STUDIES IN LO-GIC AND THE FOUNDATIONS OF MATHEMATICS VOLUME 118
Editors J. BARWISE, Stanford D. KAPLAN, Los Angeles H. ...
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STUDIES IN LO-GIC AND THE FOUNDATIONS OF MATHEMATICS VOLUME 118
Editors J. BARWISE, Stanford D. KAPLAN, Los Angeles H. J . KEISLER, Madison P. SUPPES, Stanford A. S. TROELSTRA. Amsterdam
NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD
THEORY OF RELAmONS
R.FRAISSE Universitk de Provence Marseille France
1986
NORTH-HOLLAND AMSTERDAM 0 NEW YORK OXFORD
ELSEVIER SCIENCE PUBLISHERS B.V., 1986 All rights reserved. N o part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the priorpermission of the copyright owner.
ISBN: 0 444 878653
Translationof ThLorie des relations Translated by P. Clote Published by: Elsevier Science Publishers B.V. P.O. Box 1991 1000BZ Amsterdam The Netherlands Sole distributors for the U.S.A. and Canada: Elsevier Science PublishingCompany, Inc. 52VanderbiltAvenue NewYork, N.Y. 10017 U.S.A.
Library of conppg CataloginginPubliitionData
F r a i s d , Roland. Theory of relations.
(Studies i n l o g i c a d the foundations of matheratics ; v. 118) Translation of: Thhrie des relations. Bibliography: p. Includes index. 1. Set theory. I. Title. 11. Series. QA248.FT75 1986 511.3'22 85-20701
ISBN 0-444-87865-3
PRINTED IN THE NETHERLANDS
V
INTRODUCTION Relation theory goes back t o the 1940's ; i t o r i g i n a t e s i n the theory o f order types, due t o HAUSDORFF (Grundzuge der Mengenlehre 1914), S I E R P I N S K I ( L e ~ o n ssur l e s nombres t r a n s f i n i s 1928, taken up again i n Cardinal and o r d i n a l numbers 1958), SZPILRAJN (Sur l ' e x t e n s i o n de l ' o r d r e p a r t i e l 1930), DUSHNIK, MILLER (Concerning s i m i l a r i t y transformations of 1i n e a r l y ordered sets 1940), GLEYZAL (Order types and s t r u c t u r e o f orders 1940), and t o HESSENBERG (Grundbegriffe der Mengenlehre 1906, i n t r o d u c i n g the negative and r a t i o n a l o r d i n a l s ) . A t t h a t t i m e , r e l a t i o n theory j u s t extended t o a r b i t r a r y r e l a t i o n s t h e elementary n o t i o n s o f order type and embeddability. Relation theory i n t e r s e c t s o n l y weakly w i t h graph theory, w i t h which i t i s sometimes s t i l l confused. F i r s t l y , echniques i n r e l a t i o n theory o n l y r a r e l y d i s t i n g u i s h between graphs, i.e. s j m t r i c b i n a r y r e l a t i o n s , and r e l a t i o n s o f a r b i t r a r y a r i t y . A d d i t i o n a l l y , as opposed t o graph theory, i n r e l a t i o n theory one considers e q u a l l y t h e two t r u t h values (+) and ( - ) taken on by a r e l a t i o n w i t h base
E
f o r each element o f
E2
(or o f
En
f o r the a r i t y
n ).
On the o t h e r hand, r e l a t i o n theory uses techniques e s p e c i a l l y from combinatorics, the l a t t e r which can be defined as f i n i t e s e t theory. Anything concerning r e l a t i o n s w i t h f i n i t e bases, o r counting isomorphism types o f f i n i t e r e s t r i c t i o n s o f a given r e l a t i o n , o r again the study o f permutations o f t h e base which preserves a given r e l a t i o n ( i .e. automorphisms o f t h e r e l a t i o n ) , makes use o f combinatorics. From a more t e c h n i c a l viewpoint, see t h e combinatorial lemnas i n ch.3 5 4, and the study o f the incidence m a t r i x i n ch.3 5 5. ' a
As f o r mathematical l o g i c , i t s i n t e r s e c t i o n w i t h r e l a t i o n theory i s r a t h e r import a n t . One can even say t h a t t h e two p r i n c i p a l sources f o r r e l a t i o n theory are the study o f order types, already mentioned, and l i n e a r l o g i c , i . ? . f i r s t - o r d e r o n e - q u a n t i f i e r l o g i c ; t h a t i s the study o f u n i v e r s a l formOlas (prenex formulas only having u n i v e r s a l q u a n t i f i e r s ) , and boolean combinations thereof, w i t h the p a r t i c u l a r case o f q u a n t i f i e r - f r e e formulas. From a semantic, o r model-theoretic viewpoint, t h i s i s the study o f u n i v e r s a l classes o f TARSKI,' VAUGHT 1953, and o f boolean combinations t h e r e o f .
I f one presents mathematical l o g i c from a r e l a t i o n a l t h e o r e t i c viewpoint, the basic n o t i o n i s t h a t o f l o c a l isomorphism, i . e . isomorphism o f a r e s t r i c t i o n o f the f i r s t r e l a t i o n onto a r e s t r i c t i o n o f the second one: see ch.9 5 1.4. For example, t h e f r e e i n t e r p r e t a b i l i t y o f a r e l a t i o n S i n another r e l a t i o n R w i t h the same base, i s a l g e b r i c a l l y defined by t h e c o n d i t i o n t h a t every l o c a l automorphism o f R ( l o c a l isomorphism from R i n t o R ) i s a l s o a l o c a l automorphism o f S . E q u i v a l e n t l y , f r e e i n t e r p r e t a b i l i t y i s l o g i c a l l y defined by the existence o f a q u a n t i f i e r - f r e e formula which defines S i n t h e s t r u c t u r e of R . For example, i f R i s a chain, o r t o t a l ordering, then t h e betweenness r e l a t i o n S(x,y,z) = + i f f z i s between x and y , i s defined by t h e q u a n t i f i e r - f r e e formula (Rxz h Rzy) v (Ryz A Rzx) This equivalence between algebraic and l o g i c a l n o t i o n s e x i s t s even above t h e f r e e - q u a n t i f i e r and t h e o n e - q u a n t i f i e r cases; since l o g i c a l ( o r elementary) equivalence between R and S , saying t h a t R and S s a t i s f y t h e same f i r s t - o r d e r formulas, i s equivalent t o t h e i r being (k,p)-equivalent f o r a l l i n t e g e r s k and p , which i s a p u r e l y algebraic n o t i o n : see my Course o f mathematical l o g i c 1974 vol. 2 Coming back t o t h e l i n e a r case ( o n e - q u a n t i f i e r l o g i c ) , as common notions and techniques i n both mathematical l o g i c and r e l a t i o n theory, we have those o f 1-isomorphism, 1-extension, p r o j e c t i o n f i l t e r ( a v a r i a n t o f u l t r a p r o d u c t ) : see ch.10 5 1 . And f o r each o r d i n a l o( , t h e 4 -morphism (ch.10 5 4), which i s n o t
.
.
vi
Introduction
a one-quantifier notion, b u t i s i n d i s p e n s i b l e in r e l a t i o n theory f o r t h e study of embeddability: see ch. 10 5 4 and 5 5.3. From the 1970's, an important connection appears between r e l a t i o n theory and t h e theory of permutations. See the study of o r b i t s (ch.11 § 2 ) , t h e theorem on the increasing number of o r b i t s (LIVINGSTONE, WAGNER, ch.11 5 2.8) and the theorem on s e t - t r a n s i t i v e , o r homogeneous groups (CAMERON, ch.11 5 5 . 1 0 ) . Let us mention, a l s o from t h e 1970's, some unexpected connections between r e l a t i o n theory and topology ( c h . 1 5 8 and ch.7 5 2 ) ; and even connections w i t h l i n e a r algebra (ch.11 5 2 . 6 ) . We s h a l l now b r i e f l y present the p r i n c i p a l notions s t u d i e d , by mentioning f i r s t t h a t chapters 1 through 8 concern the theory of p a r t i a l and t o t a l orderings ( o r c h a i n s ) , while chapters 9 through 12 concern t h e general study of r e l a t i o n s . In chapter 1, we review b a s i c set t h e o r e t i c a l r e s u l t s , i n general without proofs, which allow t h e reader t o know, f o r i n s t a n c e , i n which p r e c i s e sense we use the notion of f i n i t e set (TARSKI's sense r a t h e r than OEDEKIND's), o r the notion of c a r d i n a l i t y of a s e t . T h i s allows us t o p r e c i s e , throughout t h e r e s t of t h e book, which axioms a r e used f o r each proof: ZF a l o n e , t h e axiom of choice, dependent choice, t h e u l t r a f i l t e r axiom, t h e continuum hypothesis, e t c . Moreover i t seems t h a t even among l o g i c i a n s , t h e r e a r e few who a r e aware t h a t , while O1 > W i s provable in ZF a l o n e , y e t the countable axiom of choice, f o r i n s t a n c e , i s used t o prove t h a t W 1 i s r e g u l a r . Or t h a t KONIG's theorem (ch.1 0 1.8), even i n t h e very p a r t i c u l a r case of two ordered pairs of s e t s , i s not provable in ZF alone. Or t h a t t h e p o s s i b l e equivalence between the axiom o f choice and t h e statement t h a t t h e range of a function i s subpotent w i t h i t s domain, i s s t i l l an open problem, already p u t f o r t h by RUBIN 1963. Thus t h i s chapter could be useful as a memory brush-up f o r the axiomatic s e t t h e o r e t i c i a n . In chapter 2 , in a d d i t i o n t o a review of b a s i c r e l a t i o n t h e o r e t i c a l n o t i o n s , s i m i l a r in s p i r i t t o chapter 1, we introduce some notions which a r e no longer c l a s s i c a l , y e t which extend well-known concepts. For example the coherence lemma ( 5 1 . 3 ) , a not well-known version of the u l t r a f i l t e r axiom. Another example, the c o f i n a l i t y of a p a r t i a l o r d e r i n g , a s well a s t h e r e l a t e d notion of c o f i n a l height ( 5 5.4 and 5 7 ) . C l a s s i c a l l y , t h e notion of c o f i n a l i t y i s r e l e g a t e d t o t h e s i n g l e case of c h a i n s , o r t o t a l o r d e r i n g s , which while i n t e r e s t i n g i s too much r e s t r i c tive. In chapter 3 , we present RAMSEY's theorem and important refinements of i t , due t o GALVIN and t o NASH-WILLIAMS ( 5 2 ) . Furthermore, the " i n i t i a l i n t e r v a l theorem" o r GALVIN's theorem i s presented t w i c e , with very d i f f e r e n t proofs: POUZET's proof i n 5 2 and LOPEZ'S proof using t h e c l a s s i c a l Ramsey s e t s of r e a l s , i n 5 6. Then we a r e led t o the p a r t i t i o n theorems of DUSHNIK, MILLER and of ERDOS, RADO. We a l s o present a combinatorial study of the incidence matrix, w i t h the l i n e a r independence lemma due t o KANTOR. In chapter 4 , we begin t h e study of p a r t i a l o r d e r i n g s , w i t h t h e notions of good and bad sequence, of a f i n i t e l y f r e e p a r t i a l o r d e r i n g , and t h a t of a well p a r t i a l e present HIGMAN's c h a r a c t e r i z a t i o n of a well p a r t i a l ordering (the ordering. W s e t of i n i t i a l i n t e r v a l s is well-founded under i n c l u s i o n ) ; a l s o HIGMAN's theorem on words i n a well p a r t i a l o r d e r i n g , and RADO's well p a r t i a l ordering ( 5 4 ) . Also e present t h e notions of i d e a l , t r e e , dimension, bound of an i n i t i a l i n t e r v a l . W the theorem of t h e maximal reinforced chain f o r a well p a r t i a l ordering, due t o D E JONGH, PARIKH ( 5 9 ) . The chapter ends ( 5 10) w i t h POUZET's theorem on r e g u l a r ( o r f i n i t e ) c o f i n a l i t y of any f i n i t e l y f r e e p a r t i a l ordering.
Introduction
vii
In chapter 5, we consider embeddability between orderings, the well pa rtia l ordering of f i n i t e trees (KRUSKAL), the existence of immediate extensions and of faithful extensions (HAGENDORF), Cantor's theorem for pa rtia l orderings (DILWORTH, GLEASON). Then the existence of s t r i c t l y decreasing i n f i n i t e sequences of chains of reals: the denumerable sequence due t o DUSHNIK, MILLER and the continuum length sequence due t o SIERPINSKI. Finally a brief study of SUSLIN's chain and t r e e , in connection with SUSLIN hypothesis; also ARONSZAJN t r e e , SPECKER chain. In chapter 6 , we introduce the scattered chain, which does not admit any embedding of the chain Q of the r at i o n al s . Also the indecomposable, as well as the right and the l e f t indecomposable chain. We present HAGENDORF's theorem of unique decomposition of an indecomposable chain ( 5 3.3) and some connected re sults (JULLIEN, LARSON). We begin t o study the covering of a chain by r l g h t or l e f t indecomposable in te r v a l s, or by doublets of indecomposable i n t er v als. We present the hereditarily indecomposable chain with L A V E R ' S r e s u l t s , and f i n a l l y the indivisible relation o r chain. I n chapter 7 , we proof supplementary r es u l t s a b o u t f i n i t e l y free pa rtia l orderinas and t h e i r reinforcements by chains. We extend t o the s e t of i n i t i a l intervals the topology already introduced in ch.1 5 8 , and give some applications, namely BONNET'S r e s u l t s . Then we prove the following important theorem of POUZET: every directed well p a r t i a l ordering has a cofinal r es t r ic tion which i s a dire c t product of f i n i t e l y many d i s t i n c t regular alephs. The chapter ends with a short study of Szpilrajn chains (BONNET, JULLIEN); two interesting re sults due t o TUKEY and t o KRASNER are presented as exercises.
I n chapter 8 , we introduce the important notion of ba rrie r due t o NASH-WILLIAMS; the p a r t i t i o n theorem ( 5 1 . 4 ) , the theorem on the minimal bad barrier sequence ( 5 2 . 2 ) ; the forerunner and successor b ar r i er . This i s the main t o o l i n the proof of the very important theorem of LAVER: every s e t of scattered chains forms a well quasi-ordering under embeddability ( 5 4.4). In other words, there e x is t s neither an i n f i n i t e s t r i c t l y decreasing sequence nor an i n f i n i t e s e t of mutually incomparable scattered chains. LAVER proved even more, in extending his result t o chains formed from a countable union of scattered chains. However his proof has n o t yet been s u f f i ci en t l y simplified t o be presented in a textbook of a reasonable s i z e . In t h i s chapter, we also study the be tte r partial ordering, a notion due t o NASH-WILLIAMS, both f o r i t s i n t r i n s i c inte re st and for i t s applications t o chains. In chapter 9 , we begin the general theory of r el at ions, w i t h the notions of local isomorphism, free i nt er p r et ab i l i t y and free operator (which i s the re la tionist version of a logical f r ee formula, and links relation theory t o logic ). We study constant, chainable, monomorphic r el at i o n s . I n the case of a binary relation with cardinality p , we present the deep r es u l t due t o JEAN: (~-2)-monomorphy implies (p-1)-monomorphy ( 5 6 . 7 ) . We present the profile increase theorem (POUZET, 5 7 ) . Finally we extend t o ar b i t r ar y relations the homomorphic image ( 5 8 ) , and in 5 9 we introduce the bivalent t ab l e, which apparently yields d i f f i c u l t problems, one of them being very p ar t i al ly solved by L O P E Z . Most of r e l a t i o n i s t researchers seem t o be discouraged by t h i s branch of relation theory, which i s s t i l l a marginal study inside relation theory, considered i t s e l f as being marginal d u r i n g t o o long a time.
I n chapter 10, we c l as s i f y relations according t o t h e i r age: two representatives of the same age have the same f i n i t e r e s t r i c t i o n s , up t o isomorphism. This i s equivalent t o classifying relations by the s e t of universal formulas which they s a t isf y . Then we study maximalist or ex i s t en t i al l y closed relations ( 3 3.8), rich r e l a t i o n s, inexhaustible relations ( 5 4 and 5 ) , and relations which are rich f o r t h e i r age. This notion, connected t o saturated relations, leads t o the existence c r i t e r i o n of POUZET, VAUGHT ( 5 7 ) . The f i n i t i s t and almost chainable relations are presented in 5 8 and 9.
Introduction
viii
Chapter 11 i s concerned w i t h correspondence between r e l a t i o n theory and permutations, the l i n k between them being the homogeneous r e l a t i o n s and r e l a t i o n a l systems. We already mentioned t h e theorem o f i n c r e a s i n g number o f o r b i t s , due t o LIVINGSTONE, WAGNER. I n 5 3 and f o l l o w i n g , we introduce t h e c o m p a t i b i l i t y modulo a permutation group, which y i e l d s a marginal study i n s i d e permutation group theory, w i t h many open problems. The n o t i o m o f i n d i c a t i v e group and i n d i c a t o r l e a d t o FRASNAY's r e d u c t i o n theorem ( 5 4 ) . The p a r t i c u l a r case o f Q - i n d i c a t i v e groups leading t o the s e t - t r a n s i t i v e group theorem o f CAMERON. F i n a l l y we study t h e pseudo-homogeneous r e l a t i o n s , t h e prehomogeneous r e l a t i o n s w i t h POUZET's existence c r i t e r i o n ( 5 7), t h e set-homogeneous r e l a t i o n s . I n chapter 12, we introduce the bounds o f a r e l a t i o n R : f i n i t e r e l a t i o n s nonembeddable i n R b u t whose proper r e s t r i c t i o n s are embeddable i n R . We present several important theorems due t o FRASNAY: the reassembling theorem ( 5 3); t h e existence o f an i n t e g e r p such t h a t , from t h i s p o i n t on, p-monomorphism i m p l i e s c h a i n a b i l i t y ; and t h e f i n i t e n e s s o f the number o f bounds f o r a chainable r e l a t i o n . This study uses t h e method o f permuted chains, o r c o m p a t i b i l i t y modulo a permutat i o n group, already presented i n chapter 11. Proofs have been s i m p l i f i e d by using, as a powerful t o o l , the p-well r e l a t i o n due t o POUZET. The chapter, and t h e book, are ending w i t h the study o f reduction, reassembling, monomorphism and chainabil i t y thresholds: c a l c u l a t e d f i r s t by FRASNAY, they were improved by HODGES, LACHLAN, SHELAH, then proved again by FRASNAY t o have the smallest p o s s i b l e value. I n 5 6 we added some easy considerations about u n i v e r s a l classes. I n order t o keep t h i s book t o a reasonable s i z e , we suppressed two planned chapters. One about the celebrated problem o f r e c o n s t r u c t i o n , i . e . the problem t o know i n what cases a r e l a t i o n w i t h base E i s completely determined, up t o isomorphism, by the isomorphism types o f i t s r e s t r i c t i o n s t o proper subsets o f E The reader may c o n s u l t BONDY, HEMMINGER 1977, LOPEZ 1978, 1982, 1983, POUZET 1979', STOCKMEYER 1977, ULAM 1960 (see B i b l i o g r a p h y ) . The o t h g r y i s s i n g chapter concerned the n o t i o n o f i n t e r v a l i n r e l a t i o n theory: see FRAISSE 1984 i n Bibliography.
.
I would l i k e t o thank those among my colleagues - professors, researchers, students and ex-students - who solved o r c o n t r i b u t e d t o t h e s o l u t i o n o f a l l problems presented here; and t o those who, by s i m p l i f y i n g t h e i n o r d i n a t e l y long o r d i f f i c u l t p r o o f o f the o r i g i n a l paper, have made these r e s u l t s accessible, hence s u i t a b l e f o r p r e s e n t a t i o n i n t h i s textbook. T h e i r names are mentioned together w i t h t h e i r c o n t r i b u t i o n . As f o r myself, I have t h e f r e e conscience o f having accomplished my work as "chef d ' e c o l e " : namely the presentation, i n a form accessible t o a wide audience, o f r e s u l t s obtained by those who loved my research area and surpassed me.
1
CHAPTER
1
REVIEW OF AXIOMATIC SET THEORY
The purpose o f t h i s chapter i s t o s i t u a t e p r e c i s e l y "theory o f r e l a t i o n s " w i t h i n the framework o f axiomatic s e t theory, which i n i t i a l l y w i l l be t h a t o f ZERMELOFRAENKEL. The axioms f o r ZF are introduced below i n
91
and 92. Our i n i t i a l
n o t a t i o n w i l l be introduced there. I n r e f e r r i n g t o the f i r s t and sometimes second chapter, we w i l l i n d i c a t e throughout the book which statements r e q u i r e o n l y the axioms o f ZF and those which require, t o our knowledge, t h e axiom o f choice, o r r a t h e r the weaker u l t r a f i l t e r axiom (boolean prime i d e a l axiom), o r t h e axiom o f dependent choice, e t c . Most o f the proofs, as w e l l as c l a s s i c a l d e f i n i t i o n s from the f i r s t and second chapter, are l e f t t o t h e reader.
§
1 - FIRST GROUP
OF A X I O M S FOR
ZF, F I N I T E SET, A X I O M OF CHOICE,
KONIG'S THEOREM We begin w i t h the axioms of e x t e n s i o n a l i t y , p a i r , union, power s e t ( s e t o f a l l subsets o f a s e t ) and t h e scheme o f separation, a l l supposedly known t o the reader. We denote t h e empty s e t by We denote the union o f the s e t
a
0
by
, inclusion C , s t r i c t inclusion u a , and the power s e t by p ( a )
I f b s a , we designate the d i f f e r e n c e by (simply c a l l e d p a i r s ) are denoted by a}
a v {a)
of
a
o f the empty set;
,
\ a,b)
, etc.
.
.
unordered p a i r s
The successor s e t
{ 0) i s t h e successor .( 0,l) i s t h e successor o f 1 , e t c . This nota-
i s denoted by 2 = 1+1 =
. So
. Singletons,
a-b
C
a+l
that
1 = 0+1 =
t i o n coincides w i t h the n o t a t i o n f o r o r d i n a l a d d i t i o n , introduced i n fj 3 below.
1.1. FINITE SET Following TARSKI 1924', we d e f i n e a s e t set
b
o f subsets o f
a
a
t o be f i n i t e i f f every non-empty
contains an element which i s minimal w i t h respect t o
i n c l u s i o n , i . e . an element
c E b
such t h a t no
xE b
satisfies
x c c
.
Taking complements, i t i s e q u i v a l e n t t o say t h a t every non-empty s e t o f subsets o f
a
a i s f i n i t e e x a c t l y when contains a maximal element. A n o n - f i n i t e
s e t i s s a i d t o be i n f i n i t e . The empty set, a s i n g l e t o n , a p a i r are f i n i t e sets. Every subset o f a f i n i t e set i s f i n i t e .
If a
i s f i n i t e , then so i s the s e t composed o f
element. I n p a r t i c u l a r , the successor
a+l
of
a a
together w i t h an a d d i t i o n a l i s finite.
2
THEORY OF RELATIONS
f
Scheme of-injuction-for f i n i t e s e t s . If a condition i s t r u e f o r the e m p t y j e t , and i f f o r every s e t a satisfying and every s e t u , the s e t a u \ u j s m s a t i s f i e s f , t+ i s t r u e f o r every f i n i t e s e t . I f a s e t a and a l l i t s elements are f i n i t e , then the union u a i s f i n i t e . This i s often expressed in the following form called pigeonhole principle: i f we p a r t i t i o n an i n f i n i t e s e t into f i n i t e l y many subsets, then a t l e a s t one of these subsets i s i n f i n i t e .
f
1 . 2 . COUPLE OR ORDERED PAIR, CARTESIAN PRODUCT Given two s e t s a , b the couple or ordered pair (a,b) i s the s e t {{a} , {a,b]\ formed of the singleton { a ) and the (unordered) pair i a , b ) . This definition goes back t o KURATOWSKI 1921 (see also AJDUKIEWICZ). The s e t a i s said t o be the f i r s t term and b the second term of the couple. Clearly two couples are equal i f f they have the same f i r s t and same second terms. The Cartesian product a x b i s the s e t of couples (x,y) where x belongs t o a and y belongs t o b . FUNCTION, DOMAIN, RANGE A function or mapping from a onto b i s a subset f of a x b such t h a t every element x of a appears as f i r s t term in exactly one couple (x.y) belonging t o f and every element y of b appears as a second term in a t l e a s t one couple belonging t o f The s e t a = Dom f i s called the domain, the s e t b = Rng f i s the range of f . For each element x of a , the second term y of the unique couple (x,y) having f i r s t term x i s denoted y = f ( x ) or y = fx and is called the value of f on x , or the image of x under f . For every superset c 7 Rng f we say t h a t f i s a function from a c THE TRANSFORMATION f " AN0 ITS INVERSE I f uc_ Dom f , we denote by f " ( u ) the s e t of elements fx where x u The function t h u s denoted f " i s a function on the s e t of subsets of Dom f and i s called the transformation associated with f . This transformation f " ( v ) . However preserves inclusion, in the sense t h a t u c_ v implies f " ( u ) s t r i c t inclusion i s not preserved. I f v C Rng f , then the inverse image of v by f , denoted ( f - 1) " ( v ) , i s the s e t of elements x such t h a t fx belongs t o v . So we define the inverse I t preserves s t r i c t transformation associated w i t h f , denoted (f-')" inclusion as well as inclusion. INJECTION, INVERSE FUNCTION, PERMUTATION, TRANSPOSITION The function f i s said t o be a n injection o r injective function i f f x # x ' implies fx # f x ' f o r a l l x , x ' in Dom f . I f a i s the domain, b the range,
.
-
.
into
.
c
.
3
Chapter 1
t h e n an i n j e c t i o n i s s a i d t o be a b i j e c t i o n f r o m The i n v e r s e o f an i n j e c t i o n
a
i s denoted by
f-l
i n j e c t i v e , the transformation associated w i t h
f-l
f
onto
, so
.
b
t h a t i n t h e case o f
coincides w i t h
(f-l)"
i s f i n i t e , then
Rng f
f
(the
l a t t e r e x i s t s f o r every f u n c t i o n f i n i t e . For
x
f ). i n j e c t i v e o r not, i f
a
i s a b i j e c t i o n from
, the transposition
a
and y
Oom f
is
i n j e c t i v e , t h e converse i s t r u e .
f
A permutation o f
of
,
f
Given a f u n c t i o n
(x,y),
a
onto
a
.
Given two elements
i s the permutation o f
a
and i s t h e i d e n t i t y on e v e r y o t h e r element o f
a
x, y
which interchanges
.
FIXED POINT LEMMA (KNASTER 1928, g e n e r a l i z e d by TARSKI 1955) Let
a
hx
of
be a s e t and
. Suppose
a
h that
a f u n c t i o n which takes each subset h
x
f o r e v e r y x, y 5 a . Then: (1) t h e r e e x i s t s e t s x a m a j o r i z e d by
of
xc
i s i n c r e a s i n g under i n c l u s i o n :
a
t o a subset
implies that
y
hx C- hy example (2)
x
if
x
h
,in
t h e sense t h a t
x
5 hx
;for
can be t a k e n as t h e empty s e t ; i s majorizedA
h
, then
hx
&
i s majorized
u o f a l l majorized subsets-satisfies
(3) the union
h ;
hu = u
.
1.3. RESTRICTION, EXTENSION, COMPOSITION Given a f u n c t i o n & t
of
to
f
f
b
w i t h domain
a
which t h e f i r s t t e r m belongs t o Putting
, we
g = f/b
say t h a t
and a subset
, the
, denoted f!b
b
f
b
of
and
g
, with
Dom(g,f)
, we
c a l l the r e s t r i c of
f
.
i s an e x t e n s i o n o f
g
We l e a v e i t t o t h e r e a d e r t o d e f i n e t h e c o m p o s i t i o n g,f f
a
s e t o f ordered p a i r s belonging t o t o t h e domain
.
a
o f the functions
= Dom g n Rng f
EQUIPOTENCE, SUBPOTENCE
A s e t b i s s a i d t o be e q u i p o t e n t w i t h onto b . A set
b
i s s a i d t o be subpotent w i t h
A set b i s s t r i c t l y w i t h a but a i s n o t subpotent w i t h valent t o saying t h a t b i s subpotent Every s e t e q u i p o t e n t w i t h a f i n i t e s e t s t r i c t l y subpotent w i t h every i n f i n i t e one b e i n g s u b p o t e n t w i t h t h e o t h e r . I f potent w i t h
product
b
a x b
I
i s f i n i t e . If
a
i f f there exists a b i j e c t i o n o f
a
f f t h e r e e x i s t s a subset o f
a
a
a
equi-
subp k t w i t h a i f f b i s subpotent By theorem 1.4 below, t h i s i s e q u i b
.
but not equipotent w i t h
a
.
i s i t s e l f f i n i t e . Every f i n i t e s e t i s s e t . Two f i n i t e s e t s a r e always comparable, a
and
b
are f i n i t e , then the Cartesian
i s f i n i t e , t h e n so i s t h e power s e t p ( a )
.
A f i n i t e s e t i s n o t e q u i p o t e n t w i t h any o f i t s p r o p e r subsets. E q u i v a l e n t l y , i f a i s f i n i t e , then every i n j e c t i o n o f a i n t o a i s a permutation o f a 0 Suppose t h a t
a
f
i s an i n j e c t i o n s a t i s f y i n g
which i s minimal among a l l subsets
x
of
f"(a) a
c a
. Take
satisfying
.
a subset
fo(x)c x
m
. Then
of
THEORY OF RELATIONS
4
c
f"(f"(m))
f"(m)
f : t h i s c o n t r a d i c t s the m i n i m a l i t y . 0
by t h e i n j e c t i v i t y o f
DEDEKIND-FINITE SET A set
a
i s s a i d t o be D e d e k i n d - f i n i t e i f f
a
i s n o t equipotent w i t h any proper
subset o f i t s e l f (DEDEKIND 1888); i t i s D e d e k i n d - i n f i n i t e i n the opposite case. The converse w i l l be proved i n 2.6 by using
Every f i n i t e s e t i s Dedekind-finite.
the denumerable subset axiom (weaker than t h e axiom o f choice). 1.4. BERNSTEIN-SCHRODER THEOREM
Given sets
then a
a
and
b
,if
a
i s equipotent w i t h
i s subpotent w i t h
. The
b
b and
b
subpotent w i t h
a
,
f o l l o w i n g p r o o f i s i n FRAENKEL 1953 and a t t r i -
buted t o WHITAKER. It does n o t use the n o t i o n o f i n t e g e r , which i s used i n the c l a s s i c a l " m i r r o r p r o o f " ; see a1 so SUPPES 1960. 0 Let
into to
be an i n j e c t i o n from
f
a
. It suffices
a-o
a
into
t o f i n d a subset
by t h e f u n c t i o n
,or
go
b , and g be an i n j e c t i o n from b u of
a
equivalently
consider t h e f u n c t i o n which takes each subset
such t h a t u = a
-
of
a
x
b-f"(u)
into
a
i s sent
. To
g"(b-f"(u))
-
do t h i s ,
.
g"(b-f"(x))
This f u n c t i o n i s i n c r e a s i n g under i n c l u s i o n . By t h e f i x e d p o i n t lemma, the union u of all
x
such t h a t
x
5
a
-
g"(b-f"(x))
s a t i s f i e s the above.
1.5. CANTOR'S LEMMA Let
a
be a set. There i s no f u n c t i o n , i n j e c t i v e o r otherwise, w i t h domain
9 (a)
and range
( s e t o f subsets o f
a
a ).
CANTOR'S THEOREM (1) Every s e t
(2) I f a
a
9 (a) .
i s s t r i c t l y subpotent w i t h
i s non-empty, then every s e t o f m u t u a l l y d i s j o i n t subsets o f
s t r i c t l y subpotent w i t h
a
9 (a) .
1.6. EXPONENTIAL Given sets from
a
a
into
and b
However aO = 0 subsets o f
a
b
, the
. Thus
exponential o r power
i s the set o f functions
b . I n p a r t i c u l a r '0 = 1 . . For each s e t a , t h e s e t 7 ( a ) , where 2 = { O , l ) ) .
'b = {O) = 1 f o r each
f o r each non-empty s e t i s equipotent w i t h
a2
We have the f o l l o w i n g equipotences. For potent w i t h t h e Cartesian product w i t h t h e product
ab
(Ca)x(cb)
a
b
and
. Finally
c
. The
(ba)x(ca) C(ba)
disjoint, set
'(a
(b"c)a
x b)
i s equipotent w i t h
of
i s equi-
i s equipotent (bxc)a
.
1.7. CHOICE SET AND CHOICE FUNCTION Let
a
be a s e t o f non-empty m u t u a l l y d i s j o i n t s e t s
a s e t whose i n t e r s e c t i o n w i t h each element
x
of
a
x
.A
choice s e t f o r
i s a singleton.
a
is
5
Chapter 1
If a Let
i s f i n i t e , t h e r e i s always a choice s e t f o r a
be a s e t o f non-empty sets
which t o every element
x
of
a
.
x
a
( p r o o f by i n d u c t i o n ) .
A choice f u n c t i o n f o r
associates an element
f i n i t e , then there i s a choice f u n c t i o n f o r
fx
a
i s a function
of
.
x
a
If
is
.
a
AXIOM OF CHOICE (ZERMELO 1908) Every set, even i n f i n i t e , o f non-empty mutually d i s j o i n t sets admits a choice set. .____ E q u i v a l e n t l y every s e t o f non-empty sets admits a choice f u n c t i o n . An immediate consequence o f the axiom o f choice i s t h e f o l l o w i n g . Given a f u n c t i o n f
, injective
o r otherwise,
given a non-empty s e t subpotent w i t h
a
Rng f
, every
a
i s subpotent w i t h
.
Dom f
I n o t h e r words,
s e t o f mutually d i s j o i n t subsets o f
a
is
.
Problem. Does the preceding statement imply t h e axiom o f choice (problem mentioned i n RUBIN 1963 p. 5 note 1). A seemingly weaker consequence o f the axiom o f choice i s the assertion that
Dom f
i s never s t r i c t l y subpotent w i t h
. This
Rng f
does
5
1, 5 2 and 2.4; see ch.10 exerc. 2, where a FRAENKEL-MOSTOWSKI model i s constructed w i t h Dom f s t r i c t l y subpotent w i t h Rng f , a r e s u l t which i s t r a n s f e r a b l e t o ZF v i a the
n o t f o l l o w from ZF alone
i.e.
from t h e axioms mentioned i n
theorem o f JECH-SOCHOR (observation due t o HODGES).
1.8. GENERALIZED CARTESIAN PRODUCT Let a
a
be a non-empty s e t whose elements a r e non-empty. The Cartesian product o f
i s t h e s e t o f choice f u n c t i o n s which, t o each element
an element o f product
. If
ai
b x c
o f 1.2.
a
reduces t o t h e p a i r \ b , c ) , a
If
ai
of
a
associate
we have again the Cartesian
i s i n f i n i t e , i t f o l l o w s from t h e axiom o f choice
t h a t t h e Cartesian product o f
a
i s non-empty.
KONIG'S THEOREM
Let
I be a non-empty s e t o f elements
associated a p a i r o f s e t s
ai,
bi
with
i ( c a l l e d i n d i c e s ) , t o each o f which i s ai s t r i c t l y subpotent w i t h bi .
Then t h e union o f t h e product o f t h e
bi
ai ( i 6 I ) i s s t r i c t l y subpotent w i t h the Cartesian (axiom o f choice i s used).
h from u ai
0 Suppose there e x i s t s a b i j e c t i o n
the
bi
. For each
take i t s value
i and each x
(hx)(i)
. Thus we
of
ai
, take
onto Tr bi the f u n c t i o n
d e f i n e a f u n c t i o n from
ai
axiom o f choice, t h e range o f t h i s f u n c t i o n i s subpotent w i t h subpotent w i t h value o f associates
hx
bi on
ui
. Hence
t h e r e i s an element
i f o r any i s not i n
x
in
h " ( u ai)
t o see t h a t the union o f t h e
ai
ai
. The
ui
of
bi
, the
product o f
hx E 17bi
and
. By
into
bi
ai
, thus
the
strictly
which i s not the
choice f u n c t i o n which t o each
i
: c o n t r a d i c t i o n . We leave i t t o t h e reader
i s subpotent w i t h t h e product o f t h e
bi
.0
THEORY OF RELATIONS
6
Problem. Can the above theorem be proved from only the axioms of ZF in the case where the s e t I of indices i s f i n i t e w i t h cardinality greater than o r equal t o 2. Note th a t i f , in addition t o I being f i n i t e , we have f o r each index i tha t T ( a i ) i s subpotent with bi , then by CANTOR'S lemma 1 .5 , the axioms o f ZF suffice f o r the proof. For Card I = 2 , KONIG's theorem i s a consequence of ZF plus the axiom which a s s e r t s t h a t Dom f i s never s t r i c t l y subpotent with Rng f , or of ZF plus the apparently weaker axiom which as s er t s t h a t if a (resp. a ' ) i s s t r i c t l y b , b ' d i s j o i n t , then a u a ' i s s t r i c t l y subpotent with b (resp. b ' ) subpotent with b u b ' . § 2 - SECONDGROUP OF A X I O M S FOR ZF: FOUNDATION, SUBSTITUTION; O R D I N A L , INTEGER, COUNTABLE SET
INFINITY,
AXIOM OF FOUNDATION The axiom of foundation i s the statement t h a t every non-empty s e t a admits an element d i sj o i n t from a . I t follows t h at x $ x f o r any x Moreover f o r any XY Y i t i s impossible t h at x E y and Y E x , e t c . The axiom of foundation was introduced by ZERMELO 1930, inspired by a statement of von NEUMANN 1929. As t o i t s consistency, supposing t h a t a l l other axioms of ZF are consistent, see exercise 1. PREDECESSOR Given a s e t a , the successor a+l = a v j a } i s d i s t i n c t from a , since a $! a . Moreover i f a+l = b + l then a = b ; otherwise we would have a € b + l with a b , so a E b and similarly b E a , contradicting the axiom of foundation. Given a s e t c , the s e t whose successor i s c (which i s unique i f i t e x i s t s ) i s called the predecessor of c , denoted by c-1 . Finally, given a s e t a and i t s successor a t 1 , there i s no s e t x such t h a t a E x E a+l TRANSITIVE SET, TOTALLY ORDERED SET A s e t a i s t r a n si t i ve i f f , f o r every x, y , conditions y 6 x E a imply y E a If a i s t r a n si t i v e and non-empty, then every element o f a i s a proper subset of a . Also 0 E. a ( 0 i s the only element o f a which i s disjoint from a ) . Every union and intersection of t r an s i t i v e sets i s tra nsitive . I f a i s t r a n s i t i v e , then so i s a+l , A s e t a i s t o t a l l y ordered (by membership relation) i f f , f o r every x, y of a , e ith e r X E . y or y E x or x = y For example, a l l singletons are t o t a l l y ordered. However the singleton o f 1, i . e. 11) ={{O)): i s not tra nsitive . is The s e t { O , l , { l ) } i s t r an s i t i v e b u t n o t t o t a l l y ordered. The s e t {0,(1)) neither t r a n s i t i v e nor t o t a l l y ordered. Every intersection of t o t a l l y ordered s e t s i s t o t a l l y ordered. A union of such s et s i s not necessarily t o t a l l y ordered;
.
+
.
.
.
I
Chapter 1
however i f t h e s e t o f t o t a l l y o r d e r e d s e t s i s d i r e c t e d under i n c l u s i o n ( i . e . any two such s e t s a r e i n c l u d e d i n a t h i r d such s e t ) , t h e n t h e u n i o n i s t o t a l l y ordered. Finally i f
a
, then
i s t o t a l l y o r d e r e d by E
a + 1
so i s
.
2.1. ORDINAL
. For
An o r d i n a l i s a t r a n s i t i v e s e t which i s t o t a l l y o r d e r e d by E
example
,
0
1 = t o } , 2 = {0,1) . Every element o f an o r d i n a l i s an o r d i n a l . The successor s e t o f an o r d i n a l i s an o r d i n a l . The predecessor ( i f i t e x i s t s ) o f an o r d i n a l i s an o r d i n a l . The i n t e r s e c t i o n o f any s e t o f o r d i n a l s i s an o r d i n a l . An o r d i n a l
a
iff
a€ b
or
i s s a i d t o be l e s s t h a n o r equal t o an o r d i n a l
iff
a€ b
a = b ; an o r d i n a l
. Hence
If
, then
a,c b+l
a s b ac b
t o s t r i c t inclusion 0 By t r a n s i t i v i t y
a Eb
As from let
d
a& b ac b
. Let
a cb
b-a
. So e i t h e r so t h a t
d = a
. As
b
b
a< b )
(or
a
c_ b
i s equivalent
.
.
d E b-a
(yielding
u ca c
a E b
i s equivalent t o
be an element d i s j o i n t f r o m b-a
. Also
d c_ a
a € b ), o r
d ca
e b , t h i s d i s an o r d i n a l and d c b u E a-d
a
between o r d i n a l s . S i m i l a r l y
condition
. Hence
implies
Conversely, suppose t h a t
, denoted a .4( b , , denoted a < b ,
.
a = b+l
, the
b
b
> ( s t r i c t l y g r e a t e r than) a r e d e f i n e d .
or
fi
a
b
i s s t r i c t l y less than
< i s synonymous w i t h 6
( g r e a t e r t h a n o r equal t o ) and Given two o r d i n a l s
a
since
.
i s an o r d i n a l and
u
d
.
i s disjoint
I f t h e l a t t e r occurs, E b
and
d Eb
u E d o r d E u o r u = d . If U E d , t h i s contradicts I f d E u , t h e n s i n c e u E a , we have d E a which c o n t r a d i c t s I f u = d , t h e n d E a-d so d E a , again c o n t r a d i c t i n g d E b-a
,
we have e i t h e r
u E a-d d e b-a
.
.
TRICHOTOMY Given any two o r d i n a l s
a, b,
either
0 As we know, t h e i n t e r s e c t i o n
a n b = b case
or
a n b
so
a s b
a = b
a nb
b Ea
a€ b
i s s t r i c t l y included i n both or
a c b
and t h u s
a
.
a€ b
, so
that
a nb
and
a b
.
r\
. b = a
or
I n the f i r s t
A s i m i l a r conclusion i s
reached i n t h e second case. I n t h e t h i r d case, we have a 0b E b
a = b
i s an o r d i n a l . E i t h e r
.0
a
A
b
a
and
belongs t o i t s e l f , c o n t r a d i c t i n g t h e axiom o f
foundation. 0 Let
a
and
b
be two o r d i n a l s ; i f
b b a
then
b 3 a+l
or
b = a
.
We l e a v e i t t o t h e r e a d e r t o d e f i n e t h e maximum o r minimum o r d i n a l o f a s e t o f o r d i n a l s , denoted
mum:
take
b
Max, Min
belonging t o
.
Every non-empty s e t
a
and d i s j o i n t f r o m
a a
o f o r d i n a l s admits a m i n i -
.
More g e n e r a l l y we have t h e f o l l o w i n g scheme o f statements: g i v e n a c o n d i t i o n
‘8
which i s s a t i s f i e d by a t l e a s t one o r d i n a l , t h e r e i s a minimum o r d i n a l s a t i s f y i n g f Every t r a n s i t i v e s e t o f o r d i n a l s , e v e r y u n i o n o f a s e t o f o r d i n a l s i s an o r d i n a l .
8
THEORY OF RELATIONS
We leave i t t o t h e reader t o d e f i n e upper bound and lower bound o f a s e t o f o r d i nals. Given a s e t u o f o r d i n a l s , we denote the union o f u by Sup u It i s
.
If
i s an o r d i n a l and
o(
then
6 o( .
Sup u in
>
q
Given an o r d i n a l w i t h domain
.
o(
I n t h i s case
oc
i s t h e l e n g t h o f the sequence.
, the elements o r = o f
u
f o r which the f i r s t term called indices o f
EXTRACTED SEQUENCE
o( -SEQUENCE;
, an d - z e q u e x e , o r ordinal-indexed sequence, i s a f u n c t i o n
o(
Given a sequence
, or
u
u
are a l l ordered p a i r s
i i s an o r d i n a l s t r i c t l y l e s s than OC
u
i s indexed by
i
g(2), ... and a c o r r e s p o n d i n g se uence o f o r d i n a l s t ( l ) , $ ( 2 ) , each s t r i c t l y l e s s t h a n /3 , we have b x > f i 9 ( ' ) . $ ( 1 ) t $ ( 2 ) t ...
fig(*).
8
( p r o o f by i n d u c t i o n on
).
3.5. CANTOR NORMAL FORM Given
4 and f3 >, 2 , t h e r e e x i s t s a decomposition o f o( i n t o a f i n i t e sum o f f i r , 6 , w i t h c o e f f i c i e n t s $< /3 and exponents s t r i c t l y decreasing.
terms
Furthermore t h i s decomposition i s unique. I t i s c a l l e d t h e Cantor decomposition o f o( i n t o powers o f that
fi
=
0
o r Cantor normal f o r m o f 4 i n base f ,
6 are
u , the c o e f f i c i e n t s
.
I n t h e case
integers.
3.6. DECOMPOSABLE AND INDECOMPOSABLE ORDINAL An o r d i n a l o( i s c a l l e d decomposable i f f t h e r e e x i s t
o( = f l + 8
r'c.C w i t h
& < q and
; o t h e r w i s e o( i s c a l l e d indecomposable.
I f o( i s indecomposable,
then e v e r y sum o f two non-zero o r d i n a l s which i s equal t o o( has second t e r m equal t o d , and c o n v e r s e l y . A non-zero o r d i n a l o( i s indecomposable i f f
O(
i s a power o f
a .T h i s
follows
from t h e e x i s t e n c e and uniqueness o f t h e Cantor decomposition i n t o powers o f t o g e t h e r w i t h t h e a b s o r p t i o n statement (end o f 3.3).
54
-
4.1.
EQUIPOTENT WITH THE CONTINUUM
LC:
,
EQUIPOTENT W I T H THE CONTINUUM, C O N T I N U U M HYPOTHESIS, R E A L
A s e t i s s a i d t o be e q u i p o t e n t w i t h t h e continuum i f f i t i s e q u i p o t e n t w i t h ?(a), t h e power s e t o f t h e i n t e g e r s , o r e q u i v a l e n t l y w i t h " 2 on w t a k i n g values
0
s t r i c t l y subpotent w i t h Let
a
or
1
. By
9 ( IC) )
, the
s e t o f functions
CANTOR'S theorem 1.5, e v e r y c o u n t a b l e s e t i s
.
, b be two d i s j o i n t denumerable s e t s . By 1.6 we have t h a t a2 x b2 i s
equipotent w i t h
(a
" b ) 2 . Hence
t h e C a r t e s i a n p r o d u c t o f two s e t s each equi-
p o t e n t w i t h t h e continuum i s i t s e l f e q u i p o t e n t w i t h t h e continuum. The same r e s u l t h o l d s f o r t h e C a r t e s i a n p r o d u c t o f a c o u n t a b l e s e t w i t h a s e t which i s e q u i p o t e n t w i t h t h e continuum. Similarly
2)
i s equipotent w i t h
(w
w)2
.
Hence
e q u i p o t e n t w i t h t h e continuum, t h e n t h e s e t o f &-sequences
a
i s a set
w i t h values i n
a
i s a l s o e q u i p o t e n t w i t h t h e continuum.
4.2.
I f we s u b t r a c t an a r b i t r a r y denumerable subset
w i t h t h e continuum, t h e n t h e d i f f e r e n c e
c-a
a from a s e t c equipotent i s e q u i p o t e n t w i t h t h e continuum.
16
THEORY OF RELATIONS
This i s a special case o f t h e f o l l o w i n g p r o p o s i t i o n .
Let
be an i n f i n i t e s e t which i s equipotent w i t h the Cartesian product
a
c = y(a)
and l e t
. Then
t h e d i f f e r e n c e set, obtained by removinq from a
a r b i t r a r y subset which i s e q u i p o t e n t w i g 0
Since
a
i s equipotent w i t h
i s a l s o equipotent w i t h
cxc
which i s equipotent w i t h
a
f
range o f a b i j e c t i o n ordered p a i r
on
f x = (y,z)
2xa
,is
equipotent w i t h
by 1.6. Hence t h e d i f f e r e n c e o f
.
.
c
and a subset
Each element
o f elements
y, z
x of
of c
a
. Let
cxc
i s associated t o an
o f t h i s p a i r . The f u n c t i o n thus obtained has domain
cannot have range
c
?(a)
, by
and the
us associate t o each
t h e f i r s t term y
=
an
, t h e s e t c , which i s equipotent w i t h a2 ,
i s equipotent w i t h t h e d i f f e r e n c e o f a
c
,
2xa c
a
x
and
CANTOR'S lemma 1.5. Thus there e x i s t s an element
f o f an element o f a , f o r any z belonging t o c . Hence t h e d i f f e r e n c e o f cxc and f " ( a ) includes a subset which i s equipotent w i t h c , and so by BERNSTEIN-SCHRODER 1.4 i s equipotent u
of
with
c
c
f o r which
(u,z)
i s n o t the value by
.0
4.3. L e t
a
be a s e t equipotent w i t h the continuum. For every p a r t i t i o n o f
a
i n t o denumerably many subsets, one o f t h e subsets i s equipotent w i t h the continuum (uses t h e axiom of choice). 0
ai
Suppose on the c o n t r a r y t h a t there i s a p a r t i t i o n o f ( i i n t e g e r ) , and t h a t every
a
i n t o d i s j o i n t subsets
i s s t r i c t l y subpotent w i t h
ai theorem 1.8 (axiom o f choice), t h e union
a
o f the
ai
a
. Then
by KONIG's
i s s t r i c t l y subpotent
w i t h the Cartesian product o f an a - s e q u e n c e o f sets, each equipotent w i t h t h e continuum. But t h i s Cartesian product i s equipotent w i t h the continuum: contradiction. 0 4.4. CONTINUUM HYPOTHESIS, GENERALIZED CONTINUUM HYPOTHESIS The axiom c a l l e d continuum hypothesis asserts the non-existence o f a s e t which
o and y( a) ZF, and even o f ZF p l u s t h e axiom o f
i s s t r i c t l y intermediate, w i t h respect t o subpotence, between This axiom i s l o g i c a l l y independent o f
.
choice (COHEN 1963, see B i b l i o g r a p h y 1966). The axiom c a l l e d generalized continuum hypothesis asserts t h e non-existence o f a s e t s t r i c t l y intermediate, w i t h respect t o subpotence, between f o r every i n f i n i t e s e t
a
.
a
and p ( a )
When added t o t h e axioms o f ZF, t h i s i m p l i e s the
axiom o f choice (see ch.2 exerc. 1). 4.5. REAL We leave i t t o t h e reader t o r e d e f i n e p o s i t i v e and negative i n t e g e r , and then r e a l , as an ordered p a i r formed from an i n t e g e r which i s c a l l e d t h e i n t e g e r -
-
p a r t , and an i n f i n i t e s e t o f non-negative i n t e g e r s . The l a t t e r s e t w i l l be i d e n t i f i e d w i t h an W-sequence o f terms
ui
( i non-negative i n t e g e r ) w i t h
,
17
Chapter 1
ui = 0
or
1 according t o whether
i belongs t o t h e i n f i n i t e s e t o f i n t e g e r s
o r not. This sequence i s c a l l e d the b i n a r y expansion o f the r e a l , which always contains i n f i n i t e l y many occurrences o f zero. The notions o f r a t i o n a l r e a l and dyadic r e a l , i . e . r a t i o n a l whose denominator i s a power o f 2
, are assumed t o be f a m i l i a r , as w e l l as t h e denumerability o f the
set o f rationals. The s e t o f r e a l s i s equipotent w i t h the continuum: remove from the s e t o f a l l sets o f i n t e g e r s , the denumerable s e t o f f i n i t e sets o f i n t e g e r s , and use 4.2. We leave i t t o t h e reader t o d e f i n e the o r d e r i n g on t h e r e a l s : l e s s than o r equal
( 3 ) , greater
( 3 ) , and the r e l a t e d s t r i c t i n e q u a l i t i e s . Also the reader can d e f i n e the notions o f dense, c o f i n a l , c o i n i t i a l s e t o f r e a l s to
than o r equal t o
(an example being t h e r a t i o n a l s o r t h e dyadic r e a l s ) . The reader can define a closed, open, half-open i n t e r v a l o f r e a l s , an i n i t i a l , f i n a l i n t e r v a l , an _upper
bound and
lower bound o f a s e t o f r e a l s , t h e maximum, t h e minimum, a r e a l valued
sequence which i s s t r i c t l y ( o r otherwise) increasing, decreasing. Every s e t o f mutually d i s j o i n t i n t e r v a l s o f r e a l s which are n o t reduced t o s i n g l e t o n s i s countable: enumerate the r a t i o n a l s and associate t o each i n t e r v a l the f i r s t r a t i o n a l which belongs t o i t . Consequently, every s t r i c t l y i n c r e a s i n g ( o r s t r i c t l y decreasing) ordinal-indexed sequence o f r e a l s i s countable. 4.6.
DEDEKIND'S THEOREM
I f we p a r t i t i o n the r e a l s i n t o an i n i t i a l i n t e r v a l
final interval b
b
, both
non-empty, then e i t h e r
a
a
and i t s complement the
has a maximum element o r
has a minimum element.
Consequently, f o r any s e t
a
o f r e a l s , i f t h e r e e x i s t s an upper bound, then
there e x i s t s a l e a s t upper bound c a l l e d t h e supremum o f Analogous d e f i n i t i o n o f the infimum which i s denoted f o r every s e t
a
a
Inf a
and denoted
.
Sup a
.
I n o t h e r words,
o f reals, t h e r e e x i s t s a smallest i n t e r v a l ( w i t h respect t o
inclusion) including
a : the i n t e r v a l
(Inf a
, Sup a) which i s closed, open
o r half-open, i n i t i a l , f i n a l o r c o n t a i n i n g a l l t h e r e a l s , depending on the case. When u s e f u l , we w i l l use the
and p r o d u c t o f r e a l s , which the reader i s
presumed t o know.
4.7.
To see some i n i t i a l d i f f i c u l t i e s provided by the axiom o f choice, which
i n d i c a t e t h a t t h i s axiom i s n o t "obvious", note t h a t i t i s impossible i n ZF p l u s the axiom o f choice, t o define and prove uniqueness o f a choice f u n c t i o n which associates t o each non-empty s e t o f r e a l s one o f i t s elements. S i m i l a r l y i t i s impossible t o uniquely d e f i n e a choice s e t p i c k i n g one f u n c t i o n from each p a i r of r e a l f u n c t i o n s
h,-h
, where
f o r each r e a l
x
, t h e value o f -h
THEORY OF RELATIONS
18
on x
i s t h e a d d i t i v e inverse o f
h(x).
To o b t a i n a p r o o f o f uniqueness, com-
p l e t i n g the existence (which i s guaranteed by the axiom o f choice), i t i s necessary f o r example t o add t o ZF the axiom o f c o n s t r u c t i b i l i t y o f GODEL 1940.
§
5
- TRANSITIVECLOSURE,
HEREDITARILY F I N I T E SET, FUNDAMENTAL RANK,
CARD I NAL 5.1. TRANSITIVE CLOSURE For every s e t
a
, there
e x i s t t r a n s i t i v e supersets o f
a
, and
among these t h e r e
e x i s t s one which i s included i n a l l t h e others. T h i s s e t i s formed from t h e values o f a l l f i n i t e sequences xl, ... ,xh ( h i n t e g e r ) such t h a t x1 € a and xi+l E xi f o r each i (1 6 i < h) . We s h a l l c a l l t h i s s e t t h e t r a n s i t i v e closure of a For each non-empty s e t a , t h e t r a n s i t i v e closure o f a i s t h e union of a together w i t h t h e t r a n s i t i v e closures o f t h e elements o f a -
.
.
I f a s b then
(closure o f
a )
5 (closure o f
b )
,
HEREDITARILY FINITE SET A h e r e d i t a r i l y f i n i t e s e t i s a s e t whose t r a n s i t i v e closure i s f i n i t e . For i n s t a n ce, every f i n i t e t r a n s i t i v e s e t i s h e r e d i t a r i l y f i n i t e . I n p a r t i c u l a r every i n t e ger ( i . e . every f i n i t e o r d i n a l ) i s h e r e d i t a r i l y f i n i t e . The s i n g l e t o n o f 1 i s n o n - t r a n s i t i v e y e t h e r e d i t a r i l y f i n i t e . Every h e r e d i t a r i l y f i n i t e s e t i s f i n i t e , as i t i s included i n i t s t r a n s i t i v e closure which i s f i n i t e . Every element and every subset o f a h e r e d i t a r i l y f i n i t e s e t i s h e r e d i t a r i l y f i n i t e . Every f i n i t e s e t o f h e r e d i t a r i l y f i n i t e s e t s i s hered i t a r i l y f i n i t e . S i m i l a r l y f o r f i n i t e unions, f i n i t e Cartesian products, and t h e power s e t o f h e r e d i t a r i l y f i n i t e sets. A necessary and s u f f i c i e n t c o n d i t i o n f o r a s e t a t o be h e r e d i t a r i l y f i n i t e i s t h a t , f o r every f i n i t e sequence xO, ...,xh ( h i n t e g e r ) w i t h xo = a and xi+l
E xi
f o r each
i< h
, the
terms
xi
are f i n i t e .
5.2. FUNDAMENTAL RANK Let
a
be a s e t and
c
be t h e t r a n s i t i v e closure o f the s i n g l e t o n
ia 1 .
We say
t h a t t h e o r d i n a l o( i s t h e fundamental rank o f a , i f there e x i s t s a f u n c t i o n f w i t h domain c , t a k i n g o r d i n a l values 4 o( , such t h a t t h e i n i t i a l ordered p a i r (0,O) and t h e f i n a l ordered p a i r (a, cx ) belong t o f : so t h a t f ( 0 ) = 0 and f ( a ) = d ; and such t h a t i f u E c then the value f ( u ) i s t h e s m a l l e s t o r d i n a l s t r i c t l y g r e a t e r than f ( x ) f o r a l l x belonging t o u I t f o l l o w s from the axiom o f foundation t h a t every s e t has a unique fundamental
.
rank. -
.
Suppose t h a t a i s non-empty and Indeed, t h e empty s e t 0 has rank 0 t h a t every element o f a has a rank. Then by t h e preceding d e f i n i t i o n , a has rank equal t o t h e s m a l l e s t o r d i n a l which i s s t r i c t l y g r e a t e r than t h e ranks of
Chapter 1
19
a l l i t s elements. The e x i s t e n c e of rank r e s u l t s from the axiom of foundation i n the form of scheme 2.8. For every ordinal oc t h e fundamental rank i s g . 5.3. For every ordinal o( , t h e r e i s a s e t Vd o f a l l s e t s of ranks s t r i c t l y l e s s t t . 4 . Moreover , V has fundamental rank o( . 0 Obvious f o r 0 s i n c e Vo i s empty. I f t h i s i s t r u e f o r cl( , then i t i s t r u e Finally for q f o r @ + l with Vq +1 = s e t of elements and s u b s e t s of . ,V a l i m i t o r d i n a l , ,V i s t h e union of t h e Vi f o r i s t r i c t l y l e s s than o( . 0 Note t h a t f o r each ordinal & , t h e s e t Vatl - ,V of s e t s of rank o( i s nonempty, s i n c e , V and 4 belong t o t h i s s e t . For i an i n t e g e r , o r f i n i t e o r d i n a l , the s e t of sets of rank i i s f i n i t e . I t follows t h a t every i n f i n i t e s e t has rank a t l e a s t equal t o w . Note t h a t a s e t i s h e r e d i t a r i l y f i n i t e i f f i t s fundamental rank i s f i n i t e . The set of h e r e d i t a r i l y f i n i t e sets i s t h e i n t e r s e c t i o n of a l l sets which contain 0 and which, i f they contain x and y , a l s o contain x u { y ) a s an element.
5.4. CARDINAL, OR CARDINALITY Given a set a , consider sets equipotent w i t h a and among t h e s e , those of minimum fundamental rank. By t h e preceding, these form a non-empty s e t which we c a l l t h e c a r d i n a l o r c a r d i n a l i t y of a , denoted by Card a : d e f i n i t i o n from SCOTT 1955. T h u s every set has a c a r d i n a l , and two s e t s a r e equipotent i f f they have t h e same c a r d i n a l . Note t h a t every s e t a i s equipotent, not t o Card a , but t o an a r b i t r a r y element o f Card a T h i s i s only a minor inconvenience i n the d e f i n i t i o n .
.
Given two c a r d i n a l s a and b , t h e ordering of l e s s than o r equal t o , o r g r e a t e r than o r equal t o , means t h a t every s e t of cardinal a i s subpotent w i t h every s e t of cardinal b . Obvious d e f i n i t i o n of s t r i c t ordering; notations 6 , < .
5.5. CARDINAL SUM, CARDINAL PRODUCT AND EXPONENTIATION Let a and b be c a r d i n a l s ; the cardinal sum a + b i s defined as t h e cardinal e denote of the union of two d i s j o i n t sets of cardinal a , r e s p e c t i v e l y b W t h e cardinal sum by + (boldface) t o avoid confusion w i t h the ordinal sum + i n 3.1. Thus we can i d e n t i f y , i n 5 6 below, Card w w i t h W i t s e l f , and w r i t e a + 1 = W and y e t W + ~ > L S .To be rigorous, we should a l s o d i s t i n g u i s h between t h e ordering r e l a t i o n f o r c a r d i n a l s and f o r o r d i n a l s . In p r a c t i c e the context will always permit t h e d i s t i n c t i o n . Since cardinal m u l t i p l i c a t i o n and exponentiat i o n a r e denoted by a x b and a b ( n o t a t i o n s from 1 . 2 and 1.6), t h e r e w i l l be no confusion w i t h t h e operations of ordinal m u l t i p l i c a t i o n and exponentiation a.b and ba . In p a r t i c u l a r t h e cardinal notation “ i s not necessary: ~3~ w i l l be s u f f i c i e n t .
.
THEORY OF RELATIONS
20
+
The sum a b does n o t depend upon t h e choice o f d i s j o i n t s e t s o f c a r d i n a l and cardinal b Cardinal a d d i t i o n i s commutative and a s s o c i a t i v e . We have a + O = a . F i n a l l y a b a ' and b & b ' imply a + b , ( a'+b' .
a
.
The cardinal product o f a set o f cardinal
a x b i s defined as t h e c a r d i n a l o f t h e Cartesian product a with a s e t o f c a r d i n a l b . There w i l l be no inconvenience
i n using t h e same symbol f o r c a r d i n a l m u l t i p l i c a t i o n and f o r t h e Cartesian product o f two sets (see 1.2). The c a r d i n a l product does n o t depend upon the choice o f t h e sets o f cardinal a , resp. b Cardinal m u l t i p l i c a t i o n i s commutative, associat i v e , and d i s t r i b u t i v e over c a r d i n a l a d d i t i o n : ( a + b ) % c = ( a x c ) + ( b x c )
.
We have a % 0 = 0 and a x 1 = a
. Finally
.
aGa'
and
b, x2 . A
and t h e v a l u e
i s t h e c o m p o s i t i o n l a w o f t h e group. I n s t e a d o f x,y,z .
R
group i s
- when x1,x2,x3
A multirelation with base E i s a f i n i t e sequence R o f r e l a t i o n s R1, ...,Rh ( h i n t e g e r ) , each w i t h base E Each Ri (i= 1, ...,h ) i s c a l l e d a component o f the multirelation R We c a l l t h e arity o f R t h e sequence (nl, ...,nh) o f a r i t i e s o f t h e components R1, Rh We say t h e n t h a t t h e m u l t i r e l a t i o n R i s (nl, ..., n h ) - x . The l e n g t h h o f t h e sequence o f i n d i c e s can be zero: i n t h i s case, t h e m u l t i r e l a t i o n i s reduced t o i t s base E . I n s t e a d o f t h e n o t a t i o n R1, R2,R3 , o f t e n we s h a l l use R,S,T I n t h e case where h = 2 , we w i l l say
.
.
.
...,
.
n
.
THEORY OF RELATIONS
30
a b i r e l a t i o n ; f o r h = 3 a t r i r e l a t i o n , e t c . F i n a l l y , t h e base o f a m u l t i r e l a t i o n R s h a l l be denoted I R I
.
Example. An ordered group i s a (3,2)-ary
b i r e l a t i o n which i s formed o f t h e t e r n a r y
group r e l a t i o n and t h e b i n a r y o r d e r i n g r e l a t i o n . denumeraaccording t o whether i t s base i s f i n i t e , i n f i n i t e , countable, denumerable or continuum-equipotent. The c a r d i n a l o f t h e mu t i re1 a t i o n R i s the c a r d i n a l o f i t s base J R I
A r e l a t i o n o r m u l t i r e l a t i o n w i l l be c a l l e d f i n i t e , i n f i n i t e , countable o r continuum-equipotent,
.
Given two m u l t i r e l a t i o n s
E
w i t h common base
R, S
, we
c a l l t h e concatenation
o f R and S , denoted (R,S) , the sequence o f components o f R f o l l o w e d by the components o f S , i n which case f o r the l a t t e r t h e i n d i c e s a r e increased by the number of terms i n R . 1.1. n-ARY RESTRICTION, n-ARY EXTENSION R be an n-ary r e l a t i o n w i t h base
Let
E
, and
let
F be a subset o f
E
. We
c a l l t h e n - a 3 r e s t r i c t i o n o f R t o F , denoted by R/F , t h e n-ary r e l a t i o n t a k i n g t h e same value f o r each n - t u p l e w i t h values i n F . The n o t i o n o f r e s t r i c t i o n o f a f u n c t i o n i n ch.1 5 1.3, i s more general than t h a t o f n-ary r e s t r i c t i o n : t h e former would a l l o w one t o r e s t r i c t R t o an a r b i t r a r y subset o f t h e s e t 'E o f n-tuples w i t h values i n E , and n o t n e c e s s a r i l y t o a subset o f t h e form 'F w i t h F S E However i n p r a c t i c e , t h e context w i l l make t h e meaning o f t h e ad-
.
j e c t i v e "n-ary" obvious: we t a c i t l y assume t h i s . For t h e a r i t y 0, t h e r e s t r i c t i o n t o F o f t h e 0-ary r e l a t i o n
(F,+) ; s i m i l a r l y w i t h
-
Given a r e l a t i o n R w i t h base E
sion o f Let X
R
R, R '
3
to
E+
be two
and a superset
any r e l a t i o n w i t h base E+
6
n
, we
have
Given a m u l t i r e l a t i o n R = (R1,-..,Rh) we d e f i n e t h e r e s t r i c t i o n o f R t o
.
F
R/X = R ' / X
R, R '
.
of
E
-
of .
, we
E
. If
, then
9-
c a l l an
E
is
R
.
f o r every subset
R = R'
.
w i t h base E and a subset F o f E , by R/F , t o be t h e m u l t i r e l a R w i t h base E and a superset any m u l t i r e l a t i o n w i t h base E+ + where any sequence (R;,. ,Rh)
1, ... ,h)
be two m u l t i r e l a t i o n s o f common a r i t y
I f f o r each subset X base E have R/X = R ' / X , then R = R '
-
E+
w i l l be
(E,+)
, denoted
Given a m u l t i r e l a t i o n t i o n (R1/F, ...,Rh/F) E+ o f E , we c a l l an extension o f R t o Ef whose r e s t r i c t i o n t o E i s R . E q u i v a l e n t l y , + each Ri i s an extension o f Ri t o E+ ( i = Let
.
whose r e s t r i c t i o n t o
n-ary r e l a t i o n s w i t h common base
E with cardinal
F
; t h i s remains v a l i d f o r empty
(nl,
. ...,nh)
E w i t h c a r d i n a l 6 Max(nl,
.
and w i t h common
...,nh) ,
Chapter 2
31
1.2. COMPATIBLE RELATIONS Two r e l a t i o n s ( o r m u l t i r e l a t i o n s ) w i t h t h e same a r i t y a r e s a i d t o be compatible i f f t h e y have t h e same r e s t r i c t i o n t o t h e i n t e r s e c t i o n o f t h e i r bases.
Let
6%
be a s e t o f m u t u a l l y c o m p a t i b l e r e l a t i o n s ( o r m u l t i r e l a t i o n s ) :
&, , based
( 1 ) t h e r e e x i s t s a common e x t e n s i o n o f t h e r e l a t i o n s i n
on t h e u n i o n
o f t h e i r bases; ( 2 ) l e t us denote by
t h e u n i o n o f t h e bases and by
E
n
t h e common a r i t y , o r
t h e maximum o f t h e common a r i t y ( f o r m u l t i r e l a t i o n s ) ; i f each of
E
n-element subset
i s covered by one o f t h e bases, t h e n t h e common e x t e n s i o n i s unique.
1.3. COHERENCE LEMMA Consider a s e t 9 o f s e t s F f o r each o f which we have a f i n i t e non-empty s e t UF o f m u l t i r e l a t i o n s based on F ( a l l o f t h e same a r i t y ) w i t h t h e f o l l o w i n g hypotheses: (1) 3 i s a d i r e c t e d system: i f F, F ' belong t o )3 , t h e n t h e r e e x i s t s an F" i n 3 with F"? F u F' ; ( 2 ) i f F, F ' belong t o 3 and F ' C F , t h e n e v e r y m u l t i r e l a t i o n b e l o n g i n g t o
UF , when r e s t r i c t e d t o F ' , y i e l d s an element o f U F , ; i n t h i s case, t h e r e e x i s t s a m u l t i r e l a t i o n R based on t h e u n i o n o f t h e s e t s F i n 3 , such t h a t f o r each F t h e r e s t r i c t i o n R/F belongs t o UF (uses t h e u l t r a f i l t e r axiom; ZF s u f f i c e s ift h e F a r e f i n i t e and t h e i r u n i o n c o u n t a b l e ) . 0 Denote by E t h e u n i o n o f t h e F i n 3 To each F a s s o c i a t e t h e s e t VF o f e x t e n s i o n s t o E o f m u l t i r e l a t i o n s b e l o n g i n g t o UF . The supersets o f t h e VF c o n s t i t u t e a f i l t e r on t h e s e t o f m u l t i r e l a t i o n s based on E w i t h t h e g i v e n a r i t y . Indeed i f F, F ' belong t o 3 , t h e n t h e r e e x i s t s i n 3 an F " ? FuF' ; hence VFn V F 8 i s a s u p e r s e t o f VF,, . Take an u l t r a f i l t e r e x t e n d i n g t h i s f i l t e r .
.
For each F o f '3 , p a r t i t i o n t h e m u l t i r e l a t i o n s i n VF
i n t o a f i n i t e number
F o f these m u l t i r e l a t i o n s .
o f classes, each c l a s s d e f i n e d by t h e r e s t r i c t i o n t o
One and o n l y one o f t h e s e c l a s s e s i s an element o f o u r u l t r a f i l t e r : denote by
RF
t h e c o r r e s p o n d i n g r e s t r i c t i o n , so t h a t
RF
belongs t o
UF
. Hence
the
RF
a r e m u t u a l l y c o m p a t i b l e i n t h e sense o f 1.2 above: t h e e x i s t e n c e o f t h e m u l t i relation
R
stated i n our proposition follows. If
a r e f i n i t e subsets o f
E
, then
i s c o u n t a b l e and t h e
E
F
t h e u l t r a f i l t e r becomes s u p e r f l u o u s , so t h a t t h e
axioms o f ZF a r e s u f f i c i e n t . 0
1.4. The coherence lemma i m p l i e s , and hence i s e q u i v a l e n t t o t h e u l t r a f i l t e r axiom. 0
Let
e
be a s e t ,
p(e)
be t h e s e t o f subsets o f
e
, and 'bQ
a f i l t e r on
e
F be a f i n i t e s e t o f subsets o f e which i s c l o s e d w i t h r e s p e c t t o union, i n t e r s e c t i o n and t a k i n g complements ( i n e ) . To each F a s s o c i a t e t h e s e t UF o f unary r e l a t i o n s X w i t h base F which s a t i s f y t h e f o l l o w i n g c o n d i t i o n s : Let
.
THEORY OF RELATIONS
32
, i f a E 8 t h e n t h e v a l u e X(a) = + ; i f e-a F , we have o p p o s i t e values X(e-a) # X(a) ;
f o r each
a& F
f o r each
a E
i f a, b E F so a n b e F and X(a) = X(b) = + , t h e n i f a, b E F and a c b and X(a) = + , t h e n X(b) = The s e t
. The
F
i s non-empty f o r each
UF
set o f the
2 then
E
+
X(anb) =
.
+
F
X(a) = -;
;
forms a d i r e c t e d system,
so we can a p p l y t h e coherence lemma. Consequently t h e r e e x i s t s a unary r e l a t i o n
based on T ( e )
whose r e s t r i c t i o n t o each
F
belongs t o
. The
UF
subsets o f
which g i v e t h e value (+) t o t h i s unary r e l a t i o n c o n s t i t u t e an u l t r a f i l t e r on which i s f i n e r t h a n
w.0
e e
1.5. A v a r i a n t o f t h e coherence lemma i s g i v e n by RADO 1949. Consider a s e t o f
f i n i t e mutually d i s j o i n t sets consider a choice f u n c t i o n fI(a)
of
the J
a
, and
a
of
I
. Then
, and
a
fI
there e x i s t s a choice f u n c t i o n
I
f o r each f i n i t e s e t
with
f/I
f o r each f i n i t e s e t
which a s s o c i a t e s t o each o f the
equal t o t h e r e s t r i c t i o n
of
a
,
I an element
whose domain i s t h e s e t o f
f
, there
a
I o f sets
a
e x i s t s a f i n i t e superset
.
f,/I
The preceding RADO's lemma p l u s t h e axiom o f c h o i c e f o r f i n i t e s e t s i s e q u i v a l e n t t o t h e coherence lemma (BENEJAM 1970). 1.6. AXIOM OF DEPENDENT CHOICE Let
E
, such
satisfying
R(x,y)
be a b i n a r y r e l a t i o n w i t h base
R
e x i s t s a t l e a s t one
y
of
E
c h o i c e a s s e r t s t h a t , g i v e n such an ai
of
satisfying
E
R(ai,ai+l)
R =
+
, there
t h a t f o r each =
+ . The
x
of
E
there
axiom o f dependent
e x i s t s an w - s e q u e n c e o f elements
f o r each i n t e g e r
i (MOSTOWSKI 1948).
The axiom o f dependent c h o i c e o b v i o u s l y f o l l o w s f r o m t h e axiom o f c h o i c e . I t i s proved t h a t t h e dependent c h o i c e i s s t r i c t l y weaker t h a n t h e axiom o f
c h o i c e : see f o r i n s t a n c e JECH 1973 p. 122 and f o l l o w i n g . The c o u n t a b l e axiom o f choice, s t a t e d i n ch.1
5
2.5,
f o l l o w s from t h e axiom o f
dependent choice. 0 S t a r t f r o m an
take
R
R(x,y) =
w -sequence o f non-empty m u t u a l l y d i s j o i n t s e t s ai
t o be t h e b i n a r y r e l a t i o n based on t h e u n i o n o f t h e
+
i f f t h e r e e x i s t s an
i with
x
E
ai
ai
and y e ai+l
(ii n t e g e r ) by
, defined
.0
The c o u n t a b l e axiom o f c h o i c e i s s t r i c t l y weaker t h a n t h e axiom o f dependent c h o i c e : see JECH 1973 p. 119 and f o l l o w i n g . F i n a l l y , from t h e axiom o f dependent choice, assumed t o be c o n s i s t e n t , one cannot deduce t h e axiom o f c h o i c e f o r f i n i t e s e t s , s t a t e d i n ch.1
5
2.10.
The p r o o f i s
due t o MOSTOWSKI 1948 w i t h o u t t h e axiom o f f o u n d a t i o n , and t o FEFERMAN 1965 w i t h foundation.
,
Chapter 2
33
1.7. NEGATION, CONJUNCTION, DISJUNCTION Given a r e l a t i o n
, i t s negation
R
i s t h e r e l a t i o n w i t h same base and a r i t y ,
R
1
always t a k i n g t h e o p p o s i t e v a l u e . Given conjunction
R
disjunction
R v S
A
takes t h e v a l u e (+) i f f
S
w i t h t h e same base and a r i t y , t h e
R, S
and
R
t a k e t h e value ( + ) . The
S
t a k e s t h e v a l u e (+) i f f e i t h e r
or
S
, denoted
by
R
t a k e s t h e v a l u e (+).
OF A BINARY RELATION, RETRO-ORDINAL
CONVERSE
Given a b i n a r y r e l a t i o n
R
, the
converse o f
R
,i s
R-
the r e l a t i o n
f o r e v e r y x, y . I n p a r t i c u l a r w i t h t h e same base, such t h a t R-(x,y) = R(y,x) we c o n s i d e r an o r d i n a l o( as t h e b i n a r y r e l a t i o n based on t h e s e t a l r e a d y denot e d by
o(
(the set o f ordinals
QUASI-ORDERING,
P A R T I A L ORDERING,
CHAIN,
WELL-ORDERABLE SET,
WELL-ORDERING,
ORDERING,
) , and t a k i n g t h e value (+) i f f
o(
x E y
or
w i l l be c a l l e d
x ay ) . The converse r e l a t i o n o(
x = y (denoted a l r e a d y b y a retro-ordinal.
2 -
, x (mod A) o r
x < y (mod A) A(x,y)
=
or
x
y
follows
s t r i c t l y less
+ and A(y,x)
=
y
if
x
6y
l y (mod A)
or y
6x
. x
t h e base, t h e e q u i v a l e n c e c l a s s of
such
that
A(x,y)
= A(y,x)
=
+
;
; otherwise
An e q u i v a l e n c e r e l a t i o n i s a symmetric q u a s i - o r d e r i n g . Given a element x
-
; a l s o " s m a l l e r " i s synonymous w i t h " l e s s
i s comparable w i t h
i s incomparable w i t h
also write
write
x, y
precedes or i s l e s s t h a n o r
x
; i n o t h e r words
s t r i c t l y greater than
than". We say t h a t x
and y$
=
x
i s a q u a s i - o r d e r i n g and or
(mod A)
i s the s e t o f those y
of
.
A p a r t i a l o r d e r i n g i s an a n t i s y m m e t r i c q u a s i - o r d e r i n g ( n o t i o n assumed t o be
known). We a l r e a d y have t h e example o f i n c l u s i o n .
A
Given a q u a s i - o r d e r i n g
, the
e q u i v a l e n c e r e l a t i o n generated by
A , i s the
r e l a t i o n w i t h t h e same base, t a k i n g t h e v a l u e (+) i f f x < y and y 4 x (mod A ) . F o r each element x , t h e e q u i v a l e n c e c l a i s o f x (mod A) i s t h e c l a s s o f x modulo t h e e q u i v a l e n c e r e l a t i o n generated by
A
.
Take as a new base t h e s e t o f e q u i v a l e n c e c l a s s e s , and w r i t e
.
( e q u i v . c l a s s o f x)
We t h u s o b t a i n a p a r t i a l o r d e r i n g ( e q u i v . c l a s s o f y ) , i f x \ < y (mod A) c a l l e d t h e p a r t i a l o r d e r i n g generated by t h e q u a s i - o r d e r i n g A
4
Let
A
be a p a r t i a l o r d e r i n g ,
n o t i o n of maximum o f minimum, denoted
D (mod A)
Min D
. Recall
D
a subset o f t h e base
IA I
. . We
assume t h a t t h e
, denoted Max D , i s known. S i m i l a r l y f o r t h e t h a t an element i s maximal i n
D (mod A ) ,
i f it
34
THEORY OF RELATIONS
belongs t o D and there i s no element of D which s t r i c t l y follows i t . Analogous notion of a minimal element. The maximum, i f i t e x i s t s , i s maximal, b u t the converse i s f a l se . Similarly f o r the minimum. These notions extend in an obvious manner t o a quasi-ordering. Here there can e x i s t several maximums and several minimums, which are equivalent t o each other in the sense of the qenerated equivalence relation. The reader i s assumed t o know the notion of uoper bound of a s e t D (mod A ) as well as t h a t of lower bound. The supremum of D , denoted by Sup D , i f i t e x i s t s , i s the minimum in the s e t of upper bounds. Hence x > r Sup D i s equivalent t o x greater than or equal t o every element in D . If Sup D belongs t o D , then i t i s the maximum. Analogous definition of the infimum, denoted by Inf D These notions appeared already in ch.1 5 2.1 f o r ordinals, in ch.1 5 4.5 and 4.6 f o r reals.
-
.
INTERVAL, INITIAL AND FINAL INTERVAL The reader i s assumed t o be familiar with the notion of an element z between x and y (mod A) , or z intermediate between x and y , as well as t h a t of an element s t r i c t l y intermediate. An interval of A i s a subset of the base which i s closed with respect t o the notion of intermediate (mod A ) . An i n i t i a l interval o r i n i t i a l segment of A i s a subset closed with respect t o " l e s s than" . A final interval i s a subset closed with respect t o "greater t h a n "
.
2 . 1 . Let A be a p a r t i al ordering. Then every subset of the base I A 1 without a minimal element i s i n f i n i t e . Similarly for a subset without a maximal element. 0 To each element x of the subset 0 under consideration, associate the s e t Dx of elements of D which are less t h a n or equal t o x (mod A ) . None of the Dx i s minimal under inclusion (see ch.1 5 1.1, definition of a f i n i t e s e t ) . 0 2 . 2 . AMALGAMATION LEMMA
Let A , B be two p a r t i al orderings having the same re stric tion t o the intersection of the bases. Then there e x i s t s a partial ordering which i s an extension of b o t h A .-and B , based on the union of the bases. OWrite x G y when x , y G l A l and x s y ( m o d A ) , o r w h e n w e h a v e t h e s a m e condition for B , or when x belongs t o I A l , y belongs t o I B I and there (mod A) and t $ y (mod B ) , e x is t s an element t i n the intersection with x or when we have the same condition when interchanging A and B . Finally write x\y in the other cases. 0
-
st
2.3. CHAIN, ORDERABLE SET A chain, or total orderinq, i s a p ar t i al ordering whose elements are mutually
Chapter 2
35
comparable. For example, we shall denote by Z the chain of the positive and negative integers, and by Q the chain of the rationals. The previous amalgamation lemma 2 . 2 extends t o the case of two chains, the common extension i t s e l f being a chain. However, t h i s lemma does not extend t o t r e e s , defined i n ch.4 5 6. 0 Take a t r e e on a,b,c,d with a,b,c mutually incomparable, d < a , d c b and d I c ; and another t r e e on a,b,c,e with e < b , e < c and e l a . Then e i t h e r d < e < c or e Q d < a : contradiction. 0 W e say t h a t a s e t E i s orderable i f f there e x i s t s a chain based on E . Using only the axioms o f ZF, every f i n i t e s e t i s orderable (induction: see ch.1 5 1.1). ORDERING AXIOM The ordering axiom a s s e r t s t h a t every s e t i s orderable. I t follows from the ultraf i l t e r axiom, or equivalently from the coherence lemma 1.3. 0 Let E be a s e t ; t o each f i n i t e subset F of E , associate the s e t U F of chains based on F . By 1.3 there e x i s t s a relation R based on E every of whose f i n i t e r e s t r i c t i o n i s a chain; thus R i s a chain. 0 The ordering axiom i s s t r i c t l y weaker t h a n the u f t r a f i l t e r axiom (JECH 1973 p.100). The ordering axiom implies the axiom of choice f o r f i n i t e s e t s (see ch.1 5 2.10). 0 Given a s e t of mutually d i s j o i n t f i n i t e s e t s , i t suffices t o take a chain A based on the union: t o each f i n i t e s e t we associate i t s minimum (mod A ) . 0 The axiom of choice f o r f i n i t e s e t s i s s t r i c t l y weaker than the ordering axiom: see LAUCHLI 1964 f o r ZF without foundation, completed f o r ZF by PINCUS 1972. The axiom of choice f o r f i n i t e s e t s does not follow from the axiom of deoendent choice: see 5 1 . 6 . Hence the ordering axiom does n o t follow from dependent choice. 2.4. WELL-FOUNDED PARTIAL ORDERING OR QUASI-ORDERING; WELL-ORDERING
W e say t h a t a partial ordering o r quasi-ordering i s well-founded i f f every non-empty subset of i t s base has a t l e a s t one minimal element. A well-founded chain, o r t o t a l ordering, i s called a well-ordering. Every f i n i t e p a r t i a l ordering i s well-founded. Every r e s t r i c t i o n of a well-founded p a r t i a l ordering i s well-founded. Given a p a r t i a l ordering A , the reader i s assumed t o know the notion of a sequence with values in A which i s incr?asing, decreasing, s t r i c t 1 2 or otherwise.
A s a t i s f i e s the following conditions: (1) there i s no s t r i c t l y decreasing (mod A) W-sequence; ( 2 ) every t o t a l l y ordered r e s t r i c t i o n of A i s well-founded, hence a well-ordering; equivalently every non-empty t o t a l l y ordered r e s t r i c t i o n of A has a minimum. Every well-founded p a r t i a l ordering
THEORY OF RELATIONS
36
Conversely, each of the conditions ( l ) , ( 2 ) implies, hence i s equivalent t o saying t h a t A i s well-founded. This uses the axiom of dependent choice, y e t ZF suffices i f A i s countable, or i f the base I A l i s well-orderable, in the sense below. In the general case, apply dependent choice t o the relation y < x (mod A ) .
2 . 5 . WELL-ORDERABLE SET We say t h a t E i s well-orderable i f f there e x i s t s a well-ordering based on E . For example any f i n i t e or denumerable s e t i s well-orderable. A s e t E i s well-orderable i f f there e x i s t s a choice function on the s e t < non-empty subsets of E 0 Let f be a choice function on non-empty subsets. Let a. = f(E) . Let u be a non-zero ordinal, Du the s e t of a l l a i ( i i u ) Let a u = f(E-DU) , as long as possible, thus reaching a s e t DU = E . 0 WELL-ORDERING AXIOM, TRICHOTOMY AXIOM I t follows t h a t the axiom of choice i s equivalent t o saying t h a t every s e t i s well-orderable. Or again t h a t every cardinal i s an aleph, or t h a t every i n f i n i t e cardinal has the form c . ) ~ ( o( ordinal index: see ch.1 5 6.1 t o 6 . 4 ) . The axiom of choice i s equivalent t o the trichotomy axiom which says t h a t , given any two cardinals a , b , e i t h e r a < b or a = b or a > b . 0 If every cardinal i s an aleph, then trichotomy holds. Conversely, given a s e t a and the Hartogs u of a (see ch.1 5 6.2), i f trichotomy holds then necessar i l y a i s subpotent t o o( , hence a i s well-orderable. 0
-
.
.
2.6. MAXIMAL CHAIN Let A be a partial ordering and C be a t o t a l l y ordered r e s t r i c t i o n of A . The chain C i s said t o be maximal (under inclusion, mod A) i f f every t o t a l l y ordered r e s t r i c t i o n of A extending C i s identical t o C . Let E be a s e t ; denote by X any well-ordering based on a subset of E Write
.
X6 X '
.
i f f X i s an i n i t i a l interval of X ' The well-founded p a r t i a l ordering thus defined on the s e t of X will be called the interval-orderinqon E . (1) A s e t E i s well-orderable i f f there e x i s t s a maximal chain which i s a r e s t r i c t i o n of the interval-ordering on E ( 2 ) Let A be a partial ordering and C be a t o t a l l y ordered r e s t r i c t i o n of A Let U be any chain which i s both a r e s t r i c t i o n of A and an extension of C Every function f which t o each U associates f(U) , a t o t a l l y ordered rest r i c t i o n of A and extension of U , has a fixed point V such t h a t f(V) = V 0 Index by ordinals a sequence of chains U i s t a r t i n g with Uo = C ; s e t Ui+l = f ( U i ) and, f o r i a l i m i t ordinal, l e t Ui be the common extension of U j ( j < i ) t o the union of t h e i r bases. 0
.
.
.
.
37
Chapter 2
2.7. MAXIMAL CHAIN AXIOM, OR HAUSDORFF-ZORN AXIOM T h i s axiom, g o i n g back t o HAUSDORFF 1914, t h e n t a k e n up by KURATOWSKI, MOORE and
A
t h e n ZORN, i s s t a t e d as f o l l o w s . Given a p a r t i a l o r d e r i n g which i s a r e s t r i c t i o n o f
and a c h a i n
C
e x i s t s a c h a i n which i s an e x t e n s i o n o f
C
.
and maximal (mod A) By 2.6.(2)
, there
A
above, t h e axiom o f c h o i c e i m p l i e s t h e maximal c h a i n axiom. By 2.6.(1),
the
By 2.5,
t h e maximal c h a i n axiom i m p l i e s t h a t every s e t i s w e l l - o r d e r a b l e . maximal c h a i n axiom i s t h e n e q u i v a l e n t t o t h e axiom o f c h o i c e . 2.8.
The u l t r a f i l t e r axiom f o l l o w s f r o m t h e axiom o f choice.
0 Consider t h e s e t o f f i l t e r s on a g i v e n s e t , w i t h t h e comparison o r d e r i n g
" f i n e r f i l t e r t h a n " . Take a maximal c h a i n e x t e n d i n g t h e c h a i n reduced t o a g i v e n
. The
filter
u l t r a f i l t e r g i v e n by t h e u n i o n o f t h e f i l t e r s b e l o n g i n g t o t h e
F.0
maximal c h a i n i s f i n e r t h a n
The u l t r a f i l t e r axiom i s s t r i c t l y weaker t h a n t h e axiom o f choice: HALPERN, LEVY 1971 p. 83-134. 2.9. FREE SUBSET, ANTICHAIN, MAXIMAL FREE SUBSET, MAXIMAL ANTICHAIN
,a
A
Given a p a r t i a l o r d e r i n g
subset o f i t s base i s c a l l e d
i t s elements a r e m u t u a l l y incomparable (mod A) f r e e subset
D
i s c a l l e d an a n t i c h a i n (mod A )
r e l a t i o n based on
D
. The
free (mod A)
restriction
. It reduces
A/D
iff
t o such a
t o the i d e n t i t y
.
A f r e e subset, and t h e c o r r e s p o n d i n g a n t i c h a i n , a r e c a l l e d maximal (under i n c l u -
A
s i o n ) i f f t h e r e i s no p r o p e r s u p e r s e t which i s f r e e . Given a p a r t i a l o r d e r i n g and a f r e e subset
D
, there
e x i s t s a maximal f r e e subset i n c l u d i n g
axiom o f choice; ZF s u f f i c e s i f
A
0 (uses
i s c o u n t a b l e ) : a p p l y t h e maximal c h a i n axiom
t o t h e i n c l u s i o n among f r e e subsets.
3 - ISOMORPHISM^
AUTOMORPHISM, H E I G H T OF A WELL-FOUNDED P A R T I A L ORDERING, SUM AND PRODUCT OF CHAINS, HOMOMORPHIC IMAGE §
Let
n
an
be a non-negative i n t e g e r ,
n - a r y r e l a t i o n w i t h base
We say t h a t
f
transforms o r
f
i s an isomorphism o f
(=
+
o r -)
and we s e t
f"(R)
= (El,+)
n-ary r e l a t i o n w i t h base
E' ; l e t
f
takes
into
onto
f o r a l l elements
A relation R' from
R
an
R
x1 or
R'
R
be a b i j e c t i o n f r o m
iff R ' ( f x l
,...,xn R = (E,-)
in
. This
E
, denoted
,...,fx,)
. For
and t h e n
i s s a i d t o be isomorphic w i t h
R onto R '
R'
R
R'
= R(xl
n = 0
E'
onto
E 5
and
E
f"(R)
R'
.
, or that
,..., xn)
, either
f"(R) = (El,-)
R = (E,+)
.
i f f t h e r e e x i s t s an isomorphism
c o n d i t i o n i s r e f l e x i v e , symmetric and t r a n s i t i v e ,
y i e l d i n g an e q u i v a l e n c e r e l a t i o n o n e v e r y s e t o f r e l a t i o n s o f a g i v e n a r i t y .
38
THEORY OF RELATIONS
ISOMORPHISM TYPE, ORDER TYPE Modeled after the definition of "cardinal" in ch.1 0 5.4, we consider the relations isomorphic with R , and among such, those whose base has minimum fundamental rank. These form a set, called the isomorphism type of R (the order type if R is a chain, or total ordering). Thus two relations are isomorphic iff they have the same type. AUTOMORPHISM, EMPTY FUNCTION Given a relation R with base E , a permutation f of E i s called an automorphism of R iff f i s an isomorphism from R onto R . The automorphisms of R form a group of permutations of E . We adopt the convention that the empty function, which is a bijection of the empty set onto itself, is also an automorphism of each relation with empty base. In particular it is an automorphism of the 0-ary relation with empty base and value ( + ) , denoted (O,+) , and also of (0,-). However (O,+) and (0,-) are not isomorphic. These definitions and conventions extend to mu tirelations. Given a multirelation R = (R1 ,..., Rh) with base E and R' = (Ri,. .,Rb) with base E ' , a bijection f from E onto E' transforms R into R' or is an isomorphism from R onto R' , denoted R' = f"(R) , iff for each i = 1, ...,h , the function f is an isomorphism of the component Ri onto the component R; . In other words fo(R1 ,...,Rh) = (fo(R1) ... .,fo(Rh)) . 3.1. (1) Let A be a well-ordering and f be an isomorphism from A onto a restriction of A . Then fx & x (mod A ) for each element x of A . ( 2 ) Given a well-ordering, its unique automorphism is the identity. Given two well-orderings A, B , there exists at most one isomorphism from A onto B . ( 3 ) Given a well-ordering A , no proper initial interval of A is isomorphic with A . In particular two isomorphic ordinals are identical.
-
3.2. HEIGHT IN A WELL-FOUNDED PARTIAL ORDERING
Let A be a well-founded partial ordering. To each element x of IAI , associate as follows an ordinal called the height of x (mod A) and denoted Ht x If x is a minimal element, let Ht x = 0 . Let o( be a non-zero ordinal; assume that each ordinal (o( has been associated to at least one element, but that there still remain elements in the base to which no height (q has been associated. Then associate the height o( to minimal elements among these. Given a well-founded partial ordering A , there i s a unique height associated to each element of the base I A l . Moreover, for each element x of height cr( and every ordinal /3 < d , there exists at least one element < x (mod A) with height /3.
.
Chapter 2 However, g i v e n
>
elements (mod A) 0
Let
x
x
/s
o f heights and
x (y
(mod A)
, then
, it
i s p o s s i b l e t h a t no element
x
fs. a
< e x
(POUZET 1979, unpublished).
For otherwise there would e x i s t a y belong t o
and
belongs t o
x
Now consider t h e case t h a t D
A/D
then A
o f l e a s t h e i g h t (mod B)
. This
minimality o f the height o f of
x (mod A)
.
B
i s c o f i n a l (mod A)
D
x'>
in
For each non-empty subset
.
.
X
i s a minimal element (mod A/X)
with y
X
x (mod 8)
(mod B/X)
x
/ o( . E i t h e r u has a predecessor v
Ni)
=
s t r i c t i n e q u a l i t y . Suppose t h a t
I
ment o f (u = v+l)
f o r which
,
bk
o f height
7.2. For e v e r y w e l l - f o u n d e d p a r t i a l o r d e r i n g A (uses axiom o f choice; ZF s u f f i c e s i f
A
, we
. Then
i (mod A/U)
i , w i t h b k < bi (mod A) ; and hence which c o n t r a d i c t s t h e m i n i m a l i t y o f i . F i n a l l y t h e h e i g h t o f exists a
have
k
bi
. Assume
o r \ b o (mod A)
A/U
Cof H t A
there
by 5.1,
is
,
/
t
, there
, whose
e x i s t s an i n t e g e r p+>/ p such t h a t , m-element subsets a r e p a r t i t i o n e d i n t o
p p-element subset, a l l of whose
colors, there e x i s t s a
t h e same c o l o r . I t i s c a l l e d a monochromatic Consider t h e case
m
classes i s i n f i n i t e " , a t least
l/k
=
2
.
m-element subsets have
p-element subset.
Repeat t h e p r o o f o f 1.1, b u t i n s t e a d o f "one of t h e
say "one o f t h e c l a s s e s i s l a r g e " , meaning t h a t i t c o n t a i n s
o f t h e o r i g i n a l elements. 'It s u f f i c e s t o t a k e
p+ = (kp).k(kp-l)
= p.kkp i n o r d e r t o o b t a i n , a f t e r k p - 1 o p e r a t i o n s , a sequence o f l e n g t h >/ kp o f elements v , analogous t o t h o s e i n 1.1. Thus we have a l a r g e c l a s s o f v , o f
cardinality
>/
p
.0
63
Chapter 3
1.4. RAMSEY NUMBERS p+
The l e a s t such
i n t h e p r e c e d i n g p r o p o s i t i o n i s c a l l e d a Ramsey number, denoted
(p): . T h i s l o o k s i k e t h e usual Erdos-Rado n o t a t i o n , where t h e arrow w i l l be r e placed by = o r < o r > , e t c . We g i v e s e v e r a l v a l u e s .
-
Case m = 1 have
.
I f each o f t h e
k.(p-1)
a t l e a s t one c l a s s w i t h principle" : i f
k
c l a s s e s had
k(p-1)
p
+
p (p);
k
pigeonholes, t h e n
= p
.
p = m : a p-element s e t i s necessary monochromatic, t h u s
.
(3); = 6 Consider t h e elements 1,2, (b,c)
or
+ 1 t o obtain
= k(p-1)
objects.
...,6
; p a r t i t i o n t h e edges
two c o l o r s . A t l e a s t one c o n t a i n s t h r e e edges or
(p):
1 objects are partitioned i n t o
k = 1 : a s i n g l e class, thus
Calculation o f 0
elements, t h e e n t i r e s e t would
elements. T h i s argument i s c a l l e d t h e "pigeonhole
a t l e a s t one o f t h e pigeonholes has
Case Case
p-1
elements. Hence i t s u f f i c e s t o t a k e
(l,a),
(1,2)
(l,b),
.
= m
(m):
to
(1,6)
.
(1,c)
into
Either
(a,b)
(c,a)
has t h e same c o l o r , o r t h e s e t h r e e edges have t h e o p p o s i t e 2 c o l o r : t h i s shows t h a t ( 3 ) 2 i 6 . 2 To see t h a t ( 3 ) 2 > 5 , t a k e t h e usual pentagon w i t h one c o l o r , and t h e s t a r r e d pentagon w i t h t h e o p p o s i t e c o l o r . 0 Calculation o f
0
(3);
= 17
Consider t h e elements
(GLEASON, GREENWOOD 1955). 1,2,.
..,17
and p a r t i t i o n t h e 16 edges
(1,2)
to
(1,17)
(l,al), ... , ( l , a 6 ) . I t remains t o p a r t i t i o n t h e edges (ai,a.) ( i , j = 1 t o 6) i n t o two c o l o r s : hence we J 2 2 f a l l back t o t h e case ( 3 ) 2 = 6 ; t h i s shows t h a t ( 3 ) 3 6 17 . i n t o t h r e e c o l o r s . A t l e a s t one c o n t a i n s 6 edges, say
The f o l l o w i n g counterexample shows t h a t
1+1 = 0
o f t h e i n t e g e r s 0 and 1 w i t h
2 (3)3
>
16
. Consider t h e
t h e r i n g o f p o l y n o m i a l s on t h i s f i e l d w i t h t h e i d e n t i t y (0 o r 1)
composed o f 16 elements
These elements a r e e x a c t l y element i s a power
xi
+
(0 o r 1).x
x4 = x + l
.
, and
This r i n g i s
+ ( 0 o r 1).x 2 + (0 o r l ) . x 3
.
2 ,. . .,x14 (we have x15 = 1). Every non-zero
O,l,x,x
(i= O , l ,
f i e l d composed
( t h e f i e l d o f t h e i n t e g e r s modulo 2)
..., 14) , and
has i n v e r s e
x15-i
. Hence
this ring
i s a f i e l d . P a r t i t i o n the p a i r s o f polynomials i n t o three colors, according t o whether t h e d i f f e r e n c e o f these two p o l y n o m i a l s i s a cube i s o f the form
x3'+'
or
x3'+'
.
x3" ( u = O , l ,
...,4)
or
It s u f f i c e s t o see t h a t t h e sum o f two non-zero
cubes i s n o t a cube. 0 1.5. L e t
E be a f i n i t e s e t ; p a r t i t i o n i t s m-element subsets i n t o k c o l o r s i n t e g e r s p1 ,..., pk 3 m , by (pl ,..., pk)m we denote
u1 ,..., uk . Given k the l e a s t cardinal o f
E
f o r which t h e r e e x i s t s e i t h e r a pl-element
subset
THEORY OF RELATIONS
64 with color tion
(p
t h e ramsey number Calculation o f 0
.
u l , ... , o r a pk-element subset w i t h c o l o r uk
,,...,pk)m
i s symmetric. Moreover, t a k i n g (p)!
(p,.
=
(3,412 = 9
. . ,p) m .
1
=
..*
The f u n c -
pk = p, we o b t a i n
=
. .
9
We show t h a t t h i s number i s
E i t h e r among t h e 8 edges
p
(1,2)
to
J o i n up t h e i n t e g e r s 1 t h r o u g h 9 by edges. (1,9)
t h e r e e x i s t 4 edges o f c o l o r ( + ) . T h i s
t h e n y i e l d s e i t h e r a 3-element s e t w i t h c o l o r (+) o r a 4-element s e t w i t h c o l o r ( - ) , O r t h e r e e x i s t 6 edges w i t h c o l o r ( - ) , which t h e n y i e l d s e i t h e r a 3-element subset
(+) o r a 4-element subset ( - ) . Or f i n a l l y none o f t h e p r e c e d i n g cases i s r e a l i z e d f o r any o f t h e p o i n t s
1 through
9
. Then
f r o m each p o i n t t h e r e emanate e x a c t l y
3 edges (+) and 5 edges ( - ) . B u t t h i s i s i m p o s s i b l e , s i n c e we would t h e n have 3.(9/2) = 27/2 edges ( + ) . 2
>
2 C a l c u l a t i o n o f ( 4 ) 2 = 18
.
We now show t h a t t o t h e edge
.
8 Take t h e i n t e g e r s 0 t o 7, and g i v e t h e c o l o r (+) i f f t h e a b s o l u t e v a l u e o f y - x i s 3, 4, or 5. 0
(3,4)
(x,y)
Take t h e i n t e g e r s 1 t o 18. Among t h e edges emanating f r o m 1, t h e r e a r e a t l e a s t
0
9 o f t h e same c o l o r which we d e s i g n a t e ( + ) . They j o i n 1 t o t h e i n t e g e r s d e s i g n a t e d
...,
al,. . . , ag . By t h e preceding, i n t h e s e t o f al, ag t h e r e e x i s t s e i t h e r a 3-element s e t w i t h c o l o r ( + ) , o r a 4-element s e t o f t h e o p p o s i t e c o l o r ( - ) . Hence t h i s Ramsey number i s a t most 18.
\
/ 2p + q i n a a r e a l l d i s j o i n t from b , and so a l l have t h e same c o e f f i c i e n t i n o u r combination. Moreover, i f we denote these p-element sets by y , these are the i n t h e incidence matrix, o n l y ones y i e l d i n g t h e value 1 i n p o s i t i o n (a,y) w h i l e t h e m a t r i x has t h e value 0 i n p o s i t i o n (a,b) It f o l l o w s t h a t t h e i r
.
c o e f f i c i e n t i s zero, hence each row which represents a p-element s e t d i s j o i n t from b has c o e f f i c i e n t zero. The problem i s thus answered n e g a t i v e l y f o r p = 1 , since i n t h i s case the p-element sets d i s t i n c t from b are d i s j o i n t w i t h b , hence t h e above assumed l i n e a r combination does n o t e x i s t . 2 , and consider a column al representing a (ptq)-element Assume t h a t p s e t which i n t e r s e c t s b i n a unique element. Then t h e rows y f o r which t h e m a t r i x has value 1 i n (a,,y) a r e those which represen! e i t h e r a p-element s e t d i s j o i n t from b , hence w i t h c o e f f i c i e n t zero, o r a p-element s e t i n t e r s e c t i n g b i n a s i n g l e p o i n t . By t h e preceding discussion, t h e l a t t e r have t h e same c o e f f i c i e n t i n the combination. Since the m a t r i x has the value 0 i n (al,b) , t h i s c o e f f i c i e n t i s zero. The problem i s thus answered n e g a t i v e l y f o r p = 2 , s i n c e i n t h i s case t h e p-element s e t s d i s t i n c t from b have a t most one element i n common w i t h
b
. I n the
general case, by i t e r a t i n g t h e preceding
argument, we prove t h a t a l l t h e c o e f f i c i e n t s are zero, .hence t h a t t h e above assumed l i n e a r combination does n o t e x i s t . The r e s u l t f o l l o w s imnediately i n t h e case o f E i n f i n i t e . F i n a l l y , f o r t h e cnnclusion concerning t h e e x t e n d i b i l i t y o f a non-zero determinant, assume on t h e c o n t r a r y t h a t t h e r e e x i s t s a non-zero determinant which i s n o t extendible, and deduce t h a t an a r b i t r a r y row o f t h e m a t r i x i s a l i n e a r combination o f rows o f the submatrix which corresponds t o t h i s determinant,
82
THEORY OF RELATIONS
5.2. In t h e "degenerate case" where h = Card E < 2p+q , t h e number of columns i s s t r i c t l y l e s s than t h e number of rows. In t h i s case t h e columns of t h e incidence matrix a r e l i n e a r l y independent; i n o t h e r words, t h e r e e x i s t s a non-zero determinant based on t h e col umns. 0 Interchange each p-element s e t y w i t h t h e (h-p)-element s e t E-y , and each (p+q)-element s e t x with t h e (h-p-q)-element s e t E-x . Then the inclusion E-y . The r o l e of p i s played by p ' = h-p-q ; y c x i s equivalent t o t h e r o l e of p+q i s played by p ' + q ' = h-p , so t h a t q ' = q . We have 2p' + q ' = 2 h - 2 p - q < h : hence we can apply t h e l i n e a r independence lemna with rows and columns interchanged. 0
E-xc
5.3. MULTICOLOR Let E be a f i n i t e s e t , h i t s cardinal and p , q two i n t e g e r s . P a r t i t i o n the p-element subsets of E i n t o a f i n i t e number k of c l a s s e s which a r e c a l l e d colors uo, ul, ..., u ~ .- For ~ each (p+q)-element subset a of E , we c a l l t h e multicolor of a t h e function which t o each c o l o r u i ( i k ) associates t h e number of p-element s e t s of c o l o r u i which a r e included i n a When t h i s number i s non-zero, we say t h a t t h e c o l o r u i f i g u r e s in the multicolor. MULTICOLOR THEOREM I f Card E 2, 2p+q , then t h e number of m u l t i c o l o r s ' o f (p+q)-element subsets
.
of
E i s a t l e a s t equal t o t h e number of c o l o r s of p-element subsets. More p r e c i s e l y , t h e r e e x i s t s an i n j e c t i o n which t o each c o l o r u ( t o which a t l e a s t one p-element s e t belongs) a s s o c i a t e s a multicolor i n which u f i g u r e s , and t o which a t l e a s t one (p+q)-element s e t belongs (POUZET 1976). Assume f i r s t t h a t E has f i n i t e cardinal h > , 2p+q . Hence t h e number of (p+q)-element s e t s i s a t l e a s t equal t o t h a t of t h e p-element s e t s , and the rows of t h e incidence matrix a r e l i n e a r l y independent. To each c o l o r t h e r e corresponds a f i n i t e s e t of rows of t h a t c o l o r . Replace these by a unique row which i s t h e i r sum, obtained by adding the values 0 o r 1 i n each column. T h u s each new row represents a c o l o r u Each column continues t o represent a (p+q)-element s e t , and i n d i c a t e s the number of p-element sets of c o l o r u which a r e included i n t h i s (p+q)-element set. Note t h a t , i n t h e new matrix t h u s obtained, t h e rows a r e l i n e a r l y independent. I t s u f f i c e s t o see t h a t , given a matrix w i t h k independent rows ( k 3 2 ) , the replacement of two rows b and b ' by t h e i r sum y i e l d s a matrix w i t h k-I independent rows. Indeed, t h e r e e x i s t s a non-zero determinant based on the k-2 i n t a c t rows. So t h a t the only o t h e r p o s s i b i l i t y would be t h a t t h e row sum o f b and b' i s a l i n e a r combination of the k-2 i n t a c t rows. B u t then the row b , f o r example, would be a l i n e a r combination o f t h e k-2 i n t a c t rows plus
.
83
Chapter 3
the row b' , c o n t r a d i c t i n g t h e hypothesis. T h u s , i f k i s now t h e number of c o l o r s , hence of rows, we have a non-zero determinant of order k . Take i n this determinant a sequence of k ordered p a i r s ( x , y ) where x i s a column and y a row, w i t h non-zero value of the new matrix i n each considered ordered p a i r . We thus o b t a i n the i n j e c t i v e function i n the theorem. This i n j e c t i o n a s s o c i a t e s , t o two d i s t i n c t c o l o r s y. y ' two ( p + q ) element s e t s x , x ' whose m u l t i c o l o r s a r e d i s t i n c t . Otherwise we would have two i d e n t i c a l columns i n t h e determinant. Thus t h i s is an i n j e c t i o n from t h e s e t of colors i n t o the s e t of m u l t i c o l o r s . I t remains t o consider t h e case when E i s countably i n f i n i t e . I f we only have a f i n i t e number of c o l o r s , then we r e s t r i c t E t o a s e t of f i n i t e c a r d i n a l i t y a t l e a s t equal t o 2p+q and including p-element s u b s e t s of each c o l o r . The rows, which r e p r e s e n t t h e c o l o r s , a r e l i n e a r l y independent, and remain so when one takes up the e n t i r e i n f i n i t e set E . I f t h e r e a r e i n f i n i t e l y many c o l o r s , then we s t i l l have l i n e a r independence. Then a s mentioned f o r t h e l i n e a r independence lennna, every non-zero determinant i s extendible t o a non-zero determinant over one more row, hence one more c o l o r . The e x i s t e n c e of t h e i n j e c t i v e function i n the theorem follows. 0
5 6 - RAMSEY
SEQUENCE: ANOTHER PROOF OF
GALVIN'S
THEOREM
The following notion of Ramsey sequence of conditions is a form of the c l a s s i c a l Ramsey s e t : see ERDOS, RADO 1952. The connected proof of GALVIN's i n i t i a l i n t e r v a l theorem i s due t o LOPEZ 1983'. As opposed w i t h POUZET's proof i n 5 2 , here we need n e i t h e r lexicographic rank nor t r a n s f i n i t e induction. As well a s i n 5 2 , t h e axioms of ZF w i l l be s u f f i c i e n t : see 5 6.5 below.
6.1. Given two s e t s A, B of i n t e g e r s , p u t A < B o r B > A i f f every element of B i s s t r i c t l y g r e a t e r than every element of A . We adopt the convention t h a t the empty s e t i s < and > any s e t ; so t h a t < i s i r r e f l e x i v e and t r a n s i t i v e only f o r non-empty s e t s . L e t H be a f i n i t e s e t , Z an i n f i n i t e set of i n t e g e r s . A f i n i t e sequence of conditions g i ( H , Z ) ( i = 1, ..., r ) i s s a i d t o be a Ramsey sequence i f f we have the following: V H fin i n f X > H 93, i n f YCX A ( b z i n f Z c y 3rl(H,Z))V ...
VX
[
... v ( d zi n f
Z s Y
+Vr(H,Z))]
( n o t a t i o n s : f i n = f i n i t e , i n f = i n f i n i t e set of i n t e g e r s ; obvious l o g i c a l symbols).
THEORY OF RELATIONS
84
Example. P a r t i t i o n t h e p a i r s o f i n t e g e r s i n t o two c o l o r s (+) and (-). Take f o r
i n H , a l l p a i r s 1h.z) h ) " . Then alone
f(H,Z) t h e f o l l o w i n g statement: " f o r each i n t e g e r h where z belongs t o Z , have same c o l o r (depending on
,
c o n s t i t u t e s a Ramsey sequence. Another example. Take a c o n d i t i o n f and d e f i n e i g as t h e negation o f d i s o f t e n a Ramsey sequence. I t i s t h e case, for Then t h e sequence ( ff, -I instance, i f yf (H,Z) means t h a t the preceding p a i r s { h,z) have c o l o r (t). I n the case o f two such opposite conditions, the above formula means t h a t , given
.
e)
H , the s e t o f a l l i n f i n i t e Z s a t i s f y i n g C(H,Z) i s Ramsey i n the sense o f ERDOS, RADO 1952. I n other words, there e x i s t s an i n f i n i t e s e t Y o f i n t e g e r s such t h a t e i t h e r each i n f i n i t e Z C_ Y belongs t o t h e s e t defined by and H o r each i n f i n i t e Z c Y belongs t o t h e complement. Among sets o f i n f i n i t e sets o f integers, i . e . among sets o f r e a l s , i t i s known t h a t t h e f o l l o w i n g a r e Ramsey: a l l open sets (see t h e topology defined i n ch.1 exerc. 4 ) ; Bore1 sets (see GALVIN, PRIKRY 1973); a n a l y t i c s e t s (SILVER 1970). See a l s o ELLENTUCK 1974, who characterizes t h e "completely Ramsey sets" by t h e Bai r e property.
el ,..., er , we have t h e f o l l o w i n g
6.2. Given a Ramsey sequence
statement,
modulo t h e axiom o f dependent choice:
J ~ i ~ f f l ~ f i(vzinf ,, (
...
(Vz
inf
H C A A Z S A ~ Z > H ) V~1 ( H J ) ) V - - *
( H c A h Z E A A Z > H ) 3 f,(H,Z))
0 The p r o o f generalizes the f i r s t p a r t o f RAMSEY's p r o o f 1 . 1 , i n o b t a i n i n g e l e -
.
.
S t a r t from uo = 0 , Ho = { D l and Xo = s e t o f i n t e g e r s # 0 We ments vi get an i n f i n i t e Ys Xo , c a l l e d Yo and s a t i s f y i n g t h e above c o n d i t i o n i n
.
and A = {O)uYa Then l e t u1 be t h e f i r s t element { uo,ul) and XI = Yo -{ul) We g e t an i n f i n i t e
brackets, where H = Ho of Yo S t a r t from HI
.
.
=
Y1s X1
which s a t i s f i e s our above c o n d i t i o n i n brackets, where H = and A = \uo,uI) u Y1 Then s t a r t from H i = {ul) and = Y1 infinite Y1 which s a t i s f i e s our condition, where H = Ho o r and A = {uo,ul)u Yi Then l e t u2 be t h e f i r s t element o f Y i
Yi
.
.
Xi
.
Yi
and X2 = - {u2). from H2 = {uo,u1,u2) s a t i s f i e s our condition, where H = Ho o r
.
Ho
or
H1 an or H i
. We g e t HI Start
We g e t an i n f i n i t e Y2 5 Y i HI o r H i o r H2 and where
which
A = {uo,u1,u2\ u Y2 I t e r a t e , t a k i n g f o r H i , H;,.. a l l the sets w i t h l a s t element u2 , and so g e t t i n g Y h y Y i , before d e f i n i n g u3, Hg and Y3 ; and so on. F i n a l l y take f o r A t h e s e t o f ui ( i i n t e g e r ) The axiom o f dependent choice i s used f o r choosing s e t s Y . 0
...
.
6.3. L e t H, F be f i n i t e sets o f i n t e g e r s and Then t h e p a i r o f t h e f o l l o w i n g c o n d i t i o n e(H,Z) (HJ) c ( H , Z ) : 3 F fin F c Z A
a
.
b (H,F)
be an a r b i t r a r y c o n d i t i o n .
w i t h i t s negation i s Ramsec
Chapter 3
85
0 Suppose the contrary. There e x i s t a f i n i t e s e t
H
and an i n f i n i t e s e t
X 7 H
such t h a t , f o r every i n f i n i t e s e t Y 5 X , there e x i s t two i n f i n i t e subsets Z1 and Z2 with (H,Z1) and the negation iff (H,Z2) Then each f i n i t e subset F of Z2 s a t i s f i e s (H,F) Now replace Y by Z2 : there e x i s t s an i n f i n i t e subset Z i of Z2 such t h a t 'f: ( H , Z i ) . Thus there e x i s t s a f i n i t e subset F of (H,F) : contradiction. 0 5 Z2 which s a t i s f i e s
.
.
a
Zi
6 . 4 . A PROOF OF GALVIN'S INITIAL INTERVAL THEOREM be a s e t of f i n i t e s e t s of integers, assumed t o be mutually incomparable Let under inclusion, and t o s a t i s f y GALVIN's hypothesis: every i n f i n i t e s e t of integers includes a t l e a s t one element of 3 as a subset. Take 3 (H,F) t o be the following condition: "the union H u F admits an i n i t i a l interval belonging t o " ; more b r i e f l y " H,J F has i . i . " . Then by 6 . 2 there e x i s t s an i n f i n i t e s e t A such t h a t , f o r every f i n i t e subset H of A : either (1) inf ( Z g A A Z > H ) =) 3, f i n ( F c Z A H u F has F i . i . )
v,
or ( 2 )
tJz
inf ( Z 5 A
A
Z 7 H) =)
vF f i n
( F c Z
+
HuF
has no
'3 i . i . ) .
F i r s t l y we prove t h a t , assuming GALVIN's hypothesis, there e x i s t s an i n f i n i t e s e t A such the above ( 2 ) i s f a l s e : so only (1) i s true. 0 For H empty, the above conclusion ( 2 ) i s f a l s e . Indeed ( 2 ) reduces t o saying t h a t , f o r every i n f i n i t e s e t Z , there does n o t e x i s t any f i n i t e subset of Z which belongs t o Now l e t G be a f i n i t e s e t of integers. Assume t h a t the above ( 2 ) i s f a l s e f o r every subset H of G . Then i t suffices t o prove t h a t there e x i s t s an integer g 7 M a x G such t h a t every H 5 G u{g> f a l s i f i e s ( 2 ) . For t h i s purpose, i t suffices t o prove t h a t , f o r each subset H of G , there e x i s t only f i n i t e l y many integers h Z Max G such t h a t H u { h ) s a t i s f i e s ( 2 ) : indeed i t will suffice t o choose g s t r i c t l y greater t h a n a l l such h Arguing ad absurdurn, assume the existence of an i n f i n i t e sequence h l , h 2 , ..., h i , ... with Hu{hi} satisfying ( 2 ) . Take Z t o be the i n f i n i t e s e t of these h i : by hypothesis H f a l s i f i e s ( 2 ) , so H v e r i f i e s (1). Thus there e x i s t s a f i n i t e subset F of Z such t h a t H u F has an i n i t i a l interval which belongs t o 3-. Let h P (p integer) be the f i r s t element of F ; then H w{h ) f a l s i f i e s (2): contraP diction. 0
-
.
.
Secondly we obtain GALVIN's conclusion. A of 6.2, now assumed t o s a t i s f y only the above ( 1 ) . Let B be an a r b i t r a r y i n f i n i t e subset of A Let K be a f i n i t e subset of B belonging t o , and denote by H the i n i t i a l interval of B which ends with Max K . Then by ( 1 ) above, there e x i s t s a f i n i t e subset F of 6-H such
0 Consider the i n f i n i t e s e t
.
THEORY OF RELATIONS
86
.
t h a t HuF has an i n i t i a l interval which belongs t o 3 This i n i t i a l interval are mutually incomparable under cannot surpass Max K , since elements of inclusion. Consequently our i n i t i a l interval of H u F reduces t o an i n i t i a l interval of H , thus of B . 0 6.5. To end t h i s section, l e t us prove t h a t the axiom of dependent choice, used in 6.2, i s avoidable in view of obtaining GALVIN's theorem.
Come back t o our hypotheses in 6.1. Given a f i n i t e s e t H and a n i n f i n i t e set X > H : e i t h e r (1) 3 i n f x A z' i n f ( Z C y 3~ f i n (FcZ A H u F has $-i.i.))
0
ys
or ( 2 )
3,
inf Y c _ X
A
vz inf(ZcY=> 'dF f i n ( F c Z =>
HuF
has n 0 F i . i . ) ) .
Either ( 2 ) i s f a l s e ; in other words:
vy inf
Y c _ X =>
2 z inf
(ZSY
3, f i n
(FcZAHUF
has 5 i . i . ) ) .
I n such a case, take Y = X ; then take any Y c X b u t change the notation, writing Z instead of Y : we get the following 3, inf Y = X A \dz inf ( Z S X => 3, f i n (FcZ A HuF has F i . i . ) )
So we obtain (1) strengthened by the unambiguous definition
Y = X
.
.
Or ( 2 ) i s t r u e , with (1) true or f a l s e , which i s immaterial. In t h i s case, take a l l the i n f i n i t e s e t s Y which s a t i s f y ( 2 ) , and note t h a t each i n f i n i t e subset of a Y i s s t i l l a Y satisfying ( 2 ) . Let us proceed lexicographically: we take the l e a s t integer uo f o r which there e x i s t s a Y beginning with uo . Then the l e a s t u l , uo f o r which there e x i s t s a Y beginning with uo, u1 ; and so on. Finally we adopt the unambiguous definition Y = i u o , u l , . . . ) .0 EXERCISE 1 - SCHUR NUMBERS 1 - Given a p a r t i t i o n of the s t r i c t l y positive integers into a f i n i t e number of classes called columns, show t h a t a t l e a s t one of the columns contains three d i s t i n c t integers a,b,c with c = a+b (SCHUR 1916). Hint. To each column U associate the class of pairs of integers x , y such t h a t the absolute value I x-yl belongs t o U ; then apply RAMSEY's theorem. 2 - A s e t U of integers i s said t o be additively f r e e i f f the sum of any two integers belonging t o U does not belong t o U Given an integer k , show that there e x i s t s an integer k+, k such t h a t , f o r each p a r t i t i o n of the integers 1,2, ...,k+ i n t o k classes called columns, there is a t l e a s t one nonadditively f r e e column. The smallest k+ will be denoted by s ( k ) and called the Schur number r e l a t i v e t o k Show t h a t s ( 1 ) = 2 , s ( 2 ) = 5 , s(3) = 14 ( s t a r t with the column 5,6,7,8,9).
.
.
Chapter 3
3
-
87
I n 1961, Leonard D. BAUMERT (see p u b l i c a t i o n 1965) e s t a b l i s h e d , w i t h t h e a i d
o f a computer, t h a t s ( 4 ) = 45. Here i s t h e example he gave o f a p a r t i t i o n o f t h e f i r s t 44 s t r i c t l y p o s i t i v e i n t e g e r s i n t o 4 a d d i t i v e l y f r e e columns: 1 2 4 9 3 7 6 1 0 11 5 8 13 15 18 20 12 17 21 22 14 19 24 23 16 26 27 25 29 28 33 30 31 40 37 32 34 42 38 39 35 44 43 41 36 2 Show t h a t t h e Ramsey number ( 3 ) k >/ s ( k ) + l : a s s o c i a t e t o each column U t h e s e t o f p a i r s o f integers from
-
Show t h a t
s(k+l)
gers
1 to
p = s(k)-1
gers
p+l
4
2p + 1
to
.
>/
ned
& 158 , thus
I n general
s(5) 2 (3)5
k
3 ( 3 +1)/2
s(k)
k
>/
.
1
46
.
Begin
with the p a r t i t i o n o f the inte-
columns. Add a
(k+l)st
column o f t h e i n t e -
Then complete each column formed o f i n t e g e r s
Hence
2p+l+u
s ( k ) f o r which t h e a b s o l u t e v a l u e o f t h e
2 (3)4
3.s(k) into
.
integers s(5)
1 to
U ; thus
d i f f e r e n c e belongs t o
+ 159
u by t h e
>, 135 . FREDRICKSON 1979
3 134 and (3);
obtai-
. 3 (3k-4.89/2) + 1/2 f o r k >/ 4 . T h i s
and even
i n e q u a l i t y i s improved by ABBOTT, HANSON 1972 who o b t a i n , i f one r e c t i f i e s t h e i r numerical e r r o r ,
-
EXERCISE 2
1
-
Let
s(k)
3
89(k-7)/4.1201
+
1 for
k
>/
4
.
RAMSEY'S THEOREM WITH m-TUPLES
m, k
be two i n t e g e r s ; t a k e an
m-sequence o f i n t e g e r s
ai (i= 1, ..., m).
P a t t e r n e d a f t e r t h e f i n i t e Ramsey theorem, p r o v e t h a t t h e r e e x i s t s an of i n t e g e r s Fi Xm)
bi
+ ai
o f cardinality
where
subsets
xi
, for
belongs t o Fi
Gi
bi
s a t i s f y i n g t h e f o l l o w i n g . Given an
with -
Gi
i n the Cartesian product
m = 2 with
(kp).kkp
and
p
x
... x
k
Gm
m-tuples
(x
,.... ,
c o l o r s , t h e r e e x i s t s a sequence o f
, such t h a t a l l t h e m-tuples
have t h e same c o l o r . Note t h a t n o t h i n g
p a i r w i s e d i s j o i n t , which i s o f t e n convenient.
Fi al = a2 = p
. Beginning w i t h
a set
F1
o f cardinality
F2 o f c a r d i n a l i t y (p).kkp , we o b t a i n G2 5 F2 o f c a r d i n a l i t y H1 C, F1 o f c a r d i n a l kp , such t h a t , f o r e v e r y o f H1 , a l l t h e o r d e r e d p a i r s w i t h f i r s t t e r m x and second term
and an " i n t e r m e d i a t e " s e t
element
, into
o f c a r d i n a l i t y ai -
G1 x
i s changed i n assuming t h e Example:
every p a r t i t i o n o f the s e t o f Fi
m-sequence
m-sequence of s e t s
THEORY OF RELATIONS
88
c
an element of G2 have the same c o l o r . F i n a l l y we obtain G1 H1 p , w i t h t h e monochromatic Cartesian product G1 x G2 . 2 - For m = 2 and a l = a 2 = 2 , we can take bl = 3 and b2 = 9 sely. W e search f o r a symmetric s o l u t i o n bl = b2 ; show t h a t bl = i n s u f f i c i e n t . On t h e o t h e r hand, f o r the common value 5 , we always
of cardinal
, o r converb2 = 4 i s have a mono-
.
chromatic Cartesian product with Card G1 = Card G2 = 2 One could assume f i r s t t h a t t h e r e e x i s t s an element x of F1 joined t o a t l e a s t 4 elements of F2 by the same c o l o r , denoted ( + ) . Call t h i s subset of F2 of cardinal 4. Then e i t h e r t h e r e e x i s t s another element of F1 y i e l d i n g t h e c o l o r (+) w i t h two elements of , o r two elements of F1 d i s t i n c t from x y i e l d the opposite c o l o r ( - ) w i t h two elements of F; Next, we assume t h a t each element of F1 y i e l d s a p a r t i t i o n of 2 as opposed t o 3 elements of F2 f o r t h e c o l o r s . Such an element x i s c a l l e d (+)-major i t y o r (-)-majority, according t o whether t h e r e a r e 3 edges ( o r p a i r s ) emanat i n g from x with c o l o r (+), o r 3 w i t h c o l o r ( - ) . Then t h e r e e x i s t 3 elements of F1 of t h e same majority: this y i e l d s our conclusion. 3 - Attempt t o extend t h e statement in s e c t i o n 1 above t o t h e case of i n f i n i t e Note t h a t for m = 2 and al = a? = W the c a r d i n a l s a ( i = 1,. ,ml
Fi
Fi
.
,
.. . i = b2 = ~3 do not hold:
values bl we do not have a simple i n f i n i t a r y theorem analogous t o RAMSEY's theorem. Indeed, taking F1 = F2 = is , t h e s e t of i n t e g e r s , i t s u f f i c e s t o c o l o r (+) t h e ordered p a i r s ( x , , ~ , ) i f f x l \ < x2 . However, with t h e axiom of choice, ERDOS, RADO 1956 proved t h a t one can take bl countable and b2 equipotent w i t h t h e continuum, o r conversely: s e e e x e r c i s e 3 below. EXERCISE 3
-
A PARTITION THEOREM WITH POINTS, i . e . ORDERED PAIRS
Take a denumerable s e t D , a s e t C equipotent w i t h t h e continuum, and t h e Cartesian product D x C whose elements ( x , y ) s h a l l be c a l l e d points with t h e abscissa x in D and the o r d i n a t e y i n C P a r t i t i o n the p o i n t s i n t o two c l a s s e s c a l l e d t h e c o l o r s (+) and ( - ) . We s h a l l show t h a t t h e r e e x i s t s either a denumerable s u b s e t Do fo D subset Co of C equipotent w i t h t h e continuum, w i t h Do x Co e n t i r e l y of c o l o r ( - ) , o r two denumerable s u b s e t s D1 0-f D and C1 C with D1 x C1 e n t i r e l y of c o l o r (+) (uses axiom of choice; ERDOS, RADO 1956 p. 482).
.
-
1 Assume t h a t t h e r e does not e x i s t any (-)-monochromatic s e t which i s t h e Cartesian product of a denumerable s e t w i t h a continuum-equipotent set. Take an
a r b i t r a r y denumerable subset X of D and an a r b i t r a r y s u b s e t Y of C which is continuum-equipotent: by hypothesis X x Y i s never (-)-monochromatic. Call Y o t h e projection onto Y of t h e s e t of points of X x Y w i t h c o l o r (+), and note t h a t , by our hypothesis, Card(Y-Yo) i s s t r i c t l y l e s s than t h e
Chapter 3
89
continuum. Every p a r t i t i o n of a continuum-equipotent s e t i n t o countably many c l a s s e s y i e l d s a t l e a s t one c l a s s which i s continuum-equipotent (ch.1 5 4 . 3 , axiom of c h o i c e ) . T h u s there e x i s t s an element xo i n X y i e l d i n g continuum many points w i t h abscissa x o and o r d i n a t e s i n Y with t h e c o l o r (+). 2 - W i t h the s e t s X, Y a s above, f o r each point y in Y , denote by f ( y ) the s e t of a b s c i s s a s x i n X such t h a t the point ( x , y ) has c o l o r (t). Note t h a t f ( y ) i s not always f i n i t e , f o r otherwise t h e r e would e x i s t a continuumequipotent s e t of o r d i n a t e s y w i t h t h e same f ( y ) , hence the same complement X - f ( y ) and f i n a l l y a (-)-monochromatic s e t which i s t h e Cartesian product of a denumerable and continuum-equipotent s e t . Thus t h e r e exists an element y o of Y y i e l d i n g continuum many p o i n t s w i t h o r d i n a t e yo and a b s c i s s a s i n X , having t h e c o l o r (t). 3 - Beginning w i t h Xo = D and Yo = C , take x1 i n Xo y i e l d i n g continuum many points w i t h a b s c i s s a x1 and c o l o r ( + ) . Denote by Y1 t h e continuumequipotent set of o r d i n a t e s . Take y1 i n Y1 y i e l d i n g a denumerable s e t of points with o r d i n a t e y1 and c o l o r ( + ) . Denote by XI t h e denumerable set of {xl} and Y1 - { y l \ , thus a b s c i s s a s . I t e r a t e t h i s , beginning v i t h X1 obtaining an element x2 of t h e f i r s t s e t , and y2 an element of t h e second, w i t h continuum many points with a b s c i s s a x2 and c o l o r (+), and denumerably many points w i t h o r d i n a t e y2 and c o l o r (+). The f o u r points w i t h a b s c i s s a s x l , x2 and o r d i n a t e s y l , y2 have c o l o r ( + ) . By i t e r a t i o n , obtain a denumerable s e t of a b s c i s s a s x i and a denumerable s e t o f o r d i n a t e s y i y i e l d i n g t h e color (t) f o r a l l p o i n t s ( x i , y . ) ( i , j p o s i t i v e i n t e g e r s ) . J 4 - I t follows t h a t i f the plane, which i s the Cartesian product of two sets both equipotent w i t h t h e continuum, i s p a r t i t i o n e d i n t o two c o l o r s , then e i t h e r t h e r e exists a denumerable " g r i d " of t h e f i r s t c o l o r , or t h e r e e x i s t both a g r i d obtained a s t h e product of denumerably many a b s c i s s a s w i t h continuum many o r d i n a t e s , and a g r i d obtained as t h e product of continuum many abscissas with denumerably many o r d i n a t e s , both of t h e second c o l o r . On t h e o t h e r hand, using SIERPINSKI's counterexample i n 3.1, we can p a r t i t i o n t h e plane i n t o f o u r c o l o r s so t h a t every monochromatic g r i d i s a t most the product of a denumerable s e t of a b s c i s s a s w i t h a denumerable s e t of o r d i n a t e s .
-
90
THEORY OF RELATIONS
Problem. Does there e x i s t a p a r t i t i o n of the plane into two colors, or into three colors, such t h a t every monochromatic grid i s a t most the product o f a denumerable s e t with a denumerable set. EXERCISE 4 - SPERNER'S LEMMA Let E be a s e t with f i n i t e cardinality 2h or 2 h + l . Then every s e t o-f sub. s e t s of E which are mutually incomparable with respect t o inclusion has 2 (even case) ( 2 h + l ) ! / h ! ( h + l ) ! (odd case) cardinality a t most ( 2 h ) ! / ( h ! ) In other words, the largest possible cardinality i s obtained by taking the s e t of a l l h-element subsets. Beginning with a s e t Jf o f subsets of E , none of which i s included in another, by replacing i f necessary each s e t by i t s complement, we can always assume t h a t We shall prove t h a t the smallest cardinality of the elements of 4 i s p,< h i f p < h (even case) or p 4 h (odd case) , then we can injectively substitute for every subset A of minimum cardinality p a superset B of A of cardinality p + l . Indeed, i f t h i s i s possible, then the B will be d i s t i n c t
.
or
.
and of the same cardinal p + l , hence will be mutually incomparable with respect t o inclusion. Moreover, no B can be included in an element of of larger cardinality, since then the p-element s e t A from which B was obtained would i t s e l f be included in t h a t s e t . Now note t h a t SPERNER's lemma e a s i l y follows from the l i n e a r independence lemma 5.1 in the case of E f i n i t e and q = 1 For further developments on t h i s subject, see POUZET, ROSENBERG 1982.
4
.
91
CHAPTER
4
GOOD AND BAD SEQUENCE, FINITELY FREE PARTIAL ORDERING, WELL PARTIAL ORDERING, IDEAL, TREE, DIMENSION
§
1 - PARTIAL
ORDERING
OF THE INITIAL
INTERVALS,
PARTITION
NI
SLICES
1.1. INITIAL INTERVALS Let A be a p a r t i a l ordering, k a s e t of i n i t i a l intervals of A ; by an abuse of language, we confuse the notion of interval as subset of the base with t h a t o f r e s t r i c t i o n of A , and speak of the union and intersection of &! The union of i s an i n i t i a l interval o f A and i s the supremum or l e a s t upper bound of with respect t o inclusion; similarly the intersection of 4 i s an i n i t i a l interval and i s the infimum of
.
u
u.
I n the case of a chain, o r t o t a l 01 jering A , we get a total ordering of the i n i t i a l intervals of A . To each i n i t i a l interval X of A , associate the final i s said t o be interval X ' , the complement of X ; the ordered pair ( X , X ' ) a of A . The comparison ordering between cuts i s defined by inclusion of t h e i r f i r s t terms, which are i n i t i a l intervals.
cut
Separate the s e t of cuts of A into two complementary subsets '& and v , such i s l e s s t h a n every cut in v ; we have DEDEKIND's r e s u l t : that every cut in has a maximum c u t , or ?f has a minimum cut. 0 Take the union C of the i n i t i a l intervals in the cuts in , and the final interval C ' , the complement of C . Then (C,C') i s the supremum of If t h i s cut belongs t o then i t i s the maximum; i f i t belongs t o t ' then i t i s the minimum. 0
u
u
u
.
A . If A',A") i s a cut of A , then of B and A ' t o the intersection of the bases, and the common r e s t r i c t i o n B" of B and A " constitutes a cut ( B ' , B " ) of B , called the restriction, of the cut (A'"") t o B We also say t h a t (A'"") is a cut induced by ( B ' , B " ) on A Obviously there may e x i s t many such cuts.
l e t A be a chain, B the common r e s t r i c t i o n
a r e s t r i c t i o n of B'
.
. .
1 . 2 . Let E be a s e t , f a s e t of subsets of E Two d i s t i n c t elements of E are said t o be separable by f , i f there e x i s t s an element of f which contains one and n o t the other.
be a s e t of subsets of E , which we assume t o be t o t a l l y ordered under Let t o be a maximal chain with respect t o inclusion, i t i s inclusion. necessary and s u f f i c i e n t t h a t f? be closed under unions and intersections
For
92
THEORY OF RELATIONS
( f i n i t e or otherwise), and t h at any two d i s t i n c t elements of
E
be separable
txv. I n general, l e t L& be a s e t of subsets of E , closed under union and intersection. Let f? be a subset of & which i s t o t a l l y ordered by inclusion. t o be maximal among t o t a l l y ordered r es t r i ctions of LR , i t i s necessary For and intersection, and tha t any t w q and s u f f i c i e n t t h a t f be closed under union.-___ d is t i n c t elements of E which are separable by ‘4 are also separable by f? .
Assume t h a t i s a chain which i s maximal among the re stric tions of &? . Then i s closed under union and intersection, f o r otherwise we could add the unions and intersections of elements of ‘f . Moreover, i f x , y are two d i s t i n c t elements of E which are separated by an element X o f ‘4 with x belonging t o X yet n o t y , then denote by U the union of the elements of which contain neither x nor y , and by V the intersection of the elements of which contain both x and y . By the preceding discussion, U and V are . elements of Since (e i s t o t a l l y ordered by inclusion, then U i s s t r i c t l y included in V Either ‘Cr contains an element separating x and y , in which case our conclusion holds. Or every element of f i s a subset o f U or a superset of V . Hence U and V are consecutive with respect t o inclusion in the chain So ( U u X ) n V i s an element of (R , situated between U and V with respect t o inclusion, and d i s t i n c t from U and V as i t contains x b u t n o t y . This . contradicts the maximality o f Conversely, assume t h at the chain i s not maximal among the t o t a l l y ordered r e s tr i c t i o n s of & . We can thus add t o f? a subset W of E which i s an element of L & , and e i t h e r including Or included in every s e t which belongs t o which are included in W , . Denote by U the union of those elements of and by V the intersection of those elements o f which include W Then e ith e r i t i s the case t h at U or V does not belong t o : so tha t is e ith e r n o t closed under union or under intersection. Or i t i s the case that U and V belong t o ‘t? : so t h a t W i s d i s t i n c t from U and V , and hence i s properly situated between U and V . Let u be an element of W-U and v an element of V-W : the elements u and v are separated by the element W .0 of dz , and y e t are n o t separated by any element o f
0
.
.
.
1.3. For any p a r t i a l ordering A , denote by 3 ( A ) the partial ordering o f the i n i t i a l intervals o f A (with respect t o inclusion). We know tha t 9 ( A ) i s closed under union and intersection. Moreover any two elements x , y of I A I are separable by 3 ( A ) : indeed we can assume t h a t x < y o r x l y (mod A) , and i t suffices t o take the interval 5 x (mod A) t o separate x and y
.
Chapter 4
93
To every t o t a l l y ordered reinforcement B of A (see ch.2 § 4 ) there corresponds the t o t a l ordering 3 ( 6 ) of the i n i t i a l intervals of B . These intervals are chains (mod B ) , and although n o t necessarily chains (mod A ) , they remain i n i t i a l i s an element of 3(A) . intervals (mod A ) . I n other words, every element of J ( B ) Let A be a p a r t i a l ordering. If B i s a t o t a l l y ordered reinforcement of A , then 3 ( B ) i s a total ordering which i s maximal among those tota lly ordered r e s tr i c t i o n s of 'j ( A ) . Conversely, every maximal t o t a l l y ordered re stric tion of 3 ( A ) i s of the form g(B) , where B i s a t o t a l l y ordered reinforcement of A ; moreover this B i s unique (BONNET, POUZET 1969). 0 Starting with a t o t a l l y ordered reinforcement B of A , we already know t h a t
3(B) i s closed under union and intersection. Moreover, any two d i s t i n c t elements x , y of IA I are separable by J (B) : suppose for example tha t x c y x (mod 6 ) . By the preceding 1 . 2 , the total (mod B ) and take the interval ordering 3(B) i s maximal among those t o t a l l y ordered re stric tions of 3 ( A ) . Conversely, l e t f be a t o t al ordering which i s maximal among those tota lly ordered r e s t r i c t i o n s of 3 ( A ) . By 1 . 2 , t h i s kf i s closed under union and intersection, and any two d i s t i n c t elements in the base I A I are separable by an element of To obtain = j ( B ) , define the t o t a l l y ordered reinforcement B of A by the condition t h a t , when given two elements x , y of A , we p u t xd y (mod 6 ) i f f every element of which contains y as an element also contains x as an element. The antisymmetry of B follows from the fa c t t h a t , f o r d i s t i n c t x, y , there ex i s t s an element of which separates them. Moreover, since f i s a t o t al ordering, B i s also. Finally, the uniqueness of B follows from the f a c t t h at two d i s t i n c t t o t a l orderings B and B ' yield two d i s t i n c t 3 ( B ) and Y ( B ' ) . 0
.
e
1 . 4 . I n p a r t i c u l a r , l e t E be an ar b i t r ar y s e t . For every chain B based on E , the i n i t i a l intervals of B form a maximal t o t a l l y ordered r e s t ri ct i o n 3 ( 6 ) of the p ar t i al ordering of inclusion. Conversely, every chain which i s a maximal r es t r i ct ion of the pa rtia l ordering of inclusion of subsets of E i s the s e t of i n i t i a l intervals 3(B) for a certain chain B on E .
AXIOM OF MAXIMAL CHAIN OF INCLUSION This axiom asserts t h a t , f o r every s e t E , the p artia l ordering of inclusion among subsets o f E admits a maximal t o t al ordering among i t s re stric tions. I t i s considerably weaker than the general maximal chain axiom, as stated in ch.2 5 2.7: the HAUSDORFF-ZORN axiom, equivalent t o the axiom of choice. By the preceding proposition, the axiom of maximal chain o f inclusion i s equivalent t o the ordering axiom, ch.2 5 2.3.
94
THEORY OF RELATIONS
AXIOM OF MAXIMAL C H A I N ON THE SET OF INITIAL INTERVALS This axiom a s s e r t s t h a t , given a partial ordering A , the partial ordering of inclusion on the i n i t i a l intervals of A admits a maximal chain among i t s r e s t r i c .-t I t i s equivalent t o the reinforcement axiom, ch.2 5 4.2. 1.5. Starting with an a r b i t r a r y partial ordering A , there e x i s t s t r a t i f i e d partial ordered reinforcements of A (see ch.2 5 5 . 2 ) : f o r examole the chains which are reinforcements of A . We obtain as follows a minimal s t r a t i f i e d reinforcement, in the sense t h a t i f x ( y (modulo the reinforcement), then e i t h e r x c y (mod A) , or there e x i s t x ' , y ' w i t h x x' I y ' + y or with x & X I I y ' < y (mod A ) (however the converse can be f a l s e ) .
-=
LEMMA FOR PARTITION IN SLICES Let A be a p a r t i a l ordering. There e x i s t s an equivalence relation R for which the equivalence classes are free subsets (mod A ) , and a & c
on H
IAI
on the s e t of equivalence classes, such t h a t f o r any two elements x , y of I A I : (equivalence class of x ) 6 (equivalence class of y ) modulo H i f f there e x i s t two elements X I , y ' which are equivalent (mod R ) and s a t i s f y and y ' d y (mod A ) (uses axiom o f choice; BONNET, POUZET 1969).
x + x'
F i r s t of a l l , denote by R any equivalence relation on the base 1 A l , and by H any chain on the s e t of the equivalence classes of R , which s a t i s f y the following condition: (equivalence class of x ) 6 (equivalence class of y ) (mod H) i f f e i t h e r x i s equivalent t o y (mod R ) , or there e x i s t s an equivalence class U of R which i s a f r e e subset of I A l and two elements x ' , y ' of U with x & x ' and y ' b y (mod A) . Such ordered pairs ( R , H ) e x i s t : i t suffices t o take R t o be the t r i v i a l equivalence relation with one equivalence class I A I , and H the chain on the singleton of I A I . An ordered pair ( R ' , H ' ) i s said t o be finer t h a n ( R , H ) i f every equivalence class (mod R ) which i s f r e e (mod A) remains an equivalence class (mod R ' ) , and moreover every equivalence class (mod R ) i s a union of equivalence classes (mod R ' ) constituting an interval of H ' , and f i n a l l y f o r any two of these i n t e r v a l s , the t o t a l ordering induced by H ' i s identical t o the t o t a l ordering (mod H ) of the corresponding equivalence classes. The comparison " f i n e r than" defines a p a r t i a l ordering o n the ordered pairs ( R , H ) We shall prove t h a t an ordered pair ( R , H ) f o r which the equivalence classes are n o t a l l free (mod A) admits an ordered pair (R' ,HI) which i s s t r i c t l y f i n e r . For thisn take an equivalence class U of R which i s not free. Take a subset V of IJ which i s maximal free (mod A) (ch.2 5 2 . 9 ) , and p a r t i t i o n U into three d i s j o i n t subsets: V , the s e t of elements having a greater element (mod A ) in V , and the s e t of elements having a smaller element (mod A ) in V
0
.
.
Chapter 4
95
Define the equivalence classes (mod R ' ) as those (mod R ) p a r t i t i o n e d i n t o three equivalence classes (mod R ' ) as
H
, except
ordering
f o r the s u b s t i t u t i o n o f
U
.
, excepting U which i s
The chain
i s t h e same
H'
by t h e t h r e e classes w i t h t h e obvious
.
Using HAUSDORFF-ZORN axiom, consider a maximal ordered p a i r obtained from a maximal chain o f ordered p a i r s than". For t h i s maximal a l l f r e e (mod A)
(R,H)
, totally
(R,H)
, the
ordered by the comparison " f i n e r
equivalence classes defined by
R
are
.0
1.6. There e x i s t s a p a r t i a l o r d e r i n g w i t h c a r d i n a l i t y W 1 i n which every chain and every a n t i c h a i n i s countable (uses axiom o f choice; compare w i t h ch.7 exerc. 4 ) . 0
To every countable o r d i n a l
u i n j e c t i v e l y associate a r e a l r ( u )
choice). Define a p a r t i a l o r d e r i n g based on the countable o r d i n a l s setting
u u (mod A) P 9 vo = uo ,..., v = up-l , v = u q < u , and vpti = uq+i f o r every i 3 1 P P P: 1 Then v i s bad since i t i s e x t r a c t e d from u Moreover v i s a sequence l e s s u p-1 ' up ' Uq+l ' uq+2 ,... which i s e x t r a c t e d from u , and than uo as v i s d i s t i n c t from t h i s e x t r a c t e d sequence, t h i s c o n t r a d i c t s t h e m i n i m a l i t y of u . 0 l e s s than
Every a-sequence -
.
.
,...,
L e t A be a p a r t i a l ordering, u a minimal an integer, and x < u (mod A) Then x P p o s s i b l y f i n i t e l y mani. 0 Assume on the c o n t r a r y t h a t there e x i s t s a integers p < f ( 1 ) < f ( 2 ) < Cf(i)
0 and 1 ' > 1 , and f o r every i n t e g e r i an element i ' 7 0, 1, i and f i n a l l y make t h e i' m u t u a l l y incomparable. Then t h e sequence 0 ' , 1' , 2 ' ,.. . has incomparable terms and s a t i s f i e s t h e c o n d i t i o n o f t h e preceding p r o p o s i t i o n , but t h e sequence 0, 1, 2, considered. 0 2.4.
...
i s a bad sequence which i s l e s s than t h e sequence
Let A be a p a r t i a l ordering. I f u i s a minimal bad w-sequence
(mod A), then the image of i n d i c e s i s minimal bad. 0 Denote by
the terms of
u by an a r b i t r a r y permutation o f t h e s e t o f
uo t h e image o f u under a permutation o f t h e i n d i c e s . By 2.3, u
, hence
of
uo
, are
m u t u a l l y incomparable, so t h a t
uo i s
bad. Suppose t h a t uo i s n o t minimal bad. Then t h e r e e x i s t s an W -sequence v e x t r a c t e d from u0 and a s m a l l e r w -sequence w , which i s bad and d i s t i n c t from v We can assume t h a t the f i r s t terms o f v and w are d i s t i n c t , so
.
.
wo < vo (mod A) Retransform uo, v and w by t h e i n v e r s e o f our i n i t i a l permutation. Then we have again u and we o b t a i n t h e images o f v and w L e t w' denote t h e l a t t e r image. By 2.2 t h e r e e x i s t s a bad m-sequence
.
Chapter 4
99
extracted from w ' , which begins by wo (more exactly, by the term of w ' whose value i s wo ) . This contradicts the minimality of u ( t h i s proof using only ZF i s communicated by H O D G E S ) . 0 2.5. Given a partial ordering A and a minimal bad sequence u in A , the i n i t i a l interval of A generated by u i s a well-founded partial ordering (uses dependent choice; ZF suffices i f A i s countable, or obviously i f A i s well-founded). 0 Assume on the contrary t h at there ex i s t s a term u of u with a s t r i c t l y P > ai > ... (mod A) (dependent -sequence up decreasing a. > a l 7 choice). Then a. 6 a l l except f i n i t e l y many of the terms of u : see 2.3.
...
>
Hence there exists an integer q such that a. 6 uqti Finally we have ai < uq+i f o r every integer minimality of the sequence u 0
.
i
for every integer i .
, thus contradicting the
2.6. Let A be a p a r t i al ordering, u an (.J -sequence in A with mutually incomparable terms. Then u i s minimal bad i f f every bad a-sequence embeddable in u i s extracted from u 0 If u i s minimal bad, our conclusion follows immediately from the definition. Conversely, suppose t h at u has incomparable terms, so i s bad, and yet not minimal bad. Then there e x i s t s a s t r i c t l y increasing a-sequence of integers < f ( i ) < ... with a bad sequence x = (xo,xl ,..., xi ,...) which f(0) < f ( l ) < i s a d i st i n c t sequence less than u f ( o ) , u f ( l ) ,..., u f t i ) ,... . T h u s there e xists an integer p such t h a t x q u (mod A) . Consider the sequence o f the P f(P) x (j = 0,1,2, ...) This sequence i s embeddable in u and moreover i s bad p+j since x i s bad. By hypothesis, t h i s sequence i s extracted from u , hence (mod A) , which contradicts the hypothesis there e x i st s a term u = x 4 u q P f(p) t h a t the terms o f u are mutually incomparable. 0
.
...
.
The hypothesis o f incomparability of the terms of u i s necessary. Take A t o be a well-founded partial ordering with O < 0 ' and denumerably many integers 1,2,3, ... which are s e t t o be mutually incomparable, and also Then the sequence 0',0,1,2,3, ... i s bad yet incomparable with 0 and 0 ' not minimal bad (since 0 ' > 0 ) , and every bad w-sequence embeddable in i t i s extracted from i t . 0
.
Moreover, a bad non-minimal bad w -sequence u can sa tisfy the condition t h a t every bad sequence l es s than u i s equal t o u nTake an element a s e t t o be minimal, and the even integers s e t t o be mutually Take the odd integers 1,3,.. a l l to incomparable, with 0 > a , 2 > a , . be minimal and incomparable w i t h the preceding elements. Then the sequence of a l l
.
.. .
.
THEORY OF RELATIONS
100
integers
..
0,1,2,.
i s n o t minimal s i n c e
a
i s l e s s t h a n e v e r y even i n t e g e r . Yet
e v e r y d i s t i n c t sequence l e s s t h a n i t must t a k e t h e v a l u e
a
i n the position o f
an even i n t e g e r , and hence i s good. 0
A
Problem. L e t
be a p a r t i a l o r d e r i n g ,
r a b l e terms. F o r
u
u
an w - s e q u e n c e i n A
t o be minimal bad, does i t s u f f i c e t h a t e v e r y
w i t h incompa-
w -sequence
u be e x t r a c t e d f r o m u
w i t h incomparable terms which i s embeddable i n
.
2.7. THEOREM ON THE MINIMAL BAD SEQUENCE Let
A
be a well-founded p a r t i a l o r d e r i n g . F o r e v e r y bad w - s e q u e n c e
u
t h e r e e x i s t s a minimal bad sequence which i s embeddable i n choice; ZF s u f f i c e s i f
elements
x & uo
uo
of
u
by
v o c < uo w i t h
vo
.
u
,
minimal among those
f o r which t h e r e e x i s t s a bad w - s e q u e n c e which begins by
and i s embeddable i n
A
i s denumerable).
A
Replace t h e f i r s t t e r m
0
&
u
(uses dependent
Denote by
m - s e q u e n c e embeddable i n
u
.
vo, w1
w1
Replace
x
t h e f i r s t two terms i n such a bad by
w1
vl&
with
v1
minimal
among those
f o r which t h e r e e x i s t s a bad a - s e q u e n c e which begins by xBwl and i s embeddable i n u . Denote by vo, vl, w2 t h e f i r s t t h r e e terms i n
vo, x
such a bad w - s e q u e n c e embeddable i n v = vo, vl,
... which
v2,
over, f o r e v e r y i n t e g e r by
vo, vl,
...,
To see t h a t
v
vi-l,
i and e v e r y
x
vf(l)
,...,
,... .
vf(i)
, no w - s e q u e n c e b e g i n n i n g
.
u
v
... < f ( i ) < ...
vo, vl,
and a bad
which i s d i s t i n c t and l e s s t h a n
Then t h e r e e x i s t s an i n t e g e r
p
with
x
< v f(P) P
v
f(O) ’ (mod A)
v ~ ( ~ ) -xp, ~ , x ~ + ~ , . . .. T h i s sequence i s
...,
x
.
u
and so embeddable i n
sequence i s good. As t h e sequences
C f(p)
(dependent c h o i c e ) . More-
u
(mod A )
f(l)
Max(x,y)
i
, the
i n i t i a l interval
Ai
formed o f t h e
an a r b i t r a r y n a t u r a l i n t e g e r , and t h e couples
: f o r an a r b i t r a r y i n t e g e r
x+y,(i
Ai
y
j
+
i
, the
i n i t i a i intervals
a r e incomparabie w i t h r e s p e c t t o i n c l u s i o n . 0
j A second example o f a w e l l p a r t i a l o r d e r i n g w i t h t h e same incomparabie i n i t i a l i n t e r v a l s : t a k e a g a i n t h e couples o f i n t e g e r s w i t h e i t h e r x = x i and y & y ' ,
dnd y '
arbitrary.
A t h i r d example, wnich w i l l be g e n e r a l i z e d i n ch.8 e x e r c . 2: tegers with e i t h e r
4.3. L e t A
x = x'
and
y,< y '
, or
X I >
Max(x,y)
couples o f i n and y < y ' .
be a p a r t i a l o r d e r i n g and
u
a minimal bad w-sequence .in
t h e i n i t i a l i n t e r v a l o f t h o s e elements
x
of
i fin
g p
x
< ui
(mod A )
A
.
Then
I A I such t h a t t h e r e e x i s t s an i n t e -
i s a w e l l p a r t i a l o r d e r i n g (uses dependent choice;
ZF s u f f i c e s i f A i s c o u n t a b l e j .
Compare w i t h 2.5 above.
i4e prove f i r s t o f a i l t h a t , g i v e n a t e r m u ( p i n t e g e r ) o f t h e minimal bad P sequence, t h e i n i t i a l i n t e r v a l o f t h o s e elements u (mod A j i s a w e l l p a r t i a l P o r d e r i n g . By 2.5 t h i s i n t e r v a l i s well-founded (dependent c h o i c e ) . Suppose t h a t 0
p ) which a r e 6 u . Yet a l l t h e terms o f u a r e q P m u t u a l l y incomparable by 2.3: c o n t r a d i c t i o n . Suppose now t h a t t h e p r o p o s i t i o n i s f a l s e . There e x i s t s a bad w -sequence v
..
such t h a t , f o r each i n t e g e r i , t h e r e e x i s t s a j w i t h v i < uj (mod A ) : see 3.2.(2) u s i n g dependent c h o i c e . A t most a f i n i t e number o f terms o f v a r e < uo, or
< ul,
e t c . Hence t h e r e e x i s t s an (,d -sequence
e i t h e r s t r i c t i y less than
4.4. If
.
u
from
u (mod A)
w
extracted from
v
which i s
o r s t r i c t l y l e s s t h a n a sequence e x t r a c t e d
This contradicts the minimality o f
u
.0
THEOREM ON "ELL PARTIAL ORDERING OF WORDS A
i s a well p a r t i a l ordering, then t h e p a r t i a l orderinq o f embeddabilitv f o r
words i n
A
suffices i f
i s a w e l l p a r t i a l o r d e r i n g (HiGMAN 1952; uses dependent choice; A
i s countable).
ZF
Chapter 4 By 3.2.(2)
109
(dependent c h o i c e ) , i t s u f f i c e s t o show t h a t t h e r e does n o t e x i s t any
bad &J -sequence o f words w i t h r e s p e c t t o e m b e d d a b i l i t y ( d e f i n e d i n 5 2 ) . I f such a bad sequence e x i s t s , t h e n t h e r e e x i s t s a l s o a s t r o n g l y minimal bad sequence U (see 2.10).
We s h a l l show t h a t t h i s i s i m p o s s i b l e . Denote by
ith term o f
.
i s such t h a t no bad w-sequence o f words beby removing a t l e a s t one Uo element. No bad sequence b e g i n n i n g by Uo c o n t i n u e s w i t h a word s h o r t e r t h a n U1 , U
By h y p o t h e s i s ,
the
( i integer)
Ui
Uo
g i n s by a s h o r t e r word, i . e . a word o b t a i n e d f r o m
e t c . N o t i c e t h a t no
Ui
of t h e ui
U . ( j> i ) a r e
i s empty, s i n c e t h e
respect t o embeddability: t h e f i r s t term
ui
J
of
Ui
/ y : hence A/F e x i s t s an element t i n F with t 3 z and so t i s directed; in other words, a net. 0
.
5.5. For every denumerable net A , there e x i s t s 2 t o t a l l y p-de-r-d-restriction ofA , which i s cofinal and isomorphic t o LJ 1 . 0 Take an a-sequence of the elements a i ( i integer) of the base; then take bo = a. bo
and
, bl = the element with l e a s t index which i s greater t h a n (mod A ) b o t h a l , then b2 greater than b l and a 2 , e t c . 0
On the other hand, the d i r e c t product w X (*1 defined in $ 7 below, has no t o t a l l y ordered cofinal r e s t r i c t i o n . 0 T h a t would require a total ordering of order-type w 1 formed o f ordered pairs ( i , j ) with i , j increasing and i running through cu and j running through a 1 , which i s impossible. 0
§
6
- TREE
tree i f , f o r each element x in the base, the s e t of a l l predecessors of x i s t o t a l l y ordered. For example, every t o t a l ordering i s a t r e e . Every f r e e partial ordering, reduced t o the identity, i s a t r e e . Another example: beginning with a s e t a and a s e t A o f subsets of a , where any two elements of A are e i t h e r d i s j o i n t or one i s included in the other, and no element of A i s empty. Then reverse inclusion constitutes a t r e e based on A . A partial ordering i s called a
Conversely, l e t A be a t r e e with base E . To each element x in E , associate the s e t Ax of those elements >/ x (mod A ) : the Ax ordered by reverse inclusion form a t r e e , isomorphic with
A
.
Chapter 4
6.1.
If
A
113
i s a f i n i t e t r e e , t h e n e v e r y base o f a maximal c h a i n o f
A
and e v e r 1
maximal f r e e s e t , have one and o n l y one element i n common. (KUREPA 1952). Let
F
be t h e base o f a maximal t o t a l l y o r d e r e d r e s t r i c t i o n o f
i d e n t i c a l o r incomparable (mod A) t o each element o f to
, since
G
which i s d i s t i n c t f r o m
and
v
.
u ; f o r o t h e r w i s e v c o u l d be added t o belongs t o
u
.
Then
v
v
since
A/F
belongs
of
v
G
,
is a strict
F and so
i s maximal. Thus
a
G
u is
Then e i t h e r
A/F
would
u ,
i s comparable w i t h e v e r y predecessor o f
v
F
and
I n t h a t case
u (mod A)
and comparable w i t h
u
n o t be a maximal c h a i n . Thus and hence
G
i s a maximal f r e e s e t . O r t h e r e e x i s t s an element
G
predecessor o f
.
u be t h e maximum o f t h e c h a i n A/F
maximal f r e e s e t ; l e t
,
A
i s common t o
F
.
G
T h i s r e s u l t does n o t h o l d f o r an i n f i n i t e t r e e . i , a s s o c i a t e an i ' incomparable w i t h i n t e g e r s > i F i n a l l y t h e i ' a r e
Take t h e t o t a l o r d e r i n g o f t h e i n t e g e r s and t o each i n t e g e r
.
i ' >i b u t
element
s e t t o be m u t u a l l y incomparable. Then t h e s e t o f i n t e g e r s d e f i n e s a maximal chain, and t h e s e t o f "primed" i n t e g e r s i s a maximal f r e e s e t (or d e f i n e s a maximal a n t i c h a i n ) ; example due t o KUREPA. The r e s u l t no l o n g e r h o l d s f o r an a r b i t r a r y f i n i t e p a r t i a l o r d e r i n g . 0
Take f o u r elements and l e t a $ . Then there exists for instance an ordinal j < Ht y with Ht x EI j = $ ; and so an element y'< y (mod B) with Ht y' = j (see ch.2 5 3.2). Hence (x,y') < (x,y) and (x,y') belongs to J , so that (x,y) is not minimal in J . Or Ht x Q Ht y = 8 . Then for any (x',y') < (x,y) (mod A K B) we have either x ' < x (mod A ) with y'+ y (mod B) , or conversely x'< x with y ' < y . By the properties of commutative ordinal addition, this always gives Ht x'@ Ht y' < y , so that (x',y') belongs to I , and (x,y) is minimal in J . 0 (2) The first inequality is obvious. For the second inequality, notice that the height of a well-founded partial ordering A equals Sup(i+l) where i denotes the height of any element in A . Similarly for B we obtain Sup(j+l) ; and finally for A s B we obtain Sup(i @ j + l ) , taking in account the preceding (1). Then the second inequality immediately results from the following, which is a consequence of the definition of commutative sum , easily provable by the reader: Sup(i 0 j + 1) < (Sup i+l ) @ (Sup jtl ) . 0
0
7 . 3 . CONJUNCTION OF A SET OF PARTIAL ORDERINGS.
Given a partial ordering A and a set of partial orderings Bi which are all reinforcements of A with common base IAI , we say that the partial ordering A is the conjunction of the Bi if, for any x, y in I A I , we have x \< y (mod A ) iff x, y (mod B.) . J DIMENSION The dimension of a partial ordering A is the least cardinal of a set of chains, each with base IAI and whose conjunction is A . Modulo the axiom of choice, every partial ordering has a dimension; in ZF at least every finite partial ordering has a dimension. The notion of dimension goes back to DUSHNIK, MILLER 1941. We shall denote by Dim A the dimension of a partial ordering A .
Let A A
be a partial ordering, B a restriction of A ; if dimensions exist for and B , then Dim B 4 Dim A .
Every chain has dimension 1
. Given a set
E of cardinality
2 , the free
Chapter 4
117
o r d e r i n g , o r i d e n t i t y r e l a t i o n , has dimension 2 : t a k e a c h a i n based on t h e converse c h a i n ( s o we use t h e o r d e r i n g axiom; ZF s u f f i c e s i f
(1) L e t
7.4.
A
be a p a r t i a l o r d e r i n g w i t h dimension
h
E
E
and
i s countable).
( p o s i t i v e i n t e g e r ) . Add
u which w i l l be t h e minimum o f a p a r t i a l o r d e r i n g , e x t e n s i o n o f A
a new element
\Alu{u}
t o t h e new base
.
h ; similar-
Then we o b t a i n an o r d e r i n g w i t h dimension
l y i f we add a maximum. 0 Take t h e c h a i n s
element
u
(2) Let
h
whose c o n j u n c t i o n i s
Ci
be an i n t e g e r 3 2
, and
A, B
let
j o i n t bases, each one w i t h dimension & h u n i o n o f t h e bases, common e x t e n s i o n o f 0 Assume
h = 2
C, C '
conjunction i s
B
A
t h e p a r t i a l o r d e r i n g based on t h e
a d B
and i n which e v e r y element
l B l , has dimension \< h
.
of
.
whose c o n j u n c t i o n i s
, and
A
t h e two c h a i n s
D, D '
Then, on t h e u n i o n o f t h e two bases, t a k e t h e chains
whose CtD
.0
D'+C'
.
7.5. Every f i n i t e t r e e , e i t h e r i s a c h a i n , o r has dimension 2 0
t h e minimum
Ci
( t h e p r o o f immediately extends t o any g r e a t e r i n t e g e r ) . Consider
t h e two c h a i n s and
add t o each
be two p a r t i a l o r d e r i n g s w i t h d i s -
. Then
i s i n c o m p a r a e w i t h e v e r y element o f
IAl
, then
A
.0
We c o n s t r u c t t h e t r e e f r o m i t s maximal elements, by a f i n i t e sequence o f t h e
two f o l l o w i n g o p e r a t i o n s . (1) u n i o n o f two t r e e s w i t h d i s j o i n t bases, each e l e ment o f one base b e i n g incomparable w i t h each element o f t h e o t h e r ; ( 2 ) a d d i t i o n o f a minimum; f i n a l l y use t h e p r e c e d i n g 7.4. 0 7.6. L e t
ty
a
a
, is
be a c a r d i n a l ; e v e r y d i r e c t p r o d u c t o f c h a i n s whose s e t has c a r d i n a l i a p a r t i a l o r d e r i n g w i t h dimension
cement axiom s u f f i c e s i f
a
4
a
(uses axiom o f choice; r e i n f o r -
i s f i n i t e : see ch.2 Cj 4 . 2 ) . T h i s statement w i l l be
completed i n 7.9 below. 0 Denote t h e c h a i n s by
product o f t h e
Ai
.
( i € I with cardinality
Ai
, consider
i
For each i n d e x
a)
, and
let
A
be t h e d i r e c t
t h e d i r e c t product
Di
o f the
A j ( j # i) . By t h e r e i n f o r c e m e n t axiom, t h e r e e x i s t s a t l e a s t one t o t a l l y o r d e r e d Di
.
then a s s o c i a t e t h e c h a i n
Bi
reinforcement f(i)
< g(i)
Ci
(mod Ai)
of
or
To each
i a s s o c i a t e a unique
f ( i ) = g(i)
A
be a p a r t i a l o r d e r i n g and
restriction o f
A
has dimension
&
p
(axiom o f c h o i c e ) ,
. Then p
iff
(sequence o f t h e f ( j ) f o r j # i)4
with
(sequence o f t h e g ( j ) f o r j # i) (mod Ci) j u n c t i o n o f t h e c h a i n s Bi . 0 7.7. L e t
Ci
I A I , such t h a t f Q g (mod Bi)
w i t h base
t h e d i r e c t product
A
i s t h e con-
a p o s i t i v e integer. I f every f i n i t e
, then
A
has dimension
4p
(uses
THEORY OF RELATIONS
118
u l t r a f i l t e r axiom; ZF suffices i f
A
i s countable).
To each f i n i t e subset F of the base I A l , associate the s e t UF of multirelations ( C , ,..., C p ) , a sequence of p chains with cornon base F , such t h a t , f o r every x , y in F , we have X Q y (mod A ) i f f x G y (mod C1) a n d ... a n d x g y (mod C ) By hypothesis U F i s non-empty f o r every F . Moreover i f F'S F P then every multirelation which i s an element of UF , when r e s t r i c t e d t o F ' , yields an element of U F , . By the coherence lemma ch.2 5 1.3 (equivalent t o the u l t r a f i l t e r axiom), there e x i s t s a multirelation with base I A l whose r e s t r i c t i o n t o each F belongs t o UF . Hence t h i s multirelation i s a sequence of p chains, each of whose base i s I A 1 , and whose conjunction i s the partial ordering A . 0 Ll
.
I n p a r t i c u l a r , i t follows from 7 . 5 t h a t every t r e e ( f i n i t e or i n f i n i t e ) i s e i t h e r a chain or has dimension 2 . be a s e t ; t o each element a of E , associate the singleton a ' of = E-a' of t h i s singleton. Then f o r any given s e t of subs e t s of E which contains a l l the a ' and a " as elements, the partial ordering of inclusion has dimension equal t o Card E (DUSHNIK, MILLER 1941).
7.8. Let
E
a and the complement a"
Let U S denote by < any chain which i s a reinforcement of the partial ordering of inclusion. Then there e x i s t s a t most one element a of E f o r which a " < a ' . Indeed, i f we have two d i s t i n c t elements a , b with a " < a ' and b " < b ' , hence we obtain a " < a ' s b " < b ' s a " thus a = b . For each element a of E , the two s e t s a ' and a" are incomparable with respect t o inclusion. Hence among the chains whose conjunction i s the partial ordering of inclusion, one a t l e a s t s a t i s f i e s the inequality a " < a ' , with x ' < x" f o r a l l elements x # a . Associate t h i s chain t o a : then the s e t of those chains corresponding t o a l l elements of E gives the desired ordering of inclusion. To see t h i s , i t remains t o consider two subsets X, Y of E which are incomparable with respect t o inclusion. So there e x i s t an element x of E with x c X and x I# Y , and an element y w i t h y € Y and y f X . Then the chain associated with x gives Y 5 x " < x ' t, X so Y < X ; similarly the chain associated with y gives X < Y 0
0
.
The preceding statement extends as follows (POUZET, 1969'). Let A be a partial ordering with base E . To each element a of E , associate the i n i t i a l interval a ' of elements x s a (mod A ) , and the i n i t i a l interval a" of elements x < or I a (mod A ) Then f o r any given s e t of subsets o f E which contains a l l the a ' fi a " as elements, the partial ordering of inclusion has dimension equal t o the l e a s t cardinality of those s e t s of chains which a r e r e s t r i c tions of A and whose bases cover the whole s e t E (uses axiom of choice).
.
0
For each a , b
in
E
, if
a"< a'
and b " < b '
(where
c
denotes a chain
119
Chapter 4
which r e i n f o r c e s t h e i n c l u s i o n ) , then the preceding argument proves t h a t a and b a r e comparable: a d b o r b,< a (mod A ) . Hence t o each chain which reinforces the i n c l u s i o n , t h e r e corresponds a t o t a l l y ordered r e s t r i c t i o n I of A , such t h a t the elements u of 111 s a t i s f y u " < u' , w i t h however x ' < x" f o r those elements x in E - I11 . Conversely, given an a r b i t r a r y t o t a l l y ordered r e s t r i c tion I of A , t h e i n e q u a l i t i e s u"( u ' f o r each u i n 1 1 1 , a r e mutually compatible, and a r e compatible with i n c l u s i o n . We can always t a k e each I t o be maximal with r e s p e c t t o inclusion (axiom of c h o i c e ) , and then these conditions have t o be completed by x ' ( x" f o r every x in E - 111 . We end a s in the preceding proof. 0 7.9. Given a d i r e c t product of a - many chains (where a is finite or infinite), each chain being reduced t o t h e elements 0 and 1, we obtain a p a r t i a l ordering isomorphic t o inclusion f o r subsets of a , by replacing each subset b of a by
i t s c h a r a c t e r i s t i c f u n c t i o n , taking t h e value 1 f o r each chain belonging t o and 0 f o r each chain belonging t o a-b . Hence, by t h e preceding, t h e d i r e c t product just considered has dimension Consequently, t h e d i r e c t product of elements, has dimension a . I t has dimension by 7.6. 0
0
3
a
a
-
b
a
,
.
many chains, each havinq a t l e a s t two
by the preceding argument and 7 . 3 , and dimension
4
a
For several developments, e s p e c i a l l y concerning p a r t i a l orderings w i t h dimension 2 , see e x e r c i s e 5 below. Let us f i n a l l y c i t e a r e s u l t due t o HIRAGUCHI 1951 and 1955: f o r every p a r t i a l ordering w i t h f i n i t e c a r d i n a l i t y 4 , we have dimension most equal t o p/2 .
pa
§
8
at
- BOUND
Let A be a p a r t i a l ordering and B an i n i t i a l i n t e r v a l of A . An element u in the d i f f e r e n c e s e t I A l - I B I i s c a l l e d a -(more p r e c i s e l y a minimal s t r i c t upper bound) o f B (mod A ) i f every element s t r i c t l y l e s s than u (mod A ) i s an element of l B l
.
8.1. The bounds of an i n i t i a l i n t e r v a l a r e pairwise incomparable. Consequently i f A i s f i n i t e l y f r e e , then there a r e only f i n i t e l y many bounds f o r each i n i t i a l i n t e r v a l of A . 8.2. Let B be an i n i t i a l i n t e r v a l of a p a r t i a l ordering A u (mod A ) f o r every bound u of B .
x
3
. If
x 6 IBI , then
THEORY OF RELATIONS
120
if
Moreover
A
i s well-founded,
x 2 u
condition
0 Assume t h a t
A
f o r e v e r y bound
IAI
If
A
A
if
x
u
. l B l , there exists
does n o t belong t o
i s a bound. 0
then t h e previous proposition i s f a l s e .
+
t o be t h e t o t a l o r d e r i n g L3
converse o f w ) , and
B
i s e q u i v a l e n t Lo-the
which i s minimal among those elements i n t h e d i f f e -
u s x IBI : this
-
i s n o t well-founded,
0 Take
u f
i s well-founded;
a t l e a s t one element rence s e t
x e IBI
then t h e c o n d i t i o n
(where C J -
W-
i s the retro-ordinal
B t o be t h e i n i t i a l i n t e r v a l u : t h e n t h e r e does n o t
e x i s t any bound. 0 8.3. L e t quence .h)
A
o f elements w i t h
vi
. For
B an i d e a l i n A
be a p a r t i a l o r d e r i n g and ( i = 1, ..., h )
ui
, and
B
< ui
f o r each
(mod A )
e v e r y f i n i t e sevi
f o r e v e r y sequence
o f bounds o f
( i = 1, ...
i , t h e r e e x i s t s an element
t
o f t h e base I B I , s a t i s f y i n g t h e c o n d i t i o n s t 3 vi and t 2 ui f o r each i . 0 Consequence o f t h e d e f i n i t i o n s o f i d e a l and bound, t a k i n g i n t o account that
/
. Notice
that
X+
o f upper bounds (each element o f
X'
X+zX , idempotence: t h e n X+C Y + .
we have t h e i n c l u s i o n X EY
increasing: i f
to
X+
1 - Give examples t o show t h a t ( X u Y)+ ( X n Y ) + can be a p r o p e r subset o f X + A
of
X
, associate
E
X ),
e v e r y element o f
i s an i n i t i a l i n t e r -
i s a closure operation; i . e .
X
X++ = X+ ; and t h e o p e r a t i o n i s
can p r o p e r l y i n c l u d e
.
Y+
X+
u
Y+
, and t h a t
I n any case, e v e r y i n t e r s e c t i o n o f
closed sets i s closed (closed s e t = i d e n t i c a l t o i t s closure). 2 - Let
x
be an element o f
4x
t i a l interval
.
(mod A )
.
E
The c l o s u r e o f t h e s i n g l e t o n o f
I d e n t i f y each
x
x
i s the i n i -
with this i n i t i a l interval: the
s e t o f c l o s u r e s , p a r t i a l l y o r d e r e d by i n c l u s i o n , becomes a p a r t i a l o r d e r i n g e x t e n ding
, called
A
F o r each s e t
X
t h e MAC-NEILLE c o m p l e t i o n o f
, the closure
A (see MAC-NEILLE 1937).
i n c l u d e s t h e i n i t i a l i n t e r v a l generated by
X+
I n g e n e r a l , i t i s d i s t i n c t f r o m i t . Indeed, t a k e and
a Ib
and c o n s i d e r t h e i n i t i a l i n t e r v a l 4a.b)
o r d i n a l W +1 and i t s i n i t i a l i n t e r v a l G,
u
-
Let
be c h a i n s w i t h base
Ci
Ci
c a l l t h e component
Ti
The components o f a l l the chain Ci
.
x (mod A)
x
there
t e r m o f which covers t h e
p r e c e d i n g term, any two c o n s e c u t i v e terms of t h e sequence b e l o n g i n g t o the base o f a same Ai F o r t h i s i t s u f f i c e s t h a t , f o r e v e r y x and el'ery y c o v e r i n g
.
x
, there
Ai
e x i s t s an
x
containing
and y
as elements. I n o r d e r t o a v o i d i s i s o l a t e d i f i t i s incom-
E
useless c o m p l i c a t i o n s , we say t h a t an element o f
p a r a b l e (mod A) w i t h e v e r y o t h e r element: l e t us remove a l l i s o l a t e d elements. Consider t h e o r d e r e d p a i r s l e s s t h a n o r equal t o or
(x',y')
by
A"
.
.
= (x,y)
(x,y)
(x',y')
y
where
where
covers covers
y'
. We
x x'
, if
say t h a t either
(x,y)
is
y,< x ' (mod A)
N o t i c e t h a t t h i s r e l a t i o n forms a p a r t i a l o r d e r i n g denoted
The c o m p a r a b i l i t y (mod A " )
o f two o r d e r e d p a i r s i s e q u i v a l e n t t o t h e
o f a l l t h e f o u r elements which c o n s t i t u t e t h e s e p a i r s .
c o m p a r a b i l i t y (mod A)
Do be a f r e e s e t o f o r d e r e d p a i r s , which i s maximal w i t h r e s p e c t t o i n c l u -
Let
A"
sion, f o r t h e p a r t i a l o r d e r i n g
.
Then t o each element
i
of
ef
A
, such
j e c t i v e l y associate a t o t a l l y ordered r e s t r i c t i o n l e r cardinality
A
generates
. Moreover no
Ai
t r a n s i t i v e l y qenerated by t h e
Ai
Do
we can b i -
is.
A
that
s e t o f c h a i n s w i t h s t r i c t l y smal-
(see BOGART 1970). T h i s l a t t e r p o i n t i s o b v i o u s . Ai
Thus we o n l y have t o prove t h e e x i s t e n c e o f t h e c h a i n s
.
F o r t h a t purpose,
a p p l y t h e statement i n paragraph 1 t o t h e p a r t i a l o r d e r i n g o f o r d e r e d p a i r s w i t h one element c o v e r i n g t h e o t h e r , s t a r t i n g w i t h t h e f r e e s e t
.
Do
For each o f t h e
c h a i n s obtained, i t s u f f i c e s t o t a k e t h e elements i n t h e o r d e r e d p a i r s , and t o complete
transitively.
-
EXERCISE 3
CONJUGATE PARTIAL ORDERINGS
Let
E
and
B a r e conjugates i f , f o r each p a i r o f elements
y
be a s e t and
A, B two p a r t i a l o r d e r i n g s w i t h base
, then
a r e comparable (mod A)
Example:
A
the d i r e c t product r i n g d i r e c t product Notice t h a t i f
A
1 - Let
A, B
x g y
j u n c t i o n of
E
. Another
then
C
z< x
, and 6 i s
example: l e t
C
x
D-
where
D
, and
x
A
and
conversely.
be two chains;
i s a conjugate o f B-
, then
B
A
D
(see ch.2
y (mod A o r mod B)
A u B and
A
,is
. Deduce
u B-
1.7).
i s also a conjugate o f the
a chain. Notice t h a t
that
with the following condition: i f
or z 7 y (mod C)
5
.
be conjugates; show t h a t t h e d i s j u n c t i o n r e l a t i o n
iff x (
if
the free p a r t i a l
C, D
i s t h e converse o f
s i o n i s 1 o r 2 (see 7.3). O r e q u i v a l e n t l y i f f Cement
E
say t h a t
,
E
C x D d e f i n e d i n 5 7 has as a c o n j u g a t e t h e p a r t i a l o r d e -
converse p a r t i a l o r d e r i n g
by
. We
E of
t h e y a r e incomparable (mod B)
i s a t o t a l o r d e r i n g , o r c h a i n on
o r d e r i n g ( i d e n t i t y r e l a t i o n ) on
x, y
.
Au B d e f i n e d A
i s t h e con-
A
has a c o n j u g a t e i f f i t s dimen-
A
has a t o t a l l y o r d e r e d r e i n f o r -
x< y
and
x Iz
and
y I z (mod A)
,
Chapter 4
-
2
S t a r t w i t h a chain
C
and a s e t
U
129
o f intervals o f
U
i n c l u s i o n . Show t h a t one o b t a i n a c o n j u g a t e o f
C
,
p a r t i a l l y ordered by
by o r d e r i n g i n t e r v a l s which C
a r e incomparable w i t h r e s p e c t t o i n c l u s i o n , by means o f t h e p o s i t i o n i n
C
t h e i r l e f t c u t (i.e. the i n i t i a l interval o f
of
s i t u a t e d on t h e l e f t ) . One can as
w e l l o r d e r t h e s e incomparable i n t e r v a l s by t h e p o s i t i o n o f t h e i r r i g h t c u t . Deduce t h a t a p a r t i a l o r d e r i n g has dimension 1 o r 2 i f i t i s isomorphic t o t h e p a r t i a l o r d e r i n g o f i n c l u s i o n between i n t e r v a l s o f a g i v e n c h a i n . I n p a r t i c u l a r u s i n g 6.2 ( u l t r a f i l t e r axiom), e v e r y t r e e has dimension 1 o r 2, which was a l r e a d y mentioned i n 7.7.
-
3
Conversely, l e t
, followed
A
and
B
be c o n j u g a t e s . Consider t h e c h a i n
A- u B
with
A u B w i t h base E ' (an isomorphic copy o f E ) . Associate t o each element x o f t h e p a r t i a l o r d e r i n g A , hence x b e l o n g i n g t o E , t h e interval (x,x') where x ' i s t h e r e p l i c a o f x i n E ' (communicated by POUZET i n 1968, u n p u b l i s h e d ) .
base
E
EXERCISE 4 A
Let
-
by
WELL-ORDERED RESTRICTION OF MAXIMAL LENGTH (LEMMA 4.5 REVISITED)
be a w e l l - f o u n d e d p a r t i a l o r d e r i n g . Then e i t h e r t h e r e e x i s t i n f i n i t e l y
many elements, a l l o f d i f f e r e n t h e i g h t s (mod A) which a r e m u t u a l l y incomparable, o r there e x i s t s a well-ordered r e s t r i c t i o n o f
w i t h t h e same h e i g h t ( u s e s u l t r a -
A
f i l t e r axiom; POUZET 1979).
-
1
Suppose t h e f i r s t c o n c l u s i o n does n o t h o l d , and argue by i n d u c t i o n on t h e
height o f
/3 +
8
A
.
Ht A
Assume f i r s t t h a t
with
/3
non-zero and
i s decomposable, hence o f t h e form
I < /3 + J' .
Let
denote t h e r e s t r i c t i o n o f
C
A t o elements whose h e i g h t (mod A ) i s g r e a t e r t h a n o r equal t o A . For each x i n ICI , we have H t x (mod A ) = 0 + H t x (mod C) ( p r o o f by i n d u c t i o n ) . Hence
. By t h e
Ht C = C"
C
of
induction hypothesis, there e x i s t s a well-ordered r e s t r i c t i o n
with height
. Let
h e i g h t >/
/3 (see
ch.2
ordered r e s t r i c t i o n of
A
with height
2 - Assume now t h a t
5 B"
3.5).
be t h e minimum o f
u
u
t o elements s t r i c t l y l e s s t h a n
. The
C"
A
restriction o f
i s a well-founded p a r t i a l ordering
B
with
By t h e i n d u c t i o n h y p o t h e s i s , t h e r e e x i s t s a w e l l -
of
B
w i t h h e i g h t ('3.
B" +
Then
C"
i s a restriction
fi + 8 . Ht A =
c(
, an
i n d e c m p o s a b l e o r d i n a l ; and suppose t h a t t h e
f i r s t c o n c l u s i o n does n o t h o l d . By 4.5 ( u l t r a f i l t e r axiom), i t s u f f i c e s t o prove that there exists a r e s t r i c t i o n o f
A
w i t h t h e same h e i g h t d,
f r e e and hence a w e l l p a r t i a l o r d e r i n g . L e t pose d i n t o a sum s t r i c t l y l e s s t h a n o( For each
i
c. k , c o n s i d e r
Sup
o(
=
o(
the restriction
i s b o t h g r e a t e r t h a n o r equal t o
2 OC
, which
i s finitely
be t h e c o f i n a l i t y o f o( ; decorn-
( i c k ) with the
o< , and
k
(ch.2
Ai
5 of
increasing, a l l
o(
5.5). A
t o elements whose h e i g h t
( j < i) and s t r i c t l y l e s s t h a n
o(
j
130
THEORY OF RELATIONS
i) . This Ai has height H . By the induction hypothesis, and since there do not exist infinitely many mutually incomparable (mod Ai) elements with different heights, there exists a well-ordered restriction Ci of Ai with the same height M i . Let D denote the restriction of A to the union of the bases hence greater than or lCil . Firstly Ht D is strictly greater than each o( equal to o( = Sup M i ; so that Ht D = o( . Secondly, there do not exist infinitely many mutually incomparable elements in D , since they would all have different heights (mod A), which contradicts our hypothesis. Hence D is finitely free, s o that we can apply 4.5. ( j6
3 - As opposed to what happens in 4.5, we cannot require that the well-ordering having the same height as A , have one and only one element of each height (mod A). Consider the following counterexample due to POUZET. Take the points (XJ) of coordinates natural numbers with y~ x . Let (x,y) < (x',y') iff either x = x' and y < y' , or x d x ' and y' 3 y+2 . Then the height of each point (x,y) is y , hence Ht A = w . However, a totally ordered restriction of A isomorphic with w can have at most a finite sequence of points with heights 0,1,2, ... ; for example the points (u,O), ( u , l ) , ... , (u,u) (where u is an integer). After which one must pass to a point with ordinate greater than or equal to u t 2 , hence with height greater than or equal to u+2 .
EXERCISE 5
- REINFORCEMENT OF A WELL PARTIAL ORDERING (REVISITED)
1 - Given a well partial ordering A , there exists a well-ordered reinforcement of A . This is an immediate consequence of the reinforcement axiom (ch.2 5 4.2) by using proposition 3.5.(1) above. We propose to obtain this result in a more economic manner, using only the axiom of choice for finite sets (ch.1 5 2.10). For this, to each ordinal u strictly less than Ht A , associate the finite set Fu of those elements with height u in A ; then the finite set of chains with base Fu . Then the axiom of choice for finite sets associates to each u one of these chains. Finally it suffices to take the sum along the u . 2 - Notice that with the considered axiom, a partial ordering A is a well partial ordering iff: (i) every totally ordered reinforcement of A is a well-ordering, and (ii) there exists a totally ordered reinforcement of A (use 3.5.(3): ZF suffices). More precisely, the preceding proposition is equivalent to the following weakening of the axiom of choice for finite sets: "for every well-orderable set of finite mutually disjoint sets, there exists a choice set"(communicated by POUZET in 1979).
131
5
CHAPTER
EMBEDDABILITY BETWEEN PARTIAL OR TOTAL ORDERING
§
1 - EMBEDDABILITY,
IMMEDIATE
FAITHFUL
EXTENSION,
EXTENSION
EMBEDDABILITY, EQUIMORPHISM Let
be two r e l a t i o n s o f t h e same a r i t y . We say t h a t
R, S
R
i s embeddable
S o r i s s m a l l e r t h a n S under e m b e d d a b i l i t y , o r t h a t S admits an embedding o f R o r i s g r e a t e r t h a n R , i f f t h e r e e x i s t s a r e s t r i c t i o n o f S in
R ; we w r i t e R + S o r S i s s t r i c t l y embeddable i n S
isomorphic w i t h We say t h a t
.
o r s t r i c t l y smaller than
R
, or
S admits a s t r i c t embedding o f R o r i s s t r i c t l y g r e a t e r t h a n R , denoted
that by
R
R
R < S
or
We say t h a t
S > R , i f f
R$S
but
i s equimorphic w i t h
R
The comparison r e l a t i o n
4
S $ R .
S , denoted R
5 S
, iff
R 6'S
and
S
6
R
i s r e f l e x i v e and t r a n s i t i v e , hence d e f i n e s a quasi-
o r d e r i n g o n each s e t o f r e l a t i o n s . Moreover equimorphism i s symmetric and hence d e f i n e s an e q u i v a l e n c e r e l a t i o n . E m b e d d a b i l i t y i s n o t a n t i s y m m e t r i c , even up t o isomorphism: see t h e f o l l o w i n g examples. Let
be t h e c h a i n o f t h e r a t i o n a l s , and
Q
a l a s t element: t h e n
Q+l t h e e x t e n s i o n o b t a i n e d by adding
.
Q I Q+l
L e t LJ be t h e c h a i n o f t h e n a t u r a l numbers, and ch.2
5 5
1.7). Then
u-.G)E
a-
t h e converse c h a i n ( s e e
1 + ( W - . w ) (the ordinal product i s defined i n
3.7). I n t h e c h a i n o f n a t u r a l numbers, r e p l a c e each even number by Z ( t h e c h a i n of p o s i t i v e and n e g a t i v e i n t e g e r s ) and each odd number by a f i n i t e chain. We o b t a i n continuum many m u t u a l l y non-isomorphic chains, a l l o f which a r e equimorphic.
ch.2
1.1. L e t
R, S
be two equimorphic r e l a t i o n s . Then t h e r e e x i s t s a p a r t i t i o n o f
I R I i n t o two d i s j o i n t subsets D, D ' , and a p a r t i t i o n o f 1st i n t o two d i s j o i n t subsets E, E ' w i t h R/D isomorphic t o S/E and R / D ' isomort h e base phic t o where
S/E'
f
and
.
Repeat t h e p r o o f o f BERNSTEIN-SCHRODER's theorem (ch.1 g
5 1.4),
become isomorphisms from one r e l a t i o n o n t o a r e s t r i c t i o n o f
the other. The converse i s f a l s e , even f o r c h a i n s . Indeed, t h e c h a i n w o f t h e n a t u r a l numbers and t h e c h a i n
w+1
g i v e r i s e t o p a r t i t i o n s s a t i s f y i n g t h e above condi-
t i o n s . S i m i l a r l y f o r t h e incomparable c h a i n s U+c.4-
and
Z = W-+
.
.
132 1.2.
THEORY OF RELATIONS IMMEDIATE EXTENSION
Given a r e l a t i o n R , we say t h a t S i s an immediate extension o f R i f S i s an extension, and furthermore i f there does n o t e x i s t any s t r i c t l y intermediate r e l a t i o n T such t h a t R < T < S w i t h respect t o embeddability. We say a l s o t h a t S , o r any r e l a t i o n equimorphic w i t h S , i s an immediate successor o f R w i t h respect t o embeddability.
.
Moreover .if R For each r e l a t i o n R , t h e r e e x i s t s an immediate extension o f R has a r i t y >/ 1 , then t h e r e e x i s t a t l e a s t two immediate extensions (HAGENDORF 1977, p r o p o s i t i o n VI.5.6). 0 Suppose f i r s t t h a t
i s a 0-ary r e l a t i o n , say
R
R = (E,+)
: then i t s u f f i c e s t o
replace t h e base E by a s e t w i t h immediately g r e a t e r c a r d i n a l i t y : see ch.2 5 3.10. Suppose t h a t R i s a unary r e l a t i o n . L e t a be t h e c a r d i n a l i t y o f the s e t o f e l e -
R
ments g i v i n g t h e value (+) t o i t s u f f i c e s t o replace e i t h e r using ch.2 5 3.10. Suppose now t h a t R
a
has a r i t y
, and or
b t h e analogous c a r d i n a l i t y f o r (-). Then b by an immediately g r e a t e r c a r d i n a l , again
n >/ 2
. Add
t o t h e base E o f
R
a set
D dis-
t o have base E u D w i t h R+/E = R and R+/D j o i n t w i t h E , and d e f i n e R' always (t), and f i n a l l y w i t h R+ t a k i n g the value (+) f o r those n-tuples contain i n g a t l e a s t one term i n D F i n a l l y choose f o r d = Card D t h e l e a s t aleph f o r which Rt i s n o t embeddable i n R , hence R+> R L e t us prove t h a t R' i s an
.
.
immediate extension o f R ; t h e r e l a t i o n R- s i m i l a r l y defined by exchanging (t) and ( - ) , being another immediate extension, obviously incomparable w i t h R' with respect t o embeddabi 1ity. Suppose f i r s t t h a t d i s an i n f i n i t e aleph, and t h a t t h e r e e x i s t s a s t r i c t l y i n t e r mediate r e l a t i o n T w i t h R < T < R' . Consider T as a r e s t r i c t i o n o f Rt Then
.
.
D n IT1 has c a r d i n a l i t y d Indeed i f i t had c a r d i n a l i t y c d , R (more p r e c i s e l y , i n a r e s t r i c t i o n o f Rt t o E increased w i t h (c:d) many elements o f D ). Now p a r t i t i o n DnITI i n t o two the i n t e r s e c t i o n
then T would be embeddable i n
d i s j o i n t subsets, each w i t h c a r d i n a l i t y d
, say
-
D'
and D"
. Then
T
i s isomor-
p h i c w i t h i t s r e s t r i c t i o n t o I T 1 D" , so t h a t R i s embeddable i n t h i s r e s t r i c t i o n , and f i n a l l y R+ i s embeddable i n T : c o n t r a d i c t i o n . Now i t remains t o consider t h e case where d = Card D i s f i n i t e . We f i r s t see t h a t d = 1 . Indeed assume t h a t d f i n i t e and >, 2 Then by hypothesis, the extension of R obtained by adding o n l y one element u t o t h e base I R 1 , w i t h value (t)
.
f o r a l l n-tuples c o n t a i n i n g u , i s embeddable i n R , thus equimorphic w i t h R I t e r a t i n g t h i s , t h e s i m i l a r extension obtained by adding 2 elements, i s s t i l l equimorphic w i t h
R , and so on u n t i l we add d elements: c o n t r a d i c t i o n .
Now examine t h e case where singleton constitutes
D
d = 1
. Call
. Consider
u the supplementary element, whose
again the intermediate r e l a t i o n T as a
.
Chapter 5
133
.
r e s t r i c t i o n o f R+ ; obviously u belongs t o t h e base I T We say t h a t an e l e ment x i n the base i s a (+)-element (mod R ) i f f every n - t u p l e which contains x gives value (+) t o R . Analogous d e f i n i t i o n f o r a (+)-element (mod T) ; i n p a r t i c u l a r , u i s a (+)-element (mod T) . Every (+)-element (mod R) belongs t o the base I T 1 and i s a (+)-element (mod T). Indeed otherwise, i f x i s a (+)-element (mod R ) and does n o t belong t o I T I , t h e n by r e p l a c i n g u by x we could embed T i n R : c o n t r a d i c t i o n . E i t h e r t h e r e e x i s t D e d e k i n d - i n f i n i t e l y many (+)-elements (mod R ) . Then R+ isomorphic w i t h R : c o n t r a d i c t i o n . O r t h e s e t o f (+)-elements (mod R ) has D e d e k i n d - f i n i t e c a r d i n a l i t y , say
h
. Then t h e r e e x i s t a t l e a s t
is
h + l many
(+)-elements (mod T) . Consider a r e s t r i c t i o n R ' o f T which i s isomorphic with R Since we have e x a c t l y h many (+)-elements (mod R ' ) and a t l e a s t h t l many (+)-elements (mod T) , t h e r e e x i s t s a t l e a s t one (+)-element (mod
.
T) ,
.
say v , which does n o t belong t o t h e base I R ' I Then the r e s t r i c t i o n o f T t o t h e base I R'I p l u s t h e element v i s isomorphic w i t h R' : c o n t r a d i c t i o n . 0
1.3. FAITHFUL EXTENSION
.
(1) L e t R , S be two n-ary r e l a t i o n s ( n >r 1) Assume t h a t S does not Then t h e r e e x i s t s a s t r i c t l y g r e a t e r extension T admit an embedding o f R
.
.
of S which does n o t _admit an embedding o f R We c a l l i t a f a i t h f u l extensio? o f S modulo R . Moreover we can choose T t o be an immediate extension o f S : see HAGENDORF 1977. The statement i s obviously f a l s e f o r a r i t y zero.
R aqd S are unary. L e t at be 'the c a r d i n a l i t y o f t h e s e t o f elements g i v i n g the value (+) t o R , and a- the analogous c a r d i n a l i t y f o r (-); s i m i l a r l y l e t b+ and b- be t h e analogous c a r d i n a l i t i e s f o r S . Since R $ S ,
0 Suppose f i r s t t h a t
.
e i t h e r b + < a+ o r b-< aSuppose t h e f i r s t case holds, t h e argument being analogous f o r t h e second case. It s u f f i c e s t o take an extension of S i n which bt i s preserved and b- i s replaced by an immediately l a r g e r c a r d i n a l . Suppose t h a t R and S have a r i t y n 3 2 Add t o t h e base E o f S a s e t '0
.
which i s d i s j o i n t from
E
, and
d e f i n e the extension T+
of
S w i t h base
EvD',
THEORY OF RELATIONS
134
t a k i n g t h e v a l u e (+) f o r those choose
w i t h c a r d i n a l ( a l e p h ) s u f f i c i e n t l y l a r g e t o have
D+
+
w i t h t h e value ( - ) , t h u s o b t a i n i n g
and
0-
T->
.
S
.
Do t h e same
R
4:'
T+> S
We c l a i m t h a t
. Also
D
n - t u p l e s c o n t a i n i n g a t l e a s t one t e r m o f
or
, which y i e l d s o u r c o n c l u s i o n . Indeed suopose t h e c o n t r a r y , and c o n s i d e r R as a r e s t r i c t i o n o f T+ , The base o f R i s n o t a subset o f E , s i n c e R $ R $T-
hence t h e r e e x i s t s an element
u+
R
such t h a t
u+
n - t u p l e c o n t a i n i n g a t l e a s t one t e r m equal t o
S,
takes t h e v a l u e (+) f o r each
. There
e x i s t s an analogous e l e -
ment f o r t h e v a l u e ( - ) : c o n t r a d i c t i o n . 0 be a r e l a t i o n w i t h a r i t y 5 2
R
(2) Let
e x i s t s a common e x t e n s i o n o f The c o n t r a p o s i t i v e i s : i f X
3 S2 , t h e n
or
6 S1
R
and
S1
and
S1&
R
R,< X
f o r every
R 6 S2
.
X
and
El
.
E2
p a r t i t i o n o f t h e base o f e v e r y element =
R
have d i s j o i n t
S2
taking
E2
and t h e o t h e r i n
El
R.
and . ,Sh.
.
E2
or
S+
contrary; then there e x i s t s a
i n t o two non-empty d i s j o i n t subsets such t h a t , f o r
i n one subset and
u
. Suopose t h e
R
.
S1,.
f o r t h e value ( - ) . It s u f f i c e s t o see t h a t e i t h e r
S-
does n o t a d m i t an embedding o f
S-
and
S1
St be t h e common e x t e n s i o n w i t h base El
Let
there
X >,S1
Extend t h i s t o any f i n i t e sequence
t h e value (+) f o r those o r d e r e d p a i r s w i t h one t e r m i n Analogously d e f i n e
. Then
S2 & R
which s a t i s f i e s b o t h
0 Take t h e case o f a b i n a r y r e l a t i o n , and suppose t h a t
bases
and
which does n o t a d m i t an embedding o f
S2
v
i n t h e o t h e r , we have
R(u,v) = R(v,u)
=
+ ; same c o n c l u s i o n w i t h t h e v a l u e ( - ) . Note t h a t , g i v e n two p a r t i t i o n s o f t h e u, v
base, each w i t h two non-empty d i s j o i n t s e t s , t h e r e e x i s t two elements
in
t h e base which a r e separated b o t h by t h e f i r s t p a r t i t i o n and by t h e second. Thus R(u,v)
=
+ and - :
2 and non-empty base. Suppose t h a t
S
& R1
u, v
t h e r e e x i s t two elements
g i v i n g simultaneously
contradiction. 0 The p r o p o s i t i o n i s o b v i o u s l y f a l s e f o r unary r e l a t i o n s . (3) Let
S
. Then
S%R3
>/
have a r i t y
there e x i s t s a proper extension
embeddabilities
R1 ,
S+*
S+$
R2
and
S+ f S
S+$ R3
.
element Let
S2
a
and s e t t i n g
S,(a,x)
of
= S (x,a)
. Let
S =
+
S1
, say
r o l e of
. Hence
i n t h e base o f t h i s
t o f i x t h e ideas, p l a y i n g t h e r o l e o f a
and
R2
x
in
S3
. Suppose f i r s t l y
a
t h a t t h e base
in
.
Si
(i
IS1
i n t h e base S3
=
.
be o b t a i n e d w i t h
S (x,a) = - f o r e v e r y x i n I S ( , and moreover 3 w i t h t h e same c o n d i t i o n s , except t h a t S4(a,a) = - .
R
and
be o b t a i n e d by a d d i n g a new
f o r every
Suppose o u r c o n c l u s i o n i s f a l s e . Then t h e r e e x i s t two al
R1
and
an embedding o f a same
$ R2
which r e s p e c t s t h e non-
1 be s i m i l a r l y o b t a i n e d w i t h ( - ) ' i n s t e a d o f ( + ) . L e t
S (a,x) = + 3 F i n a l l y S4
S
F o r t h e a r i t y 1 o r f o r empty
base, t h e p r o p o s i t i o n i s o b v i o u s l y f a l s e , even w i t h o n l y 0 Consider t h e 4 f o l l o w i n g e x t e n s i o n s
,
S3(a,a)
1,2,3,4)
= +
.
which a d m i t
R , t h e r e e x i s t s an element
S1 and an a3 p l a y i n g t h e
I R I has c a r d i n a l i t y
32 .
Chapter 5 Then
al
and
and
R(x,a3)
a3 =
-
are d i s t i n c t , since x # a3
f o r every
.
135
R(x,al)
+
=
R(al,a3)
takes s i m u l t a n e o u s l y t h e
f o r every
value (+) and t h e v a l u e ( - ) : c o n t r a d i c t i o n . Analogous argument f o r and
S1
,
S4
S2
and
S3
,
and
S2
Suppose now t h a t t h e base I R I takes t h e v a l u e ( - ) . Since S >
,
S4
x # al
= R(al,x)
Moreover
and
S3
and
S1
,
S2
.
S4
has c a r d i n a l i t y 1, and t o f i x t h e ideas, t h a t R R and by h y p o t h e s i s S non-empty, n e c e s s a r i l y S
i s r e f l e x i v e . As p r e v i o u s l y d e f i n e e x t e n s i o n s S1, S2, S3 which now a r e a l l t h r e e r e f l e x i v e . E i t h e r R1 = R 2 = R3 = R and t h e n o u r c o n c l u s i o n h o l d s . O r R1 and possibly
R , t h u s have c a r d i n a l i t i e s
are d i s t i n c t from
R2
our conclusion i s false: then
, and
( i = 1,2,3)
R1
1.4. Consider an 13 -sequence o f r e l a t i o n s
Ria A
i
f o r every
, then
R+
suppose Si
A
Ri
( i integer)
3
o f common a r i t y
A
2
be a r e l a t i o n o f t h e same a r i t y .
t h e r e e x i s t s a comnon e x t e n s i o n o f a l l t h e
which does n o t a d m i t an embedding o f 0 Let
. Again
t h e argument t e r m i n a t e s as p r e v i o u s l y . 0
and w i t h m u t u a l l y d i s j o i n t bases. L e t
.-have
2
f o r i n s t a n c e i s embeddable i n a t l e a s t two
Ri
.
denote t h e common e x t e n s i o n o f t h e
Ri
on t h e u n i o n o f t h e bases, which
takes t h e v a l u e (+) f o r a l l t h o s e n - t u p l e s ( n = a r i t y ) c o n t a i n i n g a t l e a s t two terms t a k e n f r o m two d i s t i n c t bases. Analogously d e f i n e t h e e x t e n s i o n that
A
i s embeddable b o t h i n
, there
Ri
R+
and
R-
n e c e s s a r i i y e x i s t two elements
. Since x, y
t r a n s f o r m e d i n t o two elements i n two d i s t i n c t
A
, simultaneously
and second embedding. tience f o r an n - t u p l e c o n t a i n i n g b o t h for
I A I , which a r e
i n t h e base
lRil
. Suppose
R-
i s n o t embeddable i n any
x
and
f o r the f i r s t
, we
y
have
t h e v a l u e (+) and t h e v a l u e ( - ) : c o n t r a d i c t i o n . 0
A
5 2 - EMBEDDABILITY
BETWEEN PARTIAL
(KRUSKAL); CANTOR'S GLEASON); TOURNAMENT
OF FINITE
TREES
(DILWORTH,
ORDERINGS;
WELL PARTIAL ORDERING ORDERINGS
THEOREM FOR PARTIAL
2.1. There e x i s t i n f i n i t e l y many f i n i t e p a r t i a l o r d e r i n g s which a r e m u t u a l l y incom-parable w i t h r e s p e c t t o e m b e d d a b i l i t y . 0 Let
a'< i
be t h e p a r t i a l o r d e r i n g on 5 elements
A1 b'
A
or
and
.
Chapter 5 t h e r e does n o t e x i s t any c h a i n
X
satisfying
be a c h a i n and
U
the
immediate e x t e n s i o n o f
A
(HAGENDORF 1972).
Let
A
143
<X
/C
iff
s c a t t e r e d ; t h e general case remains unsolved.
For t h e dual o f t h e above statement, i . e . t h e e x i s t e n c e o f t h e infimum, we have t h e easy example
A = w + 1 and
5 5 - DECREASING SEQUENCES DUSHN I K, MILLER, S IERP INSKI 5.1.
Let
A, B
B = Z
w i t h t h e infimum
C =
AND SETS OF INCOMPARABLE
w . CHAINS
OF REALS:
be two chains, each o f which i s embeddable i n t h e r e a l s . Then
t h e r e a r e a t most continuum many r e s t r i c t i o n s o f
B
isomorphic w i t h
0 Consider t h e base
E
o f r e a l s , and l e t
lBl
as a subset o f t h e s e t
A
. F be
Chapter 5
IBI
a subset o f
B/F
such t h a t
For e v e r y subset
X
i s isomorphic w i t h
1 B l such t h a t
of
147
A
( i f t h e r e e x i s t s such).
i s isomorphic w i t h
B/X
f X f r o m F o n t o X , hence f r o m F t h e r e a r e continuum many s t r i c t l y i n c r e a s i n g maps f r o m F i n t o
5
8.4;
and f i n a l l y
5.2. L e t
A
X = fXo(F)
i s determined by
f X( n o t a t i o n
A
, there
into
a s t r i c t l y i n c r e a s i n g map
E
exists
. Now
E : see c h . 2 O
i n ch.1
5
1.2). U
embeddable i n t h e
be a c h a i n o f continuum c a r d i n a l i t y , which i s
chain o f the reals. (1) There e x i s t s a s t r i c t l y s m a l l e r ( w i t h r e s p e c t t o e m b e d d a b i l i t y ) r e s t r i c t i o n A which has continuum c a r d i n a l i t y (DUSHNIK, MILLER 1940; uses t h e axiom o f
of
choice). ( 2 ) Even s t r o n g e r , t h e r e e x i s t s a r e s t r i c t i o n l i t y , such t h a t no i n t e r v a l o f
,of
A
B
of
A
w i t h continuum c a r d i n a -
continuum c a r d i n a l i t y , i s ernbeddable i n
B
(HAGENDORF 1977, u n p u b l i s h e d ) . ( 3 ) F o r e v e r y denumerable c h a i n preceding ( 2 ) ) . 0 (2) Let
X
U
we have
be any subset o f t h e base
an i n t e r v a l o f
A
A $ B.U
(where
I A 1 such t h a t A/X
B
satisfies the
i s isomorphic w i t h
o f continuum c a r d i n a l i t y . For each such i n t e r v a l , by t h e p r e -
c e d i n g p r o p o s i t i o n , t h e r e a r e a t most continuum many corresponding s e t s
X
. More-
R o f r e a l s , each i n t e r v a l o f A i s t h e r e s t r i c t i o n t o I A l o f an i n t e r v a l o f R , which i s i t s e l f
over, i f we c o n s i d e r
A
as a r e s t r i c t i o n o f t h e c h a i n
d e f i n e d by i t s e n d p o i n t s . Consequently t h e r e a r e continuum many such i n t e r v a l s . X
Finally, the set o f a l l the
5
Apply ch.2 where
C
has a t most continuum c a r d i n a l i t y .
8.1 (axiom o f c h o i c e ) . There e x i s t s a s e t
and
D
=
IAl
a l l the intersections thus the r e s t r i c t i o n
-C
C
included i n
IAI ,
b o t h a r e e q u i p o t e n t w i t h t h e continuum, as w e l l as
.
C n X and D n X Consequently no X i s i n c l u d e d i n C B = A/C a d m i t s no embedding o f any i n t e r v a l o f A which
has continuum c a r d i n a l i t y . 0 0
( 3 ) Suppose
A 6B.U
.
Then t h e base
I A I i s p a r t i t i o n e d i n t o c o u n t a b l y many
i n t e r v a l s , each c o r r e s p o n d i n g w i t h an element o f
U
. At
t e r v a l s i s e q u i p o t e n t w i t h t h e continuum: see c h . 1 § 4.3, o v e r i t i s embeddable i n
B
, contradicting
l e a s t one o f t h e s e i n axiom o f c h o i c e . More-
the preceding (2). 0
N o t i c e t h a t (1) i m p l i e s t h e e x i s t e n c e , s t a r t i n g f r o m t h e c h a i n o f t h e r e a l s , o f a s t r i c t l y decreasing w -sequence o f c h a i n s . We s h a l l see t h a t such a sequence does n o t e x i s t f o r s c a t t e r e d c h a i n s , i . e . those i n which t h e c h a i n
Q
o f the
r a t i o n a l s i s n o t embeddable: see ch.8
5
5.3.
R t h e c h a i n o f t h e r e a l s . There e x i s t two
Let
subsets
E C,
be t h e s e t o f r e a l s and
D
of
E
4.4.
which a r e d i s j o i n t , e q u i p o t e n t w i t h t h e continuum, dense
,
148
THEORY OF RELATIONS
in
R and such t h a t , f o r each subset X f D , every r e s t r i c t i o n of R isomorphic w i t h R / ( C u X ) , e i t h e r has base C u X , o r i t s base contains a t l e a s t one
element o f E
-
(C u D)
.
Consequently f o r Y C X cD , we have the s t r i c t embeddability R / ( C uY)< R/(CuX). For X and Y s u b s e t s of D which a r e incomparable with r e s p e c t t o i n c l u s i o n ,
the preceding two restrictions are incomparable with respect to embeddability (SIERPINSKI 1950; see a l s o ROSENSTEIN 1982; uses axiom of c h o i c e ) . Take the s e t s C , D in ch.2 5 8.2.(2) (axiom of c h o i c e ) , where t h e f i designate a l l isomorphisms from R i n t o R , d i s t i n c t from t h e i d e n t i t y . For such an isomorphism f , i f a real x i s mapped t o f x # x , f o r example i f f x > x (mod R ) , then every real i n t h e i n t e r v a l ( x , f x ) i s mapped t o a s t r i c t l y g r e a t e r r e a l ,
0
hence f y # y f o r continuum many r e a l s y . Notice t h a t C and 0 a r e d i s j o i n t and each equipotent with t h e continuum. Moreover by t h e same proposition we have f " ( C ) # C f o r each considered isomorphism f ; s i m i l a r l y with D T h u s by ch.2 5 8 . 5 , t h e s e t s C and D a r e both dense (mod R ) . Take an a r b i t r a r y subset X of D and an isomorphism g of R / ( C u X ) i n t o R , which i s d i s t i n c t from t h e i d e n t i t y . Since C , hence C u X , i s dense, t h e r e
.
e x i s t s an isomorphism gt of R i n t o R , which extends g t o t h e domain E of a l l r e a l s : see ch.2 5 8.3. Hence g+ i s one of t h e previously considered i s o morphisms f . By ch.2 5 8.2, t h e s e t of images g"(C) = ( g + ) " ( C ) i s not i n c l u ded in C u D . 0 This immediately implies t h e e x i s t e n c e of a s t r i c t l y decreasing (with r e s p e c t t o embeddability) sequence, indexed by t h e continuum, of chains of r e a l s . Also t h e e x i s t e n c e of a s e t , equipotent w i t h t h e continuum, of mutually incompar a b l e chains of r e a l s (SIERPINSKI 1950).
§
6 - SUSLINC H A I N
AND
SUSLINTREE
in A : Given a chain A , t h e reader i s acquainted w i t h t h e notion of a s e t a subset D of t h e base f o r which, given any two elements x < y (mod A ) , t h e r e e x i s t s an element t of D w i t h x st d y (mod A ) The chain of r e a l s , and more g e n e r a l l y any chain A which i s embeddable in the chain of r e a l s , s a t i s f i e s t h e two following conditions: (1) t h e r e e x i s t s a countable s e t which i s dense in A ; (2) every s e t of mutually d i s j o i n t i n t e r v a l s of A , none o f which a r e s i n g l e t o n s , i s countable.
The condition (2) follows from ( 1 ) . I f D i s countable and dense i n A , then every non-singleton i n t e r v a l contains a t l e a s t one element of D . Two d i s j o i n t i n t e r v a l s cannot contain a same element
0
Chapter 5
of
D , so there are countably many i n t e r v a l s .
149 0
SUSLIN'S HYPOTHESIS (see SUSLIN 1920) The axiom called S u s l i n ' s hypothesis, a s s e r t s that the preceding condition (2) implies ( l ) , hence t h a t ( 1 ) & (2) are equivalent. This axiom i s neither provable nor refutable in ZF, even with the axiom of choice and even with the generalized continuum hypothesis. More precisely JECH and TENNENBAUM have proved the consistency of the existence o f a Suslin t r e e ( i . e . the negation of the axiom) with ZF (modulo the consistency of Z F ) . Whereas SOLOVAY and TENNENBAUM have proved the relative consistency of the axiom: see JECH 1978. For a detailed discussion of Suslin chains and Suslin t r e e s , as well as f o r the advanced r e s u l t s o f JENSEN, see for example DEVLIN, JOHNSBRiTEN 1974.
6.1. SUSLIN CHAIN I t i s more convenient t o work with the negation of Suslin's hypothesis, rather than the hypothesis i t s e l f . We say t h a t a chain i s a Suslin chain i f i t s a t i s f i e s ( 2 ) and n o t ( l ) , i . e . i f every s e t of non-singleton mutually d i s j o i n t intervals i s countable, y e t there e x i s t s no countable s e t which i s dense in the chain. A Suslin chain i s uncountable; moreover i t admits an embedding of t h e c h a i n Q rationals (uses axiom of choice). 0 The inexistence of any countable dense s e t implies t h a t the chain i t s e l f be
of
uncountable: i t s cardinality i s a t l e a s t W1 (axiom of choice). If Q i s n o t embeddable in i t , then e i t h e r the ordinal W 1 or i t s converse i s embeddable in i t : see 3.5. Hence there e x i s t uncountably many non-singleton mutually d i s j o i n t intervals. 0
6 . 2 . Every Suslin chain has cardinality exactly w 1 (uses axiom of choice and the continuum hypothesis). 0 Let A be a Suslin chain; we already know t h a t A i s uncountably i n f i n i t e , so has cardinality a t l e a s t w 1 (axiom of choice). Suppose t h a t A has cardinality a t least w . Replace A by a r e s t r i c t i o n o f cardinality O 2 and l e t B be a well-ordering of type o 2 on the same base. Partition the pairs of elements x, y of the base into two colors: l e t t i n g x < y (mod A ) , we say t h a t the pair has color (+) i f x < y (mod B ) , and color (-) i f x ) y (mod B ) By the ERDOS partition lemma (ch.3 5 3.4) f o r d = 0 (hence using only the continuum hypothesis), there e x i s t s a subset of the base of cardinality G, , a l l of whose pairs have a same color. Hence there e x i s t s a s t r i c t l y increasing or a s t r i c t l y decreasing 0 l-sequence, and hence wl-many non-singleton mutually d i s j o i n t intervals: contradiction. 0
.
THEORY OF RELATIONS
150
6.3. SUSLIN TREE We say t h a t a t r e e i s a S u s l i n t r e e i f i t has c a r d i n a l i t y
w1
and i f every chain
( o r t o t a l l y ordered r e s t r i c t i o n ) and every a n t i c h a i n i s countable. The existence o f a S u s l i n chain o f c a r d i n a l i t y
W1 S u s l i n t r e e ( t h e a d d i t i o n a l assumption o f c a r d i n a l i t y
i m p l i e s t h e existence o f a
w 1 allows us t o avoid
using the axiom o f choice: ZF s u f f i c e s ) . 0
Let
be a
A
S u s l i n chain o f c a r d i n a l i t y
w1 .
i ,
To each countable o r d i n a l
defined by i t s two endpoints ui < vi (mod A) , where Ai an a r b i t r a r y are d i s t i n c t . To do t h i s , begin w i t h A. = (uo,vo)
associate an i n t e r v a l all
ui
and
vi
interval. Let
i be a non-zero countable o r d i n a l , and suppose t h a t t h e
A
for
j j < i have already been defined so t h a t they are m u t u a l l y e i t h e r d i s j o i n t o r one
contained i n the other. The s e t o f endpoints uj, v j ( j < i ) i s countable: by hypothesis i t i s n o t dense i n A , hence there e x i s t two elements u, v i n t h e base o f
, between
A
.
v. ( j < i ) L e t Ai = J vi = v : t h i s i n t e r v a l must be e i t h e r d i s j o i n t o r included
which there i s no endpoint
uj
or
ui = u and A. ( j < i ) J thus obtained has c a r d i n a l i t y w 1 Reverse i n c l u s i o n The s e t o f i n t e r v a l s Ai defines a t r e e on the s e t o f these Ai . Every antichain, i . e . every s e t o f i n t e r which are m u t u a l l y d i s j o i n t , i s countable. F i n a l l y , a chain, o r s e t o f v a l s Ai (u,v)
so
.
i n each
Ai
intervals
.
which are mutually comparable w i t h respect t o i n c l u s i o n , i s w e l l f o r every p a i r o f countable o r d i n a l s A . c Ai J Such a chain i s countable; f o r i f i t had c a r d i n a l i t y w 1 , then
ordered by the o r d i n a l i n d i c e s w i t h
i, j ( i
<j ) .
using the endpoints o f preceding i n t e r v a l s , we could o b t a i n W l-many
mutually
d i s j o i n t intervals. 0 6.4. The existence o f a S u s l i n t r e e i m p l i e s , and hence i s e q u i v a l e n t w i t h t h e existence o f a S u s l i n chain (here we use the r e g u l a r i t y o f
W1
, thus
f o r example
the countable axiom o f choice). 0 Let
be a S u s l i n t r e e ; we can assume t h a t
A
A
i s a well-founded p a r t i a l orde-
A by a c o f i n a l well-founded p a r t i a l ordering. To 5.1 w h i l e n o t i n g t h a t , by hypothesis, t h e base o f A i s
r i n g , i f necessary by r e p l a c i n g see t h i s , apply ch.2
5
.
well-orderable w i t h order type w 1 More p r e c i s e l y l e t ui ( i < td 1) be an indexation o f the base; then remove each u f o r which t h e r e e x i s t s an i<j j w i t h ui u j (mod A) To see t h a t what remains i s s t i l l a S u s l i n t r e e , note t h a t
>
.
i , the removed elements u are a l l < ui (mod A) j there are countably many such; thus t h e r e remain w l-many elements.
f o r each index
We can f u r t h e r assume t h a t every non-empty f i n a l i n t e r v a l o f W
. To
see t h i s , l e t
many successors (mod A )
x
A
, and
hence
has c a r d i n a l i t y
be an element o f the base which has o n l y countably
. Those
x
o f minimal h e i g h t are m u t u a l l y incomparable,
Chapter 5
151
x and the ir successors. Neither A nor any f in al interval of A i s f i n i t e l y fre e . For otherwise, by ch.4 5 3.1 (using the regularity of LJ there would e x i s t a t o t a l l y ordered re stric contradicting our hypotheses. tion of A with cardinality w W e can assume t h a t A has a minimum, whose singleton will be denoted by Eo For each countable non-zero ordinal i , l e t Ei be the denumerable s e t of elements of height i : we require t h at t h i s s e t be i n f i n i t e . For each element x of Ei we require that there e x i s t denumerably many elements which are immediate successops of x (mod A ) . This s e t i s denoted Eitl ,X and the union of these se ts must be E i t l . For each countable l i mi t ordinal i and each x of height < i , we require t h a t there e x i s t i n f i n i t e l y many elements in Ei which are 7 x (mod A ) . Finally, for each countable l i mi t ordinal i and each t o t a l l y ordered re stric tion X of A containing elements of a l l heights < i , we require tha t there e xist either a unique element of Ei which i s a successor of a l l elements of X , or none such. These requirements are easy t o s a t i s f y . For example f o r the minimum, take an arbitrary element having W l-many successors. Then having obtained Ei , the s e t of elements x with height i , note t h at f o r each x in Ei , the final interval > x has cardinality w 1 and i s not f i n i t e l y f r e e . Hence take a denumerable free subset as E i + l , x and among the successors of x , retain only those which Finally, f o r each counare identical t o or successors of an element of Ei+l ,X table limit ordinal i and each chain X containing elements of a l l heights 4 i , i f there e x i s t elements above X , then decide t o retain one such plus the wl-many successors of t h i s element. For each element x of the base IAl with height i , t o t a l l y order the denumeNow consider the with the order type of a dense chain C i + l , x rable s e t Ei+l ,X set of a l l maximal t o t a l l y ordered r e s t r i c t i o n s , or maximal chains of A . This set i s t o t a l l y ordered by the preceding dense chains. Indeed, given two distinc t maximal chains U and V : none of the two bases i s included in the other. Moreover there e x i st s a l e a s t element u among those elements of I U 1 which do not belong t o I V I , and a l e a s t element v among those elements of I V I which do not belong t o I U I By the preceding, there ex i s t s a l a s t element x whose height will be denoted i , common t o b o t h bases of U and V , and having u and v as immediate successors. Let U < V i f u < v (mod C i + l , x ) : t h i s tota lly orders the s e t of maximal chains. Let H be the chain thus obtained. We shall prove tha t H i s a Suslin chain. =i-r+ -11 a rnt n irrhirh i c h n c o i n H rannot be countable. For i f i t were, and so there are countably many such: i t suffices t o remove these
.
.
.
.
THEORY OF RELATIONS
152
the i n t e r v a l o f maximal chains passing through of D
z
does n o t c o n t a i n any element
.
Now suppose t h a t t h e r e e x i s t o l-many non-singleton m u t u a l l y d i s j o i n t i n t e r v a l s I n each i n t e r v a l take two elements, o r maximal chains U and V . As of H before, take an element x whose h e i g h t i s denoted i and t h e elements u, v
.
immediate successors o f x (mod A) ; and take w between u and v modulo the chain Ci+,lx Then these w thus associated w i t h our d i s j o i n t i n t e r v a l s o f H ,
.
are m u t u a l l y incomparable (mod A): they must be countably many; c o n t r a d i c t i o n . 0
§
7 - ARONSZAJN
TREE,
SPECKER CHAIN
7.1. ARONSZAJN TREE
This i s a well-founded t r e e o f c a r d i n a l i t y ~3~ whose chains and h e i g h t l e v e l s are countable. I t i s n o t r e q u i r e d t h a t every a n t i c h a i n be countable. Hence every well-founded S u s l i n t r e e i s an Aronszajn tree; b u t the converse p o s s i b l y depends on s e t - t h e o r e t i c axioms: see the problem a t the end o f 7.4. The f o l l o w i n g c o n s t r u c t i o n o f an Aronszajn t r e e , using ZF p l u s choice, goes back t o KUREPA 1935 p . 96, c i t i n g a l e t t e r from ARONSZAJN i n 1934. The elements o f the t r e e w i l l be o r d i n a l sequences o f i n t e g e r s ai ( i < ) y w i t h o u t r e p e t i t i o n , where o( v a r i e s over a l l countable o r d i n a l s . We say t h a t such a sequence u precedes v o r t h a t v f o l l o w s u , i f u i s an i n i t i a l i n t e r v a l o f v . Moreover we r e q u i r e t h e f o l l o w i n g c o n d i t i o n s o f convergence and denumerability. Convergence. For each sequence ai (i< o ( ) , the sum o f the inverses l/ai is f i n i t e . Furthermore for each sequence u w i t h l e n g t h o( , each countable o r d i n a l and each p o s i t i v e r e a l number r , t h e r e must e x i s t , i n our set, a sequence w i t h l e n g t h o i + /s , f o l l o w i n g u , and such t h a t the sum of the inverses l/ai f o r o( i / A : then
i s decomposable, so
( t h . 1.12). X>,
A
.
A = B + C
3
w i t h B < A and A = B+C so C & B ; s i m i l a r -
C C A , y e t A s a t i s f i e s our conclusion. Then 8.2 ly B$C Moreover (C+B) .2 = C+B+C+B 2 A so C+B >, A = B+C and thus B ,< C o r C & B : contradiction. 0 Notice t h a t the chain Z o f the i n t e g e r s i s decomposable, y e t v e r i f i e s the condi-
.
.
t i o n t h a t every X < Z y i e l d s X.2 < o r L 2 The p r o p o s i t i o n i n ch.5 5 4 about o r d i n a l s , does n o t extend t o chains. Indeed
A = w-. i s indecomposable and w < A b u t W.2 I A , and n o t c A . I f a chain A i s indecomposable, then every i n i t i a l i n t e r v a l X < A s a t i s f i e s 2 However, the r e t r o - o r d i n a l ( 0 + W ) - i s decomposable and every i n i t i a l X.2 < A S i m i l a r l y A = Q+ cJ1 i n t e r v a l X < A i s isomorphic w i t h 0 - hence X.2 < A i s decomposable and y e t v e r i f i e s our c o n d i t i o n . S i m i l a r l y A = ( w + 1 ) -
.
.
.
-
Problem. L e t A be a chain. I f every chain X < A y i e l d s X.2 B + B 6 A+B ; then A.3 or AtBG B ; c o n t r a d i c t i o n . 0 3.5.
0 Suppose A.2
then
t
+
+
A.2 B S A.2
B >AtB B,(A+B
.
. Thus e i t h e r
A.Z,(A
(2) L e t A be a chain. I f every chain X < A s a t i s f i e s X + A S A ,.-then A is indecomposable. Moreover, there e x i s t s a r i g h t indecomposable chain equimorphic w i t h A (HAGENDORF 1976). 2 The converse i s f a l s e : the chain A = Z. (J i s r i g h t indecomposable w i t h w < A but
W2tA
3A.
0 We can suppose t h a t
A
i s i n f i n i t e . It s u f f i c e s t o prove t h a t
A
sable; t h e r e s t o f t h e conclusion f o l l o w s from 3.3,
since the case o f
l e f t indecomposable i s excluded: indeed 1 < A b l e i n one o f i t s proper f i n a l i n t e r v a l s .
1+A
C . By the preceding ( l ) ,we have 8.2 equimorphic w i t h B , so B i s indecomposable.
Chapter 6 Furthermore
C
A composable chain, and C an a r b i t r a r y chain. C 2 A+B C >/ B+A (JULLIEN 1969).
or
B
an i n f i n i t e l e f t inde--
-
and C 3 B
, then
either
onto a r e s t r i c t i o n o f C , and l e t CA be t h e C formed by those elements l e s s than (mod C) o r equal t o t h e images f x as x runs through I A I Then A i s embeddable i n every non-empty f i n a l i n t e r v a l o f CA S i m i l a r l y l e t CB be a f i n a l i n t e r v a l o f C such t h a t B i s embeddable i n every non-empty i n i t i a l i n t e r v a l o f CB Then e i t h e r CA and CB are d i s j o i n t , and hence C b A + B O r t h e r e e x i s t s an element u common t o the bases o f CA and CB Then B i s embeddable before u i n t h e i n t e r s e c t i o n CAnCB , and A i s embeddable a f t e r u : hence C>,B+A
0 Let
f
be an isomorphism o f A
i n i t i a l interval of
.
.
.
.
.
.
w + W , which i s embeddable n e i t h e r i n A = W - . W , i s however embeddable i n every common extension o f A a
3.7. The chain B = W
.
nor i n d B
(JULLIEN 1969). The chain A i s r i g h t indecomposable and B l e f t indecomposable. Thus i f a chain X 2 A and B , then e i t h e r X 3 A+B >/ k) + c3- o r X >/ B+A >/ &J +
0
3.8. Problem posed by HAGENDORF 1977.. L e t
A
W-.O
be a s t r i c t l y r i g h t indecomposable
chain. I f f o r every chain X < A , we have X + l d A , then i s A i t s e l f a w e l l ordering. A p o s i t i v e response f o r s c a t t e r e d chains i s due t o LARSON 1978. 3.9. L e t A be a s t r o n g l y s c a t t e r e d r i g h t indecomposable chain, and borhood rank.
o(
i t s neigh-
If OC. = 0 , then every proper i n i t i a l i n t e r v a l o f A i s f i n i t e . I f o( >, 1 , then every proper i n i t i a l i n t e r v a l i s a f i n i t e sum o f chains w i t h neighborhood ranks s t r i c t l y l e s s than g
.
168
THEORY OF RELATIONS
Consequently we find again 3 . 1 . ( 3 ) : A i s s t r i c t l y right indecomposable. Moreover in the case where o( & 1 : i f a proper i n i t i a l interval of A i s indecomposable, then i t has rank s t r i c t l y less than- cx . 0 If A has a minimum, then we are in the case of 2.5. Otherwise, take an element U of the base. Let B be the i n i t i a l interval of elements s t r i c t l y l e s s than u (mod A ) and C the final interval beginning with u . By hypothesis A i s equimorphic with C , so the neighborhood rank of C i s o( The i n i t i a l interval B i s embeddable in a proper i n i t i a l interval of C , hence by 2.5 i f o( = 0 then B i s f i n i t e , and i f o( >, 1 then B i s embeddable in a f i n i t e sum of chains with neighborhood ranks s t r i c t l y less t h a n o( . Thus B i t s e l f i s such a f i n i t e sum. 0 _ _ l l l _
.
5 4 - UNION
A N D I N T E R S E C T I O NOF INDECOMPOSABLE C H A I N S , C O V E R I N G BY INDECOMPOSABLE CHAINS OR BY DOUBLETS 4.1. Let A be a chain which i s the u_nion of an initicallngrval and a final interval, both having a---t l e a s t one common element and b o t h of which are right indecompoSame statement f o r " l e f t " . sable. Then A i s right indecomposable. -
-
~
Let B be the i n i t i a l i n t e r v a l , C the final interval and 0 t h e i r intersection. Then B i s embeddable in D . Either 0 = C so t h a t A = B i s right indecomposable. Or C has the form O+E and so A = B+E i s embeddable in C = D+E , hence A i s again right indecomposable. 0
0
4.2. Consider a chain which i s the union of a r i g h t indecomposable i n i t i a l intervalB and a l e f t indecomposable final interval C , b o t h i n f i n i t e and having a t l e a s t one common element. Then the intersection BnC i s both l e f t and r i g h t indecompo-.sable and admits an embedding of the chain Q of rationals (uses dependent choice; t h i s i s a strengthening of 3.1.( 1) and ( 2 ) ) . 4.3. COVERING BY RIGHT OR LEFT INDECOMPOSABLE CHAINS Let A be a chain. W e say t h a t two elements u , v of the base are equivalent with respect t o right indecomposable chains i f there e x i s t s an interval o f A which i s r i g h t indecomposable and contains the elements u and v . The condition thus defined i s reflexive and symmetric. Moreover by 4.1 i t i s t r a n s i t i v e . Analogously we define the equivalence relation with respect t o l e f t indecomposable chains; we c a l l these covering by r i g h t or l e f t indecomposable chains. There can be i n f i n i t e l y many equivalence classes of t h i s covering relation. For instance, take the converse 0 - of w : the equivalence classes f o r covering by right indecomposable chains are singletons. An equivalence class f o r covering by r i g h t indecomposable chains i s not necessar i l y a right indecomposable chain: take the chain Z of the rational integers.
Chapter 6
169
4.4. DOUBLET OF INDECOMPOSABLE CHAINS
A d o u b l e t i s a c h a i n which i s t h e u n i o n o f a l e f t indecomposable i n t e r v a l and a r i g h t indecomposable i n t e r v a l , b o t h h a v i n g a t l e a s t one common element. For example, t h e c h a i n
Z
o f the integers i s a doublet, being the union o f the
f i n a l i n t e r v a l w and t h e i n i t i a l i n t e r v a l d d - , which can be choosen t o have one o r s e v e r a l common elements. On t h e o t h e r hand, a l t h o u g h t h e p r o d u c t & - . W i s r i g h t indecomposable and i t s converse
c3. (.d -
t
W - . bJ
W. W -
i s l e f t indecomposable, t h e sum
i s n o t a d o u b l e t ; n o r even i s t h e sum
The l a t t e r example i s as w e l l a c h a i n which i s t h e u n i o n o f verse: i f we decompose we have
Z
i n t o CJ-
a sum i s o m o r p h i c w i t h
W-.
W.
Lc)-
and o and t h e n a t t a c h w -
i3
t
Z
t
Csr-. W
and i t s con-
W-. W
to
&-. W ,
, and s i m i l a r l y f o r t h e converse chains;
y e t t h e c o n s i d e r e d indecomposable chains a r e no l o n g e r i n t e r v a l s o f t h e f i n a l c o n s t r u c t e d sum. Every r i g h t o r l e f t indecomposable c h a i n i s a p a r t i c u l a r k i n d o f d o u b l e t , i n which one o f t h e i n t e r v a l s reduces t o a s i n g l e t o n . Note t h a t i t i s n o t r e q u i r e d t h a t o u r indecomoosable c h a i n s be i n i t i a l o r f i n a l i n t e r v a l s : one o f them may be a m i d d l e i n t e r v a l . I n t h e case o f a s c a t t e r e d c h a i n which i s a d o u b l e t , e i t h e r one o f t h e indecomposable c h a i n s i s a r e s t r i c t i o n o f t h e o t h e r , o r t h e l e f t indecomposable c h a i n i s an i n i t i a l i n t e r v a l and t h e r i g h t indecomposable c h a i n i s__ a f i n- a l i n t e r v a l o f t h e
_ _ _ I _ _
doublet. T h i s f o l l o w s f r o m 4.2 (uses dependent c h o i c e ) . On t h e o t h e r hand
1tQt1, where
Q
i s the chain o f r a t i o n a l s , i s a non-scattered
1tQ , a r i g h t indecomposable i n i t i a l i n t e r v a l , Qt1, a l e f t indecomposable f i n a l i n t e r v a l .
d o u b l e t h a v i n g a decomposition i n t o and
4.5. EQUIVALENCE RELATION FOR COVERING BY DOUBLETS Consider a s c a t t e r e d c h a i n and l e t u, v be two elements i n t h e base. We say t h a t u and v a r e e q u i v a l e n t w i t h r e s p e c t t o d o u b l e t s i f t h e r e e x i s t s a d o u b l e t o f indecomposable c h a i n s c o v e r i n g
u and v
. This
c o n d i t i o n i s r e f l e x i v e and sym-
m e t r i c . We s h a l l show t h a t i t i s t r a n s i t i v e (uses dependent choice); t h e conside r e d r e l a t i o n w i l l be c a l l e d c o v e r i n q by d o u b l e t s . 0 Take t h r e e elements
hand,
v
and w
u u , on
t o those elements
< u , on
of
We s h a l l p r o v e t h a t t h e
Q
t h e o t h e r hand, have o n l y r e i n f o r c e m e n t s i n which
i s embeddable. N o t i c e f i r s t t h a t t h e r e e x i s t elements to
.
t h e one hand, and t h e r e s t r i c t i o n
/ 2 )
more generally 0 . p (with p
&
n o t Szpilrajn. Take an uncountably i n f i n i t e s e t U which i s the union of uncountably many On each U k take a chain isomorphic with w disjoint denumerable subsets U k Furthermore any two elements of U which belong t o two d i s t i n c t U k , will be 0 hcomparable. Then i t suffices t o terminate as f o r 0+1
0
.
.
4.8. I t i s proved by GALVIN and MAC KENZIE in 1969 (unpublished) t h a t w i s the only denumerable Szpilrajn ordinal. I t i s proved by BONNET in 1971 (see BONNET, POUZET 1982) that: The only denumerable Szpilrajn chains ( u p t o equimorphism) a r e the following: the chain Q of rationals; the scattered chains defined as follows by ordinal products and sums: PI = 0 ; P2 = w - . CJ ; P 3 -- CJ. W - . W ; in general
.
THEORY OF RELATIONS
198
p.,+1. = p i-. (,d f o r each i n t e g e r s o r o r d i n a l . Furthermore P, =
i , and more generally f o r each countable succesP i ( i i n t e g e r ) ; more g e n e r a l l y , given any denu-
merable limit ordinal u , take any set i n u : then we s e t p U = Z. p i
-sequence o f i n d i c e s
i
forming a cofinal
.
F i n a l l y , f o r each preceding P i ( i countable o r d i n a l ) , the converse
Pi
-
ii
S z p i l r a j n ; furthermore each sum Pi- + Pi i s S z p i l r a j n . Note t h a t l + P i = P i , so the l a t t e r sum i s obviously S z p i l r a j n by 4.6 above. For any two countable o r d i n a l s i and j > i , we obviously have P i + P j S z p i l r a j n ; however t h i s i s already mentioned, s i n c e the l a t t e r sum i s equimorphic w i t h P. . J
4.9. Let C be a denumerable chain, I t h e denumerable i d e n t i t y r e l a t i o n ; in o t h e r words, the denumerable a n t i c h a i n . We say t h a t t h e ordered p a i r (C,I) i s u n i v e r s a l l y S z p i l r a j n i f , f o r every p a r t i a l ordering A in which n e i t h e r C nor I i s embeddable, t h e r e does not e x i s t any t o t a l l y ordered reinforcement of A in which C is embeddable (even when u s i n g the axiom of c h o i c e ) . For example ( Q , I ) , where Q i s the chain of r a t i o n a l s , i s universally Szpilrajn by 2.6.
If C i s a denumerable chain, and C' equimorphic with universally S z p i l r a j n , t h e n so i s (C',I) .
C , and
Given a denumerable s c a t t e r e d chain C , t h e ordered pair: (C-+C,I u n i v e r s a l l y S z p i l r a j n . Same r e s u l t w i t h C+C- . 0 First n o t i c e t h a t C-+C i s not embeddable i n C : use the sam
(C,I)
is
i s not
argument as a s i n ch.5 0 3 . 3 . Then t a k e a p a r t i a l ordering, composed of two chains C and C- w i t h d i s j o i n t bases, every element of t h e f i r s t being incomparable w i t h every element of the second. I t s u f f i c e s t o see t h a t C-+C i s a t o t a l l y ordered reinforcement, y e t not embeddable in our p a r t i a l ordering. 0
It is rable sally Pi ( i
proved by BONNET in 1971 ( s e e BONNET, POUZET 1982) t h a t ;he only denumechains (up t o equimorphism) which y i e l d with t h e i d e n t i t y r e l a t i o n a univerS z p i l r a j n p a i r , a r e t h e chain countable o r d i n a l ) ,
Q o f r a t i o n a l s and the preceding products
4.10. The reinforcement of R being defined f o r an a r b i t r a r y r e l a t i o n R , l e t us mention possible g e n e r a l i z a t i o n s of t h e notion of S z p i l r a j n chain. Using ch.5 0 2.4 and 2.7, r e c a l l t h a t R is a p a r t i a l ordering i f f R does n o t admit an embedding of: A1 = the r e l a t i o n w i t h c a r d i n a l i t y 1 and value (-) ; A2 = r e l a t i o n always (+) w i t h c a r d i n a l i t y 2 ; A3 = r e f l e x i v e binary cycle w i t h c a r d i n a l i t y 3 ; A4 = c o n s e c u t i v i t y w i t h c a r d i n a l i t y 3 Moreover R i s a chain
.
199
Chapter 7
i f f R does n o t admit any embedding of A1 or A2 or Ag o r A5 = identity relation with cardinality 2 . Now we can t r a n s l a t e as follows the definition of a Szpilrajn chain. An arbitrary binary relation R i s Szpilrajn i f f : ( i ) R does not admit any embedding of A1 or A2 o r A3 or A5 (tra nsla te : R i s a chain); ( i i ) for every relation A , i f A $ R and A does n o t admit any embedding of A1 or A2 or A3 or A4 ( t r an s l at e: A i s a partial ordering), then there exists a reinforcement of A which does n o t admit any embedding of R or A1 or A2 or A3 o r A5 . I n the so "translated" d ef i n i t i o n , there only occur general notions as embedding and reinforcement, which are defined f o r ar b i t r ar y relations. Hence we have many possible generalizations, f o r instance by replacing A1 to A5 by an arbitrary f in it e sequence of f i n i t e r el at i o n s . We do n o t know i f t h i s yields interesting problems.
EXERCISE 1
-
TUKEY'S THEOREM O N PAIRS OF DIRECTED PARTIAL ORDERINGS
Let us consider two directed p ar t i al orderings A and B . A function f from A into B i s said t o be convergent, i f every cofinal s e t (mod A ) , say X , i s transformed i n t o a cofinal s e t f"(X) , modulo B . Following TUKEY 1940, we shall prove, modulo the axiom of choice, the equivalence between (1) and ( 2 ) : (1) there e x i st s a convergent function from A i n t o B , and similarly a converpent function from B into A ; ( 2 ) there e x i st s a directed p ar t i al ordering C which i s a common extension of A and B (more exactly A and an isomorphic image of B ) , such t h a t both A and B are cofinal r es t r i ct i o n s modulo C .
-
1 - To see t h a t ( 2 ) implies ( l ) , i t suffices t o associate, t o each element of A , an element f ( x ) of B with f ( x ) >/ x (mod C )
x
.
-
Conversely, suppose t h at A and B s a t i s f y ( 1 ) . Denote by f a convergent function from A into B Consider the directed pa rtia l ordering of final intervals of A , ordered by reverse'inclusion, then similarly the directed partia l ordering of f i nal intervals of B ; then denote by C the dire c t product of these two partial orderings, which i s obviously a directed partial ordering. Now define as follows the embedding of A i n t o C To each element x in A , associate the ordered pair (Ax,Bx) where Ax i s the final interval of elements 3 x (mod A) and Bx i s the final interval of the upper bounds (mod B ) of a l l the images f ( t ) where t 3 x (mod A ) . 2
.
.
THEORY OF RELATIONS
200
Note t h e f o l l o w i n g easy lemma. Given (mod A)
such t h a t
x
f ( t ) < o r I f ( x ) (mod B )
I n o t h e r words, t h e r e e x i s t s an element So t h a t i f
v(x)
3
-
B)
such t h a t e v e r y x
and
From t h e p r e c e d i n g lemma, e a s i l y deduce t h a t
same argument by exchanging A into
, then
.
the
i n t o a subset o f t h e
f"
. A
, when mapped i n t o
previously indicated, y i e l d s a cofinal r e s t r i c t i o n i n B
i s < o r 1 u(x)
t
u ( x ) (mod A)
i s t r a n s f o r m e d under
A)
tax
s e t o f those elements
do n o t c o n s t i t u t e a c o f i n a l s e t i n A .
u(x)
i s a common upper bound o f
>/ v ( x ) (mod >, f ( x ) (mod
final interval final interval
, the
in A
and
B
. Finally
C
as
C
repeat the
and u s i n g t h e convergent f u n c t i o n from
.
A
As a c o r o l l a r y , n o t e t h a t c o n d i t i o n (1) (and o b v i o u s l y ( 2 ) ) i m p l i e s t h a t Cof A = Cof B EXERCISE 2
A
Let
-
. KRASNER'S LEMMA ON DIRECTED PARTIAL OROERINGS
be a d i r e c t e d p a r t i a l o r d e r i n g , and l e t
u = Cof A
. F o l l o w i n g KRASNER
1939, l e t us prove, modulo t h e axiom o f choice, t h a t , g i v e n a c o f i n a l r e s t r i c tion
w i t h minimum c a r d i n a l i t y
A'
Card A ' = u
, there
e x i s t s an i n j e c t i v e ,
i n c r e a s i n g f u n c t i o n whose domain i s t h e s e t o f a l l f i n i t e subsets i n under i n c l u s i o n , and whose range i s We can always suppose t h a t
A' = A
A'
, so
u
, ordered
. that
u
= Card A = Cof A
.
1 - As a p r e l i m i n a r y e x e r c i s e , c o n s i d e r t h e case where A i s denumerable. We know by ch.4 5 5.5 t h a t t h e r e e x i s t s a c o f i n a l r e s t r i c t i o n o f A which i s isomorp h i c t o W ( o r t o 1 , i n which case a l l i s o b v i o u s ) .
u
0 = s e t o f i n t e g e r s . Enumerate t h e base o f A as an u - s e q u e n c e (ii n t e g e r ) . Then d e f i n e as f o l l o w s t h e d e s i r e d f u n c t i o n f . and f(1) = al , and n o t e t h a t Take t h e i n t e g e r s 0 and 1 o n t o f ( 0 ) = a. f ( i ) = ai, w i t h i ' g r e a t e r t h a n o r equal t o i . Take t h e p a i r ( 0 , l ) onto t h e ai w i t h t h e l e a s t i n d e x i , p r o v i d e d t h a t ai i s an upper bound (mod A)
Then ao,
=
..., ai,..
a. = f ( 0 ) and al = f ( 1 ) . Then t a k e t h e i n t e g e r 2 o n t o f ( 2 ) = ai where i i s t h e l e a s t i n d e x n o t y e t used: s o t h a t t h i s i i s g r e a t e r t h a n o r equal and {1,2) Then t o 2 . D e f i n e as p r e v i o u s l y t h e images o f t h e p a i r s {0,2) t a k e t h e 3-element s e t {0,1,2} o n t o t h e element ai w i t h t h e l e a s t index i , p r o v i d e d t h a t t h i s ai i s n o t y e t used and i s a common upper bound (mod A) o f
of
.
t h e images under elements i n A
f
o f the three pairs
{O,l}
, {0,2}
,
{1,2).
. Note
that a l l
a r e f i n a l l y used; moreover t h e procedure always works, s i n c e f o r
any f i n i t e s e t o f elements
a
in A
, t h e r e e x i s t denumerably many common upper
bounds.
-
2 Now e x t e n d t h e same p r o c e d u r e t o t h e g e n e r a l case. Consider u = Cof A = Card A as an ordinal-indexed sequence, o r u-sequence o f terms i < u .
Chapter I
20 1
.
Well-order the base of A as a u-sequence ai Then repeat the previous procedure for ordinals 0 and 1 . I n general, l e t i be an ordinal index s t r i c t l y less t h a n u ; suppose t h a t f i s already defined f o r a l l f i n i t e se ts of indices j < i . Moreover suppose t h a t f ( j ) = a . with j' greater t h a n or equal t o j , J' for every such j . Take i onto f ( i ) = a i , where i ' a i i s the l e a s t index not yet used. Then take every pair i i , j j where j -z i , onto the element a with the l e a st possible index, provided t h a t t h i s a i s n o t ye t used, and i s a common upper bound (mod A) of the images f ( i ) and f ( j ) Then take every 3-element s e t \ i , j , k } where j , k < i , onto the element a with the l e a s t possible index, provided t h at a i s n o t y et used, and i s a common upper bound of the images under f of the three pairs included in { i , j , k ) ; and so on. The procedure always works, since f o r any f i n i t e s e t of elements a in A ,
.
on one hand there e x i s t u many comnon upper bounds; on the other hand the number of f in i t e s e t s of indices less than or equal t o i < u i s i t s e l f < u .
-
I n connection with the two preceding exercises, say tha t two partial orderings
A, B have same convergence type i f f condition ( l ) ,or equivalently ( 2 ) in exer-
cise 1 holds. Then the reader may be interested t o know t h a t , in the case of cofinality w , then i t i s consistent with the axioms of ZF t o suppose e ithe r that there e x i s t ( 2 t o the power ol) many d i f f er ent convergence types; or tha t there exist only three possible convergence types, namely: ( i ) the case of an increasing cofinal wl-sequence; ( i i ) the type of the dire c t product W x W1 ; ( i i i ) the type of the ordered s e t (under inclusion) of a l l f i n i t e subsets of W 1 ; t h i s r e s u l t i s due t o TODORCEVIC 1984.
EXERCISE 3
-
THE POSSIBLE COFINAL RESTRICTIONS OF A DENUMERABLE PARTIAL ORDERING
I n the particular case of a denumerable, directed p artia l ordering, we know by ch.4 5 5.5 t h a t there ex i s t s e i t h e r a maximum, or a cofinal re stric tion which i s isomorphic with I n the general denumerable case, show t h at the possible cofinal re stric tions are the following: (1) an antichain (with f i n i t e or denumerable c a rdina lity); (2) the union of components each of which i s isomorphic with w , with mutual incomparability for elements belonging t o different components; ( 3 ) the W-tomic t r e e , i . e . the t r e e with denumerably many edges from each vertex; ( 4 ) the union of 2 or 3 among the preceding components, with mutual incomparability . EXERCISE 4
-
ON SIERPINSKI'S PARTIAL ORDERING
In connection with Sierpinski's counterexample (ch.3 5 3.1), a pa rtia l ordering is called a Sierpinski p ar t i al ordering, i f we are i n the framework of ZF plus choice plus continuum hypothesis, so t h at the chain of real numbers i s equipotent
202
THEORY OF RELATIONS
. Take
w
to
a chain
o f r e a l s , such t h a t each f i n a l i n t e r v a l o f
R
nuum many elements; a l s o take an a r b i t r a r y w e l l - o r d e r i n g
. Then
morphic w i t h the o r d i n a l w t e d w i t h the b i c h a i n x
modulo
x,y
= continuum; indeed given a countable
. More
are o n l y countably many elements having an upper bound i n D
precisely a set 2
see t h a t every chain o r a n t i c h a i n i n A
i s d i r e c t e d : given two elements
A
and modulo
U
. We
Show t h a t i f
R
iff
C
.
i s a l o c a l automorphism o f
f
has continuum many elements i n A
(ch.9
5
1.7) and i f Dom f
has continuum c a r d i n a l i t y , then t h e r e e x i s t s a r e s t r i c t i o n o f
f
t o continuum
many elements which i s a l o c a l automorphism o f the associated b i c h a i n
.
(R,U)
To see it, show t h a t ( 1 ) given a countable subset H o f Dom f , there always e x i s t s an element
u
U , and f ( u )
modulo
in
-
(Dom f )
H
, such t h a t u i s an upper bound o f
i s an upper bound o f the image s e t
and ( 2 ) every common l o c a l automorphism o f phism o f
U
and A
f"(H)
H
modulo U ;
i s s t i l l a l o c a l automor-
( i t s u f f i c e s t o prove t h i s f o r an automorphism on 2 elements).
R
3 - By an easy refinement o f the argument i n ch.1 exerc. 4, show t h a t there e x i s t ( 2 t o the continuum power) many c o f i n a l sets C (mod A) having pairwise only a countable i n t e r s e c t i o n . Now take f o r
R
a r i g i d chain o f r e a l s (BONNET 1978 p. 7 ) , i n t h i s sense that
every l o c a l automorphism ( f ( x ) # x)
of
f
. Assume w i t h o u t
R
moves o n l y countably many elements
proof that
R
x
e x i s t s and can have continuum many
elements i n each r e a l i n t e r v a l (non-singleton). Consequently, given two cofinal sets
C, C '
A/C'
has o n l y a countable domain.
i n the previous f a m i l y , every l o c a l isomorphism from
A/C
into
By completion, o b t a i n a family of ( 2 t o t h e continuum power) c o f i n a l sets
such t h a t , f o r any two such sets Fc,cl
C, C '
y i e l d i n g a countable i n t e r s e c t i o n
isomorphism from
A/C
into
every c o f i n a l s e t
X
(mod A)
that the intersection
CnX
A/C'
, there e x i s t s i n CnC'n FC,!,
R
C
a f i n a l interval
. Consequently,
a local
cannot have a c o f i n a l domain. F i n a l l y f o r
, t h e r e e x i s t s a t l e a s t one be c o f i n a l
C
i n the family,
(mod A) (comnunicated by POUZET).
such
203
CHAPTER
8
BARRIER, BARRIER SEQUENCE, FORERUNNING, EMBEDDABILITY THEOREM FOR SCATTERED CHAINS, BETTER PARTIAL ORDERING
5 1-
BARRIER, B A R R I E R
P A R T I T I O N THEOREM, S U C C E S S I V E ELEMENTS,
SQUARE OF A BARRIER Let E be a denuneyable set of natural numbers empty subsets o f
E
, whose union i s
E
. The
, and
set
U a set of finite non-
i s a barrier i f f :
U
(1) t h e elements o f U are mutually non-inclusive; ( 2 ) f o r every i n f i n i t e subset X o f E , there e x i s t s a f i n i t e i n i t i a l i n t e r v a l o f X ( i n i t i a l w i t h respect t o the usual o r d e r i n g o f the i n t e g e r s ) which belongs to
u .
Examples. For
E take the s e t o f a l l n a t u r a l sumbers , t h i s case being the most
f r e q u e n t l y encountered. Then the s e t o f s i n g l e t o n s o f t h e i n t e g e r s
i s a barrier.
S i m i l a r l y , the s e t o f unordered p a i r s o f the i n t e g e r s i s a b a r r i e r , as w e l l as the s e t o f p-element subsets o f t h e integers, f o r any given p o s i t i v e i n t e g e r Another example o f a b a r r i e r : the union o f the s e t o f p a i r s whose minimum i s and t h e s e t o f 3-element subsets w i t h minimum 1, ... , and f o r each i n t e g e r
i a 2 , the
s e t o f i-element subsets w i t h minimum
i-2
the s e t o f those elements o f
U which are subsets o f
every b a r r i e r included i n
i s thus obtained, i . e .
U
o f i n t e g e r s and r e s t r i c t i n g
U
.
0
,
.
P an i n f i n i t e subset o f t h e union U U
1.1. L e t U be a b a r r i e r and
p
P
b
is
. Then
b a r r i e r . Moreover,
by t a k i n g an i n f i n i t e s e t
t o those elements which are subsets o f t h i s i n f i -
n i t e set. The elements o f
U
P are m u t u a l l y incomparable w i t h
which are subsets o f
respect t o i n c l u s i o n . Moreover, f o r every i n f i n i t e subset exists a f i n i t e i n i t i a l interval o f For the second assertion, l e t union
u V .
For each element
X
which i s an element o f
V be a b a r r i e r included i n r of
U which i s a subset
P , there
of
X
U
.
, and P be the o f P , take an i n U
P which begins w i t h r : t h e r e e x i s t s an element o f V which i s an i n i t i a l i n t e r v a l o f R , and t h i s can o n l y be r . 0 f i n i t e subset
R
of
Note t h a t , i f
U
i s a b a r r i e r and
P an i n f i n i t e subset o f u U , then t h e s e t
U does n o t n e c e s s a r i l y form a b a r r i e r . For instance, s t a r t i n g w i t h the b a r r i e r U o f p a i r s o f integers, l e t
o f intersections o f
P
w i t h t h e elements o f
P be the s e t o f s t r i c t l y p o s i t i v e i n t e g e r s (we remove zero). Then we o b t a i n
204
THEORY OF RELATIONS
b o t h the singleton of 1 , coming from the pair {1,2} in which the singleton i s included.
{ 0 , l ) , and
for example the pair
1.2. Let U be a barrier and r a f i n i t e s e t of natural integers which i s aper i n i t i a l interval of an element of U . Then the s e t of elementsof the form x - r , where x i s any element of U beginning with the i n i t i a l interval r , i s a barrier. Let V be the s e t of our difference s e t s x - r . The union of the elements of beginning with the i n i t i a l interval r i s i n f i n i t e ; so u V i s i n f i n i t e . Any two of these elements are incomparable with respect t o inclusion: t h i s subsists when removing the i n i t i a l interval r Finally, f or every i n f i n i t e subset P of the union u V , which necessarily begins with an integer s t r i c t l y greater t h a n Max r , there e x i s t s a f i n i t e i n i t i a l interval y of P , such tha t the union r L J belongs ~ t o U , so y belongs t o V 0
0
U
.
.
1.3. Every b a r r i e r i s lexicographically well-ordered: t h i s i s a particular case 0 2 . 1 . In other words, the s e t of the elements of a ba rrie r, when ordered lexicographically s t a r t i n g with the usual ordering of the integers, forms a denumerable well-ordering.
o f ch.3
Every barrier U t h u s has a lexicographic rank, in the sense of ch.3 5 2 . 1 : the order type o f the well-ordering of the elements of U when ordered lexicographical ly. f o r example, the b a r r i er o f the singletons has rank W . The ba rrie r of the p-element subsets ( p = positive integer) has rank u p The ba rrie r formed of the pairs w i t h minimum 0 , and the 3-element subsets w i t h minimum 1 , and f o r each w integer i 3 2 , the i-element subsets with minimum i - 2 , has rank 0 .
.
f o r each denumerable indecomposable ordinal, i .e. each power of a ,say , there e x i s t s a barrier of rank 8 (POUZET 1972"). More generally: The lexicographic ranks of b ar r i er s are exactly a l l the ordinals u p ( p = positiv e integer) and ( w'( ) . p , where o( i s a denumerable ordinal and p a positiv e integer (ASSOUS 1974). 1 . 4 . BARRIER PARTITION THEOREM Let U be a b a r r i e r and E be the union of U Partition the elements of U into two complementary s e t s U ' and U" Then there e x i s t s an i n f i n i t e subset H f E such t h a t the elements of U which are subsets of H , a l l belong to U ' or a l l belong t o U" (NASH-WILLIAMS 1968). Note t h a t these elements form a b ar r i er by 1.1. I n pa rtic ula r, we obtain RAMSEY's theorem by considering a positive integer p and taking f o r U the s e t of a l l
.
.
205
Chapter 8
p-element subsets o f
E
.
0 Given two d i s t i n c t elements o f
, one i s never i n c l u d e d i n t h e o t h e r , hence
U
one i s never an i n i t i a l i n t e r v a l o f t h e o t h e r : t h e theorem now f o l l o w s f r o m NASH-WILLIAM'S theorem i n ch.3 0
5
2.4.
0
We g i v e a n o t h e r more d i r e c t p r o o f , due t o COROMINAS i n 1 9 7 0 , unpublished.
A s s o c i a t e t o each b a r r i e r
> o . Assume
U be a b a r r i e r o f r a n k M ( U )
Now l e t
every b a r r i e r o f rank s t r i c t l y lesser than for
U
b a r r i e r has rank
Let
uo
where
o(
t h a t t h e theorem h o l d s f o r
(U) : we s h a l l prove t h a t i t h o l d s
. be t h e l e a s t i n t e g e r i n U . Let
n o t belong t o set
.A
i t s l e x i c o g r a p h i c r a n k c( ( U )
U
i f f i t i s a s e t o f s i n g l e t o n s ; i n t h i s case t h e theorem i s obvious.
o
x U1
i s an element o f i s a barrier.
U f o r which t h e s i n g l e t o n o f uo does
E = u
be t h e s e t o f those elements o f t h e f o r m
U1
U
beginning w i t h the i n t e g e r
We have t h e s t r i c t i n e q u a l i t y
i n t e g e r . The p a r t i t i o n o f o f elements o f
U
U
U'
and
, hence
uo
beginning w i t h
{uo}
o((U1)
< M(U)
: indeed
uo
,
which b e g i n w i t h a s t r i c t l y g r e a t e r
U
into
-
U which b e g i n w i t h
i n t h e l e x i c o g r a p h i c o r d e r i n g , a f t e r t h e elements o f we s t i l l have p o s t e r i o r elements o f
x
U"
gives a p a r t i t i o n o f the set
also a partition o f
. By t h e
U1
i s s t r i c t l y less than
i n d u c t i o n h y p o t h e s i s , s i n c e t h e l e x i c o g r a p h i c rank o f
U1 H ( U ) , t h e r e e x i s t s an i n f i n i t e subset H1 o f t h e u n i o n w U1 , such t h a t , l e t t i n g V1 be U1 r e s t r i c t e d t o t h e subsets o f HI , t h e n t h i s b a r r i e r V1
e i t h e r i s included i n
U'
or in
U"
(whose elements have o b v i o u s l y been modiuo ) . According t o t h e case, we say t h a t
f i e d by removing t h e i r minimum i n t e g e r uo Let
i s linked t o u1 Up
o r linked t o
be t h e l e a s t i n t e g e r i n
belong t o Let
U'
U
, since
U" H1 =
. i~
V1 ; t h e s i n g l e t o n o f
t h e r e e x i s t s an element o f
be t h e s e t o f t h o s e elements o f t h e f o r m
element o f
u1
does n o t
uo, u1
which begins w i t h
U
x
- {ul) , where
U which begins w i t h u1 and i s a subset o f H1
. The
. Then
x
i s an
U2
is a
.
U i n t o U' and U" g i v e s a p a r t i t i o n o f t h e s e t o f elements o f U b e g i n n i n g w i t h u1 , b a r r i e r whose r a n k i s s t r i c t l y l e s s t h a n hence a l s o a p a r t i t i o n o f
U2
. By
N(U)
partition of
t h e i n d u c t i o n h y p o t h e s i s , t h e r e e x i s t s an i n -
u U 2 , such t h a t , l e t t i n g V 2 be U2 H~ -, {,+\ = r e s t r i c t e d t o t h e subsets o f H2 , t h e n t h i s b a r r i e r V2 e i t h e r i s i n c l u d e d i n f i n i t e subset
U'
or in
integer linked t o
H2
of
(whose elements have been m o d i f i e d by removing t h e i r minimum
U"
u1 ). According t o t h e case, we say t h a t u1 i s l i n k e d t o U ' U"
or
.
I t e r a t i n g t h i s , we have a s t r i c t l y i n c r e a s i n g &-sequence each o f which i s l i n k e d w i t h e i t h e r of i n f i n i t e s e t s
Hi
U'
or
,
uo : by 1 . 2 above, t h e
o f integers ui , U" ; and t h e c o r r e s p o n d i n g sequence
, which i s d e c r e a s i n g w i t h r e s p e c t t o i n c l u s i o n . L e t
H
be
206
THEORY OF RELATIONS
f o r example t o uo, ul,
u2,
..
.
U'
,all
ui
an i n f i n i t e s e t o f these Then
o f which are l i n k e d t o t h e same subset, say
s a t i s f i e s our conclusion. Indeed, enumerate
H
here forming an e x t r a c t e d sequence o f the o l d
: the
ui renumber t h e corresponding b a r r i e r s
H
.
Vi
The elements o f
as
H
ui
, and
U which are subsets o f
remain p a i r w i s e incomparable w i t h respect t o i n c l u s i o n . Moreover, given an i n f i -
n i t e subset
of
P
, this P
H
ponding w i t h the b a r r i e r of
begins w i t h a c e r t a i n u (p i n t e g e r ) , corresP .And t h e i n i t i a l i n t e r v a l o f P which i s an element
V
P+i i s the union o f the s i n g l e t o n { u
U
P
1
and an element o f
Vp+,,hence
it
U' , t o which u i s l i n k e d . P
belongs t o the subset
,
The p r o o f i s f i n i s h e d ; however i n order t o see t h a t i t o n l y needs the axioms ZF avoiding f o r instance the axiom o f dependent choice when we d e f i n e t h e i n f i n i t e sequence o f sets me t h a t
, we
Hi
make p r e c i s e the c o n s t r u c t i o n o f each
y i e l d f o r instance a b a r r i e r included i n
U'
, choose
. Assu-
Hi
i s already defined. Among the i n f i n i t e subsets o f
Hi-l
Hi-l
which
those i n f i n i t e subsets which
begin w i t h t h e l e a s t p o s s i b l e i n t e g e r , say h t . Then among the i n f i n i t e subsets 1 hi and y i e l d a b a r r i e r included i n U ' , choose those i n f i n i t e 1 2 subsets which begin w i t h hi, hi where h: i s the l e a s t p o s s i b l e i n t e g e r ; and
which begin w i t h
so on. F i n a l l y we take f o r
t h e s e t o f a l l these
Hi
hr
( i fixed, k varies). 0
1.5. SUCCESSIVE ELEMENT
, we say t h a t r precedes i s a successive element o f r , denoted r
Given two f i n i t e sets o f n a t u r a l numbers or that
s
s
succeeds
r
, or that
s
and
s
ra s , i f s i s obtained from r by adding on i n t e g e r s which are a l l s t r i c t -
by
l y g r e a t e r than
Max r
and then by removing
For example, given two i n t e g e r s t o n of
b
iff
a Max s and s an By t r a n s i t i v i t y h i s a bad successor of f i The s e t s element o f V i belongs t o V i y e t not t o W , hence h should have been taken instead of f i + l , because i t leads t o Max s ( o r t o a l es s er integer) instead o f pi : contradiction. 0
.
.
.
.
s
.
.
.
.
216
THEORY OF RELATIONS
5 4
-
EMBEDDABILITY
(LAVER)
THEOREM FOR SCATTERED CHAINS
be a s e t o f h-indecomposable c h a i n s (see ch.6 § 5 ) , which i s q u a s i 4.1. L e t o r d e r e d under e m b e d d a b i l i t y . Assume t h a t & i s c l o s e d w i t h r e s p e c t t o t a k i n g
s
any h-indecomposable i n t e r v a l o f a c h a i n . L e t
be t h e r a n k i n g f u n c t i o n which
a s s o c i a t e s t o each c h a i n i t s neighborhood rank (ch.6 Let
U
be a b a r r i e r and
f
be a bad
V
a bad V-sequence -
which i s a successor o f
successor o f
64 which
The c h a i n s i n
2.4).
44 . Then there
U which i s n o t a s u b - b a r r i e r o f U , and
exists a barrier
0
5
U-sequence w i t h values i n
& , 6 ) (uses
(mod
f
belong t o t h e range o f
f
axiom o f c h o i c e ) .
are a l l i n f i n i t e . For other-
wise, e i t h e r t h e empty c h a i n o r a c h a i n c o n s i s t i n g o f a s i n g l e t o n would b e l o n g Rng f , and so t h e b a r r i e r sequence f would be good. P a r t i t i o n t h e elements o f U i n t o two s e t s , a c c o r d i n g t o whether t h e image under f i s r i g h t o r l e f t indecomposable. A t l e a s t one o f these s e t s i n c l u d e s a b a r r i e r ,
to
which we a g a i n denote by U : see p a r t i t i o n theorem 1.4. To f i x i d e a s , suppose t h a t a l l t h e c h a i n s a r e r i g h t indecomposable. Compose t h e f u n c t i o n
f
8
with the function
which t o each c h a i n a s s o c i a t e s i t s
8
c o f i n a l i t y : we o b t a i n a b a r r i e r sequence
.f t a k i n g o r d i n a l values, and more p r e c i s e l y values which a r e r e g u l a r a l e p h s . I t f o l l o w s t h a t 8 .f has no bad r e s t r i c t i o n : t a k e an
w -sequence of successive elements so 4 s1 4 . . . Q siQ..
.in
u
cannot be s t r i c t l y ( i n a t u r a l number); t h e n t h e values y ,,f(si) decreasing. By 2.1, t h e r e e x i s t s a b a r r i e r i s p e r f e c t . L e t us denote t h i s t h a t t h e r e s t r i c t i o n of b a r r i e r a g a i n by U .
2(
.f t o t h i s b a r r i e r i s p e r f e c t . L e t us denote t h i s
To each h-indecomposable c h a i n
8 (A)
along the ordinal
w i t h t h e c o n d i t i o n t h a t each rank of
, and
A
i n which
Ai
A
,of
, associate
in
a decomposition i n t o a sum,
Ai ( i < x ( A ) ); has neighborhood rank s t r i c t l y l e s s t h a n t h e
h-indecomposable i n t e r v a l s Ai
t h a t f o r each
i
a r e Z ( A ) many A . ( i C j < J 5. T h i s decomposition o f each A
, there
i s embeddable: see ch.6
5
(A) ) into
t h e Ai i s chosen once f o r a l l and s h a l l be c a l l e d t h e s t a n d a r d decomposition (axiom o f c h o i c e ) . Since
f
i s bad, f o r e v e r y
embeddability fs' = j
f s d fs'
A j . Since
8
.
s
and
s'
in
U
with
s d s '
, we
Consider t h e s t a n d a r d decompositions i s p e r f e c t , we have
r f s
/
I
or
I
be one o f t h e s e i n t e r -
a l l o t h e r s under e m b e d d a b i l i t y . We can assume t h a t
i s i n f i n i t e , and r i g h t indecomposable, t o f i x i d e a s . L e t i n t e r v a l s equimorphic w i t h i s equimorphic w i t h X = I(k-1) + 1
, this
X = 1.k
X
Let
of
A
A
I
,in
A ; then i f
, we have X
k , whose images j , k ' a k+l
the maximality o f
Suppose now t h a t
under
h
begin w i t h
j,k,v7
k
.
I t e r a t i n g t h i s , we o b t a i n a s t r i c t l y i n c r e a s i n g i(2)
=
k
, ...
begin w i t h
...
i(l), s = i(O),
and
i(O),
i(l), ...
,i(r) .
..., i ( r )
, i ( r ) , ..
r
,
i(1) = j
U which
have images under
h
i(0) = i
which b e g i n w i t h
By t h e d e f i n i t i o n ' o f b a r r i e r , t h e r e e x i s t s an i s an element o f
f s , i , l e t ai . denote the value taken by t h i s barrier ,J sequence a Since A i s a well p a r t i a l ordering, extract an increasing sequence from the sequence of the a 0 , j , which we renumber by aD,l ,< aD,? 4 .... , i ) . By hypothesis, the Ai form ,j a well p a r t i a l ordering under inclusion: t h e i r sequence i s good (see ch.4 5 3 . 2 . ( 2 ) ) . I n other words, there e x i s t two integers i and j > i with Ai included in A . Since the element ai belongs t o Ai thus t o A , then j ,j j j . < a j , k (mod A ) . B u t we have the conthere e x i s t s an integer k > j with ai , secutivity { i , j } a { j , k \ , so t h a t the barrier sequence a i s good. Conversely, suppose t h a t the i n i t i a l intervals of A do n o t form a well partial ordering under inclusion. Then e i t h e r t h i s p a r t i a l ordering o f inclusion i s n o t well-founded: then A i s n o t a well p a r t i a l ordering (ch.4 5 4.1, dependent choice), and there e x i s t s a bad a-sequence ui ( i integer) in A (ch.4 3 3 . 2 . ( 2 ) , dependent choice). I n t h i s case, the barrier sequence of W 2 i n t o A , defined by ai - ui f o r a l l j > i , i s bad. ,j Or the p a r t i a l ordering of inclusion among i n i t i a l i n t e r v a l s o f A i s wellfounded b u t not a well partial ordering: hence there e x i s t i n f i n i t e l y many mutually non-inclusive i n i t i a l i n t e r v a l s . Choose an w-sequence of such intervals AD, A1, . . . , Ai , . . ( i natural number ; uses denumerable subset axiom).
226
THEORY OF RELATIONS
which belongs t o i, j 7 i , a s s o c i a t e an ai ,j ( c o u n t a b l e axiom o f c h o i c e ) . I t s u f f i c e s t o p r o v e t h a t t h e
To each p a i r o f n a t u r a l numbers Ai
yet not t o
A
b a r r i e r sequence gers
i< j
a o and x + y = a. + al where al = red pairs (x,y) with x 2 a.
0
.
Card F1 ; and so f o r t h . The Given s t r a t i f i e d p a r t i a l ordering i s isomorphic with a r e s t r i c t i o n of the d i r e c t Droduct W x W ; by 6.1, t h i s i s a - b e t t e r p a r t i a l ordering. 0 6.7. Let A be a p a r t i a l ordering. If every proper i n i t i a l interval of A f i n i t e , then A i s a - b e t t e r p a r t i a l ordering (POUZET 1977, unpublished).
i2
229
Chapter 8
As in the preceding statement, t h i s proposition gives well partial o r d e r i n g whose elements have f i n i t e heights; y e t they are more varied: i t i s no longer required
t h a t each element of height
i + l be greater t h a n every element of height
i
.
Note f i r s t t h a t every non-empty subset of the base has a minimal element; f o r otherwise, t h i s would yield an i n f i n i t e proper i n i t i a l i n t e r v a l . Moreover every
0
free s e t , or every antichain, i s f i n i t e ; thus A i s a well partial ordering. By hypothesis, each element of the base has a height which i s a natural number. Moreover, i f A i s i n f i n i t e , then by our hypothesis A i s directed, thus A i s an i d e a l . For otherwise, i f a and b are two elements without any common upper bound, then e i t h e r the s e t of non-upper bounds of a , o r the s e t of non-upper bounds of b , i s an i n f i n i t e i n i t i a l interval which i s d i s t i n c t from A : contradiction. Let U be a barrier and g a function from U i n t o A . We shall prove that g i s good. Let h be the function which, t o each element in the base \ A 1 , associates i t s height (mod A ) , which i s a natural number. Let f be the compos i t i o n h o g . Since the chain (A) of the heights i s a b e t t e r p a r t i a l ordering, there e x i s t s a barrier V included in U , such t h a t the r e s t r i c t i o n f/V i s 2 perfect (see 2 . 1 above). Let V ' be the subset of the squared barrier V , formed of the unions s u t of elements s , t in V , such t h a t s 4 t and f s = f t . By the b a r r i e r p a r t i t i o n theorem 1.4, there e x i s t s an i n f i n i t e s e t H of natural numbers which e i t h e r only contains elements of V ' , or only contains 2 elements of V 2 - V ' Let W be the barrier V r e s t r i c t e d t o H ; then W i s e i t h e r included in V ' or included in V2 - V ' . I n the f i r s t case, there e x i s t s a natural number p such t h a t f s = p f o r Indeed, given two elements s , t of W , by 1.5.(2) every element s in W there e x i s t s a t h i r d element u of W with two f i n i t e sequences of successive elements of W , say s Q S ~ . Q U and t q t l 4 . . u Then we have
.
.
..
.a .
f s = f ( s 1) = ... = fu and f t = f ( t l ) = ... = fu . For each element x of the base I A I whose height (mod A) i s p , l e t W x be the subset of W formed of those s such t h a t gs = x . The elements x are mutually incomparable (mod A ) , hence there are only f i n i t e l y many such. Thus there e x i s t s an x with a barrier X included in W x . For two elements s , t in X , which we can take as success i v e , we have gs = g t ; hence g i s good. In the second case, recall t h a t f i s perfect. T h u s f o r every s , t in W , the condition s d t implies f s , < f t B u t here f s # f t and so f s < f t Let k be the function from the s e t w of the natural numbers, into (13, which t o each natural number i , associates the l e a s t j f o r which each element of height 6 i i s less (mod A ) t h a n every element of height j , thus also less than every element of height >, j . This value j = ki e x i s t s , since
.
.
230
THEORY OF RELATIONS
f o r each element a with height i , the s e t of non-upper bounds of a i s a proper i n i t i a l interval of A ; hence i t i s f i n i t e and there are only f i n i t e l y many heights of i t s elements. By 1 . 7 above, there e x i s t two elements s , t in W satisfying s a t and k f s < f t . Thus each element of the same height as g t i s greater (mod A) t h a n every element of the same height as gs ; in p a r t i c u l a r gs 6 g t (mod A): hence g i s good. 0 6.8. Let A be a well partial ordering which has f i n i t e l y many i n f i n i t e ideals. Then A i s a - b e t t e r partial ordering.
Let A be a p a r t i a l ordering which has only f i n i t e l y many i n f i n i t e i n i t i a l intervals. Then A i s a - b e t t e r partial ordering (POUZET 1977, unpublished; uses dependent choice; ZF suffices i f A i s countable). The second assertion follows from the f i r s t , since the partial ordering A under consideration i s necessarily well-founded and f i n i t e l y f r e e . Suppose f i r s t t h a t A i s a directed well partial ordering, with no other i n f i n i t e ideal t h a n i t s e l f . Then A has no i n f i n i t e , proper i n i t i a l i n t e r v a l . Indeed, every i n f i n i t e well partial ordering has as a r e s t r i c t i o n , a t l e a s t one i n f i n i t e ideal (see ch.4 5 5 . 2 , dependent choice); so by the preceding proposition 6 . 7 , A i s a -better partial ordering. I n the general case, we argue by induction. Given a positive integer p , suppose the proposition holds f o r any well partial ordering with a t most p i n f i n i t e ideals, and l e t A be a well p a r t i a l ordering with p + l i n f i n i t e ideals. Let I be an ideal of A , which i s maximal with respect t o inclusion. Partition the base I A I into the union C of those ideals d i s t i n c t from I , and the complement D of C . The r e s t r i c t i o n A/C i s a well p a r t i a l ordering having only p i n f i n i t e ideals, and the r e s t r i c t i o n A/D i s s t i l l a directed well p a r t i a l ordering, hence an i d e a l , having no other i n f i n i t e ideal than i t s e l f (provided D iS i n f i n i t e ) . Each i s thus a -better partial ordering; so by the previous 6.4, A i s a -better partial ordering. 0
0
6 . 9 . -BETTER PARTIAL O R D E R I N G OF WORDS Let A be a -better partial ordering. Then the s e t of a l l words ( i . e . f i n i t e sequences) A forms a - b e t t e r p a r t i a l ordering under embeddability- (uses dependent choice).
Suppose on the contrary t h a t there e x i s t s a b a r r i e r U and a bad U-sequence f taking as values words. The p a r t i a l ordering of words i s a well-founded p a r t i a l ordering: see ch.4 5 2 . So we can assume t h a t f i s minimal bad: see theorem 2 . 2 above, dependent choice. Since f i s bad, f o r every s in U , the word f s i s non-empty.
23 1
Chapter 8
Let s
g
and
in
h
U
, the
hs
is
value barrier
fs
of
V
gs
f s ; the
w i t h i t s f i r s t t e r m removed. By 2.1 above, t h e r e e x i s t s a subsuch t h a t
U
minimal bad, i n g o i n g f r o m hs
as f o l l o w s . F o r each
i s t h e word composed o f t h e f i r s t t e r m o f i s perfect (since
g/V
d e r i n g ) . On t h e o t h e r hand, t h e r e s t r i c t i o n p l a c e d by
U , defined
be b a r r i e r sequences w i t h domain value
to
f
h
, for
h/V
Thus t h e r e e x i s t two elements
s, t
p e c t t o e m b e d d a b i l i t y . Now, as t o e m b e d d a b i l i t y ; so
in
V
fs
b e t t e r p a r t i a l or-
in
s
f
is
, t h e word f s i s r e -
V
w i t h respect t o embeddability.
s Qt
with
and
hs,< h t
, we have f s , ( f t
gs,< g t (mod A)
i s good: c o n t r a d i c t i o n .
f
-
i s good; indeed s i n c e
each
which i s s t r i c t l y l e s s t h a n
is a
A
w i t h res-
w i t h respect
0
6.10. -BETTER QUASI-ORDERING OF ORDINAL-INDEXED SEQUENCES G e n e r a l i z e as f o l l o w s t h e p r e v i o u s p r o p o s i t i o n . Let
A
be a - b e t t e r p a r t i a l o r d e r i n g . Then any s e t o f o r d i n a l - i n d e x e d sequences
w i t h values i n A
forms a - b e t t e r q u a s i - o r d e r i n g under e m b e d d a b i l i t y (NASH-WILLIAMS
1968; t h e f o l l o w i n g p r o o f i s due t o MILNER 1984, unpublished; uses dep. c h o i c e ) . Cl L e t
B
A : this
be a s e t o f o r d i n a l - i n d e x e d sequences i n
under e m b e d d a b i l i t y (ch.4
5
2). We can assume t h a t
B
B
i s quasi-ordered
reduces t o a p a r t i a l orde-
r i n g , by r e p l a c i n g sequences by t h e i r e q u i v a l e n c e c l a s s e s under embeddability. Take t h e r a n k i n g f u n c t i o n which t o each sequence u a s s o c i a t e s t h e l e n g t h o f u.
s
Suppose t h a t and a bad
i s not a -better p a r t i a l ordering: there exists a b a r r i e r
B
U-sequence w i t h v a l u e s i n
e x i s t s a minimal bad b a r r i e r sequence the barrier
B
. By
(mod B,
.
Dom f
P a r t i t i o n t h e elements
s
of
U
theorem 3.5 (dependent c h o i c e ) , t h e r e
J' )
, say f
; we c a l l again
U
i n t o three d i s j o i n t classes, according t o the
U
t h r e e f o l l o w i n g p o s s i b i l i t i e s : e i t h e r t h e sequence
f(s)
has l e n g t h 1
, or
its
l e n g t h i s a l i m i t o r d i n a l , o r i t s l e n g t h i s a successor o r d i n a l s t r i c t l y g r e a t e r t h a n 1. Using t h e b a r r i e r p a r t i t i o n theorem 1.4, we can assume t h a t t h e e n t i r e barrier
U
reduces t o one o f t h e t h r e e c o n s i d e r e d c l a s s e s .
I n t h e f i r s t case, f o r each of
A
. Since
A
s
i n U t h e sequence
i s a -better p a r t i a l ordering, the
f(s)
reduces t o an element
U-sequence
f
i s necessa-
r i l y good: c o n t r a d i c t i o n .
Examine t h e second case where a l l l e n g t h s a r e l i m i t o r d i n a l s . Take any two successive elements
s 4 t
Since t h e l e n g t h o f
f(s)
i n t e r v a l of
f(s)
t h e square b a r r i e r
in U
, so
that
f(s)
$.
f ( t ) under e m b e d d a b i l i t y .
i s a l i m i t ordinal, t h e r e e x i s t s a proper i n i t i a l
which i s non-embeddable i n f ( t ) : see ch.4 g 2.1. Consider 2 V = U and t o each eiement v o f V a s s o c i a t e i t s i n i t i a l
s which belongs t o U and t h e f i n a l i n t e r v a l t = v minus i t s m i n i mum i n t e g e r , so t h a t sa t and v = s u t Then t o t h i s v a s s o c i a t e g ( v ) , t h e minimum p r o p e r i n i t i a l i n t e r v a l o f f ( s ) which i s non-embeddable i n f ( t ) . interval
.
232
THEORY OF RELATIONS
s
Note t h a t $ ( g ( v ) ) G ( f ( s ) ) s i n c e g ( v ) has length s t r i c t l y smaller than f ( s ) . Therefore f foreruns r~ (mod B , $ ) y e t g does not reduce t o a r e s t r i c t i o n of f F i n a l l y g i s bad: take any two successive elements v c l w i n V and t h e corresponding i n i t i a l i n t e r v a l s S Q t i n U ; then g ( v ) i s non-embeddable in f ( t ) thus non-embeddable in g(w) which i s an i n i t i a l i n t e r v a l of f ( t ) . This c o n t r a d i c t s our hypothesis t h a t f i s minimal bad. Examine t h e t h i r d case where a l l lengths a r e successor o r d i n a l s d i f f e r e n t from 1.
.
To each element s of U a s s o c i a t e the l a s t term l ( s ) of the sequence f ( s ) and t h e sequence g ( s ) = f ( s ) minus i t s l a s t term. By 2 . 1 we can replace U by a sub-barrier again c a l l e d U , such t h a t t h e U-sequence 1 i s p e r f e c t . Therefore t h e U-sequence g must be bad, s i n c e f i s bad. Take t h e square b a r r i e r V = U 2 , and t o each element v of V a s s o c i a t e t h e i n i t i a l i n t e r v a l s of v which belongs t o U . Then p u t h ( v ) = g ( s ) . Note t h a t $ ( h ( v ) ) = $ ( g ( s ) ) < S ( f ( s ) ) , t h e r e f o r e f foreruns h (mod B , I? ) y e t h i s not a r e s t r i c t i o n of f . F i n a l l y h is bad; indeed with t h e same notations than i n t h e second c a s e , h ( v ) = g ( s ) i s non-embeddable i n h ( w ) = g ( t ) . This c o n t r a d i c t s our hypothesis t h a t f i s minimal bad. 0
5 7 - EQUIVALENCEOF BOTH N O T I O N S OF BETTER P A R T I A L O R D E R I N G : CHAIN SEQUENCE I N A PARTIAL ORDERING Every - b e t t e r p a r t i a l ordering i s a b e t t e r p a r t i a l ordering ( u s e s dependent c h o t c e ) ; t h e r e f o r e both notions coincide, by 5.4. 0 Let A be a - b e t t e r p a r t i a l ordering. By the preceding 6.10, every s e t of ordinal-indexed sequences i n A forms a - b e t t e r quasi-ordering under embeddabil i t y , thus a well quasi-ordering by 6 . 1 . Using 5 . 3 . ( 2 ) , we see t h a t A is an * - b e t t e r ordering f o r each o( : i n o t h e r words a b e t t e r p a r t i a l ordering. 0 Consequently i n 6.10 we can replace b e t t e r p a r t i a l ( o r q u a s i ) ordering by b e t t e r p a r t i a l ( o r q u a s i ) ordering.
7 . 1 . CHAIN SEQUENCE, DOMAIN CHAIN Let C be a chain and A be a p a r t i a l ordering. A chain sequence i n A , o r C-sequence i n A i s a couple ( C , f ) where f i s a function whose domain i s t h e base I C I and whose range i s included i n 1 A I The chain C i s c a l l e d t h e domain chain of ( C , f ) . I f C is an o r d i n a l , we f i n d again an ordinal-indexed sequence w i t h length C , a s defined in ch.1 5 2 . 2 . In t h i s case i t i s unnecessary t o d i s t i n g u i s h between ( C , f ) and f , s i n c e Dom f i s well-ordered by the membership r e l a t i o n . RESTRICTION, ISOMORPHIC SEQUENCE, EXTRACTED SEQUENCE Return t o t h e general case of a chain sequence ( C , f ) . I f U i s a subset of the base I C l , then t h e sequence (C/U,f/U) obtained by r e s t r i c t i n g both the chain
.
Chapter 8
233
.
and the function t o U i s called the r e s t r i c t i o n of (C,f) t o U If h i s a n isomorphism taking C onto the chain D , then the chain sequence (D,f,h-l) i s said t o be isomorphic with (C,f) , and more precisely t o be the image of (C,f) under the isomorphism h . A chain sequence ( D , g ) i s said to be extracted from (C,f) i f f there e x i s t s a r e s t r i c t i o n of (C,f) which i s isomorphic with ( D , g ) I f C and D are b o t h ordinals, we find again the definition of an extracted sequence (see ch.1 0 2 . 2 ) . LESSER SEQUENCE, INF-RESTRICTION, EMBEDDABILITY BETWEEN CHAIN SEQUENCES Given the chain sequence (C,f) in A , a chain sequence (C,g) with the same domain chain C i s l e s s t h a n (C,f) i f f gx 4 fx (mod A) f o r each x in the domain I C l . Analogously f o r "greater than" For ordinal-indexed sequences, we find again the notion as defined in ch.4 0 2. A chain sequence (D,g) i s said t o be an i n f - r e s t r i c t i o n of (C,f) i f f the chain D i s a r e s t r i c t i o n of C and i f (D,g) i s l e s s t h a n the r e s t r i c t i o n (D,f/ D ) . Note the analogy with barrier sequences, see 2 . 2 above. We say t h a t a chain sequence (D,g) is embeddable in (C,f) or t h a t (C,f) admits an embedding of (D,g) i f f there e x i s t s an i n f - r e s t r i c t i o n o f (C,f) which i s isomorphic with (D,g) Equivalently, i f f there e x i s t s a chain sequence l e s s than ( C , f ) from which (D,g) i s extracted. I n t h i s case D i s embeddable in C However D can be embeddable in C without (D,g) being embeddable in (C,f) : consider two w -sequences, one having the constant value a and the other having the value b incomparable with a Every chain sequence extracted from a chain sequence (C,f) i s embeddable in (C,f) However the converse i s f a l s e : assume t h a t a < b and consider the sequences which reduce t o the singleton of a (resp. the singleton of b ). The notions of r e s t r i c t i o n , isomorphism, extracted from, l e s s than, inf-restriction and embeddability are reflexive and t r a n s i t i v e . The only ones which are antisymmetric are the notions of r e s t r i c t i o n , l e s s than and i n f - r e s t r i c t i o n .
.
.
.
.
.
.
7.2. RIGHT AND LEFT INDECOMPOSABLE CHAIN SEQUENCE Let C be a chain and f a function from I C I into a partial ordering A We say t h a t the chain sequence (C,f) i s right indecomposable i f f , f o r each nonempty final interval D of C , the chain sequence (C,f) i s embeddable in i t s Analogously define a l e f t indecomposable chain sequence. r e s t r i c t i o n (D,f/ D ) I f a chain sequence (C,f) i s right indecomposable, then the domain chain C i s i t s e l f r i g h t indecomposable. However the converse i s f a l s e : see 5 5 above. SUM OF CHAIN SEQUENCES Analogous t o the definition of the sum of chains along a chain I (which i s called a homomorphic image of the sum: ch.2 0 3.6), we define the %of chain sequences f i , where the index i runs through the base of a chain I which shall again be called the image chain.
.
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THEORY OF RELATIONS
For each fi , let Ci be its domain chain, and let C be the sum of the Ci along the image chain I , the bases lCil being taken to be mutually disjoint. Then the sum shall be the couple formed of C and the union of the functions fi in other words, the common extension f of the fi with domain ICl . HEREDITARILY INDECOMPDSABLE OR H-INDECOMPOSABLE CHAIN SEQUENCE A chain sequence in a partial ordering A is said to be hereditarily indecomposable, or h-indecomposable, iff it is obtained by induction from the following procedure. The chain sequence which reduces to the singleton of an element of the base I A I is h-indecomposable. If a is either 0 or 1 or a regular infinite aleph, and fi (i < a ) are h-indecomposable chain sequences such that for each i , the set of indices j (i < j < d ) for which f i is embeddable in f is cofinal j or along its converse are h-indecomin ~, then the sum of the fi along posable. Moreover, the only h-indecomposable chain sequences in A are those which can be so constructed. Every h-indecomposable chain sequence is right or left indecomposable, according to whether it is a sum along an infinite regular aleph or along its converse. Moreover, the considered infinite regular aleph is the cofinality (in the case of a right indecomposable chain sequence) or the co-initiality (in the left case). The empty chain and the chain sequences reduced to a singleton are the only h-indecomposable chain sequences which are both right and left indecomposable.
;
7.3. We immediately extend LAVER'S theorems (ch.6 5 5.4 and 5.5, using axiom of choice) to the case of chain sequences. Let A be a partial ordering, and f be a chain sequence in A whose domain chain i s scattered. Suppose that the h-indecomposable restrictions of f form a well quasi-ordering under embeddability. Then f is a finite sum of h-indecomposable chain sequences. Let f be a chain sequence in A whose domain chain is scattered. Suppose that f is indecomposable and that the h-indecomposable restrictions of f form a well quasi-ordering under embeddability. Then f is h-indecomposable.
7.4. GOOD, BAD BARRIER SEQUENCE AND FORERUNNING REVISITED Consider a partial ordering or quasi-ordering A . Let U be a barrier and f be a barrier sequence which to each element of U associates a chain sequence iff there with values in A . Then such a barrier sequence f is said to be fs under embeddability; exist two elements r, s of U with r q s and fr,( f is said to be bad otherwise, following 5 2 above. To each chain sequence in A whose domain chain is scattered, associate the neighborhood rank of its domain chain: let $ be the ranking function thus defined.
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Then the notion of a barrier sequence which foreruns another, or which i s a successor of another (mod A, 6 ) with respect t o forerunning i s defined a s in 5 3 . 3 above. However our present notation (mod A, $ ) replaces the notation ) where J.7 designates the s e t of the considered chain sequences which (mod .-$, take t h e i r values in A and are quasi-ordered under embeddability (mod A) . This forerunning remains reflexive, antisymmetric and t r a n s i t i v e , so tha t i t defines a partial ordering among barrier sequences which themselves take as values chain sequences in A As in 3 . 5 , we define the notion of a minimal bad b arrie r sequence under our new forerunning (mod A, ) . As before, given an ar b i t r ary bad barrier sequence f , there e x i s t s a minimal bad barrier sequence (mod A, which i s a successor of f under forerunning. Now we generalize the proposition 4.1 by replacing the s e t of h-indecomposable , of h-indecomposable chain sequences chains by a s e t , s t i l l denoted by taking t h e i r values in a given b et t er partial ordering denoted by A Our s e t i s quasi-ordered under embeddability. Moreover, we assume tha t (R i s closed under taking any interval of one of the considered chain domains, provided t h i s interval yields an h-indecomposable chain sequence. NOW our proofs 4 . 1 and 4 . 2 extend t o the present case of chain sequences in A , except the f i r s t paragraph in 4.1. Indeed given two successive elements r U s in the b a r r i e r , i f the chain sequence f r reduces to the singleton of an element in A , and i f the chain sequence f s does n o t admit an embedding of f r (since f i s assumed t o be b a d ) , then f s i s not necessarily empty nor necessarily a singleton: i t i s possibly i n f i n i t e , and formed of terms none of which i s greater (mod A) than the unique element of the singleton f r . Moreover, our proof 4.1 corresponds t o the case where A reduces to a singleton. Now i t becomes necessary t o use our hypothesis t h a t A i s a b et t er partial ordering, with possibly infinitely many elements. So t h at we must replace the f i r s t paragraph in 4.1 by the following argument. 0 Let U be a barrier, the domain of f Partition U into two disjoint subsets U ' and U" , where U' i s formed of those elements whose image under f i s a singleton, and U " i s formed of those elements whose image i s an infinite h-indecomposable chain sequence. By the b ar r i er p ar tition theorem 1.4, a t l e a s t one of these two subsets includes a barrier. I f U ' includes a ba rrie r, then t h i s would yield a r e s t r i ct i o n of f which i s a bad barrier sequence with values in the b e t t e r p a r t i a l ordering A : contradiction. Hence there e xists a ba rrie r included i n U" . S t i l l denote by U t h i s b ar r i er , and take up again the proof 4.1 beginning in the second paragraph, now knowing tha t a l l the values taken by f are i n f i n i t e , h-indecomposable chain sequences.
.
s
s)
.
d
.
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THEORY OF RELATIONS
Now we can take up again the proof 4.2 with "chain sequence in A " instead of "chain" . Moreover in this proof, instead of assuming the existence of a bad w -sequence, we can assume only the existence of a bad barrier sequence. So that we obtain the following generalized statement. Let A be a better partial ordering. Then every set of h-indecomposable chain sequences in A forms a better qug-orderjny under-embeddability (uses axiom of choice).
In view of the preceding 7.3, we obtain the following generalization of 4.3. Let A be a better partial ordering. Then every chain sequence in A with a scattered domain chain is a finite sum of h-indecomposable chain sequences. In particular, every indecomposable chain sequencea scattered domain chain is h-indecomposable (uses axiom of choice). 7.5.
7.6. Let A be a better partial ordering. Then each set o f chain sequences with domain chains is better quasi-ordered under values in A and with scattered____ embeddability (LAVER 1968, uses axiom of choice). In particular, each set of scattered chains is better quasi-ordered under embeddability: take the preceding statement where A reduces to a singleton. 0 We know by 7.5 that each set of h-indecomposable chain sequences forms a better quasi-ordering under embeddability. By 6.9 each set of words, or finite sequences composed o f such h-indecomposable chain sequences forms a better quasi-ordering. This subsists for each set of finite sums of h-indecomposable chain sequences, which yields a reinforcement of the previous quasi-ordering: see 5.1. Now by 7.5, every chain sequence with a scattered domain chain is such a finite sum of h-indecomposable chain sequences. 0 _ l _ _ l
7.7. If. A i s a better partial ordering, so i s the partial ordering 3 ( A ) of initial intervals of A (with respect to inclusion); uses axiom of choice. 0 To each initial interval X associate any ordinal-indexed sequence of elements of X , such that every element of X has an upper bound in the corresponding sequence. Then by 6.10 our sequences constitute a better quasiordering under embeddability. Therefore the corresponding initial intervals constitute, under inclusion, a partial ordering which reinforces the quasiordering o f sequences, hence a better partial ordering by 5.1. 0
231
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EXERCISE 1 - Every f i n i t e l y f r e e p a r t i a l ordering has a c o f i n a l r e s t r i c t i o n which i s a b e t t e r p a r t i a l ordering (POUZET 1979, unpublished, answering a conjecture due t o GALVIN; uses axiom of c h o i c e ) .
1 - Let A be a f i n i t e l y f r e e p a r t i a l ordering. P a r t i t i o n A i n t o a f i n i t e union of i d e a l s : ch.4 5 5 . 3 , axiom of choice. For each i d e a l , take a cofinal r e s t r i c t i o n which i s a well-founded p a r t i a l ordering, hence a d i r e c t e d well p a r t i a l ordering: ch.4 5 5.4. Then in view of 6.4 above, i t would s u f f i c e t o prove our a s s e r t i o n f o r each d i r e c t e d well p a r t i a l ordering. A be a d i r e c t e d well p a r t i a l ordering. By ch.7 5 3.11, t h e r e e x i s t s a cofinal r e s t r i c t i o n of A which i s isomorohic with t h e d i r e c t product of a f i n i t e number o f r e g u l a r alephs. T h i s c o f i n a l r e s t r i c t i o n is a b e t t e r p a r t i a l ordering by 6 . 3 and 6 . 5 above. 2 - Let
EXERCISE 2 - A well p a r t i a l ordering f o r which every b a r r i e r sequence on W good, b u t which i s not a b e t t e r p a r t i a l ordering.
is
Let A be t h e p a r t i a l ordering defined on a l l ordered t r i p l e s of natural nunbers x , y , z by the following condition: ( x , y , z ) 4 ( x ' . y ' , z ' ) i f xb x ' and y,( y ' and z 4 z ' and a d d i t i o n a l l y e i t h e r x = x ' , o r x K x ' and y < x ' , o r f i n a l l y x < x ' and z C y ' . 1 - Prove t h e t r a n s i t i v i t y of the above. Note t h a t A i s well-founded, t h e r e being only f i n i t e l y many predecessors of ( x , y , z ) Prove t h a t A i s f i n i t e l y f r e e , hence a well p a r t i a l ordering. For t h i s , note t h a t t h e d i r e c t nroduct ( x,< x ' and y,< y ' and z ,< z ' ) i s a well p a r t i a l ordering. Suppose t h a t t h e r e e x i s t s an w-sequence of t r i p l e s ( x i , y i , z i ) where i i s a natural number , which a r e mutually incomparable (mod A) . Then e x t r a c t an a - s e q u e n c e w i t h xi i n c r e a s i n g , yi i n c r e a s i n g , z i increasing. Since incomparability r e q u i r e s t h a t xi be s t r i c t l y increasing i n i , t h e r e e x i s t s an i f o r which xi > yo and xi > xo : c o n t r a d i c t i o n .
.
2 - Prove t h a t A i s not a b e t t e r p a r t i a l ordering; indeed we have the bad b a r r i e r sequence with domain i,s3 ( i . e . the set of a l l 3-element subsets of t h e s e t of natural numbers), defined a s follows: each 3-element subset { x , y , z ) ( w i t h x < y < z ) i s taken i n t o the ordered t r i p l e ( x , y , z ) . To see t h i s , l e t
, t h i s being and take an i n t e g e r r > z , and l e t t = { y , z , r ) the only possible manner t o g e t sa t Then ( x , y , z ) and ( y , z , r ) a r e incomparable (mod A) . s = {x,y,z}
.
3 - Now l e t f be a b a r r i e r sequence w i t h domain w ( t h e s e t of unordered p a i r s of i n t e g e r s ) . For each p a i r s of i n t e g e r s , l e t x ( s ) , y ( s ) , z ( s ) be t h e e s h a l l prove t h a t f i s good. coordinates of f ( s ) . W
THEORY OF RELATIONS
238
Denote by B the d i r e c t product ( x x ' and y , c y ' and z sz' ) , which i s a b e t t e r p a r t i a l ordering by 6.3 and 6.5 above. Since A and B a r e both based on the same s e t of ordered t r i p l e s , consider f a s a b a r r i e r sequence i n B and then replace f by a r e s t r i c t i o n which be p e r f e c t (mod B ) : see 2 . 1 above. A f t e r renumbering, this r e s t r i c t i o n of f s t i l l has domain u2 , and now we again consider f a s taking values in A . T h u s f o r any two p a i r s of i n t e g e r s , say s and t w i t h s Q t , we have x(s),( x ( t ) , and s i m i l a r l y w i t h y and with z . Then e i t h e r t h e r e e x i s t s , t w i t h s d t and x ( s ) = x ( t ) , in which case f i s good and we a r e f i n i s h e d .
.
Or s 4 t implies n e c e s s a r i l y t h a t x ( s ) < x ( t ) Then by RAMSEY's theorem, we can require e i t h e r t h a t f o r a l l i n t e g e r s i < j k we have y(J,i,j)) = y({i,k}) , o r . In t h e f i r s t c a s e , take t h a t we have t h e s t r i c t i n e q u a l i t y y ( j i , j j ) ( y ( { i , k ] ) a s t r i c t l y increasing Cr)-sequence of i n t e g e r s i o < i l < , so t h a t successive p a i r s give s t r i c t l y i n c r e a s i n g values f o r x . Then f o r h s u f f i c i e n t l y l a r g e
-=
...
Y(\iO'ih\)< x ( ( i h ' i h + l ) ) and obviously x ( { i O , i h ) ) < x ( i i h , i h + l j ) : so t h a t f i s good. In t h e second c a s e , take again a s t r i c t l y increasing d - s e q u e n c e of i n t e gers g i v i n g , f o r h s u f f i c i e n t l y l a r g e , z({io,il}) < y ( { i l , i h ) ) and obviously x({i,-,,il})
h ) P of c a r d i n a l i t y : 2 t o t h e power pn ; so t h a t , t h e b i j e c t i o n taking a l , ...,a into P 1,. . . , p , transforming the r e s t r i c t i o n R/ial ,. . . ,a i n t o an n-ary r e l a t i o n w i t h P base $1,. .. , p ) , these l a t t e r r e l a t i o n s a r e mutually d i s t i n c t . Then t h e r e necessar i l y exist two of our p-tuples, say ( a l ,..., a p ) and (bl ,..., b p ) , f o r which the
1
r e s t r i c t i o n s M/{al, ...,a p \ and M/{bl, ....b 1 a r e transformed i n t o the same P m u l t i r e l a t i o n w i t h base l, 1,..., p ) . I t follows t h a t the function taking a l ,..., a i n t o b l , ..., bp i s a l o c a l automorphism of M , y e t not a local automorphism P of R . In o t h e r words R i s not f r e e l y i n t e r p r e t a b l e i n M : c o n t r a d i c t i o n . 0 2.3. INTERPRETABILITY ARITY To obtain a common g e n e r a l i z a t i o n of t h e notions of dimensional a r i t y of a r e l a t i o n , as defined i n t h e above 2 . 2 , and t h a t of t h e dimension of a p a r t i a l o r d e r i n g , a s defined i n ch.4 5 7.3 and going back t o DUSHNIK, MILLER 1941, we say t h a t an o r d i nal sequence I of length a , taking natural number values, i s an i n t e r p r e t a b i l i t y a r i t y f o r the r e l a t i o n R , i f t h e r e e x i s t s an o( -sequence obtained from I by replacing each term ui i n I ( t h u s i < d ) by a r e l a t i o n Ri o f a r i t y u i and having t h e same base a s R , so t h a t R i s f r e e l y i n t e r p r e t a b l e i n t h e sequence of the R i , in t h e sense t h a t every local automorphism common t o the Ri i s a local automorphism of R Then f o r an a r b i t r a r y r e l a t i o n R , i f we r e q u i r e t h a t I have f i n i t e l e n g t h , then t h e dimensional a r i t y i s the l e a s t possible maximum of t h e sequences I which a r e i n t e r p r e t a b i l i t y a r i t i e s of R . I f R i s a p a r t i a l ordering and i f we r e q u i r e t h a t the r e l a t i o n s Ri be c h a i n s , then the l e a s t p o s s i b l e length of t h e i n t e r p r e t a b i l i t y a r i t i e s I i s obviously l e s s than o r equal t o the dimension of R i n the sense of ch.4 5 7.3. Problem. I f R i s the i d e n t i t y r e l a t i o n , whose dimension i s 2 , and i f we again
.
require t h a t t h e r e l a t i o n s Ri be c h a i n s , t h e n t h e l e a s t possible length under consideration i s 0 (the i d e n t i t y r e l a t i o n being f r e e l y i n t e r p r e t a b l e i n t h e multirel a t i o n reduced t o i t s b a s e ) . Apart t h i s c a s e , t h e dimension of a p a r t i a l ordering is equal t o t h e l e a s t p o s s i b l e length of i t s i n t e r p r e t a b i l i t y a r i t i e s by means of chains. 2.4. Let R , R' be two m-ary r e l a t i o n s and S , S ' be two n-ary r e l a t i o n s ; l e t E be the common base of R and S , and E ' t h e common base of R ' and S ' . I f every r e s t r i c t i o n of t h e concatenation R'S' 4 n elements i s embeddable i n RS , and i f S i s f r e e l y i n t e r p r e t a b l e i n R , then S ' i s f r e e l y i n t e r p r e t a b l e in R' .
245
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f'
Let
be a l o c a l automorphism o f
less than o r equal t o restriction from
.
n
(R'S')/F'
onto a r e s t r i c t i o n o f
RS
(R'S')/(f')'(F')
onto a r e s t r i c t i o n o f
RS
a l o c a l automorphism o f
. Thus
in
R
of
R'
i s f r e e l y interpretable i n
S'
(ch.2
/
2)
, by
R
constructed from two chains
s e t t i n g each element o f
R1
R1,
R2
(each w i t h car-
t o be incomparable w i t h each
246
THEORY OF RELATIONS
element o f =
R1
Take
S
and
S2 =
,yet
table i n
R
cardinal
1, 2
§
.
R2
Sl
chains
3
Ri
= converse o f
. Then
R2
i s freely interpretable i n
S/X
or
t o be t h e p a r t i a l o r d e r i n g s i m i l a r l y o b t a i n e d f r o m t h e i s not freely interpref o r each subset
R/X
CONNECTION WITH FREE INTERPRETABILITY
Given two f i n i t e sequences o f n a t u r a l numbers, say a s s o c i a t e s t o each
m-ary m u l t i r e l a t i o n
m and
an
R
, a f r e e operator
n
n-ary m u l t i r e l a t i o n
h a v i n g t h e same base. We have t h e a d d i t i o n a l c o n d i t i o n t h a t f o r any two R, R '
relations
each l o c a l isomorphism f r o m
T(R)
morphism f r o m
phism. The sequences i s an
(m,n)-ary
with
X
.0
3 - FREE OPERATOR,
9
S
into
R
into
T(R').
m, n
are called the a r i t i e s o f
o p e r a t o r . We say t h a t any
i s also a local iso-
R'
9
I n o t h e r words,
(R) m-ary
p r e s e r v e s l o c a l isomor-
9 , and we
say t h a t
7
9 .
m-ary r e l a t i o n i s a s s i g n a b l e t o
Note t h a t each f r e e o p e r a t o r i s c o m p l e t e l y determined by i t s values on m u l t i r e l a t i o n s whose base i s a f i n i t e subset o f t h e s e t ~3 o f n a t u r a l numbers: t h e v a l u e i n t h e general case f o l l o w s immediately by u s i n g l o c a l isomorphisms. As a l l t h e m u l t i r e l a t i o n s based on subsets o f an
(m,n)-ary
w c o n s t i t u t e a s e t , we can d e f i n e
o p e r a t o r as a f u n c t i o n which, t o each
f i n i t e subset o f CJ
m-ary r e l a t i o n based on a
, a s s o c i a t e s an n-ary r e l a t i o n on t h e same base, w i t h t h e
preceding c o n d i t i o n about l o c a l isomorphisms. A f t e r d e f i n i n g we complete by d e f i n i n g t h e v a l u e t a k e n by
9
f o r each
as i n d i c a t e d ,
m-ary r e l a t i o n : a l l t h i s
w i t h i n t h e framework o f t h e axioms o f ZF. Example. F o r a g i v e n n a t u r a l number each
, negation
n
i s a f r e e o p e r a t o r which takes
n-ary r e l a t i o n i n t o t h e r e l a t i o n h a v i n g t h e same base and always t a k i n g
t h e o p p o s i t e value. Another example. The symmetrizing o p e r a t o r which takes each b i n a r y r e l a t i o n i n t o the binary r e l a t i o n
S
satisfying
S(x,y)
= R(y,x)
for all
R
base. A l s o t h e o p e r a t o r which t a k e s each b i n a r y r e l a t i o n relation An
S ( x ) = R(x,x)
(m,n)-ary
f o r those Max n 3.1.
S
The m u l t i r e l a t i o n
9
i s determined by t h e ordered p a i r s R
(R,
T(R))
having a base o f c a r d i n a l a t most equal t o
R
i s freely interpretable i n
into
S
freely interpretable i n
R
i f f there exists a
R
.
Consider t h e case o f two r e l a t i o n s :
. . ,xn)
i n t o t h e unary
t h e r e a r e o n l y f i n i t e l y many f r e e o p e r a t o r s o f g i v e n a r i t i e s .
f r e e operator taking
(xl,.
R
i n the
.
f r e e operator
m-ary m u l t i r e l a t i o n s
. Hence
x, y
. F o r each
o f elements i n t h e base
R
is
m-ary and
m-ary r e l a t i o n
I X I , either
S
X
is
n-ary and
and each
S
n-tuple
t h e r e e x i s t s an isomorphism
is
2 41
Chapter 9
f
from X/\xl
,..., xn) onto
( 3X ) ( x l , ..., x n )
a r es t r i ct i o n
R/{ fxl ,... ,fxn) : then we define
S(fxl.... , f x n ) ; t h i s l a s t value does n o t depend on the chosen isomorphism. Or no such isomorphism ex i s t s : then we s e t ((?X)(x l , . . . , x n ) = t . 0 =
3.2. INJECTIVE OPERATOR # ?(R') A free operator .j-> i s said t o be injective i f R # R ' implies tha t for a l l R , R ' assignable t o 9 ; or equivalently i f for a l l R , R ' every local isomorphism from q ( R ) into T(R') i s a local isomorphism from R into R ' .
Y(R)
Every injective operator has an inverse. More precisely, i f i s inje c tive , then there e x i s t s a f r e e operator 2 such t h a t $, T(R) = R for each R assignable to .
9
9.
Then Consider the case of r el at i o n s , and l e t m , n be the a r i t i e s of given an n-ary relation Y and an m-tuple ( x l , ..., x,) of elements in the base I Y I , we define ( d , Y)(xl ,..., xm) = X(xl ,..., xm) i f the re stric tion
0
x l , ...,xm] i s the image under 9 of an m-ary relation X having the same Y ) ( x l , ...,xm) = + i f there i s no such X . 0 base; o r
Y/{
(a
Hence, if i s i n j ect i v e, then R and F(R) are each freely interpretable in the other. A converse of t h i s r es u l t shall be proved in 3.5 below.
9
COMPARISON BETWEEN ARITIES We say t h a t the a r i t y n (of a multirelation) i s greater t h a n the a r i t y m , i f each term mi of m can be associated with a term n o f n , with m i $ n j , j in an injective manner: i . e . two d i s t i n c t indices i , i ' in m correspond t o two d i s t i n c t indices j , j ' in n .
If the a r i t y n i s greater t h a n the a r i t y m , then f o r any natural number P, there are more n-ary multirelations with base having cardinality p , than there are of m-ary multirelations with the same base. his However, even i f the above condition i s t r u e f o r every natural number p , tdoes n o t necessarily imply t h at the a r i t y n i s greater than m . 0 For example, take m = (1,l) and n = (0,2) Then f o r a base of cardinality p , there are ( 2 t o the power 2 p ) many m-ary b i r ela tions, and ( 2 t o the power ( l + p 2 ) ) many n-ary b i r el at i o n s , with l + p 2 >/ 2p ; ye t n i s not greater than m.
.
Problem. If the a r i t y n i s greater t h a n m , then there obviously e xists an (m,n)-ary injective operator. Indeed t o each mi-ary component Ri of the multirelation R , i t suffices t o associate the n.-ary component S j ( m i Q n j ) J whose value only depends on the mi f i r s t terms, tha t value being equal to tha t o f Ri . Conversely i f n i s n o t greater t h a n m , then we conjecture t h a t there e xists no injective free operator with a r i t i e s ( m , n ) For example i f m = (1,l) and
.
0
THEORY OF RELATIONS
248
, then
R = (0,2)
there e x i s t s no i n j e c t i v e f r e e
w i t h a base o f three elements the value (+) o n l y f o r
a, b, c
, and
a
R2
(m,n)-ary
operator. Indeed s t a r t
and the b i r e l a t i o n
only f o r
b
. Then
(R1R2)
where
R1
takes
t h e o n l y l o c a l automorphisms
o f t h i s b i r e l a t i o n are the i d e n t i t y on each subset o f the base. On the o t h e r hand, any a r b i t r a r y b i n a r y r e l a t i o n must take the same value, f o r instance f o r and
(a,a)
, and thus admits a l o c a l automorphism o t h e r than the i d e n t i t y on a
(c,c)
subset o f t h e base. Obviously the concatenation o f our b i n a r y r e l a t i o n w i t h a 0-ary r e l a t i o n , changes nothing i n the previous discussion. F i n a l l y , the reader who i s tempted by t h e pseudo-solution which associates t o S(x,y) = R1(x) A R2(y) , w i l l note t h a t i n j e c t i v i t y takes always t h e value ( - ) , f o r example. R1
(R1R2) t h e b i n a r y r e l a t i o n i s no longer s a t i s f i e d when 3.3.
PARTIAL OPERATOR
Let
m
, and
be a f i n i t e sequence o f n a t u r a l numbers
18
a set o f
m-ary m u l t i r e l a -
t i o n s w i t h f i n i t e bases, which i s closed under r e s t r i c t i o n and isomorphism ( t o be rigorous i n the frame o f the axioms o f ZF, we assume t h a t the bases o f our m u l t i r e o f n a t u r a l numbers ) .
l a t i o n s are f i n i t e subsets o f the s e t (m,n)-ary
A
4 i s a function , associates an n-ary
p a r t i a l operator w i t h domain
m-ary m u l t i r e l a t i o n R
belonging t o (R
9
w i t h the same base, such t h a t
preserves l o c a l isomorphisms.
9 to
A necessary and s u f f i c i e n t c o n d i t i o n f o r a p a r t i a l operator
i s again t h a t every l o c a l isomorphism from ?(R) phism from
R
into
R'
which t o each multirelation T(R)
, f o r every
R, R '
into
belonging t o
$(R') Dom
be i n j e c t i v e ,
be a l o c a l isomor-
9.
CANONICAL EXTENSION Let
9
arity
be a
,in
m
(m,n)-ary p a r t i a l operator, where t h e a r i t y
9 ,and whose
extending
domain contains a l l
F i r s t , t o each term mi
, such t h a t
i s g r e a t e r than the
of
m
, associate i n an i n j e c t i v e manner a term n . o f
mi$
mi
by
m
and
.,
e i t h e r the r e s t r i c t i o n we s e t
(
The operator If
9
9=
=
i n the base R i s an element o f Dom
. . ,xn) ( T R')(xl,. . . ,xn)
R ' = R/{ xl,.
9 =R)(xl,. . .,xn)
injective.
.
IR I ,
y, and then
thus defined i s c a l l e d t h e canonical extension o f
.
6 n (replace
f o r a m u l t i r e l a t i o n ) , then the canonical extension i s t h e unique f r e e
operator extending I n general,
J by n
n j
; o r not, and then we s e t
i s already defined f o r a l l m-ary r e l a t i o n s o f c a r d i n a l i t y
by Max n
9=
m-ary m u l t i r e l a t i o n s .
n ; i n t h e f o l l o w i n g we denote j Given an m-ary r e l a t i o n R and elements xl,.. x n
n
t h e sense o f 3.2 above. We s h a l l d e f i n e as f o l l o w s an operator
9 .
can be i n j e c t i v e w i t h o u t i t s canonical extension
9=
being
2 49
Chapter 9
m
Let
9
= n = 1 and l e t
associate t o each unary r e l a t i o n t a k i n g always the
9 i s undefined f o r 9 = takes every unary
value (+),theunary r e l a t i o n always ( - ) ; y e t
unary r e l a t i o n s
taking a t l e a s t once, t h e value ( - ) . Then
relation into
the unary r e l a t i o n w i t h same base, always ( - ) ; so t h a t p = i s n o t i n j e c t i v e . 0 3.4. L e t
k
be a n a t u r a l number
. If a partial
7
operator
i s defined f o r every
m-ary r e l a t i o n o f c a r d i n a l i t y
4 k and o n l y f o r such, and i f
then the canonical extension
9=
Moreover i n t h e case where
(
9
i s injective,
i s injective.
i s an
(m,m)-ary
p a r t i a l operator, then we have
9 = ) - I - (9- I ) =
3.5.
n
greater than
i n j e c t i v e p a r t i a l operator w i t h domain
L e t m, n
be two a r i t i e s w i t h
and range
i n t e g e r f o r which
&
. Let 9
m
@ .
Let
be an (m,n)-ary
k
be the l a r g e s t
contains every m-ary m u l t i r e l a t i o n o f c a r d i n a l i t y
Then t h e r e e x i s t s a p a r t i a l operator
+
7 ,which
extending
defined f o r every m-ary m u l t i r e l a t i o n o f c a r d i n a l i t y Consequently every i n j e c t i v e p a r t i a l (m,n)-ary
6 k+l
.
& k
i s i n j e c t i v e and
(POUZET 1973).
operator, where
n
i s g r e a t e r than
m , i s e x t e n d i b l e t o an i n j e c t i v e f r e e operator: go from k t o k + l , e t c . u n t i l reaching
.
Max n
Another consequence i s the f o l l o w i n g converse o f 3.2: Let
If
R R
be an m-ary r e l a t i o n ,
and
S
an n-ary r e l a t i o n , w i t h
i n j e c t i v e f r e e (m,n)-ary
operator which takes
R
w i
t i v e p a r t i a l operator which takes every r e s t r i c t i o n o f of 0
S
n
g r e a t e r than
.
m
are each f r e e l y i n t e r p r e t a b l e i n t h e other, then there e x i s t s an
S
S : s t a r t w i t h the i n j e c i n t o the r e s t r i c t i o n
R
having the same base.
Proof o f the f i r s t assertion. I t s u f f i c e s t o consider the case of an m-ary
relation o f cardinality Let
pk be the
, which
k+l
tions with c a r d i n a l i t i e s
,
/ p >/ q has a chainable r e s t r_i c _ tion ~
.
of cardinality p 0
(1) L e t
n
be t h e a r i t y o f
IRI
chain w i t h base multirelation
RC
.
, which
R
, which
We say t h a t two
n-element subsets
equivalent iffthe r e s t r i c t i o n s
(RC)/X
( i Q n)
. There
for
i-element subsets
we can assume i s >/ 2
. Let
C
be a
i s isomorphic w i t h W ; consider the concatenated and
(RC)/X'
X, X '
o f the base, are
are isomorphic; s i m i l a r l y
are o n l y f i n i t e l y many equivalence classes
f o r t h e equivalence r e l a t i o n thus defined. Using RAMSEY's theorem (ch.3 base, i n which a l l subsets, f o r each
5 l.l),
t h e r e e x i s t s a denumerable subset
n-element subsets are equivalent, as w e l l as a l l
i6 n
.
L e t A = C/D
,a
D of the i-element
chain isomorphic w i t h W . Then every
l o c a l automorphism o f A having domain o f c a r d i n a l i t y \< n i s a l o c a l automorBy 2.1, t h e r e s t r i c t i o n R/D i s f r e e l y i n t e r p r e t a b l e i n A 0 phism o f R/D
.
.
256
THEORY OF RELATIONS
fj 1.3.0
0 ( 2 ) Analogous proof, using t h e f i n i t a r y version o f RAMSEY's theorem: ch.3
5.6.
Given the o r d i n a l
w ,
, each G)-chainable r e l a -
the chain o f n a t u r a l numbers
t i o n i s minimal w i t h respect t o embeddability, among the denumerable r e l a t i o n s ;
i.e.,
every denumerable r e l a t i o n which i s embeddable i n an W-chainable r e l a t i o n i s isomorphic w i t h
R
Let
a
.
R
be an aleph, i . e . an o r d i n a l which i s equipotent w i t h no s t r i c t l y smaller
o r d i n a l . Then each
a-chainable r e l a t i o n i s minimal w i t h respect t o embeddability,
among the r e l a t i o n s o f c a r d i n a l i t y
.
a
The proposition 5 . 5 . ( 1 ) above asserts t h a t , f o r every denumerable r e l a t i o n R , R there e x i s t s a denumerable minimal r e l a t i o n which i s embeddable-i! This r e s u l t does n o t extend t o t h e c a r d i n a l i t y o f t h e continuum. Indeed by DUSHNIK,
.
MILLER, ch.5 fj 5.2.(1), dable i n t h e r e a l s
every chain w i t h continuum c a r d i n a l i t y and which i s embed-
has a r e s t r i c t i o n o f continuum c a r d i n a l i t y which i s s t r i c t l y
l e s s e r , w i t h respect t o embeddability. Problem. Does there e x i s t a r e l a t i o n w i t h continuum c a r d i n a l i t y , which i s minimal among r e l a t i o n s w i t h continuum c a r d i n a l i t y , y e t which i s n o t chainable by mean o f the continuum aleph ( i . e . the smallest o r d i n a l w i t h continuum c a r d i n a l i t y ; the axiom o f choice being used). 5.7.
For each ordered p a i r of n a t u r a l
numbers
n
the S t i r l i n g number, o r number o f p a r t i t i o n s o f an
and
r< n
, l e t S:
n-element s e t i n t o
denote r
non-
empty classes. Zn o t h e r words, the number o f equivalence r e l a t i o n s w i t h cardinality
n
We have
and having e x a c t l y r non-empty equivalence classes. 0 So = 1 : the equivalence r e l a t i o n w i t h empty base i s supposed t o e x i s t ;
0
obviously i t has e x a c t l y each
n 31
, since
non-empty equivalence classes. We have
an equivalence r e l a t i o n on
$=0
for
n elements has a t l e a s t one
non-empty equivalence class. We have the
S;
= S i = 1 f o r each s t r i c t l y p o s i t i v e i n t e g e r
S:
reCUrSiOn e q u a l i t y
Suppose t h a t the s e t
= Sn-'
r-1
{ 1,2,
and p u t aside the element
n
+ r.S:--l
... ,n} . Then
f o r every
n ; moreover we have r
i s partitioned i n t o
( 1 6 r , n
relation with cardinality a t
i s c h a i n a b l e . More s p e c i f i c a l l y , f o r each
R i s a (6p)-monomorphic
n
exists a least
n-ary r e l a t i o n w i t h c a r d i n a l i t y
chainable. By t h e above 6.3, we can suppose s i m p l y t h a t
R
there exists a such t h a t i f
q>/ p
>, q , t h e n R is
is
p-monomorphic,
( Lp)-monomorphic.
instead o f
On t h e o t h e r hand, f o r each i n t e g e r w i t h i n f i n i t e base, such t h a t 0 To each
e x i s t s an i n t e g e r
R
, there
n >, 2
e x i s t s an
& (4n)-monomorphic
n-element s e t , a s s o c i a t e a unique
n-tuple,
n-ary r e l a t i o n
R
.vet n o t c h a i n a b l e .
formed o f t h e elements o f
t h i s s e t w i t h o u t r e p e t i t i o n , i n such a way t h a t t h e s e p a r t i c u l a r
n - t u p l e s do n o t
form a c h a i n , f o r which t h e y would be t h e r e s t r i c t i o n s o f c a r d i n a l i t y
n
. Then
R t a k e t h e v a l u e (+) f o r o u r p a r t i c u l a r n - t u p l e s , and t h e v a l u e ( - ) f o r
let
a l l other
n-tuples. 0
-
Problem. Does t h e r e e x i s t a p o s i t i v e i n t e g e r each n a t u r a l number
yet not
p
, there
e x i s t s an
no
such t h a t f o r each
n-ary r e l a t i o n which i s
n >,no
and
p-monomorphic
( f r o m t h e p r e c e d i n g i t f o l l o w s t h a t such r e l a t i o n s
(p+l)-monomorphic
would necessary be f i n i t e ) . 6.5. The n o t i o n of a tournament r e l a t i o n was d e f i n e d i n ch.5
5
2.7.
Every r e s t r i c -
t i o n o f a tournament t o a 3-element s e t , i s e i t h e r t h e c h a i n o f c a r d i n a l i t y 3
, or
the b i n a r y c y c l e o f c a r d i n a l i t y 3 : more e x a c t l y t h e r e f l e x i v e b i n a r y c y c l e o f card i n a l i t y 3 : f o r s h o r t we s h a l l c a l l i t t h e
u
Given an element
3-cycle.
i n t h e base o f a b i n a r y r e l a t i o n , we s h a l l say t h a t a
which i s a r e s t r i c t i o n o f o u r r e l a t i o n , passes t h r o u g h
Let
If
be a tournament and
A
A
.
u
the element
&
(p-2)-monomorphic,
(1) f o r each element
u
independent o f passinq throuqh (2) l e t
(x,y)
u of
i t s base, o f f i n i t e c a r d i n a l i t y
E
u and
and
v
(x',y')
E
, t h e number o f 3-cycles p a s s i n q t h r o u q h u
i s independent o f
1 1
u, v o f E , t h e number o f 3-cvcles
u, v
;
be any two a r b i t r a r y o r d e r e d p a i r s o f elements i m x # y
f
be an isomorphism f r o m A/(E-{x,y))
z
in
E-\x,y)
pa 5 .
then
; f o r each p a i r o f elements
which s a t i s f y t h e c o n d i t i o n s
A/{T,y' , z ' )
3-cycle
u i f i t s base c o n t a i n s
,
x'
# y ' , A(x,y)
onto
A/(E-{x',y'))
= A(x',y')
=
+
E
; and l e t
; t h e n f o r each element
and f o r i t s image z' = f z , t h e r e s t r i c t i o n s A/-(x,y,z)= a r e e i t h e r b o t h 3 - c y c l e s , o r b o t h c h a i n s (POUZET 1977, unpublished).
262
THEORY OF RELATIONS
(1) L e t
0
.
u, v be two d i s t i n c t elements o f E
Since
u
dent o f
and
v ;let
k2
denote t h i s number.
u
number o f 3 - c y c l e s which do n o t pass t h r o u g h
5
4.1.(1)
with
Now l e t
k
p = 3
is
(o-2)-monomorphic,
Given an element
p t q = (Card E)-2 ; l e t
kl
of
u
u , there e x i s t
hl
= k-kl
, the
E
u : a p p l y ch.3
i s independent o f
denote t h i s number.
denote t h e number o f a l l t h e 3 - c y c l e s which a r e r e s t r i c t i o n s o f
F o r each element
.
A
u
many 3 - c y c l e s which pass t h r o u g h
.
u and v d i s t i n c t , t h e r e e x i s t kl many 3 - c y c l e s which do n o t pass t h r o u g h
Given v
and
A
u n o r t h r o u g h v , i s indepen-
t h e number o f 3-cycles which pass n e i t h e r t h r o u g h
, and
u e i t h e r , hence (kl-k2) u without passing through v . F i n a l l y there e x i s t many 3 - c y c l e s which pass t h r o u g h u and t h r o u g h v . 0
among these
k2 many which do n o t pass t h r o u g h
many which pass t h r o u g h h = k-2kl+k2
( 2 ) F o r elements
in E , let
x,y,z
o f t h e 3-cycles p a s s i n g t h r o u g h 3-cycles which pass through
and
y
We r e t u r n t o t h e two o r d e r e d p a i r s fz
z
Let
h(-x,y,z)
denote t h e number o f
without passing through
(x,y)
and
(x',y')
, etc.
x
and t h e elements
i n o u r statement. The number o f 3 - c y c l e s which pass t h r o u g h
z
z
and
i s equal t o
by t h e above (1). On t h e o t h e r hand, t h i s number i s equal t o t h e f o l l o w i n g sum:
hl
h(x,y,z)
+
through
x
+ h(x,-y,z) + h(-x,-y,z) . The number o f 3-cycles which pass z i s h2 by t h e above ( l ) , and i s a l s o equal t o h(x,y,z) + S i m i l a r l y t h e number o f 3 - c y c l e s p a s s i n g t h r o u g h y and z i s h2
h(-x,y,z) and
.
h(x,-y,z) and equals
.
+ h(-x,y,z)
h(x,y,z)
s u b t r a c t i n g t h e f i r s t , we o b t a i n
Adding t h e second and t h i r d e q u a l i t i e s and t h e n h(x,y,z)
same r e s u l t when s u b s t i t u t i n g x ' , y ' , z ' phism f r o m A/(E-{x,y})
6.6. p 2 5
Let
A
A
A
that
&
hl
+
, the
numbers
= h(x',y',z')
y'
due t o POUZET 1977)
,is
. We
h(-x,-y,z)
. Now,
since
f
have t h e
i s an isomor-
h(-x,-y,z)
and
.0
E i t s base, which i s f i n i t e and o f c a r d i n a l
(p-2)-monomorohic. x # y
onto
morphism f r o m A/(E-{x,y)) y
h(x,y,z)
-
x,y,z
i s a chain; o r e l s e , f o r any two o r d e r e d p a i r s
satisfying the conditions x'
= 2h2
for
A/(E- { x ' , y ' } )
be a tournament and
. Suppose
Then e i t h e r
onto
a r e e q u a l : so t h a t
h(-x',-y',z')
0
denote t h e number, e i t h e r 0 o r 1,
h(x,y,z)
.
x,y,z
,
x' # y'
,
A(x,y)
= A(x',y')
, when
A/(E-{x',y'j)
(x,y)
and
= +
, every
extended by t a k i n g
( x SY') iso-
x
to
(JEAN 1969; t h e f o l l o w i n g p r o o f i s
an automorphism o. f A
.
E i t h e r a l l r e s t r i c t i o n s of
A
w i t h c a r d i n a l i t y 3 a r e chains: t h e n
c h a i n . O r t h e r e e x i s t s a t l e a s t one 3 - c y c l e . By t h e p r e c e d i n g 6.5, o f d i s t i n c t elements t h e r e passes a t l e a s t one 3-cycle.
A
is a
f o r each p a i r
Suppose f r o m t h i s p o i n t
on, t h a t we a r e i n t h i s case. Let
(x,y)
element i n
and
(x',y')
E-{x,y}
, and
be t h e two o r d e r e d p a i r s i n o u r statement, and z'
t h e image o f
z
z
under t h e isomorphism i n o u r
an
263
Chapter 9
statement. Suppose t h a t
A(x,z)
=
+
: we s h a l l prove t h a t
towards a c o n t r a d i c t i o n i n supposing t h e c o n t r a r y , t h a t that
A(z',x')
= t
A(x',z') A(x',z')
+ by a r g u i n g
= =
-
and hence
.
i s n o t a c y c l e : by t h e p r e c e d i n g 6.5.(2), t h e r e s t r i c The r e s t r i c t i o n A/jx,y,z) t i o n A/\x',y',z') i s n o t a c y c l e ; hence A ( z ' , y ' ) = + . L e t u be an element such t h a t A/(x,y,u} A(y,u) = A(u,x) A(z',u')
i s a c y c l e : we know t h a t such an element e x i s t s . Then we have
+ . We c l a i m t h a t A(u,z) = + . I f n o t , t h e n A(z,u) = + , so + as w e l l , where u' i s t h e image o f u under t h e isomorphism i n o u r
=
=
z'
h y p o t h e s i s . Then no 3 - c y c l e p a s s i n g t h r o u g h (recall that
nor through y ' Yet
z
A/{x',y',u'J
u'
and
would pass through
x'
i s a c y c l e , by t h e p r e c e d i n g 6 . 5 . ( 2 ) ) .
would be a c y c l e ; so t h a t t h e number o f 3-cycles p a s s i n g through
A/{x,z,u)
u would be s t r i c t l y g r e a t e r t h a n t h e number o f 3 - c y c l e s p a s s i n g through z ' and u' , c o n t r a d i c t i n g 6.5.(1). Thus A(u,z) = A ( u ' , z ' ) = + . It now f o l l o w s t h a t A(z,y) = + ; f o r i f n o t , t h e n no 3 - c y c l e p a s s i n g t h r o u g h z and u would pass t h r o u g h x n o r through y , and and
yet
A/{y',z',u'j
would s t i l l be a c y c l e , c o n t r a d i c t i n g 6.5.(1). Thus o u r h y p o t h e s i s A(x,z) = A ( z ' , x ' ) = + a l l o w e d us t o determine t h e values o f t h e tournament
A
f o r a l l pairs included i n
We t e r m i n a t e t h e p r o o f by t a k i n g an element we know t h a t such an element e x i s t s . Thus Moreover, no 3 - c y c l e p a s s i n g t h r o u g h
1 x,y,z,u]
z, t
Problem communicated by POUZET, 1978. L e t
E with f i n i t e cardinality
x'
phic, then are
A
A'
and
o r by y '
A
A'
and
.
p a 5
u
.
(under
t
. Hence z', t'
0
be two tournaments h a v i n g
I f f o r each subset
E , t h e r e s t r i c t i o n s A/X
by removing two elements f r o m
and from
on t h e one hand, and t h r o u g h
on t h e o t h e r hand, i s d i f f e r e n t , c o n t r a d i c t i n g 6.5.(1).
t h e same base
i s a cycle:
z' and t h r o u g h t h e image t ' o f
t h e isomorphism i n o u r h y p o t h e s i s ) , can be completed by t h e number o f c y c l e s p a s s i n g t h r o u g h
A/{x,z,tj
i s d i s t i n c t from y
t
.
and i n { x ' , y ' , z ' , u ' l
f o r which
t
and
A'/X
X
obtained
a r e isomorphic,
isomorphic.
2
p>/ 5. If A A & (p-1)-monomorphic. Moreover, i f A i s a n o n - t o t a l l y o r d e r e d tournament, t h e n f o r any two elements x & a x ' i n t h e base, t h e r e e x i s t s an automorphism o f A which takes x i n t o x ' (JEAN 1969).
6.7.
Let
A
be a b i n a r y r e l a t i o n w i t h f i n i t e c a r d i n a l i t y
(p-2)-monomorphic,
then
~
-
0 Since t h e r e l a t i o n
by 6.3 ( i n d e e d p ideas, suppose t h a t
A
5
A
is
, so
A
(p-2)-monomorphic, that
2
< p-2
) . Since
i s r e f l e x i v e . Since
A
i s also A
( 4 2)-monomorphic
i s 1-monomorphic, t o f i x o u r
i s 2-monomorphic,
t r i c t i o n s t o 2-element s e t s a r e symmetric, i n which case
A
e i t h e r i t s res-
i s .a c o n s t a n t r e l a -
t i o n , hence monomorphic. O r t h e r e s t r i c t i o n s t o 2-element s e t s a r e o r i e n t e d , i n which case
A
i s a tournament. By t h e p r e c e d i n g 6.6, e i t h e r
A
i s a chain, and
THEORY OF RELATIONS
264
A
hence monomorphic, o r
i s a n o n - t o t a l l y o r d e r e d tournament.
I n t h e l a t t e r case, l e t of
.
E
, and
A
be t h e base o f
E
E i t h e r t h e r e e x i s t two o t h e r elements
let
x, x '
with
y, y '
be two elements
A(x,y)
into
x'
. Thus
into y'
and y
- {XI) and A/(E
A/(E
O r t h e r e do n o t e x i s t two such elements
x
e v e r y ordered p a i r w i t h f i r s t t e r m
,let
l a r g e r than 3 A(x,t)
x'
which takes takes
, while
x
x
into
into
t
A(x',x)
. By
p
x ' : thus
elements i s
= A(x',t)
is
A
(p-1)
Another example: f o r nality
has c a r d i n a l i t y s t r i c t l y
E
x
and
x'
-
=
2
, taken
x
A(x,x')
=
and takes
; s o we have an automorphism o f
(p-1)-monomorphic.
but not
p = 2q
. Then
composition, we have an automorphism o f
I n o p p o s i t i o n t o t h e preceding: f o r each i n t e g e r on
t a k e s t h e v a l u e (+) f o r
A
t a k i n g t h e v a l u e ( - ) f o r e v e r y orde-
x ' ( t o f i x i d e a s ) . Since
. Similarly
+ ; x
(p-1)-monomorphic.
. Then
+ ; t h u s we have an automorphism o f A which preserves
=
into
t
is
A
y, y '
be an element d i s t i n c t f r o m
t
=
taking
A
we have an isomorphism o f t h e r e s t r i c t i o n s
- i x ' ) ) ; so t h a t
r e d p a i r w i t h f i r s t term
A(x',y')
=
t h e n by t h e p r e c e d i n g p r o p o s i t i o n , t h e r e e x i s t s an automorphism o f
A
A
which
0
p a4
, t h e b i n a r y c y c l e based
(p-2)-monomorphic.
, t h e p a r t i a l o r d e r i n g formed o f q c h a i n s o f c a r d i -
t o be m u t u a l l y incomparable.
p a 6 , i f i t i s (p-3)-mono( < 3)-monomorphic, hence c h a i n a b l e : see 6.3 and 6.4. I n g e n e r a l , f o r r a 3 , e v e r y b i n a r y r e l a t i o n based on p + r+3 elements, which i s (p-r)-monomorphic, i s ( 43)-monomorphic, and t h u s c h a i n a b l e .
Note t h a t , f o r a b i n a r y r e l a t i o n w i t h c a r d i n a l i t y morphic, t h e n i t i s
Problem (JEAN 1976, u n p u b l i s h e d ) . L e t based on
p>, 2n+l
elements, which i s
n 3 2
and
k,< n
(p-k)-monomorphic,
.
Is e v e r y
n-ary r e l a t i o n
necessarily
(p-k+l)(,< k)-monomorphic by 6.3. F o r n > / 6 , i s e v e r y n - a r y r e l a t i o n based on p>/ 2n+l elements, which i s (p-6)-monomorphic, n e c e s s a r i l y c h a i n a b l e ( t h i s i s connected t o Jordan h y p o t h e s i s about p e r m u t a t i o n groups ; see ch.11 5 2.2).
monomorphic. Note t h a t i t i s
A
6.8. L e t
If 0
be a tournament w i t h c a r d i n a l i t y
A i s n o t a chain, t h e n
Let
Let
be t h e base of
E
q
p = 3
, modulo
4
p
and which i s
(p-2)-monomorphic.
.
A , o f c a r d i n a l i t y p ; and l e t u be an element i n E u (mod A) ; i n o t h e r words, t h e num-
denote t h e number of successors o f
ber o f
x
such t h a t
A(u,x)
= +
.
u' , A taking u o f successors i s preserved. The p r o d u c t p.q i s
I f we r e p l a c e
u
by a n o t h e r element
t h e n by t h e p r e c e d i n g d i s c u s s i o n t h e r e e x i s t s an automorphism o f into
u'
, hence
t h e number
q
t h u s equal t o t h e t o t a l number o f o r d e r e d p a i r s o f elements g i v i n g t h e v a l u e (+) to
A ; hence equal t o t h e number
q = (p-1)/2
p.(p-1)/2
o f unordered p a i r s o f elements. Thus
, which a l r e a d y shows t h a t p i s odd.
.
Chapter 9
Given an element the s e t
F of F
u u
phism preserves
F and
v
, partition x
(elements
. By the
which preserves cessors o f
E
, and
u
such t h a t
A(x,u)
the s e t =
G
.
v
into
v'
.
p-3
r
(p-l)(p-3)/8
i s a multiple o f
2-set-transitive
be two
A
.0 5
. The
v'
, this
automor-
F which are sucproduct
r.(p-1)/2
F g i v i n g value (+); hence
o f unordered p a i r s i n
4
u
o f elements i n
remains the same number when passing t o
6.9. A n t i c i p a t i n g the notions o f ch.11 is
(p-1)/2 many
+ ) . L e t v and v '
Since i t preserves
Thus the number
i s thus equal t o the t o t a l number o f ordered p a i r s i n and thus
of
G
preceding discussion, there e x i s t s an automorphism o f
and takes
equal t o t h e number
u into
the s e t o f elements d i s t i n c t from
(p-1)/2 many successors o f
predecessors o f elements i n
in
u
265
F . Hence
r = (p-3)/4
,
2.2, we say t h a t a group o f permutation
i f ff o r any two unordered p a i r s o f elements i n t h e base, there
e x i s t s a permutation o f the group, which takes the f i r s t p a i r i n t o the second. The above p r o p o s i t i o n 6.6 then takes the f o l l o w i n g form: i f morphic, n o n - t o t a l l y ordered tournament w i t h c a r d i n a l i t y p automorphisms o f A i s 2 - s e t - t r a n s i t i v e (p i n t e g e r 3 5 ) . Groups which are
A
is a
, then
(p-2)-mono-
the group o f
2 - s e t - t r a n s i t i v e have been s t u d i e d i n p a r t i c u l a r by DEMBOWSKI
1968, p. 96 note 2, under the name o f
2-homogeneous groups. The term " s e t - t r a n s i -
t i v e " i s e q u a l l y used and w i l l be employed i n t h i s book t o avoid confusion w i t h homogeneous r e l a t i o n s (see ch.11
5
1).
From t h e above c i t e d work, i t f o l l o w s t h a t f o r every l y ordered tournament based on
prime congruent t o 3 (mod 4)
p a5
elements,
this
(p-2)-monomorphic, p
non-total-
i s an odd power o f a
.
Thus, a f t e r t h e b i n a r y c y c l e on 3 elements, we have a tournament on 7 elements which i s 5-monomorphic, and hence 6, a z 2 , and 1-monomorphic, y e t n e i t h e r 3
nor
4-monomorphic; t h i s tournament has chains o f c a r d i n a l i t y 3 and cycles o f cardina-
l i t y 3 as r e s t r i c t i o n s . To c o n s t r u c t i t , s t a r t w i t h a heptagon, o r polygon w i t h 7 v e r t i c e s C y c l i c a l l y o r i e n t the edges heptagon
acegbdf
i n the d i r e c t i o n
second s t a r r e d heptagon ae
.
ab, bc,
adgcfbe
ac, ce,
a,b,c,d,e,f,g
. C y c l i c a l l y o r i e n t the s t a r r e d ... , d f , f a . C y c l i c a l l y o r i e n t the
... , f g ,
ga
i n the "opposite" d i r e c t i o n
da, gd,
... , eb,
I n view o f the r o t a t i o n a l symmetry, i t s u f f i c e s t o v e r i f y the isomorphism
between the t h r e e sub-tournaments o f c a r d i n a l i t y 5 : one which is obtained by removing two consecutive v e r t i c e s , a second obtained by removing two v e r t i c e s which are separated by one intermediate vertex, and a t h i r d obtained by removing two v e r t i c e s which are separated by two intermediate v e r t i c e s .
.
THEORY OF RELATIONS
266
§
7 - PROFILE
Let
OF A RELATION,
be a r e l a t i o n w i t h base
R
f i n i t e number
f(p)
f
for all
R
Examples. I f
. To
p
each n a t u r a l number
f ( 0 ) = 1 and i f
.
R
( n o t i o n due
R
f(h) = 1
>h .
i s a chain, o r i n general a monomorphic r e l a t i o n , then the p r o f i l e p & Card E ) .
i s a unary r e l a t i o n t a k i n g the value (+) f o r a f i n i t e number
R
and ( - ) on a l l o t h e r elements, then the p r o f i l e increases from
a
o f elements
f(0) = 1 to
, and then remains s t a t i o n a r y a t t h i s l a t t e r value.
f ( a ) = a+l If
, associate the
p
p-element r e s t r i c t i o n s o f
Card E = h ( f i n i t e ) , then
f u n c t i o n has constant value equal t o 1 (when If
(POUZET)
THEOREM
thus defined i s c a l l e d the p r o f i l e o f
t o POUZET 1972). Note t h a t f(p) = 0
E
INCREASE
o f isomorphism types o f the
The numerical f u n c t i o n and
PROFILE
i s unary and takes t h e value (+) and t h e value ( - ) , each on an i n f i n i t e
R
set, then the p r o f i l e i s
f ( p ) = p+l
.
By taking, f o r instance, the c o n s e c u t i v i t y r e l a t i o n on the n a t u r a l numbers
, we
o b t a i n a p r o f i l e f u n c t i o n w i t h a f a s t e r growth r a t e . I n the case where every f i n i t e r e l a t i o n w i t h the same a r i t y the p r o f i l e o f
R
i s maximum, hence f o r each
phism types o f r e l a t i o n s o f t h e given a r i t y
p
i s embeddable i n
R
,
i s equal t o the number o f isomor-
with cardinality
p
.
7.1. PROFILE INCREASE THEOREM Let
p, q
equal t o
be two n a t u r a l numbers and 2p+q
. Then
to -
R
a relation with cardinality a t least
the number o f isomorphism types o f the r e s t r i c t i o n s o f
R
p+q elements i s a t l e a s t as g r e a t as the number o f isomorphism types o f the p elements. restrictions o f R
2
More p r e c i s e l y , there e x i s t s an i n j e c t i v e f u n c t i o n which, t o each isomorphism type u -
of a restriction of
triction to 0
p+q
R
5
p
elements, which i s an extension o f
This f o l l o w s from the m u l t i c o l o r theorem, ch.3
types on
p
5
u
(POUZET 1976).
5.3, where the isomorphism
elements p l a y the r o l e o f the c o l o r s o f the p-element sets, and (p+q)-element r e s t r i c t i o n s have the same m u l t i c o l o r . 0
two isomorphic
Consequently, i f a r e l a t i o n has an i n f i n i t e base, then i t s p r o f i l e i s increasing. 2h
I f a r e l a t i o n has even c a r d i n a l i t y
integers l e s s than o r equal t o
h
.
, then
i t s p r o f i l e i s increasing f o r
I f a r e l a t i o n has odd c a r d i n a l i t y
i t s p r o f i l e i s i n c r e a s i n g f o r i n t e g e r s l e s s than o r equal t o
h+l
.
2h+l
, then
For t h e case o f an i n f i n i t e base, an a l t e r n a t i v e p r o o f w i l l be given i n ch.10
5
9.9,
about "almost chainable" r e l a t i o n s .
Note t h a t t h e f i r s t o f t h e preceding p r o p o s i t i o n s
i s stronger than the second.
For example, consider a r e l a t i o n on 7 elements. Not only does i t s p r o f i l e increase
267
Chapter 9
f o r integers value f o r 7.2. L e t
Card E
0
p, q
>/
tion of
to
, but
4
i t s value f o r
2 ; and i t s value f o r
, and l e t
t o p+q -
i s g r e a t e r than o r equal t o the
5
i s g r e a t e r than o r equal t o the value f o r
be two n a t u r a l numbers and
2p+q R
6
a r e l a t i o n w i t h base
R
be a permutation o f
f
.
E
E
, where
I f the image o f every r e s t r i c -
f ) i s an isomorphic r e s t r i c t i o n o f
elements (under
R
5
to
p
then t h e image of every r e s t r i c t i o n o f
-
p
1
,
R
elements i s an isomorphic r e s t r i c -
tion.
0
Take an a r b i t r a r y r e s t r i c t i o n o f
have the c o l o r
.
U
i f i t s image under
Now a
of
R/f"(a)
U
U
.
a
5
denote i t s
i s s a i d t o have the c o l o r
E
a
of
Thus
a
4.3.(2).
, the
E
and i t s image
restrictions
f"(a)
V
V
Hence
. It f o l l o w s f
R/a
have the same number
includes t h e same number o f
as w e l l as o f c o l o r
are i d e n t i c a l , by ch.3
U
.
(p+q)-element subset
p-element subsets o f c o l o r
and V
U
are isomorphic, hence
ment subsets o f c o l o r
elements, and l e t
E w i t h t h i s isomorphism type s h a l l
p-element subset o f
has the c o l o r
f
By hypothesis, f o r each and
R
p-element subsets o f
isomorphism type. A l l
p-ele-
t h a t the c o l o r s
takes each
U
p-element subset
i n t o another o f the same c o l o r . 0
§
8 - HOMOMORPHIC
I M A G E OF A N A R B I T R A R Y RELATION
I n t h i s paragraph, we attempt t o generalize t o a r b i t r a r y r e l a t i o n s the c l a s s i c a l n o t i o n o f homomorphic image
among chains (see ch.2
Starting with a relation
w i t h base
let
U1
,
... , Un
R
with the variables
element relations
Ui
x
3 of Ui
and ( - ) i f
...
x,y,z,
meaning E
3.6) o r among groups.
"
(and),
A
o f subsets o f
E"
x
rJ
v (or),
representing elements o f
" f o r every element
of
E
E "
where
belongs t o
E-Ui
Ui(x)
(if
, with
and
E
,
...then),
etc.;
the q u a n t i f i e r s
"there e x i s t s an
; and f i n a l l y w i t h predicates the r e l a t i o n
( i = 1, ...,n) x
5
and w i t h a s e t
be a f i n i t e sequence o f these subsets. Construct a l o g i c a l
formula w i t h the connections* 3 ( n o t ) ,
v x and
E
R
takes t h e value (+) i f
and the unary x
belongs t o
.
The usual semantic t r u t h value o f such a formula being obvious, we say t h a t an n-ary r e l a t i o n
R"
w i t h base
E"
i s a homomorphic image o f
a l o g i c a l formula whose t r u t h value i s through ED . 8.1.
F i r s t example. L e t
R
Ro(U1,
...,Un)
when
be a p a r t i a l o r d e r i n g w i t h base
R i f f there e x i s t s U1
E
, ... , Un , and l e t
run
E"
be
the s e t o f a l l subsets o f E . Then the unary r e l a t i o n R" such t h a t R"(U) = + i f f U i s an i n i t i a l i n t e r v a l o f R i s a homomorphic image o f R , v i a the
268
THEORY OF RELATIONS
vx,y (U(x)
formula
.
3 U(Y)
R(y,x))
A
S i m i l a r l y , we express the r e l a t i o n t a k i n g (+) i f f
vx 3 y R(x,y)
v i a t h e formula
E"
Now t a k i n g f o r
, whose arguments are denoted by
E"
vx U(x) =7
v i a t h e formula
R
i s a c o f i n a l subset (mod R)
t h e s e t o f i n i t i a l i n t e r v a l s (mod R), t h e p a r t i a l o r d e r i n g o f
i n c l u s i o n w i t h base p h i c image o f
U
.
U(y)
A
Second example. Suppose t h a t
R
,
U
V
,
i s a homomor-
.
V(x)
i s a chain w i t h base
E
, and
that
i s a set
E"
o f pairwise d i s j o i n t i n t e r v a l s . Then the corresponding homomorphic image o f i n the usual sense o f ch.2
vx,y(u(X)
A v(Y))
5
3.6,
3 R(x,Y)
.
Problem. Returning t o t h e p a r t i a l o r d e r i n g
w i t h base
R
, and l e t t i n g
E
be
E"
E ; then we conjecture t h a t t h e r e does n o t e x i s t any
the s e t o f a l l subsets o f
l o g i c a l formula o f the preceding kind, which would express t h a t the subset
of
U
i s a c o f i n a l subset (mod R) w i t h minimum c a r d i n a l i t y ( s o d e f i n i n g the c o f i n a l i -
E
ty of
R ).
8.2. Take
R
t o be a t e r n a r y r e l a t i o n o f a group based on
s e t of a l l subsets o f "
,
R
i s expressed by t h e formula:
the r e s t r i c t i o n
R/U
E
. We
i s a subgroup
are homomorphic images o f
R
. Also
"
and
R/U
"
, and
E"
for
i s a normal subgroup
the b i n a r y r e l a t i o n
i s an equivalence class modulo U
group and V
E
the
l e a w i t t o the reader t o see t h a t the unary r e l a t i o n s
"
,
i s a normal sub-
R/U
"
"
.
0 , and f o r E" take the s e t o f equivalence 0 : the reader w i l l see t h a t the usual homomorphic image, o r
Now take a f i x e d normal subgroup classes modulo
q u o t i e n t group, defined on the s e t o f these equivalence classes, i s represented by a l o g i c a l formula o f the preceding kind. Problem o f t r a n s i t-i v i t y o f c e r t a i n homomorphic images. S t a r t w i t h a r e l a t i o n
.--
R
E , and w i t h a R , w i t h base E" . I t e r a t e , by t a k i n g a s e t E"" o f ( n o t n e c e s s a r i l y mutually d i s j o i n t ) subsets o f E " , and a homomorphic image R o o o f R" , w i t h base En" . Associate t o each element o f
w i t h base
E
and a s e t
E"
of m u t u a l l y d i s j o i n t subsets o f
l o g i c a l formula d e f i n i n g a homomorphic image
E"" E""
i t s union, which i s a subset o f
. Note
of
t h a t , t o two d i s t i n c t elements o f
t h e r e correspond two d i s t i n c t unions, because o f t h e above d i s j u n c t i o n of
elements o f E
E
R"
, and
E"
. Hence we
have a b i j e c t i o n from
t h i s b i j e c t i o n transforms
Ron
E""
onto a s e t o f subsets o f
i n t o a r e l a t i o n on subsets o f
r e l a t i o n thus transformed, n e c e s s a r i l y a homomorphic image o f
R
.
E
.
I s the
Chapter 9
§
9
- BIVALENT
269
TABLE
A b i v a l e n t t a b l e i s t h e system formed by two d i s j o i n t sets: t h e s e t
E
o f columns
and t h e s e t
F o f rows; and a f u n c t i o n which, t o each element i n the Cartesian
product
F
E
Y
, associates t h e value (+) o r the value ( - ) . T on E x F and T' on E'x F ' , we say t h a t
Given t h e t a b l e s
, or
i n T'
f
injection all
x
either X
there e x i s t s an i n j e c t i o n e
into
F
and y
in
F'
F
.
, preserving
, or
.
Xt
there e x i s t s an
Otherwise, i f
X+
We l e a v e i t t o t h e reader t o see t h a t the t a b l e X
E'
and an
T ' ( e x , f y ) = T(x,y)
X
for
t o rows, i f
by adding a row, such
f o r every
X+
obtained from
T i s inextensive by X ( r e l a t i v e l y t o rows). X
l e f t ) , which i s n o t embeddable i n the t a b l e r i g h t ) , i s i n e x t e n s i v e by
X+
T,
,X and Z>/ Y
.
R
and an i n t e g e r
n a l i t y l e s s t h a n o r equal t o For c e r t a i n r e l a t i o n s
R
p
, this
p
, the
R
p = 0
p = 1
and
set o f restrictions o f
X,Y
o f cardi-
R
s e t i s also directed: f o r instance i f
5
R
is a
4 and 6.
i s t h e c o n s e c u t i v i t y r e l a t i o n on t h e i n t e g e r s , t h e n e x c e p t f o r
Given a r e l a t i o n
,this R
s e t i s not directed.
and a p o s i t i v e i n t e g e r
o f c a r d i n a l i t y g r e a t e r t h a n o r equal t o
embeddabi 1 it y
4X
i s c l o s e d under e m b e d d a b i l i t y .
c o n s t a n t r e l a t i o n , o r a c h a i n , o r a monomorphic r e l a t i o n : see ch.9 However i f
Y
(J&.
, there
Given a r e l a t i o n
R
Dom f
are both
i s an e x t e n s i o n o f
S
then there e x i s t s a r e l a t i o n isomorphic w i t h
A set i n (R
f
i f it i s i s extendi-
s i n c e t h e n e g a t i v e i n t e g e r (-1) i s s t r i c t l y l e s s t h a n 0
be an e x t e n s i o n o f
subset
f
.
S
(mod S ) , and t h e r e e x i s t s no element i n
S
1-morphism,
(1,p)-isomorphism
be t h e c h a i n o f t h e n a t u r a l numbers
i d e n t i t y on t h e s i n g l e t o n o f
Let
is a
S
I n o t h e r words i f
t h e i d e n t i t y on any subset o f t h e base
t h e p o s i t i v e and n e g a t i v e i n t e g e r s , so t h a t even a
.
p
i s extendible t o every f i n i t e superset o f
f
every f i n i t e superset o f S
R
.
Dom f
1-isomorphism,
I n o t h e r words i f
If
from
f o r e v e r y n a t u r a l number
p p
, the
set o f f i n i t e restrictions o f
i s d i r e c t e d b u t n o t c l o s e d under
.
AGE, REPRESENTATIVE RELATION
Let
R
be a r e l a t i o n . Then t h e s e t o f f i n i t e r e s t r i c t i o n s o f
R
i s c l o s e d and
d i r e c t e d under e m b e d d a b i l i t y . Considered up t o isomorphism, t h i s s e t s h a l l be called the R
of
R ; we s h a l l say t h a t
i s a r e p r e s e n t a t i v e o f t h e age.
R
r e p r e s e n t s t h i s aqe, o r a g a i n t h a t
279
Chapter 10
R i s the s e t o f isomor-
For the reader d e s i r i n g a rigorous d e f i n i t i o n , the age o f
R , which are based on f i n i t e subsets
phic copies o f the f i n i t e r e s t r i c t i o n s o f o f the s e t o o f the i n t e g e r s .
For a l o g i c i a n , an age i s a u n i v e r s a l theory o f f i r s t order predicate calculus w i t h i d e n t i t y ; see f o r example KRAUSS 1971. Since ages are countable sets, they can be compared under i n c l u s i o n . For example the age o f a l l f i n i t e chains, which i s represented by an i n f i n i t e chain, i s included i n the age o f a l l f i n i t e p a r t i a l orderings, which i t s e l f i s represented by those i n f i n i t e p a r t i a l orderings, i n which every f i n i t e p a r t i a l o r d e r i n g i s embeddable.
R, S o f the same a r i t y , R i s younger than S ,or S & R (see 1.2 above) i f f t h e age o f R i s included i n the aqe o f S . The r e l a t i o n s R and S have t h e same age i f f each i s o l d e r (and younger) than Given two r e l a t i o n s o l d e r than the o t h e r . To each age
@
there corresponds t h e negation age o f
each element o f
@,
value: see ch.2
5
, obtained by r e p l a c i n g
by i t s negation ( i . e . t h e r e l a t i o n t a k i n g always the opposite 1.7); o r e q u i v a l e n t l y by r e p l a c i n g a representative by i t s
negation. For example, the negation o f the age o f a l l f i n i t e chains, o r t o t a l orderings, i s the age o f a l l s t r i c t f i n i t e t o t a l orderings ( w i t h
2 , t h e n i t o f c a r d i n a l i t y (Card F) t 2n , i n o r d e r t o i n s u r e
designates the a r i t y o f
s u f f i c e s t o consider sets
R'
1.1); by c o n s i d e r i n g two f i n i t e subsets
o f t h e base as e q u i v a l e n t , i f b o t h i n c l u d e F-isomorphic. I f
2.4. Hence
, and F be a f i n i t e subset
E D
G e n e r a l i z a t i o n o f ch.9
t h e n a p p l y RAMSEY's theorem (ch.3
0
ch.9
.0
F'
Then t h e r e e x i s t s a denumerable subset
R/D
, by
...)
(A',U;,
almost c h a i n a b l e w i t h a k e r n e l i n c l u d e d i n
R
F-chainability. 0 9.8. A necessary and s u f f i c i e n t c o n d i t i o n f o r a denumerable r e l a t i o n almost c h a i n a b l e , i s t h a t t h e p r o f i l e o f ch.9 § 7 ; s u f f i c i e n c y 0
Let
uses t h e u l t r a f i l t e r axiom).
be almost c h a i n a b l e , w i t h base
R
and k e r n e l
E
R when r e s t r i c t e d t o p
t h e isomorphism t y p e o f intersection o f
R
be bounded ( f o r t h e p r o f i l e , see
R
F w i t h t h e s e t o f these
p
. F o r each
F
integer p
,
elements, o n l y depends on t h e
elements. Since t h e k e r n e l i s f i n i t e ,
t h e number o f these isomorphism types i s bounded by t h e number o f r e s t r i c t i o n s of
.
R/F
Conversely, l e t
be a r e l a t i o n w i t h denumerable base
R
c h a i n a b l e . To prove t h a t t h e p r o f i l e i s unbounded, l e t we s h a l l c o n s t r u c t
h
restrictions o f
, which i s n o t almost
E
h
be an a r b i t r a r y i n t e g e r ;
R , a l l o f t h e same f i n i t e c a r d i n a l i t y and
m u t u a l l y non-isomorphic. S t a r t w i t h a denumerable, c h a i n a b l e r e s t r i c t i o n The r e l a t i o n
i s n o t younger t h a n
R
Hence t h e r e e x i s t s a f i n i t e subset in
Ro
.
Take a denumerable,
, no
p 3 Card Fo
ger
any r e s t r i c t i o n o f The r e l a t i o n R1
.
to a
F1
of
to a h
to a
Ro
Ro
, are
of
nor than
p
. For every
Ro
R2
of
i s isomorphic w i t h
. Hence R
there exists a
Ro
. For every
with cardinality
Fo
, and
inte-
.
Fo R1
i s n o t embeddable R
i s n e i t h e r embeddable i n
, a restriction o f F1
R/FO
R1
with cardinality
p-element subset i n c l u d i n g h
that
p-element subset i n c l u d i n g
p-element subset i n c l u d i n g
times, we o b t a i n
, such
F1-chainable r e s t r i c t i o n
p 3 Max(Card Fo,Card F1) R1
E
, such t h a t R/F1
E
Take a denumerable,
triction of
of
i s n e i t h e r younger t h a n
R
f i n i t e subset
Fo
(see 9.7 above ) .
R
u s i n g t h e u l t r a f i l t e r axiom.
Fo-chainable r e s t r i c t i o n
restriction o f
R1
of
Ro
Ro : see 9.6,
p
nor i n integer
, and
a restriction o f
m u t u a l l y non-isomorphic.
a resR2
Iterating this
r e s t r i c t i o n s which a r e m u t u a l l y non-isomorphic. 0
THEORY OF RELATIONS
310
9.9.
The f o l l o w i n g remark due t o POUZET, proves t h a t t h e p r o f i l e o f an i n f i n i t e
r e l a t i o n i s i n c r e a s i n g , w i t h o u t u s i n g t h e i n c i d e n c e m a t r i x o r t h e m u l t i c o l o r theorem ( p r o f i l e i n c r e a s e theorem, ch.9 Given a denumerable r e l a t i o n
R
5
t h e base, such t h a t e v e r y r e s t r i c t i o n o f in
R/F
.
Then t a k e an
R
, at
R
5.3).
, take
p
a f i n i t e subset
with cardinality
F-chainable r e s t r i c t i o n o f
t i o n i s t h e same as t h e p r o f i l e o f
5
7.1, u s i n g ch.3
and an i n t e g e r
p
F
of
i s embeddable
R : the p r o f i l e o f t h i s r e s t r i c -
l e a s t up t o t h e v a l u e
p
.
Hence i t
s u f f i c e s t o prove t h a t t h e p r o f i l e i n c r e a s e s i n t h e p a r t i c u l a r case where
R
almost c h a i n a b l e . Now assume to
p
that
R
is
F-chainable. To each isomorphism t y p e o f a r e s t r i c t i o n
elements, a s s o c i a t e a
and t h e i n t e r s e c t i o n
G
associate t o t h i s
G
taking
G with
p-element subset
F nG
H =
t h e isomorphism t y p e o f c a r d i n a l i t y
p l u s an element n o t b e l o n g i n g t o
. Thus
F
t i o n which, t o each isomorphism t y p e o f c a r d i n a l i t y type o f c a r d i n a l i t y
R/G
p+l
having t h i s type,
h a v i n g t h e l e a s t p o s s i b l e c a r d i n a l i t y . Then p+l
, obtained
p
, associates
an isomorphism
.
To see t h e i n j e c t i v i t y : i f t h e same isomorphism t y p e o f c a r d i n a l i t y ned from two e q u i p o t e n t subsets belonging t o
by
we d e f i n e an i n j e c t i v e func-
H, H '
of
F
, each
p+l
i s obtai-
augmented by elements n o t
F , t h e n e v e r y isomorphism o f t h e f i r s t r e s t r i c t i o n o n t o t h e second H i n t o H ' . F o r o t h e r w i s e H ' would n o t be t h e i n t e r s e c t i o n
r e s t r i c t i o n , takes of
F with least possible cardinality.
9.10.
Let
with
We a r e now i n a p o s i t i o n t o prove t h e converse o f 8.5 R
E ; i f e v e r y r e l a t i o n w i t h base E , .-which is . R , R i s finitist.
have base
R
,is
isomorphic w i t h
(1,p)-equivalent
then
~
0 We g i v e an argument f o r t h e case t h a t
E
i s denumerable. Take a r e p r e s e n t a t i v e
R
o f each isomorphism t y p e o f a r e s t r i c t i o n o f
to
6
p
elements, and l e t
F
F i s f i n i t e , by 9.7 which extends R/F and i s
be t h e u n i o n o f t h e bases o f these r e p r e s e n t a t i v e s . Since above, t h e r e e x i s t s a denumerable r e s t r i c t i o n o f
R
F-chainable. By h y p o t h e s i s , t h i s r e s t r i c t i o n i s isomorphic w i t h suppose t h a t Assume t h a t
R
R
R : hence we can
i s almost c h a i n a b l e . is
(F,A)-chainable,
where
A
e i t h e r isomorphic w i t h G, or i s o m o r p h i c w i t h
i s a c h a i n which can a r b i t r a r i l y be Q
. From t h i s
p o i n t on, t h e argu-
ment i n ch.9 e x e r c i s e 2, s l i g h t l y m o d i f i e d t o t a k e account o f t h e k e r n e l included i n transposition
F ) , proves t h a t f o r e v e r y p a i r o f elements (x,y)
preserves
R
. Hence
R
is
x, y
in
E-F
F
(or
, the
F-finitist. 0
EXERCISE 1 - THE EXISTENCE OF A R I C H RELATION I N THE UNCOUNTABLE CASE
Modulo t h e axiom o f c h o i c e and t h e continuum h y p o t h e s i s , we s h a l l p r o v e as f o l l o w s t h e e x i s t e n c e , f o r each i n t e g e r
n
, of
an n-ary r e l a t i o n o f c a r d i n a l i t y W 1
,
31 1
Chapter 10
o r e q u i v a l e n t l y continuum c a r d i n a l i t y , in which every n-ary r e l a t i o n of same card i n a l i t y i s embeddable. Let E be t h e base s e t , which we i d e n t i f y w i t h the ordinal GJ1 . There exist t d l many countable subsets of E , denoted by D i ( i < W 1 ) . To each Di , we a s s o c i a t e a subset Ci of E w i t h c a r d i n a l i t y W 1 , so t h a t a l l these Ci a r e mutually d i s j o i n t . This is p o s s i b l e , s i n c e t h e Cartesian product of W 1 by i t s e l f i s equipotent with w1 . S t a r t w i t h an a r b i t r a r y n-ary r e l a t i o n Ro w i t h base Do . Consider a l l possible manners of extending Ro t o a new element, t h e r e b e i n g a t most continuum many, thus cdl many such manners. To each of these p o s s i b i l i t i e s , a s s o c i a t e an element x of Co , and c o n s t r u c t the corresponding extension on t h e base Do u { x] . This i s p o s s i b l e , s i n c e t h e r e a r e u1many elements i n Co , hence a l s o W 1 many such elements which do not belong t o t h e countable set Do . More g e n e r a l l y l e t u be a countable ordinal index. Suppose t h a t f o r each i < u , we have already defined an n-ary r e l a t i o n Ri cn D i , these Ri being mutually compatible ( i . e . w i t h same r e s t r i c t i o n t o any i n t e r s e c t i o n of t h e i r b a s e s ) . Moreover suppose t h a t by using t h e Ci , we have already ensured a l l possible extensions f o r each Ri t o an additional element. Then define RU on Du , i n an a r b i t r a r y manner with t h e only requirement of c o m p a t i b i l i t y w i t h preceding Ri and t h e i r extensions. Now consider a l l possible extensions of R u t o an additional element. For each p o s s i b i l i t y , use an element x i n Cu . This i s p o s s i b l e ; indeed a l l the n-tuples t o which a value (+) o r ( - ) has already been assigned, have a l l their terms belonging e i t h e r t o a Di o r t o t h e corresponding C i ( i < u) Yet Cu i s d i s j o i n t from these Ci , and t h e union of t h e s e D i i s countable. So t h a t t h e r e remain G) many elements x i n Cu , such t h a t no value has been assigned t o any n-tuple having x among i t s terms. F i n a l l y we obtain on t h e e n t i r e base E , an n-ary r e l a t i o n R such t h a t each countable r e s t r i c t i o n of R i s a r b i t r a r i l y e x t e n d i b l e t o i t s base plus an a d d i t i o nal element: thus every n-ary r e l a t i o n w i t h c a r d i n a l i t y W1 i s embeddable in i t .
.
EXERCISE 2 - FINITIST STRUCTURE AND FRAENKEL-MOSTOWSKI MODEL, IN CONNECTION WITH A FUNCTION WHOSE DOMAIN IS STRICTLY SUBPOTENT WITH ITS RANGE 1 - S t a r t w i t h the set of natural numbers, which we consider a s urelements, i . e . as copies of t h e empty s e t , each having no element. On t h e set N o f these i n t e g e r s , consider t h e s e t of f i n i t i s t r e l a t i o n s , i n t h e sense of 5 8 above. More g e n e r a l l y , t o each ordinal o( , a s s o c i a t e the set of f i n i t i s t s t r u c t u r e s with rank 4 on N , defined a s follows by induction. Each element o f ti , or urelement, i s a s t r u c t u r e w i t h rank 0 . A s t r u c t u r e of rank o( >/ 1 i s a s e t of f i n i t i s t s t r u c t u r e s of ranks s t r i c t l y l e s s than o( , such t h a t t h e r e e x i s t s a f i n i t e s u b s e t F of N , w i t h the condition t h a t every
312
THEORY OF RELATIONS
.
permutation of N-F preserves the s e t or structure of rank o( The reader can verify t h a t these f i n i t i s t s;ructures constitute a mcdel of FRAENKEL-MOSTOWSKI's s e t theory: see FRAiSSE 1958. More precisely t h i s model s a t i s f i e s the axioms of ZF excepting extensionality, which must be weakened as follows: any two non-empty s e t s which have the same elements are identical. One can easily transform the model so as t o s a t i s f y the f u l l extensionality axiom, b u t while abandonning the foundation axiom. Indeed i t suffices t o consider each urelement a as equal t o the singleton of a i t s e l f . In t h i s case, we add the empty s e t t o the elements of N , and we give t o each of them the rank 0 , the rank 1 being given only t o those s e t s which contain several elements of N o r one element of N plus the empty s e t . 2 - Construct the s e t o f words, or f i n i t e sequences without repetition on the s e t N of the urelements; note t h a t t h i s s e t i s f i n i t i s t with F empty. Consider the function f which t o each non-empty word u without repetition associates the word obtained from u by removing i t s l a s t term. This function i s a f i n i t i s t s t r u c t u r e , with F empty. Dom f i s the s e t of non-empty words, Rng f i s the s e t of a l l words (without r e p e t i t i o n ) , including the empty sequence. W e shall prove t h a t in the considered model, Dom f i s s t r i c t l y subpotent with Rng f (example communicated by H O D G E S ) . 0 Indeed a bijection from Rng f o n t o Dom f cannot be f i n i t i s t . For such a bijection h , l e t t i n g 0 denote the empty sequence, consider the a - s e q u e n c e 2 of successive i t e r a t e s 0 , h ( O ) , h ( 0 ) = h ( h ( O ) ) , ... . If t h i s w-sequence were f i n i t i s t , with a f i n i t e subset F of N , then a l l i t s terms would be sequences of elements of F . Since h i s a bijection, these terms are a l l d i s t i n c t , so t h a t F should be i n f i n i t e : contradiction. 0 Problem. Is i t possible t o generalize t o structures the lemmas in ch.9 0 1 . 2 and 1.3, so t h a t we could define a f i n i t i s t structure by using only transpositions instead of general permutations of N-F . Consequently the intersection of two f i n i t e subsets F should be an F , so t h a t each f i n i t i s t structure should have a minimum F called i t s kernel.
313
11
CHAPTER
HOMOGENEOUS RELATION, RELATIONAL SYSTEM, CONNECTION WITH PERMUTATION GROUPS, ORBIT
§
1 - HOMOGENEOUS
Let
p
AMALGAMABLE SET AND AMALGAMABLE AGE
RELATION;
be a n a t u r a l number ; a r e l a t i o n
e v e r y l o c a l automorphism o f
R
morphism o f
.
Every r e l a t i o n i s
i s s a i d t o be
R
d e f i n e d on
R
p
o-homogeneous, if
elements i s e x t e n d i b l e t o an auto-
0-homogeneous, s i n c e t h e empty f u n c t i o n i s
e x t e n d i b l e t o t h e i d e n t i t y mapping on t h e e n t i r e base.
( S p)-homogeneous r e l a t i o n . i s s a i d t o be homogeneous, i f R i s p-homogeneous f o r e v e r y natu-
Obvious d e f i n i t i o n o f a A relation
R
.
r a l number p Example. A b i n a r y c y c l e on a t l e a s t 5 elements
is
1-homogeneous b u t n o t
2-homo-
f o u r c o n s e c u t i v e elements; t h e n t h e mapping which t a k e s a,b,c,d a,c i n t o a,d i s a l o c a l automorphism and i s n o t e x t e n d i b l e . The c h a i n on 2 elements i s n o t 1-homogeneous, b u t i t i s 2-homogeneous, s i n c e geneous: c a l l
t h e o n l y l o c a l automorphism on 2 elements i s t h e i d e n t i t y t r a n s f o r m a t i o n . The f o l l o w i n g b i n a r y m u l t i r e l a t i o n ( w i t h b i n a r y and unary component r e l a t i o n s ) 2-homogeneous y e t n o t
3-homogeneous.
0 S t a r t with the binary r e l a t i o n
.
&
R
g i v e n i n ch.9 exerc.1, on t h e 10 elements
f o r each ordered A such t h a t A(x,y) = + e x a c t l y f o r those o r d e r e d p a i r s (x,y) which a r e an image o f (a,s) under one o f t h e t h r e e f o l l o w i n g p e r m u t a t i o n s : ( 1 ) t h e mapDing which preserves a and c
a,b,c,d,r,s,t,u,i p a i r , say
,j
(a,s)
d
2-homogeneity:
f o r i n s t a n c e , add a b i n a r y r e l a t i o n
and i n t e r c h a n g e s and
I n o r d e r t o ensure t h e
(b,d),
and i n t e r c h a n g e s
(r,u), (s,t), (a,c),
(r,s),
mappings (1) and ( 2 ) , which p r e s e r v e s
( i , j ) ; ( 2 ) t h e mapping which preserves
(t,u),
b
( i , j ); ( 3 ) t h e c o m p o s i t i o n o f
i and j
and i n t e r c h a n g e s
(a,c),
(b,d),
(r,t), ( s , ~ ) . Here t h e images o f (a,s) a r e ( a , t ) , ( c , r ) , (c,u) and o b v i o u s l y i t s e l f , t h e i d e n t i t y p e r m u t a t i o n b e i n g added t o ( 1 ) , ( 2 ) , ( 3 ) . (a,s) The o b t a i n e d m u l t i r e l a t i o n i s n o t 3-homogeneous: as a l r e a d y mentioned i n t h e exerc i s e , t h e l o c a l automorphism which p r e s e r v e s
a
and
b
and i n t e r c h a n g e s
(i,j)
i s n o t e x t e n d i b l e t o an automorphism on t h e e n t i r e base. 0 The p r e c e d i n g example i s n o t
1-homogeneous: f o r examule map
a
into
b ;y e t
i t becomes
1-homogeneous i f we add, f o r i n s t a n c e , a unary r e l a t i o n t a k i n g value
(+) f o r
and
a
c
, and
a n o t h e r t a k i n g (+) f o r
i and j
.
THEORY OF RELATIONS
314
Example of homogeneous r e l a t i o n s . The r e l a t i o n always (+); another example, the chain Q of the r a t i o n a l s . Another homogeneous r e l a t i o n . The equivalence r e l a t i o n w i t h f i n i t e l y many, o r w i t h In c o n t r a s t with t h e two preceding denumerably many c l a s s e s having c a r d i n a l i t y 2 examples, here homogeneity does not s u b s i s t when we remove an element from the base. Problem. For each a r i t y n , does t h e r e e x i s t a threshold s ( n ) above which every n-ary r e l a t i o n which i s ( 4 s(n))-homogeneous, i s homogeneous.
.
1.1. Let E be a denumerable s e t . A r e l a t i o n R with base E i s homogeneous i f f , E , f o r any l o c a l automorphism f R w i t h domain f o r any f i n i t e subset F F , and f o r any element u iJ E-F , t h e r e e x i s t s a local automorphism of R which extends f t o the domain F U {u 1 .
0-f
of
R i s homogeneous, then i t obviously s a t i s f i e s our condition. Conversely, suppose t h a t t h e condition holds. Enumerate the elements of E as a i ( i i n t e g e r ) . S t a r t w i t h an a r b i t r a r y local automorphism f o of R , with f i n i te domain. Add a. t o t h e domain, i f i t does not y e t belong t o this domain, thus obtaining a local automorphism f i extending f o . S i m i l a r l y add a. t o the range of f b , i f i t does not y e t belong t o t h i s range, thus obtaining a local automorphism f l extending f; , t h u s extending f o I t e r a t e t h i s , going from f i t o f i + l ( i i n t e g e r ) by adding a i t o t h e domain and then t o t h e range. F i n a l l y the common extension of these f i i s an automorphism of R 0
0 If
.
.
This proposition does not extend t o t h e case of any uncountable r e l a t i o n . Consider t h e chain Q+R , where Q i s t h e chain of r a t i o n a l s and R i s isomorphic t o t h e chain of r e a l s . Then t h e condition i n our preceding statement i s sat i s f i e d ; y e t mapping an element of the i n i t i a l i n t e r v a l Q i n t o an element of the f i n a l i n t e r v a l R , we cannot extend t h i s l o c a l automorphism t o the e n t i r e base. 0
0
1 . 2 . Note t h a t the r i c h denumerable r e l a t i o n R defined in ch.10 5 4 . 1 s a t i s f i e s t h e following condition. Each f i n i t e r e s t r i c t i o n of R i s a r b i t r a r i l y extendible t o i t s base plus an additional element, which we can f i n d i n t h e base of R We immediately see t h a t R s a t i s f i e s t h e condition i n our preceding statement. Hence f o r each i n t e g e r n , t h e r e e x i s t s a denumerable n-ary r e l a t i o n which i s r i c h and homogeneous.
.
1.3.(1) Any two denumerable homogeneous r e l a t i o n s , with same a r i t y and same age, a r e isomorphic. ( 2 ) Let R be a denumerable homogeneous r e l a t i o n . Then every denumerable r e l a t i o n which i s younger than R , i s embeddable i n R . 0 (1) Let
R and
R'
be denumerable homogeneous r e l a t i o n s w i t h r e s p e c t i v e bases
315
Chapter 1 1
E and E ' , and the same age. Enumerate the elements of E as ai , the elements of E ' as a; ( i i n teg er ) . Embed the r es t r i ct i o n R/{ao) into R ' , thus obtaining a local isomorphism f o from R i n t o R ' with domain { ao\ . Augment the range of f o by adding ah . By the preceding proposition, f o i s extendible t o a local isomorphism go from R i n t o R ' , whose domain contains a. and whose range contains ah . I t er at i n g t h i s f o r each i , we obtain a local isomorohism f i whose domain contains the elements ao, ...,ai and which i s extendible t o a local The union of the isomorphism gi whose range contains the elements a h , . . . ,a! f i (or equivalently the gi ) i s an isomorphism from R onto R ' . 0
.
( 2 ) Same argument, b u t only in one direction, using the ocal isomorphisms from the younger relation i n t o the homogeneous relation R 0
0
.
Consequently, f o r the age of chains, the chain Q of the denumerable homogeneous chain, u p t o isomorphism.
ationals i s the only
1.4. If we remove an element from the base of a denumerable relation which i s rich and homogeneous, then we obtain an isomorphic relation. Similarly i f we change the value of an a r b i t r a r y n - t u p l e ( n = a r i t y ) 0 Indeed, the f i r s t and second operations both preserve age, since t h i s age i s constituted by a l l f i n i t e n-ary relations ( u p t o isomorphism). Moreover the condition t h a t each f i n i t e r es t r i ct i o n be a r b i t r a r i l y extendible, i s preserved. 0
.
Many other relations are isomorphic t o t h e i r r es t r i ction a f t e r removing an arbitrary element. For example the chain Q (which i s homogeneous), the chain W (which i s not homogeneous). We have already seen t h at the equivalence relation with classes of cardinality 2 , i s homogeneous b u t n o t isomorphic with such a re stric tion. Problem. Let R be a denumerable n-ary r el at i o n , which i s isomorphic with any relation obtained by changing the value of an ar b i t ra ry n-tuple; then i s R an homogeneous r e l a t i o n . Note t h at necessarily every n-ary f i n i t e relation i s embeddable in R : i f R i s homogeneous, then i t i s rich. 1.5. AMALGAMABLE SET, AMALGAMABLE AGE A s e t 6;L of f i n i t e relations of the same a r i t y i s said t o be amalgamable i f , for any relations A , B, C belonging to'R, f o r any isomorphism f from A onto a r e s t r i c t i o n of B and any isomorphism g from A onto a re stric tion of C , there e x i s t s a relation D belonging t o 61. , an isomorphism f ' from B onto a r e s t r i c t i o n of D and an isomorphism g ' from C onto a re stric tion of D , such t h a t f o r each element x of the base I A I , we have ( f ' , f ) x = ( g ' o g ) x Obviously the relations in the considered s e t are defined up t o isomorphism. A s e t (R of f i n i t e relations i s said t o be strongly amalgamable i f , f o r any relations B, C belonging t o 6!,and C ' isomorphic w i t h C , with a common
.
316
THEORY OF RELATIONS
r e s t r i c t i o n t o the intersection of the bases l B l and I C ' l , then there e x i s t s a relation D in &, ( u p t o isomorphism), which i s a common extension of B and C ' . For example, the age of a l l f i n i t e chains, and also the age of a l l f i n i t e p a r t i a l orderings, are strongly amalgamable: see ch.2 fj 2 . 2 and 2.3. The age of those f i n i t e unary relations which take the value (+) f o r a t most one element, i s amalgamable y e t n o t strongly amalgamable. The age of a l l f i n i t e trees i s not amalgamable: the example given in ch.2 fj 2.3 contradicts ordinary, as well as strong amalgamability. 1.6. AMALGAMATION THEOREM Given an age 61 , there e x i s t s a countable homogeneous representative of d?, i s amalgamable.
&
iff
Let R be a countable homogeneous relation. Let A , B be two f i n i t e r e s t r i c tions of R , where B i s an extension of A , and l e t g be an isomorphism from A into a t h i r d f i n i t e r e s t r i c t i o n C of R . Then by hypothesis g i s extendible t o an automorphism of R . Let f ' denote the r e s t r i c t i o n of t h i s automorphism t o the base I B I . Let f be the identity on I A l , l e t g ' be the identity on I C l and f i n a l l y l e t 0 be the r e s t r i c t i o n R / ( [ C I y ( f ' ) ' ( I B l ) ) We thus have the condition of amalgamation. Conversely, l e t @, be an amalgamable age. Let B be a f i n i t e relation belonging t o & , and f a local automorphism of B . Let A denote the r e s t r i c t i o n of B t o Dom f . W e shall construct an extension D of B , s t i l l belonging t o bt , which has a local automorphism extending f t o the domain B . For t h i s , l e t C = B , and denote by g the identity on I A I = Dom f . Now using the amalgamabil i t y condition, obtain an extension 0 of B with an isomorphism g ' from B onto a r e s t r i c t i o n of D , such t h a t f o r every x in Dom f , we have g ' x = ( g ' , g ) x = ( f ' , f ) x = fx : so t h a t g ' extends f . S t a r t with an w -sequence of f i n i t e relations Ai ( i i n t e g e r ) , the s e t of whose r e s t r i c t i o n s gives the amalgamable age &. Let Bo = A. Using the preceding, replace Bo by i t s extension B1 belonging t o & , such t h a t every local automorphism of Bo (there are only f i n i t e l y many) has an extension which i s a local automorphism of B1 and whose domain i s Bo Moreover we require t h a t A1 be embeddable in B1 , which i s possible since every age i s directed. , such t h a t Iterating t h i s , we obtain an 0 -sequence of elements Bi of each Bi+l extends B i , and Ai i s embeddable in Bi f o r each integer i Take the common extension R of the Bi t o the union of bases. Then every local automorphism of R with f i n i t e domain i s extendible, with alternatively a domain and a range containing every element in every base I S i ( : the homogeneity i s proved. 0
0
.
.
.
61.
.
Chapter 11
317
5
1.7. Given an amalqamable aqe, t h e c r i t e r i o n o f ch.10
7.6 i s s a t i s f i e d , and
unique denumerable homogeneou? r e p r e s e n t a t i v e o f o u r a q t (up t o isomorphism) i s u e s a t u r a t e d r e l a t i o n o f ch.10
5
7.3 and 7.6.
R be a denumerable homogeneous r e l a t i o n , S a r e l a t i o n r e o r e s e n t i n g t h e same age, and f a l o c a l isomorphism f r o m R i n t o S , w i t h f i n i t e domain F
0
Let
.
Take an a r b i t r a r y f i n i t e s u p e r s e t from
onto a r e s t r i c t i o n o f
S/G
phism
h,f
kh,-'
i s an isomorphism f r o m
G R
of
.
Since
There e x i s t s an isomorphism h
i s homogeneous, t h e l o c a l automor-
R
i s e x t e n d i b l e t o an automorphism
Replacing G
.
fa(F) k
of
R
. Then
onto a r e s t r i c t i o n o f
S/G
k-l0h
is a
(see t h e d e f i n i t i o n s i n ch.10
1-morphism f r o m
5
S
into
R
extends
.
f-l
IS1 ) , t h e same argu-
b y any f i n i t e s u o e r s e t ( i n c l u d e d i n t h e base
ment shows t h a t
t h e composition
, and
R
. Hence
R
i s saturated
7 and 7.3). 0
1.8. I t i s proved by HENSON 1972, t h a t t h e r e e x i s t continuum many m u t u a l l y nonisomorphic denumerable homogeneous r e l a t i o n s . Hence continuum many amalgamable ages. F o r o t h e r r e s u l t s on homogeneous r e l a t i o n s , see JONSSON 1965 and CALAIS 1967.
2 - RELATIONAL SYSTEM: ORBIT: ADHERENCE OF A PERMUTATION GROUP: THEOREM OF THE I N C R E A S I N G NUMBER OF O R B I T S (LIVINGSTONE,WAGNER) 2.1. RELATIONAL SYSTEM Given a s e t
E
, we
say t h a t a r e l a t i o n a l system w i t h base
i s any o r d i n a l sequence whose terms a r e r e l a t i o n s Letting
ni
denote t h e a r i t y o f
s a i d t o be t h e
Ri
, the
Ri
E
, or
based on
( i o r d i n a l ) based on
o r d i n a l sequence o f i n t e g e r s
arity o f t h e system, and each
Ri
ni
,
E
.
E
is
i s s a i d t o be a component o f t h e
system. By t a k i n g a f i n i t e sequence o f r e l a t i o n s based on based on
E
, we
f i n d the m u l t i r e l a t i o n
E (see ch.2 § 1).
The n o t i o n s o f r e s t r i c t i o n o f a system t o a subset o f t h e base, e x t e n s i o n t o a s u p e r s e t of t h e base, isomorphism, automorphism, l o c a l isomorphism o r automorphism, a l l e x t e n d immediately t o t h e case o f
r e l a t i o n a l systems.
However, we have an i m p o r t a n t d i f f e r e n c e which p r o h i b i t s c e r t a i n g e n e r a l i z a t i o n s . Indeed t h e number o f r e l a t i o n a l systems o f a g i v e n a r i t y and o f a g i v e n f i n i t e base, i s i n general i n f i n i t e . I n p a r t i c u l a r RAMSEY's theorem, used when p a r t i t i o ning the
p-element subsets o f t h e base i n t o a f i n i t e number o f classes, c o r r e s -
ponding t o d i f f e r e n t isomorphism types, can no l o n g e r be used s y s t e m a t i c a l l y . The same remark h o l d s f o r t h e coherence lemma. p-HOMOGENEOUS SYSTEM, HOMOGENEOUS SYSTEM The n o t i o n o f homogeneity i n t r o d u c e d i n 5 1 above, can be extended t o r e l a t i o n a l systems. A system R i s s a i d t o be p-homogeneous, i f e v e r y l o c a l automorphism
THEORY OF RELATIONS
318
of
, d e f i n e d on p elements, i s e x t e n d i b l e t o an automorphism o f
R
.
R
A system i s s a i d t o be homogeneous, i f i t i s p-homogeneous f o r e v e r y i n t e g e r
2.2. ORBIT OF AN Let
n-TUPLE, OR OF AN
be a s e t , and
E
Given an i n t e g e r o r b i t o f the
n
and an
Given an If
n - t u p l e o f elements
n - t u p l e (mod H )
permutation belonging t o n-element s e t
s e t images o f
n-ELEMENT SET
a s e t o f permutations o f
H
, n o t n e c e s s a r i l y a group.
...,an
in
n - t u p l e images o f
H
.
n - t u p l e i n t o another, i s an e q u i v a l e n c e r e l a t i o n between
for
n-element s e t s .
n-TRANSITIVE GROUP,
under any
, we c a l l t h e o r b i t o f F (mod H) t h e s e t o f n-element
i s a group, then t h e e x i s t e n c e o f a p e r m u t a t i o n b e l o n g i n g t o
H
c a l l the
... ,an
al,
one
H
, which
takes
n-tuples. S i m i l a r l y
n-SET-TRANSITIVE GROUP
A group o f p e r m u t a t i o n s o f
E
i s s a i d t o be
there e x i s t s a permutation belonging
n - t r a n s i t i v e , i f f o r any two
t o t h e group and t a k i n g one
o t h e r . I n o t h e r words, i f t h e r e e x i s t s o n l y one o r b i t o f
E
.
E has f i n i t e c a r d i n a l i t y a t l e a s t equal t o n+2 ,
Another example, i f t h e base
then t h e a l t e r n a t i n g group formed o f a l l even p e r m u t a t i o n s , i s n - t r a n s i t i v e group i s
n-tuples
n-tuple i n t o the
n-tuples.
F o r example, t h e symmetric group formed o f a l l p e r m u t a t i o n s o f
An
, we
E
.
H
F
the set o f
E
al,
under any p e r m u t a t i o n b e l o n g i n g t o
F
.
p
m - t r a n s i t i v e f o r any
R e c a l l JORDAN'S h y p o t h e s i s , 1893: f o r
n
m
3 6 ,
6
n
every
n-transitive.
. n - t r a n s i t i v e group i s
symmetric o r a l t e r n a t i n g . A f f i r m a t i v e s o l u t i o n ; see f o r example CAMERON 1981 p. 9. A group o f p e r m u t a t i o n s on
i s s a i d t o be
E
n-set-transitive,
i f any two
n-ele-
ment s e t s a r e t r a n s f o r m a b l e one i n t o t h e o t h e r by a c e r t a i n p e r m u t a t i o n o f t h e group. I n o t h e r words, i f t h e r e e x i s t s o n l y one o r b i t o f
n-element s e t s .
2.3. ADHERENT PERMUTATION, GROUP CLOSED UNDER ADHERENCE Let in
be a s e t ,
E E
into
n
an i n t e g e r and
fan
.A
E
permuta-
.
H
n-adherent t o
o f permutations o f
permutation o f under
a s e t o f permutations o f
of
Every p e r m u t a t i o n which i s A set
H
E i s s a i d t o be n-adherent t o H , i f f o r any elements al,. .. ,a n , t h e r e e x i s t s a p e r m u t a t i o n o f H t a k i n g al i n t o fal , ... , and an
f
tion
E which i s
n-adherence,
then
H
E
,is
H
m-adherent f o r any
i s s a i d t o be c l o s e d under
n-adherent t o
H
, belongs
to
m gn
.
i f every i s closed
n-adherence,
H
.
If H
i s c l o s e d under m-adherence f o r each
m>,
n
.
A p e r m u t a t i o n i s s a i d t o be adherent t o H , i f i t i s n-adherent f o r e v e r y n . A s e t H o f p e r m u t a t i o n s i s s a i d t o be c l o s e d under adherence, i f e v e r y adherent p e r m u t a t i o n belongs t o
H
.
Chapter 11
2.4.(1)
Let
a-4
be a r e l a t i o n a l system w i t h base
R
. Then
n
319
, formed
E
t h e group o f automorphisms o f
o f components a
i s c l o s e d under
R
x
n-adherence.
( 2 ) F o r e v e r y r e l a t i o n a l system, t h e group of automorphisms i s c l o s e d under adhe-
rence. Consequences 2.5.(1)
Let
rence.
5
o f ch.9
1.7.
be a group o f p e r m u t a t i o n s o f
G
, which i s c l o s e d under n - m -
E
(6n)-homogeneous
Then t h e r e e x i s t s a
r e l a t i o n a l system, formed o f G (uses axiom o f c h o i c e when
components, whose group o f automorphisms i s
n - u
E is
uncountable). (2) Let
G
E , which i s c l o s e d under adherence.
be a group o f p e r m u t a t i o n s o f
Then t h e r e e x i s t s a homogeneous r e l a t i o n a l system whose group o f automorphisms i s G
(same remark).
( 3 ) Assume t h a t
tions o f
E
-
has f i n i t e c a r d i n a l i t y
E
group o f automorphisms i s 0
h
. Then
f o r e v e r y group
G
o f permuta-
, t h e r e e x i s t s a homogeneous m u l t i r e l a t i o n of maximum a r i t y h , whose
(1) To each
.
G
n - t u p l e o f elements o f
, associate the o r b i t , i.e. the class o f
E
n - t u p l e s which can be o b t a i n e d f r o m i t , by t a k i n g i t s image under any p e r m u t a t i o n in
G
. Then
t o each o r b i t , a s s o c i a t e t h e
t h e v a l u e (+) f o r those
o f choice, t a k e an o r d i n a l sequence Every p e r m u t a t i o n b e l o n g i n g t o automorphism
g
of
t h e same o r b i t . Thus
G
t a k e s each
R
E , which takes
n - a r y r e l a t i o n based on
n - t u p l e s i n t h e o r b i t , and ( - ) o t h e r w i s e . U s i n g t h e axiom
g
belongs t o
R
of t h e s e r e l a t i o n s , which f o r m a system.
i s an automorphism o f
R
, into
an
n-tuple i n
E
, since
G
G
. Conversely,
every
n-tuple belonging t o
i s c l o s e d under
n-adherence.
(6n)-homogeneous. Given a l o c a l automorphism f o f R , d e f i n e d on a domain of c a r d i n a l i t y 6 n , t a k e an a r b i t r a r y n - t u p l e c o n t a i n i n g a l l t h e elements al, ..., an o f F , w i t h p o s s i b l e r e p e t i t i o n s . T h i s n - t u p l e It remains t o see t h a t
and i t s image
fal,
permutation o f phism o f
R
is
R
..., f a n
belong t o t h e same o r b i t of
E which extends
and belongs t o
.0
( 2 ) Analogous p r o o f , b u t where
under adherence, i n s t e a d o f 0
f
G
G
. Thus
, hence
n t a k e s a l l i n t e g e r values,
there exists a
which i s an automor-
G
being closed
n-adherence. 0
( 3 ) P a r t i c u l a r case, where
G
i s c l o s e d under
h-adherence, and w i t h f i n i t e l y
many o r b i t s . 0 By t h e p r e c e d i n g p r o p o s i t i o n s , t o e v e r y r e l a t i o n a l system
a homogeneous system S
R
, there
corresponds
on t h e same base, h a v i n g t h e same automorphisms, and
w i t h o u t augmenting t h e maximum a r i t y o f t h e components. Mote however t h a t , s t a r t i n g from a simple r e l a t i o n
R
, we
can end up a t a homogeneous r e l a t i o n a l system S
w i t h i n f i n i t e l y many components. For example, s t a r t w i t h
R = W (chain o f natural
numbers), whose o n l y automorphism i s t h e i d e n t i t y . Then we end u p w i t h t h e
THEORY OF RELATIONS
320
sequence o f s i n g l e t o n unary r e l a t i o n s o f a l l the i n t e g e r s . 2.6. Here are two examples o f a group o f permutations which i s closed under
0
Let
be a s e t o f c a r d i n a l i t y
E
, and
n+2
an a r b i t r a r y t r a n s p o s i t i o n interchanging two elements o f (n+l)-adherent t o
position
such t h a t
t
G
.
Then
and
g
g
belongs t o
.
Let
Let
G
, where t i s
f,t
g
be a permuta-
G : indeed, there e x i s t s a t r a n s -
are i d e n t i c a l on
f,t
E
.
E
an odd permutation o f
f
be the group o f even permutations generated by the permutations tion
(n+l)-
n-adherence ( n n a t u r a l number ); POUZET 1979, unpublished.
adherence b u t n o t under
n+l
elements o f
E
, and
. Since f i s odd, i t does not belong t o G : it suffices t o see that f is n-adher e n t t o G . Indeed, f o r any x1 ,..., xn belonging t o E , l e t y, z be two d i s -
hence are s t i l l i d e n t i c a l on the
(n+2)nd
and l a s t element o f
t i n c t elements which are a l s o d i s t i n c t from the
.
(y,z)
... , (f,t)xn
= fxl,
(f,t)xl
Then
x
. Let
, which
= fxn
t
E
be the t r a n s p o s i t i o n
proves t h e
n-adherence
f . n
of
This example extends t o the case where in
i s i n f i n i t e , by t a k i n g
E
Another example. L e t
be t h e s e t o f a l l
E
The group
G
i s n o t closed under
tation of
E
w i t h negative determinant, hence
arbitrary points
.. ,an
al,.
in
n-adherence.
respect t o a hyperplane passing through into
On t h e o t h e r hand, mutation
Indeed, l e t
f
al,
i s closed under
G
G
(n+l)-adherent t o
. Take
f
be a l i n e a r permu-
does n o t belong t o
...,an . We G
.
fan
G
G
. Take
, then compose f w i t h the symmetry w i t h
E
w i t h p o s i t i v e determinant, hence belonging t o an
( n i n t e g e r >/ 2). L e t
n+l
E w i t h p o s i t i v e determinants.
be the group o f l i n e a r permutations o f
and
elements
( n + l ) - t u p l e s o f r a t i o n a l s , which we
s h a l l c a l l the r a t i o n a l vector space o f dimension
n
n+2
and repeating the preceding c o n s t r u c t i o n .
E
o b t a i n a l i n e a r permutation
, which
takes
(n+l)-adherence. n+l
points
xl,
al
into
Indeed, l e t
... ,
g
, ...
fal
be a per-
E ,
x ~ + belonging ~ to
which are l i n e a r l y independent. The images gxl,. . . ,g~,+~ are by hypothesis t h e images o f the x under a l i n e a r permutation w i t h p o s i t i v e determinant; hence they are l i n e a r l y independent. Let
u
be any p o i n t i n
t o the p o i n t s v = xl, sis
u 1. x 1 +
...,xn,v of
x
... +
.
Let
u ~ , . . . , u ~ + ~be the coordinates o f 1 1+
.= u .x
... +
are l i n e a r l y dependent. Since t h e i r images under
(n+l)-adherence G
. Similarly
u
relative
~ ~ + ~ .. Now x ~d e f+i n e~
. The
and w = ~ ~ + ~ ,. sox t~h a +t ~u = v + w
un.xn
t a t i o n belonging t o
+ un.(gxn)
E
, so t h a t we have u
g
points
are by hypothe-
i d e n t i c a l t o t h e i r images under a c e r t a i n l i n e a r permu-
, the w
+
dependence r e l a t i o n i s preserved:
and
QW = ~ ~ + l . ( g x , + ~ ); s i m i l a r l y
gv = ul.(gxl) x ~ +are~ l i n e a r l y dependent, s o t h a t we have
gu = gv + gw
=
ul.(gxl)
+
... +
~,.,+~.(gx,,+I)
.
...
321
Chapter 11
Letting
u
vary, we see t h a t
g
. Hence
i n t o gxi ( i = l,.,.,n+l) longs t o G [7
.
i s a l i n e a r permutation which takes each
xi
g has p o s i t i v e determinant, so t h a t
g be-
I n t h e preceding p r o o f , t h e hypothesis n >/ 2 i s used t o go from u = v + w t o gu = gv + gw ; f o r t h i s , the 3-adherence o f g i s required. On the o t h e r hand, take n = 1 thus n + l = 2 , so t h a t E i s the s e t o f the r a t i o n a l p o i n t s , o r vectors i n the plane, and G i s the group o f l i n e a r permutations o f E w i t h p o s i t i v e determinants. Then the permutation g which takes each vector (0,v) i n t o (0,2v) and which preserves each (u,v) when u # 0 ( w i t h r a t i o n a l s ) , i s 2-adherent t o G ; y e t t h i s g does n o t belong t o G
u, v
.
Example, due t o t h e same author, o f a group o f permutations closed under adherence, y e t n o t under n-adherence f o r any n 0 For each i n t e g e r n , take a s e t En w i t h c a r d i n a l i t y n , where the En are mutually d i s j o i n t , and l e t E be t h e i r union. Take G t o be the group o f those
.
permutations o f
E whose r e s t r i c t i o n t o
En
, for
each
n
,is
an even permutation
i s necessarily an'eleo f En . Then a permutation o f E which i s adherent t o G ment o f G . However, given an a r b i t r a r y i n t e g e r n , a permutation f i s n-adherent t o f o r each
G provided t h a t i t s r e s t r i c t i o n t o Ei i s an even permutation o f Ei i& n t l , and an a r b i t r a r y permutation o f Ei f o r each i >/ n t 2 . 0
2.7.(1) L e t p be an i n t e g e r , and R be a p-monomorphic, p-homogeneous r e l a t i o p - s e t - t r a n s i t i v e . Indeed, n a l system. Then t h e group o f automorphisms o f R f o r any two p-element subsets a, b o f t h e base, t h e r e e x i s t s an isomorphism from R/a onto R/b , which i s e x t e n d i b l e t o an automorphism o f R
2
.
.
( 2 ) L e t E be a set, and G be a p - s e t - t r a n s i t i v e group o f permutations o f E Then the closure 'G o f G under adherence i s again p - s e t - t r a n s i t i v e , and conversely. Moreover, there e x i s t s a homogeneous r e l a t i o n a l system whose automorphism group i s
-
-
, and such a r e l a t i o n a l system i s always p-monomorphic. T h i s f o l l o w s from 2.5 above; i f E i s uncountable, then t h i s uses t h e axiom o f choice. G+
2.8. THEOREM ON THE INCREASING NUMBER OF ORBITS
( 1 ) L e t p, q be i n t e g e r s , E a f i n i t e s e t w i t h c a r d i n a l i t y a t l e a s t equal tc , and G a group o f permutations o f E Then t h e number o f o r b i t s (mod G) of t h e (p+q)-element subsets o f E , i s g r e a t e r than or equal t o t h e number o f o r b i t s o f the p-element subsets (LIVINGSTONE, WAGNER 1965).
.
2p + q
-
0 Associate t o t h e group
group i s
G
a homogeneous m u l t i r e l a t i o n R whose automorphism p-element s e t s a, b belong t o the
G : see 2 . 5 . ( 3 ) above. Then two
same o r b i t (mod G) i f f the r e s t r i c t i o n s R/a and R/b are isomorphic: indeed by homogeneity, every isomorphism o f one r e s t r i c t i o n onto another i s e x t e n d i b l e t o an
322
THEORY OF RELATIONS
.
automorphism of R Same r e s u l t f o r the (p+q)-element s e t s . T h u s our proposition follows from t h e f a c t t h a t t h e number of isomorphism types of r e s t r i c t i o n s t o p+q elements i s g r e a t e r than o r equal t o the number of isomorphism types of the restrictions t o
p elements: see ch.9
5
7.1. 0
( 2 ) Let E be a denumerable s e t , and G be a group of permutations of E . To each i n t e g e r p a s s o c i a t e the countable number of o r b i t s of p-element subsets of E . Then t h i s number increases w i t h p (POUZET 1976). G+ of G under adherence, and note t h a t f o r each p , t h e o r b i t s of t h e p-element s e t s (mod G ) a r e the same a s t h e o r b i t s of the p-element s e t s (mod G'). Take a r e l a t i o n a l system R whose automorphism group i s G+ :
0 Consider t h e c l o s u r e
see 2 . 5 . ( 2 ) above. The proof terminates a s before, using t h e p r o f i l e i n c r e a s e theorem, ch.9 5 7 . 1 . However we must note t h a t t h i s theorem extends t o t h e case of a r e l a t i o n a l system with denumerably many components. Indeed, the multicolor theorem i n ch.3 5 5.3 includes the case of i n f i n i t e l y many c o l o r s , hence here of i n f i n i t e l y many isomorphism types f o r c e r t a i n values of
p
.0
( 3 ) In p a r t i c u l a r , l e t E have countable c a r d i n a l i t y a t l e a s t equal t o 2 p + q , and l e t G be a group of permutations of E . I f G i s ( p + q ) - s e t - t r a n s i t i v e , then G i s p - s e t - t r a n s i t i v e . 0 Indeed p - s e t - t r a n s i t i v i t y means t h a t a l l the p-element subsets of E belong t o t h e same o r b i t . Another proof i s obtained from ch.9 5 6 . 3 , i n view of t h e i n e q u a l i t y : p l e s s than o r equal t o Min(p+q,(Card E)-p-q) ; by using a l s o 2 . 7 . ( 2 ) above. 0 2.9. Let p , q be two i n t e g e r s , and E be a s e t of c a r d i n a l i t y g r e a t e r than o r equal t o 2 p + q ; l e t G be a group of permutations of E Then every oermutat i o n of E which preserves the o r b i t s o f (p+q)-element s e t s (mod G ) , a l s o preserves t h e o r b i t s of p-element s e t s .
.
Let G+ be t h e c l o s u r e of G under adherence, and l e t R be a homogeneous r e l a t i o n a l system whose automorphism group i s G+ : see 2 . 5 . ( 2 ) above. Let f be a permutation of E which t a k e s each (p+q)-element s e t i n t o another i n t h e same o r b i t (mod G ) , hence again i n the same o r b i t (mod G'). Then f t a k e s each r e s t r i c t i o n of R t o p+q elements i n t o an isomorphic r e s t r i c t i o n . Hence f t a k e s each r e s t r i c t i o n of R t o p elements i n t o an isomorphic r e s t r i c t i o n : see ch.9 5 7.2. Since R i s homogeneous, f o r any p-element s e t a , t h e r e e x i s t s an automorphism of R , hence an element of G+ , hence an element of G , which t a k e s
0
R/a
i n t o the isomorphic r e s t r i c t i o n
R/f"(a)
.0
2.10. Let E be a s e t of c a r d i n a l i t y g r e a t e r than o r equal t o 2p + q , and l e t G , H be two groups of permutations of E . I f every o r b i t of (p+q)-element sets
Chapter 11
p-element s e t s (mod G)
, then
(p+q)-element s e t s (mod H)
i s i n c l u d e d i n an o r b i t o f
(mod G)
of
323
i s i n c l u d e d i n an o r b i t o f
every o r b i t
.
p-element s e t s (mod H)
T h i s r e s u l t was c o n j e c t u r e d by BERCOV, HOBBY 1970, and a weaker v e r s i o n was proved by them; t h e p r e s e n t r e s u l t i s due t o POUZET 1976. 0
Let
a, b
be two
p-element s e t s b e l o n g i n g t o t h e same o r b i t (mod G)
e x i s t s a permutation tion
g
g
belonging t o
p r e s e r v e s a l l t h e o r b i t s (mod
, such G) , and
that
G
.
g"(a) = b
. There
T h i s permuta-
i n particular the o r b i t s o f the
(p+q)-element s e t s (mod G)
, hence by h y p o t h e s i s g preserves t h e o r b i t s o f t h e
(p+q)-element s e t s (mod H)
.
o r b i t s o f the o r b i t (mod H )
§
p-element s e t s (mod H)
.
3 - CHAINS
.
Hence
a
and
b
g
preserves t h e
belong t o t h e same
0
MODULO A PERMUTATION
COMPATIBLE
BY CHAINS;
By t h e p r e c e d i n g p r o p o s i t i o n ,
GROUP GENERATED
GROUP;
DILATED GROUP, CONTRACTED GROUP
COMPATIBILITY MODULO A GROUP m
Let
be a n a t u r a l number
1,..., m
A, 6
Two c h a i n s
.
A group
G
o f p e r m u t a t i o n s on t h e s e t o f i n t e g e r s
m-ary group, o r a group w i t h
i s s a i d t o be an
arity m
. , i f for < a2 < ...
G-compatible, o r c o m p a t i b l e modulo G
a r e s a i d t o be
every s e t o f
m elements i n t h e i n t e r s e c t i o n o f t h e bases, say
< am
, t h e p e r m u t a t i o n u which r e o r d e r s these elements a c c o r d i n g t o
(mod A)
c
au(l)
au(2)
We see t h a t
< . . . < au(,,,)
(mod B)
, belongs t o t h e group
G
al
.
G - c o m p a t i b i l i t y i s r e f l e x i v e and symmetric. I f t h e chains
t h e same base, then
A, B
have
G - c o m p a t i b i l i t y i s t r a n s i t i v e , hence i s an equivalence r e l a -
tion.
I n t h e general case, n o t e t h a t two c h a i n s a r e
G-comoatible when t h e i n t e r s e c t i o n
o f t h e bases has c a r d i n a l i t y s t r i c t l y l e s s t h a n i s no t r a n s i t i v i t y , s i n c e f o r example i f has base d i s j o i n t f r o m t h e base o f
B
and
on t h e o t h e r hand, a r e
A
n o t t h e case f o r If
m
= 2
and
G
and
A
A
, then
and
A
m ( a r i t y o f G ) . Hence t h e r e A ' have t h e same base and B and
G-compatible f o r e v e r y
on t h e one hand, and
B
G
A'
, which i s o b v i o u s l y
A ' , assumed t o be d i s t i n c t .
reduces t o t h e i d e n t i t y on t h e s e t { 1 , 2 )
, then
G-compati-
b i l i t y means t h a t t h e r e s t r i c t i o n s o f b o t h c h a i n s t o t h e i n t e r s e c t i o n o f t h e i r bases ch.2
i s t h e same. Then we
5 1.2;
f i n d t h e n o t i o n o f c o m p a t i b i l i t y i n t h e sense o f
and t h e r e e x i s t s a common e x t e n s i o n which i s a c h a i n based on t h e
u n i o n o f t h e two bases.
I f m i s an a r b i t r a r y i n t e g e r , and G i s t h e m-ary symmetric group, o r group o f 1,. . ,m , t h e n a l l c h a i n s a r e G-compatible. Two c h a i n s a r e G-compatible i f f t h e i r r e s t r i c t i o n s t o t h e i n t e r s e c t i o n o f t h e a l l permutations o f
bases a r e
G-compatible.
.
1
THEORY OF RELATIONS
324
Let A, B be two chains with the same base E , and l e t G be an m-ary group; l e t n be such t h a t m 6 n Card E Then A and B are G-compatible i f f , f o r each n-element subset X of E , the r es t r i ct i o n s A/X and B/X are G-compatible.
, m' . 4.6. REDUCTION THEOREM (FRASNAY 1965) Given an m-ary group G , there e x i s t s a maximumnndicative m-ary group H ( G ) included in G . Moreover f o r n 3 m we have H ( G n ) = ( H ( G ) ) n . Taking n m such t h a t G n i s indicative, then we have H ( G ) n = Gn and H ( G ) = (G')),
.
§
5 - Q-BICHAIN,
SET-TRANSITIVE
Q-INDICATIVE GROUP THEOREM
GROUP;
FIVE
Q-INDICATIVE
GROUP THEOREM;
(CAMERON)
Q-BICHAIN, Q-INDICATIVE GROUP We shall c a l l a Q-bichain, any birelation both o f whose components are chains each isomorphic with Q . We say t h a t a group i s Q-indicative, i f i t i s generated by a Q-bichain. We see t h a t the f i v e following groups are Q-indicative, f o r each a r i t y m : the identity I,,, ; the group J m ( i d e n t i t y and r e f l e c t i o n ) ; the group Tm of translations; the dihedral group Dm generated by the union of Jm and Tm ;
and f i n a l l y the symmetric group Sm . We call these temporarily the canonical groups. The group generated by the union of two canonical groups i s canonical. 0 Indeed, the only case where two of these groups are non-inclusive, i s the case of Jm and Tm , whose union generates Dm . 0 Consequently, f o r each group G , there e x i s t s a maximum canonical qroup included in G -
-
.
THE FIVE Q-INDICATIVE GROUP THEOREM The five canonical groups are the only Q-indicative groups. To prove t h i s , we shall show t h a t i f G i s Q-indicative, then i t i s equal t o the maximum canonical group included in G This i s obvious f o r the case of the symmetric group S, ; so we shall consider the four cases I,, Tm, J m , Dm in the following propositions 5 . 1 t o 5.8.
.
33 1
Chapter 11
.
5.1. Consider a Q-bichain with components A , B We say that a pair of elements x , y of the base, i s preserved or inverted, accordin9 t o whether x and y are in the same order modulo A and modulo B , o r in the opposite order. Let AB be a Q-bichain having -a t l e a s t one preserved pair and one inverted pair. AB , f o r every Then e i t h e r the group Jm i s included in the group generated bym , Tm i s included in the group, f o r every m .
or
Let u and v denote two elements such t h a t u < v (mod A ) and v < u (mod B ) . Since A i s isomorphic with Q , there e x i s t i n f i n i t e l y many elements x between u and v (mod A ) . For each such x , we have v < x or x < u (mod B ) , hence there e x i s t i n f i n i t e l y many x s a t i s f y i n g , f o r example, x < u (mod B ) . Using RAMSEY's theorem (ch.3 g l . l ) , e i t h e r there e x i s t i n f i n i t e l y many of these x which form mutually inverted p a i r s , in which case the aroup Jm i s included in the m-ary group generated by AB , f o r each m . Or there e x i s t i n f i n i t e l y many of these x forming preserved pairs. Then since u i s less t h a n (mod A) and greater t h a n (mod B ) these elements, the group Tm of translations i s included in the group generated by AB , f o r each m 0
0
.
5 . 2 . Let G be the m-ary g r o u p generated by a given Q-bichain. If the maximum canonical group included in G I,,, , & t G i s identical t o I,,,
.
Either there only e x i s t preserved pairs in the given bichain, in which case G = I . Or there only e x i s t inverted p a i r s , in which case G = Jm . Or f i n a l l y m there e x i s t s a t l e a s t one pair of each kind, in which case Jm or Tm i s included in G . Only the f i r s t case i s possible under our assumption t h a t I,,, i s the maximum canonical group included in G . 0
0
.
Suppose t h a t 5.3. Let AB be a Q-bichain w i t h base E ~subset U Lf E a l l of whose pairs are preserved, hence Let m be a positive integer; suppose t h a t the group Tm m-ary group generated by the bichain. Then e i t h e r G = Tm symnetri c g r o u p .
there e x i s t s an i n f i n i t e A/U = B / U . i s included in the or G = Sm , the
Suppose t h a t mg 2 , since the case where m = 1 i s obvious. The s e t U i s d i f f e r e n t from E , since Tm i s included in G . For each element x of E-U , l e t xA denote the cut defined on A/U by the i n i t i a l interval of those elements of U l e s s than x (mod A) , and the complementary f i n a l i n t e r v a l . Let xB denote the cut analogously defined with B F i r s t suppose t h a t f o r each x in E-U , we have xA = xB By hypothesis, there e x i s t s an inverted p a i r , say { x,y} which i s thus included in E-U , which then implies t h a t xA = xB = yA = yB I f , before t h i s c u t , we have i n f i n i t e l y many elements of U , then G contains the transposition (m-1,m) , which together with T, generates S, Similarly, i f there e x i s t i n f i n i t e l y many elements
.
.
.
.
THEORY OF RELATIONS
332 greater than t h i s cut, then Tm generates
with
From t h i s p o i n t on, we are i n t h e case where t h e r e e x i s t s an xA # xB
. Suppose t h a t
before
A/U
on U
, and
l i e s before i t , then
G
defined by t h e c y c l e
(2,3
.
E-U
with
a t l e a s t one o f these i n t e r v a l s i s i n f i n i t e . xA and
xB
, and
i f the o t h e r non-empty i n t e r v a l
contains the permutation which preserves
. By composition
,..., m)
S i m i l a r l y i f the i n t e r v a l between
(1,Z
with
1 and which i s
,...,m) , we
generate
xA and xB i s i n f i n i t e and the o t h e r
non-empty i n t e r v a l l i e s a f t e r i t . I f the i n t e r v a l between and t h e i n t e r v a l l y i n g before i s i n f i n i t e , then which, together w i t h t h e t r a n s l a t i o n
(m-1,m)
in
, nor the f i n a l c u t . Then t h e r e e x i s t a t l e a s t two i n t e r v a l s bounded
I f i t i s t h e i n t e r v a l between
Sm
x
a t l e a s t one o f these cuts i s n e i t h e r the i n i t i a l c u t l y i n a
xA and xB
by
, which together
(1,2)
contains t h e t r a n s p o s i t i o n
G
.
Sm
G
xA and
xB
is finite
contains the t r a n s p o s i t i o n
(1,2,
...,m)
, generates
Sm
. Same
r e s u l t i n t h e case o f an i n f i n i t e i n t e r v a l l y i n g a f t e r . xA and
Suppose now t h a t , i f
xB
are d i s t i n c t , then they are extremal ( i . e . one
o f them i s the i n i t i a l cut, and the o t h e r i s the f i n a l c u t ) . We can r e q u i r e t h a t any two elements
x, y
which give t h e same non-extremal c u t
xA = xB = yA = yB
,
form a preserved p a i r . Indeed otherwise, we o b t a i n again t h e t r a n s p o s i t i o n (1,2) o r (m-1,m) Thus augment U by a l l these x corresponding t o non-extremal
.
cuts. From t h i s p o i n t on, every E i t h e r there e x i s t
x, y
. Then
yA f i n a l c u t
identical with
Sm
G
in
x
in with
E-U
E-U
xA and yB i n i t i a l c u t
contains the t r a n s p o s i t i o n
. Then
x
in
G
contains t h e permutation which transforms
are preserved, i n which case
+ A/(E-U)
5.4. L e t
in
G
G
2
, hence G
xA
i s i n i t i a l and
12 3 Sm
. Or
i s t h e sum A/(E-U)
A
= Tm
be generated by a
Tm,=
and
xB and
and hence again i s xB
f i n a l f o r each
e i t h e r there e x i s t s an i n v e r t e d p a i r among these; i n which case
and t h i s together the t r a n s l a t i o n generates A/U
(1,m)
.
O r we are i n the case where, f o r instance,
E-U
y i e l d s two extremal cuts.
... m-1
m into 3 4
...
m 2 1
f i n a l l y a l l the pairs i n
+
A/U
and
B
E-U
i s the sum
.0 Q-bichain. I f the maximum canonical group i n c l u d e d
G=Tm.
Because o f RAMSEY's theorem, there e x i s t s an i n f i n i t e subset
U of E , all of
U , a l l o f whose p a i r s are i n v e r J m i s included i n G , and since by hypothesis Tm i s included i n G as w e l l , we have t h e dihedral group Dm included i n G ,
whose p a i r s are preserved, o r an i n f i n i t e subset t e d . I n t h e second case, t h e group
i s the maximum canonica.1 group included i n Tm by the preceding statement, G = Tm o r G = S, , t h i s l a s t case contra-
c o n t r a d i c t i n g the assumption t h a t G
. Thus
d i c t i n g our assumptions. 0
333
Chapter 1 1
5.5. L e t
be a
AB
of
U
t -s
Q - b i c h a i n w i t h base
.
E
Suppose t h a t t h e r e e x i s t s an i n f i n i t e
, a l l o f whose p a i r s
E
m
a l l o f whose p a i r s a r e i n v e r t e d . Then f o r each AB
i s t h e symmetric group
0 Note f i r s t t h a t
existence o f
and
U
m-ary group generated by ~
.
Sm
have a t most one element i n common. Moreover, t h e
V
shows t h a t t h e group
V
, the
Jm o f t h e r e f l e c t i o n , i s generated. Using
5.3 above, i t s u f f i c e s t o prove t h a t t h e group
Tm o f t r a n s l a t i o n s i s generated,
s i n c e t h e n t h e e n t i r e generated group cannot be reduced t o t h e s i n g l e group Tm and i s t h e n i d e n t i c a l w i t h Sm . F i r s t suppose t h a t f o r each i n t e g e r h , t h e r e e x i s t s an x i n V f o r which t h e cuts
xA
least
h
and
xB
d e f i n e d by
elements o f
.
U
x
, are
A/U = B/U
on t h e c h a i n
I n t h i s case, t h e t r a n s l a t i o n
(l,Z,
separated by a t
...,in)
i s obtained,
hence o u r p r o p o s i t i o n h o l d s . Suppose now t h a t t h e r e e x i s t s an i n t e g e r a t most
h
elements i n
o f elements i n
U
, which
V
h
such t h a t f o r each
between t h e c u t s
xA
and
. Take
xB
i s f o r example d e c r e a s i n g (mod A ) ,
x
in
V
, we have
an w-sequence
hence decreasing
. Then f r o m some p o i n t on, t h e c u t s xA become i d e n t i c a l , as w e l l as t h e become i d e n t i c a l . Thus t h e r e e x i s t i n f i n i t e l y many elements i n U which a r e
(mod B ) xB
e i t h e r a l l g r e a t e r t h a n these c u t s , o r a l l l e s s t h a n these c u t s . T h i s y i e l d s f o r i n s t a n c e , f o r each ..,p
t h e symmetric group 5.6. L e t
pair.
m
and each
and which i n t e r c h a n g e s
AB
be a
Then f o r each
Sm
, t h e p e r m u t a t i o n which p r e s e r v e s 1 2 , ...
p d m
(p+l,m),
(p+Z,m-l),
etc.;
t h i s s u f f i c e s t o generate
.
(I-bichain h a v i n g a t l e a s t one p r e s e r v e d p a i r and one i n v e r t e d , m
,t h e group Tm
i s i n c l u d e d i n t h e group generated by
AB
.
u < v (mod A) w i t h v < u (mod B) . Take i n f i n i t e l y many u and v (mod A ) . Then e i t h e r t h e r e e x i s t i n f i n i t e l y many o f them which a r e > u (mod B ) . I n t h i s case, by RAMSEY's theorem, e x t r a c t an i n f i n i -
0 Consider two elements
elements between
t e subset o f t h e s e elements, a l l o f whose p a i r s a r e preserved, o r a l l o f whose p a i r s a r e i n v e r t e d . I n t h e case where t h e p a i r s a r e preserved, by o u r c h o i c e o f t h e element obtain again
, we
v
t e d , t h e group
Jm Tm .
o b t a i n t h e group
T,
O r t h e r e e x i s t i n f i n i t e l y many elements
(mod A)
. Then
.
I n t h e case where t h e p a i r s a r e i n v e r -
i s obtained; by t h i s r e f l e c t i o n and by o u r c h o i c e o f
, 3 , the binary cycle of cardinality p
.
THEORY OF RELATIONS
35 2
R be a relation. For any f i n i t e relation X , we have X embeddable i f f no bound of R i s embeddable in X . ( 2 ) Let R , S be two r el at i o n s , a t l e a s t one of which i s f i n i t e . Then RG S iff no bound of S i s embeddable in R
1.1.(1) Let
in
R
.
( 1 ) Follows from ch.4 5 8.2, since embeddability between f i n i t e relations i s a well-founded p a r t i a l ordering. However l e t us give a d i r ect proof. If X < R , then no bound of R i s embeddable in X Conversely, i f X 3 ; R , then e i t h e r X i s a bound of R . Or there e x i s t s a r e s t r i c t i o n X1 of X t o i t s base minus one element, such t h a t R Iterating t h i s , a f t e r a f i n i t e number h of s t ep s , we obtain a re stric tion Xh of X which i s a bound of R 0 0 ( 2 ) If R i s f i n i t e , we find ( 1 ) . Suppose t h at R i s i n f i n i t e and S f i n i t e , hence R $ S . Replace R by a r es t r i ct i o n R ' whose cardinality i s f i n i t e b u t S , hence R ' admits an s t r i c t l y greater than the cardinality of S . Then embedding of a bound of S , by our ( 1 ) . This bound i s thus embeddable in R 0 0
.
XB I .
.
R'4
.
On the other hand, i f R and S are both i n f i n i t e , then statement ( 2 ) does n o t necessary hold. 0 Take R t o be the chain of the natural numbers. Take a sequence of f i n i t e relations Ai ( i integer) with d i s j o i n t bases, and such tha t every f i n i t e binary relation is isomorphic t o an Ai . Take S t o be the common extension of the Ai , which takes the value (+) f o r every ordered pair whose terms belong t o the bases of two d i s t i n c t Ai . Then S has no bound; hence no bound of S i s embeddable in R , and y e t R i s non-embeddable in S . 0
f i n i t e or S f i n i t e , i f every bound of S i s a bound of R , Indeed no bound of S i s embeddable in R (2) Any two f i n i t e relations having the same bounds, are isomorphic. (3) If R i s f i n i t e and R < S , then there ex i s t s a bound of R which i s embeddable in S ; hence there e x i s t s a bound of R which i s n o t a bound of S Indeed S i s not embeddable in R , hence S admits an embedding of a t l e a s t a (2). bound of R , by the preceding 1.l. ( 4 ) If e i t h e r R o r S i s f i n i t e , and i f the s e t of bounds of S i s properly included in the s e t of bounds of R , then R < S Follows from ( 1 ) and ( 2 ) . 1.2.(1) For
then
R
.
.
RQ S
-
.
-
.
>/ 1 has a t l e a s t 2 bounds. ( 2 ) Every non-empty f i n i t e relation of a r i t y >/ 2 has a t l e a s t 4 bounds.
1.3.(1) Every f i n i t e relation of a r i t y
-
.
be f i n i t e ; by the preceding 1 . 2 . ( 3 ) , there e xists a bound A of R We have R By ch.5 5 1.3.(1), assuming t h at the a r i t y i s n o t zero, there e x is t s an extension S 7 R respecting the non-embeddability S . By the preceding 1.2.(3), there e x i s t s a bound B of R with B & S , hence non-isomorphic
0
(1) Let
R
AS .
A4
353
Chapter 12
.0
A
with
0 (2) Let
be a non-empty f i n i t e r e l a t i o n w i t h a r i t y
R
.
2
Keeping t h e two
bounds A, B a l r e a d y o b t a i n e d and u s i n g ch.5 § 1.3.(3), t h e r e e x i s t s a p r o p e r e x t e n s i o n R+ o f R r e s p e c t i n g A $ R+ and B * R + . By t h e p r e c e d i n g 1 . 2 . ( 3 ) , t h e r e e x i s t s a bound with
nor with
A
of
C
.
B
such t h a t
R
obtain another proper extension o f i s n o t isomorphic w i t h any o f 1.4.(1)
Let
There does n o t e x i s t any (2) Let
S
be a
e x i s t s no of
T
S
Then
of
t i n c t and T
o f cardinality
S
.
p+l
i s a bound o f
S
, we
R
D o f R which
be t h e b i n a r y c y c l e o f c a r d i n a l i t y
and add an element
S
which i s
S
b, c
p-monomorphic,
, for
p
Let
.
p+l
which t h e r e
be a r e s t r i c t i o n
R
and e v e r y bound o f
R
p
S
T(b,b')
= T(b,a)
=
whose base c o n t a i n s
Note t h a t t h e c y c l e
b
t o i t s base. To o b t a i n an T(b,a) S
i n the cycle
= T(a,c)
. Then
=
a, b, b '
. Let
+
are dis-
+ , which makes i t i m n o s s i b l e f o r any r e s t r i c t i o n , t o be isomorphic w i t h t h e c y c l e S . 0
a, b, b '
o f cardinality
S
a
i t i s necessary t h a t t h e r e e x i s t
I S 1 with
o f t h e base
be t h e element c o n s e c u t i v e t o
of
p
i s obviously
(p-1)-monomorphic;
thus
s a t i s f i e s t h e hypotheses o f o u r ( 2 ) .
S 0
.
p-1
and
(communicated by POUZET 1978).
R
two d i s t i n c t elements b'
, and
p-monomorphic e x t e n s i o n o f
(1) S t a r t w i t h t h e c y c l e
extension
0
p-monomorphic e x t e n s i o n o f c a r d i n a l i t y
i s a bound o f 0
B, C .
(p-1)-monomorphic r e l a t i o n o f c a r d i n a l i t y
with cardinality
S
A,
i s isomorphic n e i t h e r
C
A, B, C
and f i n a l l y a f o u r t h bound
R
be an i n t e g e r 3 3
p
, hence
R+
C,
.
Let Ri ( i positive integer) be our f i n i t e r e l a t i o n s , which are l i s t e d by increasing values of p . We shall modify our construction in the preceding 2 . 2 , as follows. Take a sequence of a l l the f i n i t e relations U . ( j positive integer) with 3 the same a r i t y as the Ri , and l e t k br the f i n i t e cardinality of U j j' Replacing, i f necessary, each Ri by an isomorphic copy, we can suppose the following. For each i , i f U1 i s embeddable in Ri , then U1 i s embeddable in the r e s t r i c t i o n of Ri t o { 1 , 2 , ..., k l ) Again f o r each i , i f U 2 i s embeddable in Ri , then U 2 i s embeddable in the r e s t r i c t i o n of Ri t o the s e t {1,2 ,... , k l + k 2 ) ; and so forth. Now, construct relations Sr ( r integer) as in the preceding 2 . 2 , and then take t h e i r common extension S . Then A 1 , . . . , A h are a l l bounds of S . I t remains t o prove t h a t S has no other bound. Suppose t h a t B i s a bound of S d i f f e r e n t from A 1 , . . . , A h Then f i r s t l y , each proper r e s t r i c t i o n of B i s embeddable i n S , hence in the Sr f o r a l l r greater than some r ( 0 ) ; hence in a l l the Ri which extend S r(O) . Secondly B cannot be a bound of Ri , f o r i s u f f i c i e n t l y large, so t h a t the integer p associated by hypothesis with Ri i s larger than the cardinality of B . Hence B i s embeddable in a l l the Ri which extend S and whose r(O) index i i s s u f f i c i e n t l y large. Finally there e x i s t s r(1) >/ r ( 0 ) such t h a t B i s embeddable in a l l those Ri which extend S r(1) . From the f i r s t paragraph, there e x i s t s an integer k f o r which, i f B i s embeddable in Ri , then B i s s t i l l embeddable in the r e s t r i c t i o n Ri/{1,2 ,...,k ) . Hence B i s embeddable in Sm , where m i s the maximum of k and r ( 1 ) . Thus B i s embeddable in S : contradiction. 0
0
.
.
§
3 - WELL
MULTIRELATION:
REASSEMBLING
THEOREM
(FRASNAY)
&,
We say t h a t a multirelation R i s i f the s e t of i t s f i n i t e r e s t r i c t i o n s , when p a r t i a l l y ordered under embeddability, forms a well partial ordering. In other words, i f any s e t of f i n i t e r e s t r i c t i o n s of R , mutually incomparable under embeddability, i s f i n i t e .
For example, every chain i s well. Every t r e e i s well, by KRUSKAL's theorem (ch.5 5 2.3). The consecutivity relation on the natural numbers i s well. Indeed, each f i n i t e r e s t r i c t i o n can be represented by a f i n i t e sequence of positive integers, each
360
THEORY OF RELATIONS
integer i representing a component of i consecutive integers. Then the embeddab i l i t y between two f i n i t e sequences of integers, implies the embeddability between the two corresponding f i n i t e r e s t r i c t i o n s . Now i t suffices t o recall t h a t embeddabil i t y between f i n i t e sequences, or words of integers, i s a well partial ordering by HIGMAN's theorem ch.4 5 4.4.
p Given a natural number concatenating R with any mu1 t i re1 ation.
rf p-well.
3.1.
R
2
, we p
say t h a t a multirelation R i s p - e , i f upon unary relations with the same base, we obtain a well
p-well, then every multirelation freely interpretable in. R
Let S be freely interpretable in R , the base. I f S/F augmented by p unary augmented by p unary relations A ' on embeddability, then the same i s true f o r augmented by the A ' .
0
&
and l e t F, F ' be two f i n i t e subsets of relations A based on F , and S/F' F ' , are incomparable with respect t o R/F augmented by the A and R/F'
The consecutivity relation on the natural numbers i s well, b u t i s n o t 1-well. i , take a sequence of i+2 consecutive elements, and define a unary relation t o take the value ( - ) f o r the f i r s t and the l a s t element, and the value ( + ) between. 0
0 For each integer
Problems communicated by POUZET in 1972. (1) If a multirelation i s 1-well, then i s i t 2-well, and even p-well f o r every integer p ( 2 ) S t a r t with a multirelation R Take the concatenations RX with X an arbit r a r y unary relation; then the union of the ages of these concatenations, where R i s fixed and X varies. This union i s not in general an age. If t h i s union i s well partial ordered under embeddability, then R i s 1-well. Is the converse t r u e . Same question f o r 2 , 3 , ... unary relations added.
.
.
3.2. Every chain, and consequently every chainable multirelation, i s each integer p (POUZET 1 9 7 2 ) .
p-well f o r
t o see t h a t the multirelations ( A , B l , . . . , B p ) , where A i s a f i n i t e chain and the B are unary r e l a t i o n s , form, up t o isomorphism, a s e t which i s well p a r t i a l l y ordered under embeddability. To each multirelation, associate the f i n i t e cardinality h of i t s base. Then associate a word of length h , obtained by replacing each i = 1, ,h by the sequence of values B1(x) , ... , Bp(x) , where x designates the i t h element of the base, ordered modulo A . We say t h a t two of these sequences are considered t o be incomparable i f f they are d i s t i n c t . T h u s we have a well partial ordering of
0 I t suffices
...
i d e n t i t y , with
2p elements, or sequences, mutually incomparable. By HIGMAN's
36 1
Chapter 12
theorem (ch.4
5
4.4 f i n i t e case, p r o v a b l e i n ZF), t h e s e t o f words formed o f t h e
p r e c e d i n g elements, c o n s t i t u t e s a w e l l p a r t i a l o r d e r i n g , under t h e usual embeddabil i t y o f words. T h i s g i v e s a w e l l p a r t i a l o r d e r i n g , under e m b e d d a b i l i t y , o f o u r
f i n i t e multirelations. 0 3.3. THEOREMS ON FINITE NUMBER OF BOUNDS
Let
If
be a m u l t i r e l a t i o n ; denote by
R
g
R
2m-well,
then
t h e maximum o f i t s components' a r i t i e s .
m
has f i n i t e l y many bounds (POUZET 1972).
R
Consequently e v e r y c h a i n a b l e m u l t i r e l a t i o n has f i n i t e l y many bounds (FRASNAY 1965) 0 Let
al
be a bound o f
U
in
, and
E
let
t h e r e e x i s t s an
and
R
w i t h base
S< R
F ut a l i . Note t h a t when t h e a r i t y
m = 1
condition. For otherwise
S
I f t h e r e e x i s t s no such
relation:
El and
,..., Vm,A1
(S,V1
V
; where t h e
o n l y on
Consider t h o s e subsets
F o f El
E
, which
U
coincides w i t h
, then ,..., A),, A
on
f o r which and on
El
, t h e n no subset F , even empty, s a t i s f i e s t h i s
, hence U&
U
a s s o c i a t e t o t h e bound where
and
S,< R
.
R
the following multi-
U
coincides w i t h
t a k i n g t h e value (+) on
V1
a r e unary,
S
U
b e i n g t h e s i n g l e t o n unary r e l a t i o n o f
A1
an a r b i t r a r y element
.
would be i d e n t i c a l w i t h
and t h e
, and
El
F
. Take
U
be t h e base o f
E
- {al)
El = E
al
El
on and
V
; the other
t a k i n g always t h e value ( - ) .
A
Now suppose t h a t t h e r e e x i s t subsets
s a t i s f y i n g o u r p r e c e d i n g c o n d i t i o n . Take
F
F o f maximum c a r d i n a l i t y , say
such an
E2
.
This
i s a p r o p e r subset o f
E2
El
, t h e n we would have U,< R . L e t a2 be an a r b i t r a r y element o f El - E2 . Consider those subsets F o f E2 f o r which t h e r e e x i s t s an S,< R w i t h base E , which c o i n c i d e s w i t h U on El and on E2 u {al\ and on Fu{al,a2). Note t h a t when t h e a r i t y m = 2 , t h e n no such F , even empty, e x i s t s . F o r o t h e r , wise, t h i s would y i e l d t h a t S c o i n c i d e s w i t h U on t h e union E2u{al,a2) for if
E2 = El
contradicting the maximality o f relation: El
(S,V1
and on
E2u{al\
(+) o n l y on singleton o f
F
; where
E2u{al\ a2
E2
.
, t h e n a s s o c i a t e t o t h e bound U t h e f o l l o w i n g m u l t i ,..., Vm,A1 ,...,Am) where S 4 R and S c o i n c i d e s w i t h U on
I f t h e r e e x i s t s no such
V1
takes t h e v a l u e (+) o n l y on
A1
; the relation
, and
i s the singleton o f
V
f i n a l l y the other
A
and
El
al
, and V2 takes
Em
and elements
Thus t h e r e e x i s t s an E2 \al) tion S
al S,
/
n
(finite or infinite);
and f o r each
AF w i t h base F , where t h e G-compatible; t h e n t h e r e e x i s t s a c h a i n based on E , which i s
subset
a chain
n-element
AF a r e m u t u a l l y G-compatible w i t h
(FRASNAY 1965; t h e a r i t y o f a group and t h e G - c o m p a t i b i l i t y a r e
d e f i n e d i n ch.11
0
3; o u r p r o o f uses u l t r a f i l t e r axiom; ZF s u f f i c e s i f
E
countable).
Chapter 12
0
Suppose f i r s t t h a t
and d e f i n e t h e
i s f i n i t e . Let
E
363
be t h e c h a i n o f t h e n a t u r a l numbers,
A
m-ary r e l a t i o n
R
on t h e s e i n t e g e r s , by s e t t i n g
i f f e i t h e r a t l e a s t two o f t h e
x
are i d e n t i c a l , o r i f the
x
,. . . ,xm)
R(xl
+
=
are a l l d i s t i n c t
xl,. . . ,xm i n i n c r e a s i n g o r d e r C l e a r l y e v e r y l o c a l automorphism o f t h e c h a i n A i s a
and t h e p e r m u t a t i o n which r e o r d e r s t h e sequence belongs t o t h e group R
is
A-chainable.
.
G
R ; i n o t h e r words
l o c a l automorphism o f
p ; and by t h e p r e c e d i n g theorem 3.3,
R
is
p - w e l l f o r each i n t e g e r
has f i n i t e l y many bounds.
R
F
of
Let
,let
E
be t h e
S
E
has c a r d i n a l i t y a t l e a s t
be a c h a i n on
AF
F
, where
m-ary r e l a t i o n based on
e i t h e r a t l e a s t two
x
n AF
the
. F o r each
are i d e n t i c a l , o r i f the
G-compatible
...,xm)
... ,xm
xl,
G ; where
taining
the value o f F
S
o f the
AF
onto a r e s t r i c t i o n o f
, so
n
no bound o f
i s embeddable i n
S
would be a bound o f of
R
chain
.
B
, with
. Let
x
according the chain AF
f o r each
t h e case t h a t Now suppose non-empty s e t
f
E
UD
.
R
S/F
,
.
F o r o t h e r w i s e some r e s t r i c t i o n o f
...,x m )
=
+
S
S
onto a r e s t r i c t i o n
i f f e i t h e r two
A x
into a are i d e n t i -
a r e d i s t i n c t and t h e p e r m u t a t i o n which r e o r d e r s these
B , belongs t o
.
G
n-element subset
F
It f o l l o w s t h a t of
E
. The
B
is
x
G-compatible
p r o o f i s now achieved f o r
i s finite.
E i n f i n i t e . F o r each f i n i t e subset D o f E , t h e r e e x i s t s a UD o f c h a i n s based on D , each b e i n g G-compatible w i t h t h e AF
, when r e s t r i c t e d t o
. For a D'
subset
, gives
D'
of
an element o f
0
based on
, whose r e s t r i c t i o n t o each f i n i t e s e t
E
F
S : o t h e r w i s e , such a
, hence i n R
lemma (ch.2 is
con-
c a r d i n a l i t y o f each bound o f
takes a r e s t r i c t i o n o f t h e c h a i n
S(xl,
g i v e n chains, s t i l l denoted by ging t o
. The
A
be an isomorphism f r o m
f-l
the r e s u l t t h a t
cal, o r i f a l l the with
R
The i n v e r s e f u n c t i o n
E
f o r any two
.
F
i s embeddable i n
R
bound would be embeddable i n a r e s t r i c t i o n It f o l l o w s t h a t
n-element
AF
S/F i s embeddable i n R : i t s u f f i c e s t o t a k e an
restriction
AF
i s less than
, belongs
n-element subset o f
G-compatibility o f the
does n o t depend on t h e chosen
, the
isomorphism f r o m R
d e s i g n a t e s an a r b i t r a r y
... ,xm . Because
xl,
F o r each
F
+ iff
=
a r e a l l d i s t i n c t and
t h e p e r m u t a t i o n which r e o r d e r s t h i s sequence a c c o r d i n g t o t h e c h a i n t o t h e group
assume
n-element subset
are mutually
, such t h a t S(xl,
E
. We
R
be s t r i c t l y g r e a t e r t h a n t h e c a r d i n a l i t i e s o f bounds o f
Let n a m that the f i n i t e set
A , hence
i s freely interpretable i n
R
Using 3.2 above, we see t h a t
D
, every UD,
c h a i n belon-
. By t h e
coherence
1.3, e q u i v a l e n t t o t h e u l t r a f i l t e r axiom), t h e r e e x i s t s a c h a i n
G-compatible w i t h a l l t h e
AF
D
belongs t o
.0
UD
. This
chain
3.5. CHAINABILITY THEOREM Let
m
be a p o s i t i v e i n t e g e r . There e x i s t s an i n t e g e r
m-ary r e l a t i o n w i t h c a r d i n a l i t y >/ p
, which
is
p a m such t h a t
(6p)-monomorphic,
e
x
i s chainable
(FRASNAY 1965; uses u l t r a f i l t e r axiom; ZF s u f f i c e s f o r a c o u n t a b l e r e l a t i o n ) .
THEORY OF RELATIONS
364
Consider a l l groups of a r i t y m , and l e t n be the maximum of the integers associated t o these groups in the preceding reassembling theorem. By ch.9 5 5 . 5 . ( 2 ) , there e x i s t s an integer pa n such t h at every m-ary re la tion with cardinality 3 p has a chainable r es t r i ct i o n with cardinality n Let R be a ( 4p)-monomorphic m-ary relation with base E o f cardinality a t le a s t equal t o p , Then a l l the r es t r i ct i o n s o f R with cardinality n are isomorphic, hence they are a l l chainable. To each n-element subset F of E , associate a chain AF based on F , such t h at the re stric tion R/F i s AF-chainable. Moreover, f o r any two n-element subsets F, F ' of E , take A F , t o be the image of AF under one of the isomorphisms of R/F onto R/F' . Let H be an m-element subset of the base E , and l e t F , F ' be two n-element subsets, each of which includes H . The permutation of H which takes AF/H i n to AF,/H i s an automorphism of R/H . Indeed, take the image H ' of H under the isomorphism from AF onto A F , ; then H ' i n t o H by preserving the order of elements (mod A F , ) and using chainability by A F , . Designate each element o f H by i t s rank (mod AF/H) . Then the group o f automorphisms of R/H becomes an m-ary permutation group. The preceding isomorphisms show t h a t G depends neither on H nor on the choice of the n-element s e t F including H . By the preceding, AF and A F , are G-compatible for any two n-element s e t s F and F ' Now apply the reassembling theorem: there ex i s t s a chain A based on E , which i s G-compatible with every AF . W e shall prove t h a t R i s A-chainable. Let H, H ' be two m-element subsets of E : we shall f i r s t prove t h a t the isomorphism from A/H onto A / H ' takes R/H i n t o R / H ' . We can assume t h a t 2m , hence t h a t there e x i st s an n-element subset F of E including both H and H ' . The desired isomorphism can be obtained by composing three isomorphisms: from A/H onto AF/H , from AF/H onto AF/H' , then from AF/H' o n t o A/H' The f i r s t and the t h i r d isomorphisms belong t o G , once each element i s designated by i t s rank (mod AF/H o r AF/H') . These are respectively an automorphism of R / H and an automorphism of R / H ' . The second i s an isomorphism from R/H onto R / H ' by the definition of AF . Now l e t K, K ' be two r-element subsets of E , with r s t r i c t l y le ss t h a n m Suppose f i r s t t h a t there ex i s t s an m-element subset H of E including both K and K ' Take an n-element s e t F including H ; transform K and K ' by the isomorphism of A/H onto AF/H (which belongs t o G and hence i s an automorphism of R/H ) ; we see t h at the isomorphism from A / K onto A / K ' takes R / K into R / K ' If no such m-element s e t H e x i s t s , then we take an m-element s e t H including K and another H ' including K ' , and by the isomorphism from A/H onto A/H' we are in the preceding case. 0
0
.
.
na
.
.
.
.
365
Chapter 12
3.6. Let m be an i n t e g e r , p t h e i n t e g e r 3 m defined by the preceding proposit i o n , and R an m-ary r e l a t i o n of c a r d i n a l i t y s t r i c t l y g r e a t e r than p . A s u f f i c i e n t (and necessary) condition f o r R t o be chainable is t h a t each rest r i c t i o n of R t o p + l elements i s chainable, o r even i s ( s p)-monomorphic (uses t h e u l t r a f i l t e r axiom; ZF s u f f i c e s i f R i s countable). This follows from t h e preceding proposition and from ch.9 § 6.1.
§
4 - REDUCTION
THRESHOLD, REASSEMBLING THRESHOLD, G - C H A I N
4.1. REDUCTION THRESHOLD Given an m-ary group G of permutations, we d e f i n e t h e reduction threshold of G denoted by s(G) , a s t h e l e a s t i n t e g e r s such t h a t the d i l a t e d group Gm+s i s i n d i c a t i v e : see ch.11 5 4.5. In p a r t i c u l a r s(G) = 0 i f f G i s i n d i c a t i v e . Given an i n t e g e r m , we define t h e m-ary reduction t h r e s h o l d , denoted by s(m) , a s being t h e maximum of t h e s(G) f o r a l l m-ary groups G . These d e f i n i t i o n s a r e due t o FRASNAY 1965, who o b t a i n s t h a t s ( 1 ) = s ( 2 ) = s ( 3 ) 2 = 0 , and s ( 4 ) = 2 , and f o r m >/ 5 , t h e i n e q u a l i t i e s 16 s(m) 5 (3m-8) - m + l ( I b i d . p . 493-494). The upper bound is improved t o s(m) 6 m-3 (again f o r m a 5 ) by HODGES, LACHLAN, SHELAH 1977. F i n a l l y i t i s proved by FRASNAY 1984 t h a t s(m) = m-3 f o r m 3 5 . More p r e c i s e l y , t h e value s(G) = in-3 i s reached by taking G t o be t h e group on { 1 , 2 , ,m} which preserves t h e e x t r e m i t i e s 1 and m Indeed t h e f i r s t d i l a t e d group 6"'' preserves 1 , 2 and m , m t l ; then Gm+2 preserves 1 , 2 , 3 and m , m + l , m + 2 and f i n a l l y t h e d i l a t e d group G2m-3 preserves 1,2,...,in-2 and m , m + l , ..., 2m-3 , hence i t i s t h e i d e n t i t y group, which i s obviously i n d i c a t i v e .
.
...
4.2. REASSEMBLING THRESHOLD Given an m-ary group G , we define t h e reassembling threshold of G , denoted by t ( G ) , as t h e l e a s t i n t e g e r t such t h a t n = m + t s a t i s f i e s t h e reassembling theorem 3.4. I f G i s not i n d i c a t i v e , then t(G)4 s(G) + 1 (FRASNAY 1965 p . 500). 2 However, f o r t h e i n d i c a t i v e group J 4 on {1,2,3,4) generated by ( 1 , 4 ) , ( 2 , 3 ) and the two t r a n s p o s i t i o n s ( 1 , 2 ) and ( 3 , 4 ) , we have s = 0 and t = 2 ( I b i d . p . 500). Given an i n t e g e r m , we define t h e w a r y reassembling t h r e s h o l d , denoted by t ( m ) , as being the maximum of t h e t ( G ) f o r a l l m-ary groups G We have t(1) = 0 , t ( 2 ) = t ( 3 ) = 1 , t ( 4 ) = 2 , and f o r m a 5 we have 14 t ( m ) 4 s(m) + 1 (FRASNAY 1965 p . 500 and JULLIEN 1966). Hence t(m)4m-2 f o r m a3 , i n view of the preceding improved upper bound of s(m) Finally, i t is proved by FRASNAY 1984 t h a t t ( m ) = s(m) + 1 = m - 2 f o r m 3 5 , by
.
.
,
THEORY OF RELATIONS
366
t a k i n g a g a i n t h e group on 41,. . . ,m} Problem. F o r each group (G,A)-CHAIN,
4.3.
, do
G
we have t h a t
A
w i t h base
E
, and an m-ary group
i n s p i r e d by CLARK, KRAUSS 1970, we d e f i n e t h e
R
. . ,xm
xl,.
.
s(G),< t ( G )
G-CHAIN
Consider a c h a i n relation
.
1 and m
which preserves
based on
and
E
G
.
F o l l o w i n g FRASNAY 1973,
as b e i n g t h e
(G,A)-*
A-chainable, such t h a t
...,)x,
R(xl,
s
a r e a l l d i s t i n c t and t h e r e e x i s t s a p e r m u t a t i o n
m-ary iff
= t
belonging t o
G
x ~ ( ~ )x <s ( 2 ) < ... < x s(m) modulo A . I n t h e p a r t i c u l a r case where G i s t h e b i n a r y group which reduces t o t h e i d e n t i t y ,
with
t h e n we have a g a i n t h e usual c h a i n is a
G-chain i f f
, more p r e c i s e l y t h e s t r i c t c h a i n < (mod
A
We say t h a t
R
there e x i s t s a chain
(G,A)-chain
( G-rangement i n FRASNAY's t e r m i n o l o g y ) .
A
such t h a t
S t a r t i n g from the chain w
o f the integers, consider the
denote by
m
G
group
i s t h e group o f automorphisms o f any r e s t r i c t i o n o f
Gn
the a r i t y o f
. Then
(G, w ) - c h a i n
we see t h a t , f o r each
, the
n a m
to
R
A).
i s the
R
R
and
dilated n
elements
( o b v i o u s l y we r e p l a c e each element by i t s r a n k , which i s an i n t e g e r ) . Consequently, becomes an i n d i c a t i v e group f o r
Gn
More g e n e r a l l y , c o n s i d e r an n a m R
, denote by G(n)
to
n
n &s(G)
m-ary r e l a t i o n
the
which i s LJ -chainable;
f o r every
n - a r y group o f automorphisms o f any r e s t r i c t i o n o f
elements. Then t h e r e e x i s t s an
~ r o u pG(n)
(reduction threshold). R
n
such t h a t , f r o m t h i s p o i n t on, t h e
i s i n d i c a t i v e (HIGMAN 1977).
Problem, posed by FRASNAY 1984. F o r
n >/ s(m)
, the
group
G(n)
i s i t always
indicative . 4.4. The reassembling t h r e s h o l d l e d FRASNAY 1965 ( p . 517) t o t h e f o l l o w i n g r e s u l t
(l), and t h e n POUZET 1981 (p. 307) t o t h e f o l l o w i n g ( 2 ) and (3). Let
G
be an
m-ary group. There always e x i s t s an
m-ary r e l a t i o n
f r e e l y i n t e r p r e t a b l e i n t h e c h a i n o o f t h e i n t e g e r s , such t h a t o f automorphisms o f t h e r e s t r i c t i o n o f
R
t o an a r b i t r a r y
ordered by i n c r e a s i n g v a l u e s ( f o r i n s t a n c e we can t a k e c h a i n ) . Then d e n o t i n g a g a i n by
(1) t h e maximum b(R) b(R)
\
/ m + l : we can always pass from an n-element s e t X t o another Y by a f i n i t e number of intermediate n-element s e t s Xi , each of which includes the i n t e r s e c t i o n XnY , and each of which has a t l e a s t m common elements with t h e preceding one. Thus f o r these successive Xi , t h e
P(DX) , then y(DX)and y(Dy)have i i n t e r s e c t i o n Xn Y . images
a common r e s t r i c t i o n t o t h e
.
Under t h e s e c o n d i t i o n s , t h e r e e x i s t s a common extension S of these ? ( O x ) Moreover, each r e s t r i c t i o n of S t o a t most n elements i s t h e image of a chain under 9 , hence i s embeddable i n R . Additionally n 3 b(R) , hence S admits no embedding of any bound of R I t follows t h a t S i s i t s e l f embeddable in R . Thus t h e r e e x i s t s a chain D w i t h base E , such t h a t S = P(D). This D i s G-compatible w i t h each DX , thus the reassembling theorem holds. 0
.
Problem. Existence of a " t e r a t o l o g i c a l " w -chainable m-ary r e l a t i o n R whose bounds have maximum c a r d i n a l i t y b ( R ) = m and y e t w i t h t ( G ) = 1 where G denotes t h e group of automorphisms of any r e s t r i c t i o n of R t o an m-element s e t .
4 . 5 . Given t h e i n t e g e r m , the maximum c a r d i n a l i t y of t h e bounds of a l l m - 3 chainable r e l a t i o n s with i n f i n i t e base is equal t o t h e m-ary - reassembling t h r e s hold, in i t s complete form m + t ( m ) (FRASNAY 1973; t h i s r e s u l t i s extended
-
THEORY OF RELATIONS
3 68
by POUZET 1981 t o chainable and almost chainable m u l t i r e l a t i o n s w i t h maximum a r i t y equal t o m ) . In view of the preceding p r o p o s i t i o n , i t s u f f i c e s t o c o n s t r u c t an m-ary r e l a t i o n R f r e e l y i n t e r p r e t a b l e i n w , which admits a t l e a s t one bound of c a r d i n a l i t y m + l . Take the ( G , d ) - c h a i n where G i s t h e i d e n t i t y ; i n o t h e r words, take t h e r e l a t i o n x l < x2 < . . . 4 xm (mod L S ) ) . Then consider t h e binary cycle C w i t h c a r d i n a l i t y m+l ; on t h e base of C , define t h e m-ary r e l a t i o n U which t a k e s t h e value (+) i f f x1,x2, ...,xm a r e consecutive modulo C : t h i s U i s a bound
0
of
R . 0
4 . 6 . I f an m-ary r e l a t i o n R (m+t(m))-monomorphic and i f i t s c a r d i n a l i t y i s i n f i n i t e , o r f i n i t e but s u f f i c i e n t l y l a r g e , then R is chainable (FRASNAY 1965 p. 508 prop. 12.1.1 and POUZET 1981 p . 311 proo. V.3.8). 0 By the f i n i t a r y form of RAMSEY's theorem (ch.3 5 1 . 3 ) , i f t h e c a r d i n a l i t y of the base i s i n f i n i t e o r f i n i t e b u t s u f f i c i e n t l y l a r g e , then t h e r e e x i s t s a r e s t r i c t i o n of R which is chainable and of c a r d i n a l i t y m + t ( m ) . By monomorphism, a l l r e s t r i c t i o n s with c a r d i n a l i t y l e s s than o r equal t o m + t ( m ) a r e chainable.
S t a r t i n g w i t h such a r e s t r i c t i o n , and extending t h e chain i n which i t i s f r e e l y i n t e r p r e t a b l e t o t h e chain w , we obtain an m-ary r e l a t i o n S , which i s f r e e l y i n t e r p r e t a b l e i n G, , and a l l of whose r e s t r i c t i o n s 'to a t most m + t ( m ) elements a r e embeddable in R . We s h a l l show t h a t every f i n i t e r e s t r i c t i o n of R i s embeddable i n S , which w i l l prove t h a t t h e s e r e s t r i c t i o n s a r e chainable, and consequently t h a t R i s i t s e l f chainable. Suppose t h a t every r e s t r i c t i o n of R t o a t most k elements i s embeddable in S , and suppose k >/ m + t ( m ) . I f t h e r e e x i s t s a r e s t r i c t i o n with c a r d i n a l i t y k + l which i s not embeddable i n S , then t h i s r e s t r i c t i o n i s a bound of S , c o n t r a d i c t i n g t h e preceding proposition. 0
§
5 - MONOMORPHISM
THRESHOLDS, C H A I N A B I L I T YTHRESHOLD
5.1. MONOMORPHISM THRESHOLD-PAIR, THRESHOLDS p AND q Given an i n t e g e r m , we define t h e monomorphism threshold-pair a s being an ordered p a i r ( p , q ) of i n t e g e r s such t h a t : (1) every m-ary, (6p)-monomorphic r q l a t i o n w i t h c a r d i n a l i t y a t l e a s t equal t o q i s chainable; ( 2 ) t h i s i s no longer true i f we replace p by p ' < p o r q by q ' < q Consequently we have two d i s t i n c t monomorphism thresholds: the threshold p(m) i s the l e a s t i n t e g e r p f o r which every m-ary p-monomorphic r e l a t i o n w i t h i n f i n i t e o r s u f f i c i e n t l y l a r g e f i n i t e c a r d i n a l i t y i s chainable
.
( r e c a l l t h a t , f o r a s u f f i c i e n t l y l a r g e cardinal of t h e base,
p-monomorphism
369
Chapter 12
( 4 p)-monomorphism by ch.9 5 6.3);
implies
the threshold
q(m)
i s the least integer
r e l a t i o n o f cardinal & q
q
f o r which e v e r y
m-ary monomorphic
i s chainable.
The e x i s t e n c e o f these t h r e s h o l d s f o l l o w s f r o m 3.5 above (see FRASNAY 1965 p. 513) By t h e p r e c e d i n g 4.6, we have
p(m)
\
/ 3
\