w t e semigmp which is also a topological space and Ieti E (1.2.3.4). 7 l r c m c n c n s t a p i r ( q i . y i S ) ~ h t h o r (1) Q i ( y i 9 .

(2) qi is a continuou~homomorphismfrom S to yi S. (3) qi [Sl is dense in yi S. (4) qi[Sl G N y i S ) , and (5) given ony D such that \Iri(D)and any continuous homomorphism g : S + D with g[Sl G A(D), them ainr a continuous homomorphism p :yiS + D such tlurtpoqi=g. So thefollowing diagmm commutes:

S

-

D.

Proof: Pick a set Fi as guaranteed by Lemma 21.3 and let T = X ( / , o + F ; C. Define qi : S + Tby qi(s)(f. C) = f ( s ) . LetyiS = ctrqi[Sl. Byfheonm2.22T(withthe product topologj. and coordinatewise operations) is a compact right topological semigroupand w[Sl A(T). If i € {2,3,4)oneeasily verifies theremainingrtquimnents needed to establish that *,(TI. By Exercise 2.3.2 yiS is a subsemigroup of T and, if i = 4 , by Exercise 2.2.3 yiS is a topological p u p . Consequently Oi(yi S). For each ( f , C) E F,, n(~,c) o qi = f so qi is continuous. Given s , t E S and ( f ,C ) 6 Fi.

s

so qi is a homomorphism.

Trivially qi [Sl is dense in yi S and we have already seen that qi[Sl G A ( T ) so that 4i[S1 G A(yiS). Finally, let D be given such that Wi(D) and let g : S + D be a continuous homomorphism. Pick by Lemma 21.3 some ( f , C ) E Fi and a continuous one-to-one homomorphism (p : C -r D such that (p a f = g. Let p = p a q1.c).Then. given s E S,

We now o b s m e that the compactificationswhose existence is guaranteed by Theorem 21.4 an essentially unique.

.

Theonm 215. Ler S be m injinite semigroup which is also a ropological space and let i E [I. 2.3.4). Let (qi. yiS) be as guaranteed by Theorem 21.4. Assume that also the pair (p, T) satisfies (1) (2) (3) (4) (5)

*i(T), q is a continuous homomorphismfmm S to T ,

q[Sl is dense in T,

dS1 E NT). and given any D such that rlri(D) and any continuous homomorphism g : S + D wirh g[S] g A(D), then exins a continuous homomorphism p : T -+ D such rharpoq=g.

Then then is afunction 8 : yiS + . Y which is both an isomorphism and a homeomorphism such that 6 takes qi[S] onto q[S] and 6 o q = q.

Pmof: Let 8 : yiS -+ T be as guaranteed by conclusion (5) of 'Ihaorem 21.4 for T and q and let p : T -+ yiS be as guaranteed by conclusion (5) above for yiS and qi. Then 6 and p ate wntinuous homomorphisms, 8 o q; = q , and p o q = qi. Now G[yiS] is a compact set containing q[S] which is dense in T so 8[yiS] = T: Also @ o 6 o qi = p o q = qi so it o 6 agrees with the identity on the dmse set qi[S] and thus p o 6 is the identity on yiS. Consequently @ = 6-' so 6 is a homeomorphism and an isomorphism. 0 As a consequence of Theorem 21.5 it is reasonable to speak of "the" C M @ compactification, and so forth.

LMlnition 21.6. Let S be an infinite semigroup which is also a topological space and for each i E (I. 2.3.4) let (q;. yiS) be as guaranteed by Theorem 21.4. (a) (I,. ytS) is the GMC-compactification of S and LMC(S) = n S. (Q. n S ) is the WAP-compactification of S and W A P ( S ) = n S . (c) (m. n S ) is the AP-compactification of S and A P ( S ) = nS. (d) (q4, y4S) is the 8AP-compactification of S and I A Q ( S ) = y4S.

(b)

Notice that by Theorem 4.8, if S is a discrete semigroup, then (1.6s) is another candidate to be called "the" PMC-compactification of S. We remind the rrader that semigroup compactifications need not be topological compactifications because the functions (in this case qi) are not required to be embeddings. In Exercise 21.1.1. the reader is asked to show that if it is possible for qi to be an embedding. or just one-to-one, then it is. Exerripe 21.1.1. Let S be an infinite semigroup which is also a topological space and let i E (1.2,3,4). Let Y he a semigroup with topology such that $i(Y) Let r : S -+ Y be a continuous homomorphism with r[S] 2 A(Y). (a) Rove that if r is one-to-one, then so is qi. (b) Prove that if r is an embedding. then so is qi.

21.2 Right Topological Compaetiflcations

429

Exercise 21.13. If Sis a group (whichis alsoa topalogicalspaceXshow that AB(S) b A B ( S ) . (Hint: Rovc that A P ( S ) is a group and apply Ellis' 'Iheorem (Comllary 2.391.) Exercisetl.l.3. Let C denote any of the semigmups W A P ( N ) . A P O , or b A P ( N ) . Let x E C and let m and n be distinct positive integers. Show that m+x#n+x. Exerdse 21.1.4. Show that each of the compac!ifications W A P ( W ) , A P ( N ) and b A P ( N ) has ZC elements. (Hint: By Kronecker's Theorem. FJ can be densely and homomorphically embedded in a product of c copies of the unit circle.)

21.2 Right Topological Compactifications The requirements inconclusions (4) and (5)of Theorem 21.4 referring to the topological center seem awkward. They arc redundant exceptwhen i = 1, when the property being considered is that of being a right topological semigroup. We show in this section that without the requirement that g[S] 5 A ( D ) , there is no maximal right topological compactification of any infinite discrete weakly rightcancellative semigroup in the sense of Theorem 21.4.

Theorem 21.7. k t S bean infinitediscme weakly rightcancelkztivesemigmup and let K bean infinite cardinal. There exist a compact right topological semigroup T and an injective homomorphismT : S + T such that any closedsubsemigroupof T containing r [ S ]has cardinalify at least K .

Pmof Let 00 be apoint not in K x S and let T = ( K x S)U {m).(Recall that the cardinal is an ordinal, so that K = {a : a is an ordinal and a < K).) Definean openlion . on T as follows. (0, S ) . (ti.5') = (1'. 5s'). (1, s ) .(t', s') = (I' r, s ) if t # 0, and

K

+

(t,s).w=w.(t,s)=a,-00=00,

+

where r' t is ordinal addition. We leave it as an exercise (Exercise 21.2.1) to verify that the operation is associative. Now fix an element a E S and define a topology on T as follows.

.

(1) Each point of ( K x (S\(a))) U {(O, a ) )is isolated. (2) Basic open neighborhoods of a,arc of the form

where I' c K and F is a finite subset of S with a E F.

(3) If t = r'

+ 1. then basic opn neighborhoods of (r, a ) an of the f o m W. all U @'I

X;S\F))

w h F is a finite subset of S with a E F. (4) If r is a nolimit ordid. then basic open neighbarhoods of (1, a ) at of the form {(t. a)) u ({r" :r' c t" < r) x S) U ([r'] x (S\F))

wh~r'ctandFisafinitewbsetofSwithaEF. That is to say. a basis for the topology on T is 9)

= [((r. s)I :t < K ands E S\{al} U (((0. d l ) ~ ( ( mU) ((1'' :I' < t" < K) x S) U ({f'] x (S\F)) : ~ ' c K , F ~ 9 f ( ( S ) , a n d aF ~]

+ I. a ) ] U ((1')

~(((1'

x (S\F))

:I' < K. F E 9,(S), and a E F)

u{{(t,a)] U ({t" :t' < rn < I] x S) U ((1') x (s\F)) : t is a limit, t' c r < K. F E 3',(S), and a E F).

The verification that 9) is a basis for a Hausdorff topology on T is Exercise 21.2.2. To sce thal with this topology, T is compact, let P1 be an open cover of T. Pick

U

E P 1 s u c h t h a t m ~ U a n d p i c k f 1< ~ s u c h t h a t { r " : t ~ < t " c w ] x S ~ U .If a finite subfamily of P1 covers (t : r 5 11) x S, then we arc done. So assume that no

Iinite subfamily of e( covers {f : r 5 111 x S and pick the last to such that no finite subfamily of W.covers (r : f j to] x S. Pick V E e( such that (to 1,a) E V and pick F E Pf(S) such that {to] x (S\F) C V. Pickfinite $ G V such that {to] x F S U $. Assume first that to = t' + 1 for some r' and pick finite P G U such that {r : r 5

+

r')xS~UF.ThenFU~U(VJeovm{r:rs$]xS,acmnadjctMn. Thus ro is a limit ordinal. If ro = 0, then $ U (V] coven (0) x S. Thus # 0. Pick W E P1 such that (10, a ) E W and pick r' < ro such that (1'' :r' c r" c to] x S G W. Pick finite 3 E e( such that {r : r 5 t'] x S U 3. Thcn P U $ U (V, W] coven x S, a contradiction. Thus T is compact as claimed. Wemify now that T is aright topo1ogical semigroup. Since p, is constant it is continuous. Let (r', s') E K x S. Trivially p(,t,q is continuous at each isolated point of T, so we only need to show that p(,r,,) is continuous at m and at each point of (K\[O]) x {a). To see that p(,,..r) is continuous at m , l b U be a neighborhood of w and pick u < K such that (r" : u c r" < K J x S E U. Let ( t : t j 10)

W = {m) U ({r" : u

+ l c r" < K ) x S ) u ({u + I] x (S\{a])). +

Then p(,,,,r)[W] c U since r" 5 r' r" for dl r" c K. Next let 0 c r < K and let U be a neighborhood of (r. a ) . (rl,s') = (r' + r, a). A s s u m c h t that1 = to+] andpick F E Pf(S) witha E F suchthatfr'+to)x (S\F) E

21.2 Right Topologial Compactif~cati~~

431

U. t c t G = { s E S : s s ' € F)wdaotiathat,sinrrSis~eaLlyright-rrIlui~~ is finite. Let W = Kt. a)lU(Ifol x (S\(F u G))). Then&?,7[WI E U.(Ifto = Oands E S\G,tbenssf f F.) with a E F Now assume that t is a Limit ordinal and pick to < t and F E Pf(S) smb that {(tf+t,a))U({t":t'+to

< t " < t l + t ) x S ) ~ ( [ t ' + t o } x (S\F)) E U

and let

W = ((t.a)} U ((t" :to

4

1' < t) x S)U ((to) x

(S\F)).

Then ~(z.+)[WlE (1.

Thus T is a compact right topological semigroup. Define T : S -t T by r(s) = (0. s). Trivially r is an injective homonmphisrn. Now let H be iclosed subsemigroup of T containing r[S]. Weclnimthatr x {a) E H. Suppose not. and pick the least r c r such that (t. a ) f H. Notice that t > 0. Pick a neighborhood U of ( I , a ) such that U n H = 0. Then there exist some t' < t and some F E Pf(3such thal (t'l x (S\F) S U.Then (r'. a ) € H so

.

((0)x S) (ti, a ) = (1') x Sa S H.

Since S is weakly right canceIIative. So is infinite, so ((['I x S a ) n U # 0, a contradiction.

0

The foUowing corollary says in tbe strongest tnms that without the requirement that r[S] E A(T) (or some o k requirement), t k c could not be a maximal right topological compactification of S. Notice that the corollary docs not wen danand any relationship between the topology on X and the semigroup operation, nor are. any requirements, either algebraic or topological, placed on the function (p. (Although, presumably. the kind of result one would be looking for would have X as a compact right topological semigroup and would have cp as a homomorptusm with (p[S] dense in X.) Corollsv 21.8. Let S be an infiite discrete weakly right wuellative semigmup, let X be a sernigmup with a compact topology. and let (p : S -r X. There exist a compact right topological semigmup T and an injective homomorphism r : S -r T such thnt there is no continuous homomorphism q : X + T with q o (p = r. Pmof: Pick a cardinal K > 1x1and let T and r be as gusnurtced by Theorem 21.7 for S and K. Suppose one has acontinwus homomorphism q : X + T such that q o (p = r. Then q[X] isaclosedsubJnnigroupof T containing r[S] andthusr > 1x1 2 q[X1 2 x . a contradiction.

0

F%erdse 21.2.1. Rove that the operation on T defined in the proof of lkomn 21.7 is associative.

Exerdse 21.2.2. Let 91 and T be as in the prclof of T h e m m 21.7. Rove that 33 is a basis for a Hausdorfftopology on T.

213 Periodic Compacti6cations as Quotients Given a semigroup S with topology, the WAF-. 047-.' and dA9-compactifications are all compact right topological semigroups that are equal to their topological centers andthus, by Theorem4.8 each is a quotient of BSd, where Sd denotes S with the discrete topology. In this section we shall identify equivalence relations on /3Sd that yield the W A P and AP-compactifications. We shall use the explicit descriptions of these compactifications as quotients to characterize the continuous functions from a semitopological semigroup S to a compact semitopological or toplogical semigroup that extend to the W A P - or AP-compactifications of S. We shall also show that, if S is commutative, BAP(S) is embedded in WAP(S) as the smallest ideal of WAP(S). We first give some simple 'well-known properties of quotient spaces defined by families of functions. De6nition 21.9. Let Xbe a compact space and let g (f, Y) be a statementwhich implies that f is a continuous function mapping X to a space Y. We define an equivalence relation on X by stating that x I y if and only i f f ( x ) = f ( y ) for every f for which @( f, Y) holds for some Y. Let X / i denote the quotient of X defined by this relation. and let n : X + X / E denote the canonical mapping. We give X/I the quotient topology. in which a set U is open in XP if and only if n - ' [ U ] is open in X. Ofcourse, the space X / idepends on the statement& although wehavenot indicated this in the notation.

Lemma 21.10. k t X be a compact space and let #( f, Y)be a statement which implies that f is a continuourfwtction mapping X to a space Y. Then X/c has the following properties: (1) Ifg : X -* Z is a continuousfunction and if8 : X/I + Z is afunction for which g = p o n, then p is continuous. (2) X/3 is a compact Hausdo#space. (3) A net ( R ( X < ) ) converges ~E~ to n(x) in X/= ifand only if(f (x,)),,, conveges to f(x) in Y forevery (f, Y) forwhichg(f, Y)hol&. Pmof. (1) If V is an open subset of Z. then n-I[p -'[V]]= g-'[VJ, which is open in X and s o p -'[V]is open in X/I. (2) Since X/I is the continuous image of a compact space, it is compact. To see that it is Hausdorff, suppose that x , y E X and that x(x) # n(y). Then there is a p i r

21.3 Periodic Compneti6cationsas Quotients

433

( f , Y ) such that $ ( f , Y ) holds and f ( x ) jC f (y): For every u , n E X , a @ ) = n ( v ) implies f ( u ) = f (v). So there is afunction f : X/i+ Y for which f = f o a. By (I). f is continuous. I f G and H are disjoint subsets of Y which are neighborhoods o f f ( x ) and f ( y ) rrspcctively, then -'[GI and f -'[HI are disjoint subsets of X / I which arc neighborhoods of n ( x ) and n ( y ) respectively. So XI= is Hausdofl. (3) As in (2). for each ( f , Y ) for which $(f, Y ) holds, let f : X/= Y denote the continuous function for which f = f o a . The set ( f -'[[I] : $(f, Y ) and U is open in Y )is a subbase for a topology r on X /= which is cthan the quotient topology and is therefore compact. However, we saw in ( 2 )that, for every x , y E X , n ( x ) # ~ ( y ) implied that n ( x ) and n ( y ) had disjoint r-neighborhoods. S o T is Haudorff and is therefore equal to the quotient topology on XI=. Let ( x J I E tbe a net in X and let x E X. I f ( f (&)),GIconverges to f ( x ) in Y whenever $ ( f , Y) holds, then (z(x,)),,r converges to n ( x ) in the topology r. So (x(x,)),,l converges to n ( x ) in the quotient topology. Now assume that (n(x,)),,r converges to n ( x ) and!et ( f , Y ) be given such that @( f. Y ) holds. I f U is a neighborhood of f ( x ) . then f -'[Ul is a neighborhood of n ( x ) . Thus ( X ( X ~ ) ) , is ~ Ieventually in j - ' [ ~ ]and so ( f ( x , ) ) , , ~ is eventually in U.

-

We mall that we defined in Section 13.4 a binary operation o on /IS,+by puning x o y = lim lim st, where s and t denote elements of S. Note that, in defining x o y, t-+"*-.Z

we have reversed the order of the limits used in wr definition of the semigroup (pd, .). The semigroup (BSd,o) is a left topological semigroup while (BSd. .).as we defined it, is a right topological semigroup. The following lemma summarizes the basic information that will be used in our conshuction of the W A P - and APcompactificationsof a semitopologicalsemigroup.

Lemma 21.11. k t S be a semitopological semigmup and let B ( f , C ) be a starentent which implies that f is a continuousfunction f mapping S too compact semitopological semigmup-C. Let $(g. C ) be the statement that there exists f for which B ( f , C ) holds and g = f : BSd + C. Suppose that, for every ( f , C )for which B( f , C ) is m e , we have: (1) B(f o I., C ) andB(f op., C ) holdforeverya E S(where A..p. (2) f o r d l x . y E B S ~.T(x. , Y) = .T(x o y).

:S

+ S)and

Let B S d / r and n be defined by the statement @ as dkcribed in Definirion 21.9. We dejine . on pSd/= by purring n ( x ) - n ( y ) = n ( x - y).

(a) The operation. on BSd/= is well dcfined. (b) With this operation. @Sd/=is a compact semitopological semigmup and n is a continuous homomorphismfmm (BSd, .) andfmm (BSd. 0). (c) The restriction xp is continuous. P m f : Notice that for x , y ~ i 3 S . dT. e have x r y if and only if for every pair ( f , C ) satisfying B( f , C). one has f ( x ) = f ( y ) .

-

-

(a)Givena € S a n d x , x ' € p S d w i t h x gnx'.weclaimthatx.a = x l . a a n d a x I a x'. To see this, let (f. C) be given satisfying B(f, C). Then B(f o pa. C) %dB(f o b , C ) h o l d . Sjf op%= T o & and f o L = f o G wehavethat f (x .a) = f (x' .a) and f ( a . x) = f (a. x') as requid. Now assume that x = xi and y E BSd. We claim that x y I x' y. Recall rhat we denote by 1, the continuous function from BSd to pSd d e w by &(z) = x o z. Now foreacha E S and each (f, C) for which B(f. C) holds.

.

.

-

-

70- 70 .

.

and so L, and L,, are continuous functions agreeing on S. hence on pSd. Thus f (X.y) = f (x' y). Similarly (using p,(z) = z x), y - x Iy x'.

.

.

Nowletx~x'andy=y'.Thenx.y~x'.y~x'.y'. @) By Lmuna 21.10, )%?dl' is compact and Hausdorff. By (a), x is a homomorphism from @Sd, .), and by (a) and (Z), n is a homomorphism from (BSd. 0). By Exercise 2.2.2. since p S d l r is the continuous homomorphic image of @Sd, .),it is a right topological semigroup, and since it is the continuous homomorphic image of (BSd, 0). it is a left topological semigroup. (c) To see that n l j : S + &I= is continuous, k t s E S and let (s,),,, be a net in S converging to s in S. To see that ( ~ ( s , ) ) ,converges ~r to n(s) it suffices by Lemma 21.10 to let (g. C) be given such that &(g. C) holds and show that (g(s,)),,l converges to g(s). Pick f such that f?( f. C) holds and g = f . Then f :S + C is continuous so ~l to g(s) as required. (f (s,)),~Iconverges to f (s). That is. ( g ( ~ , ) ) ,converges The numbering 62 and & in the foUowingdefinitionis intended to cornpond to the function 02 : S + WAP(S). The statem& 82 and & depend on the semigroup S. but the notation docs not reflect this depeodence. Delinition 21.12. Let S be a semitopologid scrnigrwp. &(f, C) is the statement "C is-a compact s_emitopologicalsemigroup, f is a continuous function from S to C, and f (x o y) = f (X .y) for every x. y E &". &(g. C) is the statement "them exists (f, C) for which &(f, C) holds and g = f :BSd += C". Lemma 21.13. Let S be a semiropologicalsemigroup. If&(f, C) holds and a E S, then 6M f o I., C) and &(f o pa. C) holdas well ( w k n A., p. :S + S).

-

Proof Since A,, is continuous(bsause S is a semimpological semigroup). we have that f o A. is continuous. Notice also that f o Aa = G. Now. let x, y E PSd. W. sineeaES.aox=a.xandso

-

70

foAa(xoy) = f " ( a o ( ~ ~ ~ ) ) = ~ ( ( a ~ ~ ) ~ ~ ) = ~ ( ( a . x ) o ~ = ?((a. x) Y) = j ( a (x y)) = f=(x . y).

.

Similarly &(f o pa. C) holds.

. .

The m d w may wo& why we now rrqaire S to be a semitopological semigroup. after pointing out in Section 21.1 thst we w e n not demanding this. -The reason is that we need this fact forthe validity of Lrmma 21.13.

Theorem 21.14. Ler S be a semitopological semigroup. kt q5 =

andlet flSd/= and n : flSd + f l S d / ~be defied by Definition 21.9. Then (ny,flSd/=) b a WAQcompactifiation ofS. '?'hat is: (a) (b) (c) (d)

*2'2(Bsd/'), nls is a continuous homomorphismfmm S to !S~/S. n [ q is dense in BSdlzs and given m y D such that Wz(D)and any continuo^^ homomorphism g : S + D. then aLns a continuoushomomorphismfi :f l S d / ~+ D such that r onp = g.

P m J Conclusions (a).

(b), and (c) are an immediate conssqumce of Lemmas 21.11

and 21.13. Letg and D be given such that V+(D) and g is a continuous homomorphism from Sto D. We claim that &Or. D) holds. So let x. y E BSd. Notice that g'is a homomorphism r ~ Thus from (flSd. .) to D b i ~ o r n l l a 4.22.

L*fi : f l S d / ~-f D denote the function for which won = % Then fi iscontinuous, by Lemma 21.10. and is easily seen to be a homomorphism. 0 We now turn o w attention to c h w t i o n s of functions that extend to WAQ(S).

Theorem 21.15. Ler S be a semitopological semigroup, let C be a compocr semitopological semigroup, and lct f be a continuousfunctionfrom S to C. +re isa co@mous function g : WAQ(S) + C for which f = g o qz ifandonly i f f (x y) = f (x o y) for every x. y E flSd.

.

P m $ Necessity. Pick amtimousfunctiong : WAB(S) + C such that f =gem. Then

Sufficimcy. Define zs as in 'Iheaem 21.14. Then by Th-s 21.5 and 21.14. it suffices to show that t h exists ~ a continuous function g : f l S d / i -r C for which = g o nls. Since &(f, C ) holds, t h m is a function g : f l S d / ~+ C for which f = g o n . ByLemma21.10,giscontinuous. 0

t

If S is commutative, we obtain a characterization in terms of (BSd. .) alone.

436 ,

21 Ocher Semigroup Compactifiatims

Corollary 21.16. Let S be a commutative semitopalogical sernigmup, let C be a compact semitopological semigmup, and let f be a continuousfmctionfmm S to C. %re i~ a c o n t i n y s function g : W J I P ( S ) -P C for which f = g o qz if and only if f ( x . Y) = f ( .x)forrvery ~ x. y E BSd. Pmof This is an immediate consequence of Theorems 13.37 and 21.15.

0

RecaU that C ( S )denotes the algebra of bwnded wntinuous complex-valued functions defined on S.

Definition 21.17. Let S be a semitopological semigroup and let f E C(S). (a) The function f is weakly almostperiodic if and only if the& is a continuous g : W A P ( S )+ C such that g o w = f . (b) The function f is almost periodic if and only if there is a continuous g : JI.?'(S) + C such that g o q3 = f .

Theorem 21.18. Let S be an infinite semitopological semigmup and let f E C(S). The following statements are equivalent: ( 1 ) Thrfunction f is weakly almostperiodic. (2) Wheneverx. y E BSd, f"(x. y ) = f"(x o y). (3) Whenever (a& and (b,,):, are sequences in S and all indicated limits exist, lirn lirn f (a, .b,) = lirn lim f (a, .bn). n-rmn-m tt-mm-m

PmoJ By Theorem 21.15, (1) and (2) are equivalent. We shall show that ( 2 ) and (3) are equivalent. (2)implies (3). Let ( a , ) z l and (b,):, be sequences in S for which all the limits indicated in the expressionsm ; mn!m f (a,. b.) and ;irnmrnnw f (a, .b.) exist. Let x and y be Limit points in BS,j of the sequences ( a , ) Z 1 and (b.)$, respectively. Then lim lim f (a, - b.) = f"(x 0 y ) and lim lirn f (a. .b,) = f ( x .y). So (2)implies n-rrnm-m n-mn-m that these double limit$are equal. (3) implies (2). Let x.y E BSd. let k = y ( x . y), and k t C = a x Q y). For each nENletA.=[a~S:Jf(a~y)-k]c$)andlet~.={b~~:1f(x~b)-t1c$). Since (BSd. .) is a right topological semigroup. (BSd. o)is a left topological semigroup, andx.b=xobforbES,A.~xandB. ~ y . Choose a , E A1 and bl E B I . Inductively, let r E N and assume that we have chosen ( a l ,az.. .. a,) and { b ~bz, . .. b,] so that forall m , n E (1.2.. . rl

.

(i) a, E A, and b,, E B. (ii) if m > n. then ( f(a, - b.)

. .

- Ll c i, (ii) if n > m , then If (a, . b,,) - kl < A. (iv) if m > n , then If(am . b,) - f"(x. bJ c 4, and (v) i f n > m, then If (am. b.) - f"(a,,, . y)l c ;I.

..

A}

Nowforeachn E (1.2. .... r1.b. E B.so[a E S: (f(a.b.)-tl < E X . ALso. foreafhn E (1.2. r ) . ( a s S : lf(a.b,)-?(x.b.)l < $1 ~ x s o p i c k a , + l s u c h thatforeachn E (1,2. rl, I f (a,+~.b.)-el < :andl f(a,+l.b.)-f"(x.b,,)l -= Similarly. choose

..., ....

&.

f (am . b.) = ?(x .b.) and given m E N o m has lim f (a, b.) = ?(am .y). Also Aimm "5%f (a, - b.) = k and lim tim f (a,, . mm -. "-mm-m b.) = I . SO e = k. o Now. given n E

R,w e has , I ,

.

Theorem 21.19. Let S be a commutative semitopological semigroup. Then W A P ( S ) is also cornmutorive. F u n h e m r e , ($2. WAP(S)) is the mnrinral commutative semigmup compactificarion of S in rhefollowing sense: If(v. T) is any commutative semigmupcompactificationof S. then there isa conrinuoushomomorphismp : W A P ( S ) + T for which v = p o $2. Pmof By Theorems 21.5 and 21.14, we may assume that W A P ( S ) = bSd/l and $2 = q s . Now given (fLC) such that @(f, C), one has by Theorem 13.37 that for all X, Y E bSd. f (X - y) = f (y . x) and thus x(x). s ( y ) = x ( y ) . ~ ( x ) . Now, if (v, T) is a commutative semigroup compactification of S. then T is a semitopological semigroupso there exists acontinuous homomorpsm p : W A P ( S ) + T for which v = p o m. 0

Theorem 2130. LQI S be an infinite commutative semiropologicalsemigmup. Then K ( W A P ( S ) ) can be identified with dAP(S).

Proof Let K = K (WAP(S)). By Theorem 21.19 and Corollary 2.40. K is acornpact topological group. Let e denote the unique minimal idempotent of WAP(S). We define a continuous homomorphism r : W A P ( S ) + K by r(x) = xe and define a continuous homomorphism p : S + K by p = r o w. Let T be a compact topological group and let v : S + T be acontinuous homomorphism. By the defining property of WAP(S). there is a continuous homomorphism 6 : W A P ( S ) + T for which 6 o $2 = v . We note that &(el is the identity of T and so $Ix o p = v. This shows that we can identify BAP(S) with K (by Theorem 21.5). We now discuss some simple properties of WAP(PI). We use Theorems 21.5 and 21.14 to identify WAP(H) with @N/= and $2 with nlm If S = NU (oo), the one point compactificationof N. and oo+x = x +m = oo for all x E S, then S is a semitopological semigroup. Thus, by Exmise 21.1.1, $2 : N + WAP(N) is an embedding. Consequently. we shall regard N as being a subspace of WAP(N). Lemma 21.21. Lcr x be in rhe interior in p N of BN\(N*

+ N*). Then la(x)l = 1.

P m f . Let y E BW\(x). A n (N*+N*) = 0.

We can choose a subs* A of Nforwhich x E Si. y !+ .

.

X,and

Ln f = X A : N + (0.1)sothat j = x S i . ~ e c l a i m t h o c f " ( ~ + q ) = j ( ~ + ~ ) f o t 4 l p . q e p N . I n d 9 i f p ~ N o r qsN.thenp+q =q+p.whileifp,q ~ W , t h e n f ( p q) = 0 = f (q + p). Thus by Corollary 21.16, thae is a continuous function g : W A P + {0,1) such that f = g o q~: Then f and g o R-a continuousfunctions from BN to {O,l) agreeing on N and so f = g o R. Since f ( x ) = 1 and f ( y ) = 0, ~ ( x#) ~ ( y )l l. n t is, y $ n ( x ) .

+

Theom 2122. Them isadenseopensubser U ofN.with tkpmpenythat In(x)l= 1 for every x E U.

-

P m f . By Ex&i 4.1.7, if (x~):=~is a squence in N fm wbichx.+l x. -r ca and if A = {x. :n E R),then (ctgn A ) r l (N* +PI*) = 0. Now any infinite sequence in W contains a subsequence with this property, and so the= is a dense open subset U of N ' for which U n (N* W ) = 0. 0

+

Theorem 21.23. I f x is a P-poi# in PI*, then n ( x ) is P-point in WAS(W)\N.

+

N*). We shall show that x E (I. Our P m J Let U denote the interim of BN\(N* claim will then follow from Lemma 21.21 and the observation that n [ U ]is open in W A P ( M because R - ' [ n [ U ] ] = (I. For each k E W. we can choose a set A* E x with the property that la a'( 2 k whenever a and a' are distinct integers in At. F i s can be seen from the fact that there exists i E (0. 1,2, .. k 1 ) for which Nk i E x.) Since x is a P-point in N*, there is a set A E x for which A\Ak is finite for every k. It follows from Exercise 4.1.7 that --

-

.

.-

+

Arl(W*+N*) =0.

0

We now tum our attention to representing A$'($)

ss a quotient of BSd.

Definition 21.24. Let S be a semitopological semigroup. & ( f , C) is the statement "C is a compact semitopologifal semigroup, f is a continuous h t i o n from S to C. and there is a continuous function f' : pSd x BSd + C such that for all s. r E S. C) is thestatement "there exists ( f , C ) for which & ( f , C) f(s . t ) = f*(s,t)". holdsandg= f : p S d + C " .

+&!.

We show next that h(f. C ) implies &(f. C).

Lemma 21.21 Let S be a semiropological smigmup. I f @ ( f . C) bI& and a E S, then&( f ope. C ) a d h (f oA.,, C )ho[daswell(whem &, Aa :S S' S). Funkmrore, fyisasguamnteed by this srmemenr, thenforall x , y E BSd, 7 ( x - y ) = f ' ( x , y) = f(x or)Pmof: Suppose that &(f, C ) holds aad that a € S. Let g : BSd x BSd + C be defined by g ( x , y) = f ' ( a x , y). Then g is continuous and g(s, t ) = ( f o A.)(s. r ) for every s, t E S. So &(f o A,, C ) holds. Similarly, &(f o pa. C ) holds.

Let x , y E pSd. Then

For the remainder of this section, we reuse the notations = and x for a different quotient. Theorem 2136. Ln S be a smiropological smigmup. Let and n be defined byDefinition21.9 with4 = &. Foreach x. y E pSd. l r t n ( x ) .nLy) = n ( x . y). Then (nls.flSd/') is an AP-compactificaliOn of S. That is. (a)

'h(@sd/r).

(b)

nls is a continuous homomorphismfrom S to &%ill.

(c) n [ S ]is dense in pSd/=. and

(d) given any D such that W3(D)and m y continuous hmomorphimr g : S + D. + D such that p onls = g. there exists a continuous homomorphism p :f&/i Pmof: It follows immediately from Lemmas 21.11 and 21.25 that x is wclldefined. Conclusions (b) and (c) are also immediate from these kmmas, as is the fact that ( B S d / r , .) is a compact semitopological semigroup. To complete the verification of (a). we need to show that multiplication in f i S d / t is jointly continuous. I ( y , ) , , ~are nets in @Sdfor which Suppose that x , y E BSd and that ( x , ) , ~and ( R ( X , ) ) , ~ I and (X(Y&I converge t o n ( x ) a n d n ( y )respectively in pSd/=. Weclaim I to x ( x ) . n ( y ) ,thatis that ( n ( x , y,)),,~converges to that ( n ( x , ). H ( ~ , ) ) , ~converges X ( X . y). By passing to a subnet, we may presume that (x,),,; a d (y,),,; converge in pSd to limits u and w respectively. Thenn(u) = n ( x ) a n d x ( v ) = n ( y ) . To see that ( n ( x , ;y,)),,l converges to x ( x .yJ = n ( u u), it suffices by Lcmma 21.10 to show that ( f (x, . y,)),.~converges to f (u . u) for evay ( f . C ) for which & ( f , C ) holds. So assume that & ( f , C) holds and let f'be the function guaranteed k t h i s statement. Then ( f ' Q , , y,)),,; converges to f ' ( u . v ) and so by Lemma 212.5. ( f (x, .YAEI COnVergCS 10 f ( u . U) as required. To v d y ( d ) , let g and D be given such that Y 3 ( D ) and g is a continuous homomorphism from S to D. We claim that 03(g. D) holds. To see this, d e k g* : pSd x OSd + D by g*(x,y) = z ( x ) . F(y). Then g* is continuousand. givens. t E S. one has ga(s,1 ) = g(s) .g(r) = g(s t). Let p : pSd/= + D be the function for which p o n = Then p is continuous 0 (by Lemma 21.10) and is easily seen to be a homomorphism

.

.

.

z.

Theorem 21.27. Lei S be a semitopologicalsemigmup, let C be a compact topological semigmup, and let f be a continuousfwction fmm S to C. There is a continuous fincrion g : b P ( S ) + C for which f = g a 73 ifand only ifthere is a continuous fwction f.:BSd x pSd + C such that f . ( s , 1 ) = f ( s . I ) foralls. t E S.

Piuof Necessity. Pick a conlinuous function g : A P ( S ) -+ C such that f = g o q,. Let : /3Sd + A P ( S ) denote the continuous cxtmsion of ~ 3 Definc . f . ( x . y) = g G ( x ).GCY)). Then givens, r E S one has g(ijj(s) . G ( t ) ) = g ( m ( s ) &)) = 8 q 3 b . 0 )= f ( s . 0 . To see that f is continuous, let x . y E pSd and let W be a neighbotbood of g ( G ( x ) - rji(y)). Pick neighborhoods U and V of % ( x ) and G ( y ) in A P ( S ) such that U .V c g-I[Wl. Then f . [ i j j - ' [ ~x] rl?-l[~l]c_ W . SuRiciency. Since & ( f , C) holds. there is a function g : oSd/= -+ C such that f = g o n . B y Lemma 21.10, g is continuous.

.

I

Theorpm 2128. Ln S be a semitopological semigmup and let f

E

C ( S ) . 771efollow-

ing statements are equivalent (1) Ikfunction f is almostperiadic.

( 2 ) There is a conrinwurfuncrion f': /3Sd x pSd + @ such thmfor all s , t E S, f'(3.t) = f ( s . t ) . (3) For every 6 > 0, there is an equivalence relation * on S with finitely many equivalence classes such that If ( s . I ) f (s' .f')l c r whenever s * sf and t 4 1'.

-

Pm$ The equivalence of (1) and (2)is a special case of Theorem 21.27. (2)implies (3). Let r r 0 be given. Each element of /3Sd x pSd has aneighborhood oftheformLrxV.whcreU andVareclopensubseuofflSdandIfYx, y)- f ' ( u , v)l < < whenever (x, y). ( u , u ) E U x V. We can choose a finite family 9 of sets of this form which covers BSd x pSd. Let W = ( U : U x V ~ F f o r s o m e V ) U ( V : U xV ~ 3 f a s o m e U ) . Define an equivalence relation 4 on S by s * t if and only if for all W E W, either (s,f ) E W or {s. f 1 t l W = 0. Then is an equivaknce relation with Wtely many equivalence classes. Now assume that s * s f and t * r'. Pick U and V such that U x V E 3and (s. r ) E U x V. Then sinces a s'andt * t', (s', r') E U x V andso If*(s. t ) - f ( s 1 , t')l c c. That is. If (s t ) f (s' .r')l < r . (3) implies (2). Define f ( x , y) = f"(x .y ) for all x , y E BSd. m e n hivially f'(s.1) = f ( s .t)fordls,r E S. To see that f' is continuous, let x , y E BSd and let E > 0 be given. Pick an equivalence relation * on S with finitely many equivalence classes suchthat If ( s .t ) f (s' . r')l c 5 whenever s 4 s' and t % t'. Let $2 be the set of *-equivalence classes and pick A. B E X such that x E ; iand y E We claim that for all ( u , v ) E ; ix 3. one has I f ( x , Y ) - f % , v)l c r. Since f (x y) = x-limy-lim f ( s . t ) , we can

. -

2.

.

-

rc s . . .

IFZ

choose (9, t ) E A x B such that f ( x .y) f ( s . r)l c 5. Similarly. we can choose ( s f ,t') E A x B such that 1 7 ( u.v ) f (s' . r')l c 5. Since s 4 S' and I 4 t', one has I f ( s . I ) - f(sr.t')l < 5 andso$(x .y)-?(u. v)l < r .

-

Exercise 2131. Show that WAP(N)\N contains a dense open set V disjoint from (WAP(PI)\W) + (WAP(N)\M). Exercise 2132. Show that WAP(N)\N contains weak P-points. Exercise 2133. Show that WAP(N) is not an F-space. (Hit: An F-space cannot contain an infinite compact right topological group. See [182. Comllary 3.4.21.)

Exercisc213.4. We have seen that K ( W A P O ) can be identified with 6 A P ( N ) (by Theorem 21.20). Show that the same statement holds for K(AS)(N)). Exercise 21.35. Let S be a semitopological semigroup and suppose that f E C(S) is almost periodic. Show that every sequence (s.) in S contains a subsequence (s.,) for which the sequence of functions t I+ f (s.,t) converges unifmly on S. (Hint: Use Theorem 21.28.)

21.4 Semigroup Compactifications ai Spaces of Filters We have seen that if S is a discrete semigroup, then fJS is the 1Me-compactification of S. We have also found it useful throughout this book to have a specific representation for the points of fJS.namely the ultrafilters on S. As we noted in Section 21.3, the W A F - , AP-. and dAP-compactifications are all quotients of ,SS. The points of these compactificationscorrespond t n closed subsets of fJS.Since the closed subseuof p S correspond to filters on S by Theorem 3.20, each of these compactifications -indeed any semigroup compactification can be viewed as a set of filters. In this section we characterize those sets of filters that are,in a natural way, semigroup compactifications of S.

-

Remark 21.29. Lct S be a discrete space, Ict X be a compact space. and let g be a continuousfwtctionfmm pS onto X. Then given any x E X andany neighborhood U n ~s E ~ E - ' [ I X J I . O~X g ,- l [ ~ The first step in the characterization is to define an appropriate topology on a set of filters.

Definition 2130. Let S be a discrete space, let W be a set of filters on S, and let A g S. (a)An=[(A€W:A€AJ. (b) The quotient topology on W is the topology with basis (AD: A S SJ. Observe that if A. B E S, then A ¶ nBG(A nB)# so that [An : A S] does form a basis for a topology on W. That the term "quotient topology" isappropriate is part of the content of the following theorem. Recall that, given a set W of sets, a choice function for W is a function f : W -+ W such that for all A E %, f (A) E A.

U

Tbcomn 21.31. Let S be a ducme space, Iet X be a compact space, and let g be a continuousfun&n/mn BS onto X. Let R = ( n g-'[[x]] :x e X). Then d is o set offilters on S sarisfving (a) given any choice function f for R fhere uisfs 5 E Pf(R) such thar S = U A ~fT(A) @) given distinct A and S in B, there exist A E A and B E S such that whenever C E LR, either S\A E C or S\B E e.

Further: with the quotien! topology on LR, thefunction h : X + 94defined by h(x) = ng-l[(x)] is a homeomorphism.

Pmof: Given x E X, g-'[[x)] is a set of ulbafiltm whose intersection is thenfore a filter. l&l f be a choice function for- and suppose lhat for each 3 € Pf(W),

s\ Ulh3f (A) #

'.

Then-[S\ f (A) : A E 8 ) has the finite intersection propcay so pick p E pS such that (S\ f ( A ) : A E W) S p. Let x = g(p) and let A = n8-'[[x)]. Then f (A) E A G p and S\ f (A) E p, a contradiction. Now let A and P) be distinct m e m h of R and pick x and y in X such that A = ng-'[[x)] and P) = ng-l[(y)]. Pick neighborhoods U of x and V of y such t h a t c e u n c c v = 0 . LetA = g - ' [ U ~ n s a n d l e t B =g-'[V]nS.ThmbyRemark 21.29, A E A and B E P). Now let C E Rand pick r E X such that C = ng-'[(z)]. Then without loss of generality r $ CL U so if C = g-'[X\cl U] n S we have by Remark 21.29 that C E C. Then C n A = 0 so. since C is a filter, S\A E C. Finally define h : X + % by h(x) = ng-'[[x)]. Then trivially h is one-to-one and onto !X. Since X is compact, in order to show that h is a homeomorphism it suffices to show that % is Hausdorff and h is continuous. To see that %is Hausdorff, let A and 33 be distinct members of 31 and pick A e A and B E 8 as guaranteed by (b). Then A m and B' are disjoint neighborhoods of A and 8 respectively. To see that h is continuous, let x € L a n d let A S S such that A"S a basic open neighborhood of h(x). Let U = X\g[S\A]. Now g[S\A] is the continuous image of a compact set so is compact. Thus U is an open subset of X. We claim that XLU and h[Ul G A#. For the first assertion suppose instead that we have some p E S\A such that g(p) = x. Then h(x) G p so A E p, a contradiction. Now let y E U. Then by Rernark21.29. wehave g-'[U]nS E h(y). Since g-'[U]n S E A wehave A E h(y) o so h(y) E Ad as required. The following thmrern tells us that the description of quotients pmvided by Theo-

rem 21.31 is enough to characterize them. Theorem 2132. Lrr S be a discrete space and let R be a set offilters on S satilfyng

(a) given any choice fwrcrion f for R there exists 5 E Pf(W) such thaf S = UAESf (A) and (b) given distinct A and 8 in R ! , *re exist A E A and B E 8 such thar whenever C E R either S\A E C or S\B E C'.

-.

Then, w.th the quotient topology, R is a compact Hadoflspace. Further:for each p E flS there is a unique A E R such thar A E p and IheJuncriDng : fiS + W defied by g(p) C p is a continwuc sujection.

P w J That 'R is Hausdorff is an immediate consequence of (b). That W is compact will follow from the fact that W is a continuous image of flS. Now let p E flS and suppose first that for no A E R is A E p. Then for each A E Wpick f (A) E A\p. By (a) pick some 5 E P,(W) such that S = UASS f (A). Then S E p so pick some A E 5 such that f (A) E p. a contradiction. Next suppose that we have distinct A and 8 in W with A E p and 8 E p. Pick A E A and B E 8 as guaranteed by (b). Then S \ A E S E . p and A E A E p. a contradiction. The function g is trivially onto R ! To see that g is continuous. let p E flS and let A E Swithg(p) E An. Foreach23 E W\(g(p))pickby (b)some f(23) E Sandsome D ( 8 ) E g(p) so that for all e E W, S\ f ( S ) E C or S\D(S) G C. Let f ( g ( p ) ) = A. Pick by (a) some 3 E PI(%) such that S = USES f ( 8 ) and let 8 = w(g(p)). Then S\A E Uspaf ( 8 ) . Let D = D(21). Then D E g(p) s p. We claim that g[Dl E Au. So let q E and suppose that A $ g(q). Pick by Comllary 3.9 some r E fiS such that g(q) U (S\A) E r and pick 8 E 8 such that f ( B ) E r. Then one cannot have S\ f (21) E g(r) = g(q) so S\D(S) E g(q) E 9, amtradiction.

nS,,

. .

Now we bring the algebra of S into play. The operation + on filtas is a natural generabization of the operation. on ultrafilters.

Definition 2133. Let S be a semigroup and let A and 8 be f i l m on S. Then A*S=(A~S:(XES:X-'AE~)EAJ.

Remark 2134. Lcr S be a semigmup and let A and 8 befihcrs on S. Then A * 8 is a filter on S.

Mnition 21.35. LM S be a semigroup and let Wbe a set of filters on S. If for all A and 21 in 8,there is a unique C E R such that C A * 8.then define an operation .

Theorem 21.36. Ln S be a discrete semigmup and let ((p, Y) be a semigmup com5-'-' [ ( x ) ] : x E Y}. Then W is a set offilters on S pactijcation of S. Let W = satisfying

{n

(a) given any choicefunction f for R there exists 5 E PI(%) such that S = UI~S f (4).

.(b)

given distinct A and 8 in R ! then &?.A

E

A and B .E 8 such that whenever

C~ReitkrS\A~CorS\B~e,and

(c) givmany A M d S i n W t k r e i s a w r i q w C E WsuchthatC cA*8. Funhes with the quotient topology and the operation .on !R thefunction h : Y + W F-'[[x]]is an isomorphism anda homeomorphism.

defined by h ( x ) =

n

Pmof: Conclusions(a)and (b) follow fromTheorem21.31. To establish(c),let A, 8 E 14. We note that it is enough to show that there exists C E W such that C g A * 8. (For then, i f 9)E W aod D # C pick by (b) some C E C such that S\C E D . Then C E A * 8 so S\C E D\(A * 8).)Pick x. y E Y such that A = n F - ' [ { x ) ] and 8 = F - L y ) ] and let C? = P L [ ( x Y ) ]To . see that C G A * 33. let A E C . Let U = Y\aS\Al. Then U isopenin Y andxy E U . (Forifxy $ U picksomep 6 S\A such that F(p) = xy. Then A $ p so A 4 F-' [ [ x y ) ]a, contradiction.) Since xy E U and Y is right topological, pick a neighborhood V of x such that V y G U . Then by Remark 21.29, (p-'[V] = ? ' [ V ] n S E A. W e claim that (p-'[v] c 1s E S : S-'A E 8 )SO that A E A * 8 as required. To this end, let s E (p-'[Vl. Then (p(s)yE U.Since b y , is continuous pick a neighborhood W of y such that (p(s)W G U. Then by Remark 21.29. (p-'[Wl E 8 so it sufficesto show that (p-'[~E l K I A . Lett E (p-'[Wl. Then st) = (p(s)(p(t)E U so st 6 S\A, i.e., st E A as required. B y Theorem 21.31, the function h : Y + W defined by h ( x ) = n p ' [ ( x ) ]is a homeomorphism from Y onto W. To see that it is a homomorphism. let x , y E Y . W e havejustshownthath(xy) = nF-l-'[(xyll E h(x)*h(y)sothath(x).h(y)= h(xy).o

n

n

n

The following theorem tells us that the description o f semigroup compactifications provided by (a), @). and a weakening o f (c) o f Theorem 21.36 in fact characterizes semigroup compactifications. Theorem 21.37. Let S beadiscretesemigroupMdIrfW be aset -of

(a) given any choice function f for W there exists 5 E PI(%) such that S = UA~I f ( 4 (b) given distinct A and 8 in IS there exist A E A and B E 8 such that whenever C E R either S\A E C or S\B E C. and (c) given any A and 8 in W there exists C E Wsuch that C E A * 8. Then infoct for any A and 8 in W there exists a unique e E W such that C E A * 8. Also, with the quotient topology and the operation 3 ' is a compact right topological semigroup and thefunetion g : fJS + W &fined by g ( p ) p is a continuous homomorphismfrom fJS onto W with g[Sl A(W) and g[S]dense in %. In particular: ( g p . W ) is a semigroup compactifcation of S.

.,

Proof: As in the pmof o f Theorem 21.36 we see that statements (b) and (c) imply that for any A and 8 in W there exists a unique C 6 W such that C G A * 8. so the operation . is well defined. B y Theorem 21.32, with the quotient topology W is a

.

21.5 Uniform Compactifications

445

mmpact Hausdorff space and g is a (well defined) continuous function from BS onto

W. We show mxt that g is a homomorphism. To this e n d let p, q E 0.7. To see that E ( P . q ) = g ( p ) g(q). we show that g ( p ) g ( q ) G P .q. Let A g ( p ) .d q ) . A E g ( p ) g ( q ) so B = (s E S : S - ' A E g ( q ) ) E g ( p ) G p and if s E B , then S - I A E g(q) E q SO ( S E S :s-IA E q ) E p. That is, A E p .q as required. Since g is a homomorphism onto W. we have immediately that the operation. on W is associative. By Exercise 2.2.2,'Jt is aright topological semigmup. Lets E S. To see that A,(,, is continuous, note that since g is a homomorphism g o As = A,(,, o g. Let U be open in W and let V = A (,,-'[U]. Then g-'[V] = (g o A,)-'[u] is open in pS. Then V = 'Jt\g[f3~\~-' is open in W. Since S is dense in pS and g is continuous, g [ S ] is dense i n s .

.

.

[vP]

21.5 Uniform Compactifications Evny uniform space X has a compactification (@, y,X) defined by its uniform structure. For many uniform spaces X which are also semigroups. (@. y,X) is a semigroup compaclilicationof X. It has the property that the map ( a ,x ) r @ ( a ) xis acontinuous map from X x y.X to y.X, and y,X is the maximal semigroup compactification of X for which this holds. This is m e for all discrete semigroups and for all topological groups. If X is a discrete semigroup, then y.X % pX and so the material in this section generalizes our definition of pX as a semigroup. We first remind the reader of the definition of a uniform structure. X x X, we define U-I and U V by DeBnltion 21.38. Let X be any set. If U. V U-' = [ ( y . x ) : ( x . y ) E U) and U V = ( ( x ,y ) : ( x , z) E V and (z, y ) E U forsome z E XI. We may use U' to denote UU. A will denote the diagonal ( ( x ,x ) :x E XI. A uniform srmcfure (or uniformity) PI on X is a filter of subsets of X x X with the following properties: (1) A U forevery U E P1: (2) for every U E e(, U-' E P1; and (3) for every U E U ,there exists V E PI for which v2G U . Let PI be a uniform StrucNre on X and let U E PI. For each x E X, we put U ( x ) = ( y E X : ( x , y) E U),and,foreach Y g X,weput U[Yl = U y s V U ( y ) . P1 generates a topology on X in which a base for the neighborhoods of the point x E X are the sets of the form U ( x ) ,where U E PI. If X has this topology, (X. P1) is called a uniform space and X is called a uniformizable space. If (X. P I ) and (Y,V ) are uniform spaces, a function f : X + Y is said to he uniformly continuousif, foreach V E V .thereexists U E P1 suchthat (f ( X I ) . f (x2)) E V whenever ( x ~ . x zE) U .

Mehic spaces and topological groups pmvide impntant examples of uniformizable spaces. If ( X , 4is a mMic space, the filter which has m base the sets of the form ( ( x , y) E X x X :d ( x , y ) < r ) ,wherer r 0,isauniformsaumveonX. Thisexampleincludes all discrete spaces. If X is disaete, it has the trivial uniform structure P1 = (U E XxX:AGU). If G is a topological group. its topology is defined by the right uniform shuchtrc which has as base the sets ( ( x ,y) E G x G :xy-' E V J .where V denotes ancighborhood of the identity. In this section. we shall assume that we have assigned this uniform srmclure to any topological group to which we refer. It is precisely tfK completely regular topological qaces which are uniformizable. Suppose that X is a space and that C a ( X ) denotes the subalgebra of C ( X ) consisting of the real-valued functions in C ( X ) . X is said to be completely regular if. for every closed subset E of X and every x E X\E. there is a fuoction f E C a ( X )for which f ( x ) = 0 and f [El = (I). For each f E C a ( X ) and each E z 0,we put UI.. = ((1,y) E X x X : If ( x ) - f (y)I -z E ) . 'Ihe finite intersections of the sets of the fam UJ,, then provide a base for a uniform s t ~ c t u r eon X. In paaicular, every compact space X is uniformizable. In f a X has a unique uniform structure given by the filter of neighborhoods of the diagonal in X x X (see Exercise 21.5.1). In the next lemma we establish a relation between compactifications of X and subalgebras of Ca(X).

Lmma 21.39. Let X be any topological space and let A be a norm closed subalgebm of C d X ) which contains the constantfunctions There is a compact space Y and a mnrinuoutfunnion 9 : X + Y with the pmperty thor @[XI is dense in Y and A = ( f E C=(X) : f = g o @ for some g E CR(Y)}. Thc mopping @ is an embedding 8 for every closed subset E of X and every x E X\E. there exlrrr f E A such that f ( x ) = 0 and f [El = (1).

Pmqf For each f E A, let IJ = (t E P : It1 _I Ilf 11). Let C = X f e A . l / and let 9 : X -r C be the evaluation map defined by # ( X I ( f ) = f ( x ) . Let Y = cCc 41x1. For each f E A, let n, : C -+ I f be the projection map. If f E A, we have f = r r f l y o 9. Conversely. let G = [g E C R ( Y ) : g o 9 E A). By Exercise 21.5.2, G is a closed subalgebra of C d Y ) which contains the constant functions. We claim that G also separates the points of Y . To see this, let j and Z be distinct points in Y and choose f E A such that y ~ # 21. Since n ~ l oy @ = f, we have nfi, E G and n J l y ( j ) # xfly(Z). So G separates the points of Y and it follows from the StoncWeietshass Theorem that G = CR(Y). Thus A = ( f E C a ( X ) : f = g o 9 for some g E CR(Y)]. Suppose now that, for every closed subset E of X and every x e X\E, there exists f E A such that f ( x ) = 0 and f [El = ( 1 ) . Then $I is clearly injective. To see that 9 is an embedding, let E be closed in X and k t y E @[X]\@[E].Pick x E X such that y = # ( X I and pick f E A such that f ( x ) = 0 and f [ E l = ( 1 ) . Pick g E C ( Y ) such that f = 8 09. Then g ( y ) = g ( 9 ( x ) )= Oand g [ 9 [ E l ]= (1) soy 4 c!@[El.

The following r e d t estabfishes the existence of the ~ t o d s compactification h f a any completely regular space. (Of course, since any subset of a compact space is completely regular. this is the greatest possible generality for such a result.)

Tbto~lll21.40. Let X be a completely regular space. Then X can be embedded in a compacr space B X which has thefollowing universal property: whenever g : X + C is a continuous fwtction fmm X ro a compact space, there is a conrinuousfwrction : BX + C which is an e x t e ~ ~ i 4 o n8 .

Pmof: Let A = Cn(X). By Lemma 21.39. X can be densely embedded in a compact space BX with the popclty that every function in A extends to a continuous hrnction in C&X). We shaU regard X as being a subspace of BX. Let C be a compact space and let F denote the set of a l l continuous functions £mm C to [0,1]. We define anatural embedding i : C -r [O, 11 by putting i ( u ) ( f )= f ( u ) foreveryu ~ C a n d e v q ' Ef F. Suppose that g : X + C is continuous. Thm. for every f E F, f o g has a continuous extension ( f o g r : pX + [O. I]. We define h : BX -+ I] by h(y)(f) = ( f o g r ( y ) for every y E p X and every f 6 F. We observe that h ( x ) = i(g(x))whenevirx E X. Since Xis dense in B X . hfX1 isdense in h[pX].So h[BX]c i [ C ] ,because h [ X ]E i[C]and i[Cl is compact. We can therefore dcfim g by i-'oh.

'

a=

(p

Recall that a topological compactification ((p. Y ) of a space X has the p p m y that is an embedding of X into Y.

Theorem 21Al. Let ( X , P1) be a uniform space. There is a topological compacnJcation (#. y.X) ofX such thar it is precisely the uniformly com'nuousfunctionr in CR(X) which have continuous extensions to y.X. (That is, [f E C R ( X ): f = g o q5 for some g E Cx(y,,X)) = [f E C R ( X ): f is uniformly continuous).)

Pmof: It is a routine m a w to prove that the unifody continuous functions in CR(X) are a norm closed subalgebra of CR(X). The conclusion thm follows from Exercise 215.3 and Lemma 21.39. 0 S i c e q5 is an embedding. we shall regad X as beiig a subspace of y.X. Ihe compactification yX . will be called the uniform compactiyicationof X . It can be shomthatall possible topologicalcompactificationsof acompletely regular space X arise in this way as uniform compactifications. We now see that y,X may be very large.

Lemma 21.42. Let ( X , U)be a uniform space. Suppose thar there exist a sequence (x.)~,inXnndaserU~Usuchtharx,~U(x.)whencverm n.f Letf :N-+X BPd + yX . is an embedding. be defined by f ( n ) = x. Then j:

P m f : We claim that, for any two disjoint subsets A and B of N . cCyMxf [ A ] n cL,x f [ B ] = 0. To see this, we observe that U[(x. : n E A)] and (x. : n E 8 )

are disjoint, and hence that thcrc is a uniformly wntinuous function g : X + [O, 11 E A ) ] = (0) and g[(x. : n E B ) ] = [I) (by Exercise 21.5.3). By 'lbeorem 21.41. g can be extended to a continuous fuoftion : yX . + B. Since g[ccfix f [ A ] ] = (01 andZ[~&x f [ B I ]= (1). we have C&.X f [A1n C&X f [ B l = 0. Now suppose that u and v are distinct elements of PN.-We can choose disjoint subA and B of N such that A E-u and B E v. Since f ( u ) E c&x( f [ A ] )and F(u) E d,x( f [ B ] ) it , follows that f ( u ) f (v). So f is injective and is therefore an embedding.

for which g[{x. : n

a

+

We now assume that X is a semigroup and that ( X , V )is a uniform space. We shall give sufficient conditions for (6.y.X) to be a semigroup compactification of X. We observe that these conditions are satisfied in each of of the following cases: (1) X i s discrete (with the trivial uniformity); (2) ( X ,d ) is a metric space with a metric d satisfying d(xz, yz) 5 d ( x , y ) and d ( z x , zy) I:d ( x , y ) for every x , y. z E X: ( 3 ) X is a topological p u p . Notice that requirement (ii) below is stronger than the assertion that p, is uniformly continuous fw each a E X.

Theorem 21.43. Suppose that X is a semigroup a d that ( X , P1) is a uniform space. Suppose rhar tk nvofollowing conditions are satisficd: (i) For each a E X, An : X + X is uniformly continuous and (ii) for cach U E U, them aists V E V such that for every (5, t ) E V and every a E X , one has (sa, t a ) E U .

Thrn we can define a sernigmup operation on yX . for which (6. y.X) is a semigroup compactificarion of X.

P m f For each a E X, thcrc is a continuous extension 1 . : y.X -P y.X of A. (by Exercise 21.5.4). We put ay = M y ) for each y E y.X. Given y E yuX, we define ry : X -r y,X by r y ( s ) r sy. We shall show that ry is uniformly continuous. We have observed that the (unique) uniformity on y,,X is generated by [Up., : g E Cdy,,X) and 6 r 01, where Ud.< = [(u,U) E y,,X x yUX: Ig(u) g(u)l c 6). So let g E C R ( ~ . X )and 6 w 0 be given. Now glx is uniformly continuous by Theorem 21.41 so pick W E P1 such that Ig(s) g(r)l < 5 whenever (s, t ) E W . Pick by conditia (ii) some V E P1 such that for all a E X and all (s, t ) E V , one has (so, t a ) E W . Then. given (s, 1 ) e V , one has 6 lg(syl- g(ty)l= lim (&a) g(ta)( 5 -.when a dewtes anelement of X. andso a-Y 2 (ry(s),~ ~ E ( Ug.r. 0 ) from Exercise 21.5.4 that ry can be extended to a continuous function -r y :Ity.Xfollows -c y.X.

-

-

We pow define a binary operation on yX . by putting x y = f y ( x ) for every X , y E yuX. We observe that for every a E X, the mapping y H ay from yX . to itself is ~ ~ ~ t i n u oand, u s ; for every y E yuX. the mapping x I+ x y from y.X to itself is continuous, because these are the mappings and Ty respectively. It follows that the operation defined is associative because. for every x , y, z E y.X,

where s. t , and u denote elements of X. So x(yz) = (xy)z. Thus y.X is a semigroup compactification of X. Wenow show that the semigroup operation on y.X

0

isjointly continuous on X x y.X.

Theorem 21.44. Let ( X , U ) satisfy the hypotheses of Theorem 21.43. Then the map (s,x ) H sx is a continuous mapfmm X x y.X to y.X.

Pmof:Lets E X and x E yuX. Choose any f E &(y.X) and any e > 0. Since f i x is uniformly continuous, there exists U E e( such that If ( s ) - f @')I c whenever (s. s') E U. By condition (ii) of Theorem 21.43. there exists V E 91 such that (st. s'r) E U for every (s. s') G V and every t E X. I f x' E y,X. then If (sx')-f (s'x') 1 = lim, If (st)- f (s't)l 5 c whenever (s. s') E V . Thus, ifs' E V ( s ) I-,

-

andx' E l ; ' [ ( y E yX . : If(=) f(y)I c ell. we have I f ( s x ) If (sx') - f (s1x')J5 E . SO If ($1) - f ( S ' X ' ) ~ < 2e.

- f(sx')I

c sand 0

Theorem 21.45. Suppose that ( X . 91) satis~5esthe condiIiom of Theorem 21.43. Suppose that Y isa compact right topological semigmup and that h : X -t Y isa uniformly continuous homomorphism with h [ X ]g A ( Y ) . Then h can be wtendedtoacontinuous homomorphism ii : y.X + Y.

Proof:By Exercise 21.5.4. h can be extended to a continuous function Si : yX . For every u , v E y.X, we have

+ Y.

where s and r denote elements of X. So ii is a homomorphism. We now show that, if X is a topological group, yX . is maximal among the semigmup compactifications of X which have a joint continuity propem. Theorem 21.46. Let X be a topological gmup and let (0. Y ) be a semigmup compactificafion ofX. Supposerhnrthcmapping ( x . y) r B(x)yfrmn X x Y to Y iscontinuous. Then there is u continuous homomorphism : y,X -t Y such that 0 = &.

Pmof: We shall show that 9 is uniformly continuous. Ln e denote the identity of X and let 4 be the set of neighbob& of e. Suppose that 9 is not uniformly continuous. Then there exists an open neighborhood U of the diagonal in Y x Y such that. for every V E &. tlmc are points sv. tv E X satisfying svt;' E V and (B(sv).O(tv))$ U. The net ((O(sv),B(rv)))vc.ue has a limit point ( y ,z ) E (Y x Y)\U. Now sv = evtv. where. ev E V. and so B(sv) = B(ev)B(tv). Since e v + e, 0(ev) + O(e). and B(e) is easily seen to be an identity f a Y . So our continuity'assumption implies that y = z, a contradiction. Thus 0 is uniformly continuous and therefore can be extended to a continuous homomorphism g : y.X -t Y (by Theorem 21.45). Thcorem 21.45 shows that y. X can be identifiedwith pX if X is a dimete semigroup with the hivial uniformity. However, there are many familiar examples in which y .X and pX IT different Suppose. for example. that X = (R. +). We shall see in Theorem 21.47 that pIR cannot be made into a semigroup compactification of (R, +). It is natural. given a nondiscrete topological semigroup such as (R. to anempt to proceed as we did with a discrete semigroup in Theorem 4.1. And, indeed, one can d o the first part of the extension the same way. That is, given any s E S,one can define t , : S + S E .pS by &(r) = st. Then by Theorem 21.40, there is a continuous function L, : pS + pS such that L,IS = L, so one can define for s E S and q G fiS. sq = L&). As before one still has the function r, : S + pS defined by rq(s) = sq. However. in order to invoke Theorem 21.40 to extend rq. OM would need rq to be continuous. We now see that this condition does not hold in many familiar semigroups, including (8,+). Note that in the following theorem we do not assume thatthe operation on pS is associative.

+).

Theorem 21.47. Let S be a semigrnup which is also a metric space with a metric 8 for which 8(sr,st') = 6 ( t . t') and 8(ts, 0 s ) = b(t.1') for every s , t , r' E S. Suppose that S contains a sequence ( y , ) z l such that 8(y,,,, y.) > 1 whenever m # n. Let. be a binary opemtion on BS which Mends the smigmup operation of S and has the pmperty that the mapping € n x .{ from pS to itself is continuousfor every x E S. Then then is apoim p E BS such that the mapping x n x .p is discontinuous at every non-isohfed point x of S. Pmof Lclpbeanylimit pointof ( y . ) z , andletx € Sbe thelimitofasequence (x,)z, of distinct elements of S. We may suppose Ihatd(x,. x.) e for every m , n E N. For every m , m', n , n' E N with n # n', we have

4

~(x,Y..x,+Y.~ 2 S(X,Y..X~Y,~)

-s(x,Y.,.x,,Y,,)

=

w~..Y.,) - ~ ( x ~ . x>~T.I , )

Let A = (x. y. : m < n and m is even) and B = (xmyn: m c n and m is odd). We claim that A and B have no points of accumulation in S. This can be seen from the fact thah for any s E S,[t E S : 8(s, t ) < cannot contain two points of the form xmy. and xm,y.r with n # n', and can therefore contain only a finite number of points

a]

inAUB.ThuscCsA=A,clsB=BandclsAncLsB=0.

21.5 Uniform ~

45 1

f i c 1 1 t i ~

lt f o l l m from Urysahn's Lnnmathat h ? acolltinums fimction f : S + 10.11 for which f [A] = (01 and f [Bl = [I]. Let f :B S [O,11 denote t k continuous exteasion o f f . lhen 7(x, p) = 0 if m is even and f (x, . p) = 1 if m is odd. So the sequence (x, p):, cannot converge to x . p. 0

.

-

-

We shall aow look at snm of the proprtiuof y.P when Rdmotes the topological grwpformedby therePlnumbersunderaddition.Wefirstnotethat,9Zcanbennbedded topologically andalgebraicallyinyYW.By 'lbeorcm21.45.theinclusimmapi : Z B can be extended to acontinums homomorphism 7 :BZ -+ y.B. We can show that 7 is injective and therefore an embedding, by essentially the same argument as the one used in the proof of Lemma 21.42. We shall therefore assume that BZ G y.B by identifying BZ with &9Z]. This identifies BZ with ce,~(Q, because = ce,~(i[4). The following theorem gives an expmsion for an element of y,R analogous to the expression of a real number as the sum of its fractional and integral parts. llnorrm21.48. Every x E y.B can be expressed uniquely ar x = t~[O.l)andz~BZ

r

+ z, where

-+ T = R/Z denote the canonical map. Since n is uniformly continuous, n can be extended to a continuous homomorphism if : y.B + T. We claim that F(x) E 0 if and only if x E BZ. On the one hand, T(x) = 0 if x E BZ, because n[Z] = (0). On theother hand. suppose that x $! BZ. Thm there is a continuous function f : y.R -+ 10, I] for which f (x) = 1and f [Zl = (0). Since fiz is uniformly continuous, there exists 8 s 0 such that f [UnGz[n - 8, n 611 g [O, f]. Sox f C&R (UnEZ[n 8, n 81 and therefore x E ct,~(U,,~(n a, n 1 6)) and E(x) # 0, because 0 $? CLT n UeEZ(n 6, n 1 a)]. x. Let r denote the unique number in [O. 1) for which n(t) = F(x). If z = -t d m F(z) = 0 and so z E BZ. Thus we have an expression for x of the typc required. To pmve uniqueness. suppose that x = t' z', where t' E [O. 1) and z' E BZ. Then A(*) = n(rl) and so t' = t and therefore z' = -t x = z.

Pnwf Let n : R

-

-

+

I

+

+

+

+ -

+ + -

+

+

Theorem 21.48 allows us to analyze the algebraic structwe of y.R m terms of the algebraic shoeturr of BZ.

C o m w 21.49. The 1 4 ideals of yJi have theform R + L, where L is a lefi ideal of PZ; and the right ideals of y.R lurve thrform R

+ R. where R is o right ideal of BZ.

Pmof: llis follows easily from Theorem 21.48 and the observation that R is contained in the center of y.R. 0

We omit the proofs of the following comUaries, as they arc easy consequences of Theorem 21.48.

Corollary 2151. Every idempotem of y.R is in p Z C m l l a r y 2152. y.R has 2' minimof Ieji ideals and 2' minim01 right ideals, each conmining 2' idempotmrs.

P m f This follows from Corollary 21.49 and Theorem 6.9. Exerdse 21.5.1. Let ( X , U ) be a uniform space. Show that every U U is a neighborhood of the diagonal A in X x X. If X is compact, show that 'U is the filter of all neighborhoods of A in X x X. . Exercise 21.52. Let G and Y be as in the proof of Lemma 21.39. Rove that G is a closed subalgebra of Q ( Y ) which contains the constant functions.

Exercise 21.53. Let ( X . U ) be a uniform space. Suppose that E and F are subsets of X and that U [ E ln F = 0 for some U E U.Show that there is a uniformly continuous function f : X + [O. 11 for which f [El = ( 0 )and f [ F ]= (I). in U by putting Uo = U i l U-' and choosing (Hint: Choose a sequence (U.).,, U,, to satisfy U. = LS;' and U: E U.-1 for every n > 0. Then define subsets E, of X for each dyadic rational r c [O, 11 with the following properties:

Eo= E , El = X\F. and for each n E Wand each k

E

(0, 1,2....,2"

- I), U,[E r1 i

Er

I 4-.

This can be done inductively by putting Eo = E and El = X\F, and then assuming that E r has been defined for every k E {0,1,2,. . ..2"). If k = 2m 1 form E F (0.1, . ,2" - I], E l can be defined as U.+I [ E $ ] . Once the Sets E, have been

..

+

*"+I

constructed. define f : X -+ [O, I] by putting f ( x ) = 1 if x E F and f ( x ) = inf{r : x E E,) otherwise.)

Exerdse 21.5.4. Suppose that ( X , U )and (Y,V )are uniform spaces and that 8 : X + Y is uniformly continuous. Show that 8 has a continuous extension : yX . + y.Y. (Hint: Let A = C&.Y). The mapping b : y.Y + *W defined by 9 ( y ) (f ) = f ( y ) is then an embedding. Iff E A, f i r is uniformly continuous and so f o 8 is uniformly continuous and has a co?tinuous extension f : y.X -r W. Let $ : y.X + @[y.Y] be defined by $ ( x ) ( f ) = f ( x ) . Forevery x E X. @ ( x ) = +(8(x)).)

Exerdse 215.5. A uniform space ( X , U ) is said to be totally bounded if, for every U E U , there exists a finite subset F of X such that X = U ,, U ( x ) . It ( X . U) is not totally hounded. show that y.X Lemma 21.42.)

contains a topological copy of BN. mint: Apply

Exercise 215.6. Let X be a topological p u p . Show that f E C R ( X )is uniformly continuous if and only if the map x H f o A, fmm X to C R ( X )is continuous. where the topology of C R ( X )is that defined by its norm. (For this reason, y,X may be called the left uniformly continuous compactification of X and denoted by P'UC(X).)

Notes

Notes It is customary to define the LAC. W A P . A P , and 4 A P compactifications only for semitopological semigroups S. Of course. if S is not semitopological. then the map 41 cannot be an embedding (because qtCSl is semitopological). It was shown in [I501 (a result of collabwation with P.Milnes) that there exists a completely regular semitopological semigroup S such that 41 is neither one-to-one nor open and that there exist completely regular xmigroups which are neither left nor right topological for which 4, is one-to-one and other such semigroups for which 41 is open as a map to VIISI. In keeping with our standard practice, we have assumed that d l hypothesized topological spaces are Hausdorff. However. the results of Section 21.1 remain valid if S is any semigmup with topology, without any separation axioms assumed. We defined weakly almost periodic and almost periodic functions in tmns of extendibility to the W A P - and AP-compactifications. Itis common to define them in terms of topologies on C(S). One then has that a function f € C(S) is weakly almost periodic if c.!{ f o p, :s E S) is compact in the weak topology on C(S) and is almost periodic provided ct[ f o p, : s E S) is compact in the norm topology on C(S). For the equivaknce of these c h ~ ~ ~ ~ t e r i z a with t i o nour s definitions see [MI. Theorem 21.7 and Corollary 21.8 are due to J. Berglund, H. Junghenn, and P.Mines in [39], where they &it the "main idea'' to J. Baker. Thequivalence of statements (I) and (3) inTheorem 21.18 is due to A. Gmthendieck [I 121. Theorem 21.22 is due to W. Ruppcrt [217]. Thealgebraic propertiesof W A P ( N ) are muchhardermanalyzethanthoseoffi(N). It is difficult to prove that W A P ( N ) contains more than o m idempotent. T.West was the first to prove that it contains at least two [245]. It has since been shown, in the work of G. Brown and W. Moran (541. W. Ruppert [220] and B. Bordbar [Sl], that WAP(N) has 2' idempotents. Whether the set of idempotents in WAPfW) is closed, was an open question for some time. It has recently been answered in the negative by B. Bordbar and I. S. Pym [531. The results of Section 21.4 arc from [37] and were obtained in collaboration with I. Berglund. In 1371 a characterization of the WAP-compactification as a space of filters was also obtained. Theorem 21.40 is due (independently) to E. &ch [601 and M. Stom [227]. See the notes to Chapter 3 for more information about the origins of the ~tone&ch compactification. Theorem 21.48 is due to M. Filali [90], who proved it for the more general case of

YX. Suppose that S is a semitopological semigroup and that f E C(S). It is fairly easy to prove that f is a W A P function if and only if {( f a A,)- : s E S] is relatively compact in {F:g E C(S))for the topology of pointwise canvqence on C(pSd). It follows from [81. Theorem IV.6.141 that this is equivalent to (f o I, : s E S) being weakly relatively compact in C(S).

Bibliography

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PI J. B*,

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L i s t of Symbols

A-s76 Ax-' 76 A* 55 An 441 A* 76.93 A'(p) 76.77.93 [A]" 91 [A]

Algebra in the Stone-Čech compactification: theory and applications

Algebra in the Stone-Čech compactification: theory and applications

Algebra in the Stone-Čech compactification: theory and applications