Orders: Description and Roles

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ORDERS: DESCRIPTION AND ROLES

z

1982

0 W

c

z

W

pe W LL

z

0

c,

LYON

annals of Ccnerd Editor Peter L. H A M M E R , Rutgers University, New Brunswick, NJ, U.S.A. Advisory E ditors C. B E R G E , Universitk de Paris M. A . H A R R I S O N , University of California, Berkeley, C A , U.S.A. V. KLEE, Universityofwashington, Seattle,WA. U.S.A. J . H . VAN LINT, California Institute ofTechnology, Pasadena, C A . U.S.A. G.-C. ROTA, Massachusetts Institute ofTechnology, Cambridge, M A , U.S.A.

NORTH-HOLLAND -AMSTERDAM 0 NEW YORK

OXFORD

NORTH-HOLIAND MATHEMATICS STUDIES

99

Annals of Discrete Mathematics(23) General Editor: Peter L. Hammer Rutgers University, New Brunswick, NJ, U.S.A.

ORDERS: DESCRIPTION and ROLES in SetTheory, Lattices, Ordered Groups,Topology, Theory of Models and Relations, Combinatorics, Effectiveness, Social Sciences. Proceedings of the Conference o n Ordered Sets and theirApplication\ Chliteau dc IaTourcttc, I’Arbrcsle,JulyS-l I , 1982

ORDRES: DESCRIPTION et ROLES enTheorie des Ensembles, deslieillis, des Groupes Ordonnks: enTopologie,ThCorie des Modeles et des Relations, Cornbinatoire, Effectivite, Sciences Sociales. Actes de la Conference sur ies Enseniblcs Ordonnds et leur Applications Ch~teatidelaTourette. I’Arhresle. juillet S-11. 19x2

edited by

Maurice POUZET and

Denis RICHARD Laboratoire dMlgebre Ordinale Departement de Mathematiques Universite Claude Bernard Lyon I France

1984

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD

ISBN: 0 444 87601 4

Pitblislirr:

ELSEVIER SCXENCE PUBLISHERS R.V. P.O. BOX I9YI 1000 BZ AMSTERDAM ‘THE NETHERLANDS .Solct/i.sir.ihiriot.~fi~r ilic

U.S.A . rtritl C(tti(r(l(t:

ELSEVIER SCIENCE PUBLISHING COMPANY. INC 52 VAN I) E RB I LT AVENUE NEW YORK. N.Y. 10017 U.S.A.

Lattice adapted from Figure 16 of Cherlin and Rosenstein, Ho-categorical groups, J. Algebra 53 (1978), 188-226 .

Library of Congress Cataloglng In Publlcatlon Data

Conference on Ordered Sets and Their Applications (4th : 1982 : L’Arbresle, France) Orders--description and roles. (Annals of discrete mathematics ; 23) (North-Holland mathematics studies ; 99) 1. Ordered sets--Congresses. I. Pouzet, M. 11. Richard, Denis, 1942111. Title. IV. Title: Ordree--description et r8les. v. Series. VI. Series: North-Holland mathematics studies ; 99. QA171.48.C66 1982 511.3’2 84-13749 ISBN 0-444-87601-4

.

PRINTED IN THE NETHERLANDS

DCdie au Professeur COROMINAS

This Page Intentionally Left Blank

vii

PREFACE

The 27 papers in this volume survey various aspects of the theory of order. These have been grouped into nine sections illustrating some of the main mathematical themes and applications in the theory. These papers were written for the “Conference on ordered sets and their applications” (I’ARBRESLE, july 1982). This international meeting, the fourth devoted to the theory of order (following BANFF, 1981; MONTEREY , 1959, CHARLOTTESVILLE, 1938) - will shortly be succeeded by two others (BANFF and LUMINY, 1984). This continuing activity, and other signs such as the appearance of the new journal ORDER, suggests that there is an increasing recognition of the importance of order and an acceleration in the development of its theory. All this calls for some interpretation of the role of order in the general landscape of mathematics. Editing the present work has also led us to consider this question, and we offer here some of our thoughts on this matter to the reader. If we imagine the theory of order as a river, we discover many contributory sources. The principal source is undoubtedly the ordinal arithmetic of G. CANTOR, but other important ideas come from the work on real analysis by G. CANTOR and R. DEDEKIND, the contributions to the theory of equations and groups by E. GALOIS and C. JORDAN (solvable groups, Jordan-Holder Series) and in the work on the theory of rings by E. NOETHER, to mention but a few important examples. From these diverse origins a number of modern current have developed such as lattice theory, boolean algebra, topology, etc. The theory of lattices which began with studying the subgroups of a group, has achieved several significant results that are now considered classical (e.g. the representation theorems of BIRKHOFF and STONE) and remains, after several decades, one of the principal preoccupations of mathematicians working in order theory. In a similar way set theory, even at the most primitive axiomatic level (the axioms of ZERMELO, KURATOWSKI, ZORN, ...), is intimately concerned with questions of order. The neighbouring disciplines of descriptive set theory, topology, measure theory, model theory, logic and combinatorics have all contributed to and enlarged the theory of order. For example, problems in model theory, from categoricity to stability, are already realised in the structure of chains, and conversely the models constructed by A. EHRENFEUCHT and A. MOSTOWSKI allows one to represent part of the complexity of chains in arbitrary structures. Even a description of the models of Peano arithmetic involves ordinal concepts (final or cofinal extensions, initial segments, the indicator functions of KIRBY & PARIS), and these sometimes completely determine the structure, for example the saturation of a model reduces to that of its order structure. It is quite possible that the real nature of the logical and combinatorial content of the independence results of J. PARIS and L. HARRINGTON may have an order-theoretic basis. Having crossed the paradise of the infinite, with its inaccessible summits (the problems of consistency) and fertile valleys (lattice theory, order groups, boolean algebra, noetherian rings, pointrset topology, problems concerning duality, representation and generation in Universal Algebra), the course of the river leads back to the realm of the finite. A frequent connecting link between these territories is compactness (e.g. from the finite to the infinite version of DILWORTH’s theorem and from the infinite to the finite form of

viii

Preface

RAMSEY’s theorems). Here, however, the soil is more difficult to cultivate. Due to the rigidity and boundedness of the objects there are very few general techniques available (consider, for example, the RAMSEY numbers or the many famous unsettled conjectures of number theory). However, it is here that we find the extensive theory of finite graphs which is very rich in applications (flows in networks, optimization, etc.). Today, the external world, with its social problems, technological advances and new sciences influences the course of our river, which the mathematician might naively have thought was simply there to be discovered, He is now required to add to his role of explorer that of engineer; he must help forge new tools. In so doing he has found himself in mathematical domains that he might not otherwise have considered. This is the case, for example, in the social sciences: the CONDORCET paradox is the beginning of the theory of social choice where one spectacular result is the theorem of ARROW. The rapid advance of computer science has led mathematicians to reconsider problems with a view to effectiveness. Recursion theory (K. GODEL, A. CHURCH) and complexity (M. RABIN) in dealing with these new problems from computer science (e.g. sorting) have extended into the new domains of algorithmic complexity (time and space), automata theory and formal languages. This dual activity as builder - as was J. Von NEUMANN - and discovered - W. SIERPINSKI referred to himself as “Explorateur de I’infini” - continues. But a panoramic view of the theory of order is still missing. A first survey (ORDERED SETS, D. REIDEL, 1982) was edited by I. RIVAL. In this volume we illustrate the appearance and the role played by order in set theory, lattice theory, topology, logic (model theory, theory of relations, Peano arithmetic), ordered groups, combinatorics, computer science and the social sciences. The editors intented that the reader of this book should pass from the infinite to the finite, from the descriptive view to the applications. Apart from the survey articles appearing in this volume, DISCRETE MATHEMATICS will separately publish the research papers presented a t the la TOURETTE conference which give the most recent advances in the theory of order. The texts of all these articles are written either in French or English together with an abstract o r introduction in the other language. This preface would not be complete without an expression of thanks to the many referees for their excellent criticisms and suggestions. Among these were some of the present authors and also the following colleagues: MM. A. ACHACHE ; R. ASSOUS ; J.P. AURAY ; B. BANASCHEWSKI ; H.J. BANDELT; P. CEGIELSKI ; Ch. CHARRETTON ; M. CHEIN ; R. DEAN ; J.P. DOIGNON ; P. DWINGER ; M.R. GAREY ; S. GRIGORIEFF ; L. HARPER ; S.S. HOLLAND ; J. ISBELL ; M. JAMBU-GIRAUDET ; H. KOTLARSKI ; D. LASCAR ; A. LASCOUX ; L. LESIEUR ; R. MAYET ; E.W. MAYR ; C. S t J. A. NASH- WILLIAMS ; E. NELSON ; G. Mc NULTY ; M.PARIGOT ; G. de B. ROBINSON ; L.SHEPP ; B.SIMONS; A. R. STRALKA ; L. SZABO ; T. TROTTER ; D. W. WEST ; S. WOLFENSTEIN, M. YASUHARA . Finally, a word of thanks to J.G. ROSENSTEIN for the motif appearing o n the cover. This is a reproduction of the needlepoint work he completed during the Conference.

Maurice POUZET and Denis RICHARD

PREFACE

Vingt sept textes Qcrits dans une perspective de synthdse constituent ce volume sur la thkorie de I’ordre. 11s sont regroupbs en neuf parties choisies parmi de grands thdmes mathbmatiques concernant les ordres ou le rble qu’ils y jouent. 11s ont ktk blaborbs i I’occasion de la “Confbrence sur les ensembles ordonnbs et leurs applications” (I’ARBRESLE, juillet 1982). Cette rencontre internationale - quatridme des grands congrds consacrks aux ensembles ordonnbs (i la suite de ceux de BANFF, 1981 ; MONTEREY, 1959 ; CHARLOTTESVILLE, 1938) - ainsi que les deux prochaines aujourd’hui annonckes (BANFF e t LUMINY, 1984) et d’autres faits - comme la parution de la revue ORDER tout semble indiquer I’importance croissante de la thkorie des ordres e t I’accelkration de son dkveloppement. Cette constatation appelle une interprktation du cours de la thkorie des ordres dans le paysage mathbmatique. L’bdition du prksent ouvrage nous conduisait aussi i une telle rbflexion; nous en soumettons quelques klbments au lecteur. Si I’on veut bien imaginer la thkorie des ordres comme un fleuve, on lui trouve de nombreuses sources. La thkorie des ordres vient en effet de I’arithmktique ordinale de G. CANTOR mais aussi des travaux en analyse rbelle de G. CANTOR e t R. DEDEKIND, e t encore de la thborie des kquations et des groupes avec E. GALOIS e t C. JORDAN (groupes rksolubles, suites de JORDAN - HOLDER) e t encore de la thborie des anneaux (E. NOETHER), ceci pour ne citer que quelques exemples importants. De ces diverses origines naissent plusieurs courants tels la thkorie des treillis, les alg&bresde Boole, la topologie, etc..., drainant les eaux vers le fleuve en formation. Issus de I’btude de I’ensemble des sousgroupes d’un groupe, la thborie des treillis, courant fkcond en rbsultats aujourd’hui classiques (e.g. les thkordmes de reprksentation de BIRKHOFF et STONE), est ainsi, depuis plusieurs dkcennies, une des pn5occupations principales des mathbmaticiens travaillant sur I’ordre. De mSme, e t ne serait-ce que dds I’abord axiomatique (Axiomes de ZERMELO, KURATOWSKI, ZORN, ...), les courants impbtueux de la thborie des ensembles ne pouvaient kviter d’affluer et de se mdler aux questions ordinales. Venus de contrees voisines, les apports de la thkorie descriptive des ensembles, de la topologie, de la thkorie de la mesure, de la thkorie des moddles, de la logique e t de la combinatoire grossissent la th6orie des ordres de leur flux: ainsi la problbmatique de la theorie des moddles - de la catkgoricitk i la stabilitk - se trouve dbji inscrite dans I’ktude des chahes; mais inversement, les moddles construits par A. EHRENFEUCHT et A. MOSTOWSKI permettent de reprbsenter partie de la complexitk des chaines dans des structures arbitraires; mSme en arithmktique, la description des moddles de Peano fait place aux notions ordinales (extensions finales ou cofinales, sections initiales e t indicatrices de KIRBY - PARIS), qui parfois gouvement compldtement les moddles puisque, par exemple, la saturation des moddles de I’arithmktique du premier ordre se reduit i celle de leur structure d’ordre (on peut mSme penser que le contenu logique et combinatoire des rbsultats d’indkpendance de J. PARIS et L. HARRINGTON est de nature ordinale). Ayant travers6 le paradis infinitiste oc se trouvent i la fois des sommets inaccessibles (e.g. tous les probldmes se ramenant a la consistance) e t des vallkes fecondes (les grandes structures: treillis, groupes ordonnks, algdbres de Boole, algdbre noethbrienne, topologie

Preface

X

ensembliste ... et ce qui se rattache i I’alggbre universelle avec la dualitk, les questions de roprbsentation et d’engendrement), le cows du fleuve se porte maintenant vers le territoire de la finitude. La liaison est parfois facilitbe par le canal de la compacitk (le passage de la version finitiste du thkoreme - si fondamental - de DILWORTH i sa version infinitiste et le passage de la version infinitiste du theorgme de RAMSEY i sa version finitiste). Cependant, ces terres sont, souvent plus difficiles i cultiver puisque I’on y substitue i la souplesse des concepts de I’infini e t i des outils bien blaborks, le caract6re born6 d’objets pour lesquels peu de mbthodes d’btude existent encore (que I’on pense aux nombres de RAMSEY), ou la rigiditk des nombres (dont les conjectures les plus cbl6bres de I’arithmbtique donnent une idbe). C’est dans ce domdne que I’on trouve I’immense thborie des graphes finis si riche d’applications (problemes de cheminement, rbseaux de transport, ...). A ce jour, le monde extkrieur par le biais des besoins sociaux, ou des avancbes technologiques, ou des sciences nouvelles, inflbchit le cours d’une rivigre dont le mathbmaticien pouvait narvement penser qu’elle n’btait la que pour qu’il la dbcouvre. On lui demande d’ajouter a son rble d’explorateur celui d’ingbnieur. I1 doit aider crber des outils; ce faisant, il se trouve, ou se retrouve, dans des domaines mathbmatiques qu’il n’imaginait pas forcbment. C’est le cas dans les sciences sociales oh la formalisation des situations proposbes conduit i des rbsultats e t des problemes d’ordre: le paradoxe de CONDORCET est l’origine de la modblisation des prbfbrences dont un rbsultat spectaculaire est le thboreme de ARROW, hi-m6me point de depart de toute une thborie. La dbcouverte du continent informatique a m h e le mathbmaticien constater le manque d’effectivitk de rbsultats qui lui semblent naturels ou simples. Le courant logique de la recursivitk (K. GODEL, A. CHURCH) et de la complexitk (M. RABIN) rejoint les problemes nouveaux poses par I’informatique au niveau le plus immbdiat (problgmes de tri, par exemple), et se prolonge dans les nouveaux domaines de I’algorithmique, de la complexitk concrete (temps et espace de calcul), de la thborie des machines et de celle des langages.

a

a

Cette double activitk de bitisseur - au sens ou le fut J. Von NEUMANN - et de dbcouvreur - W. SIERPINSKI ne se disaibil pas lui-mbme “Explorateur de I’infini” ? - se poursuit. Mais il manque encoreun panorama complet de la thborie des ordres. Une premiere synthese (ORDERED SETS, D. REIDEL, 1982) a btk bditke par I. RIVAL. Nous prbsentons ici une illustration de la prbsence e t du rble de I’ordre en thborie des ensembles, en thborie des treillis, en topologie, en logique (thborie des moddles, thborie des relations, arithmktique), dans les groupes ordonnbs, en combinatoire, en informatique thborique e t dans les sciences sociales. Les Qditeurs ont voulu, qu’au fur e t a mesure de son parcours, le lecteur aille de situations infinitistes i des situations de plus en plus finitistes, e t qu’il passe de la mdme faqon du point de vue descriptif des ensembles ordonnbs leurs utilisations. Les numbros spbciaux de la revue DISCRETE MATHEMATICS contiendront les articles de recherche des confbrenciers rbunis 6, la TOURETTE faisant &at de rbsultats les plus rbcents sur les ensembles ordonnbs et prolongeant les syntheses de I’ouvrage prbsentk ici. Tous ces textes issus de la Confbrence sont bcrits soit en franqais, soit en anglais, chacun rbdigb dans une langue b t a n t prbcbdb d’une introduction ou d’un rbsumk exprimb dans l’autre.

a

Cette preface serait incomplhte si nous ne remerciions pas tous les arbitres qui ont bien voulu nous aider, pour I’excellence de leurs critiques e t de leurs suggestions. Parmi ces arbitres, se trouvent certains des auteurs de ce livre, qui se reconnaftront ici, e t nos collegues Qtrangers ou franqais: MM. A. ACHACHE ; R. ASSOUS ; J.P. AURAY ; B. BANASCHEWSKI ; H.J. BANDELT ; P. CEGIELSKI ; Ch. CHARRETTON ; M. CHEIN ; R. DEAN ; J.P. DOIGNON ; P. DWINGER ; M.R. GAREY ; S. GRIGORIEFF ; L. HARPER ; S.S. HOLLAND ; J. ISBELL ; M. JAMBU-GIRAUDET ; H. KOTLARSKI ; D. LASCAR ; A. LASCOUX ; L. LESIEUR ; R. MAYET ; E.W. MAYR ; C. St J. A. NASH-WILIAMS ; E. NELSON ;

Preface

xi

G. Mc NULTY; M. PARIGOT ; G. de B. ROBINSON ; L. SHEPP ; B. SIMONS ; A.R. STRALKA; L. SZABO ; T. TROTTER ; D.W. WEST ; S. WOLFENSTEIN ; M. YASUHARA . Grand merci enfin a J. G. ROSENSTEIN pour le motif figurant sur la couverture, reproduisant la tapisserie qu’il a brod6e t o u t en kcoutant les confbrences.

Maurice POUZET

et

Denis RICHARD

xiii

TABLE OF CONTENTS

Preface Preface Concerning the Conference on Ordered Sets and their Applications Retour a la Conference sur les Ensembles Ordonnes et leurs Applications * List of Participants (Liste de Participants) * Conference Programme (Programme Scientifique) * Short Communications (Programme des Communications)

SOMMAIRE

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ix xv xvi xvii xxiv xxvii

PART I - ORDER AND SOME SET-THEORETICAL INVARIANTS Partie I - Ordre et Invariants Ensemblistes Eric C. MILNER, Recent results on the cofinality of ordered sets . . . . . . . . . . . . . . . . . . 1 J. Donald MONK, Cardinal functions on boolean algebras ........................ 9 PART I1 - ORDER AND LATTICES Partie II - Ordre et Treillis H.A. PRIESTLEY, Ordered sets and duality for distributive lattices . . . . . . . . . . . . . . . 39 Michael MISLOVE, When are order scattered and topologically scattered thesame? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Dwight DUFFUS, Maurice POUZET, Representing ordered sets by chains . . . . . . . . . . 81 Gunter BRUNS, Orthomodular lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 George GRATZER, David KELLY, The construction of some free m-lattices onposets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 3 I.G. ROSENBERG, D. SCHWEIGERT, Compatible orderings and tolerances oflattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

PART I11 - ORDER AND ORDERED GROUPS Partie I l l - Ordre et Groupes Ordonnes W. Charles HOLLAND, Classification of lattice ordered groups . . . . . . . . . . . . . . . . . . 151 A.M.W. GLASS, The isomorphism problem and undecidable properties 157 for finitely presented lattice-ordered groups ...........................

Table of Contents

x iv

PART IV - ORDER AND TOPOLOGY Partie IV - Ordre et Topologie Egbert HARZHEIM, On topological properties of Cartesian products of linearly ordered continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171

Karl Heinrich HOFMANN, Order aspects of the essential hull of a topological To -space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193

PART V - ORDER AND THEORY O F MODELS AND RELATIONS Partie V - Ordre, Thkorie des Modeles et Thkorie des Relations 207 Wilfrid HODGES, Models built on linear orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . Claude FRASNAY, Relations enchahables, rangements et pseudo-rangements . . . . . . 235 James H. SCHMERL, H, -categorical partially ordered set ...................... Denis RICHARD, The arithmetics as theories of two orders ....................

269 287

Roland FRAISSE, L’intervalle en thkorie des relations; ses gknkralisations; filtre intervallaire et cl6ture d’une relation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

313

PART VI - ORDER AND CHAIN CONDITION ON CLASSES O F STRUCTURES Partie VI - Ordre et Conditions de Chaine sur des Classes d e Structures

P.D.SEYMOUR, Neil ROBERTSON, Some new results on the well-quasi orderingofgraphs

...............................................

343

PART VII - ORDER AND COMBINATORICS Partie VII - Ordre et Combinatoire Ivan RIVAL, Linear extensions of finite ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Michel HABIB, Comparability invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Andrd BOUCHET, Codages et dimensions de relations binaries . . . . . . . . . . . . . . . . . . .387 Jerrold R. GRIGGS, The Sperner property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Gkrard VIENNOT, Chain and antichain families, grids and young tableaux . . . . . . . . . 409 PART VIII - ORDER AND EFFECTIVENESS Partie V I I I - Ordre e t Effectivite Joseph G. ROSENSTEIN, Recursive linear orderings . . . . . . . . . . . . . . . . . . . . . . . . . . Ir6ne GUESSARIAN, Maurice NIVAT, About ordered sets in algebraic semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J.K. LENSTRA, A.H.G. RINNOOY KAN, Two open problems in precedence constrained scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

465 477 509

PART IX - ORDER AND SOCIAL SCIENCES Partie I X - Ordre et Sciences Sociales Jean-Pierre BARTHELEMY, Bruno LECLERC, Bernard MONJARDET, Ensembles ordonnks et taxonomie mathkmatique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

523

xv

CONCERNING THE CONFERENCE ON ORDERED SETS AND THEIR APPLICATIONS This Conference, held under the auspices of the Centre National de la Recherche Scientifique (C.N.R.S.) and the Mathematical Society of France (S.M.F.), was organized by the Ordinal Algebra group a t the UniversiG Claude Bernard (LYON 1) and the group of French mathematicians comprising R.C.P. 6 9 8 of the C.N.R.S. in cooperation with I’Ecole Nationale Supkrieure des T616communications*. The Conference proceedings were dedicated to Professor E. COROMINAS founder of the Ordinal Algebra group - to mark the occasion of his election to Professor Emeritus.

-

The Conference brought together 125 participants from more than fifteen different countries. The meeting began with a lecture from Professor I. RIVAL - organizer of the 1981 BANFF meeting - and concluded with a memorable talk from Professor P. ERDOS. More than 9 0 papers were given at the conference including 34 invited addresses, 15 presentations a t the special sessions, 20 contributed papers and 20 problems given a t the Problem Sessions (Cf. The Scientific Programme). We express our thanks to the resident Dominicans for their warm welcome to the large group of mathematicians who for seven days invaded their serene setting a t the Chiteau de la Tourette in the hills above I’ARBRESLE near LYON. During the rare non-mathematical moments we were able to see GANCE’s magnificent production of Napolkon, and also to enjoy a fine concert by R. JAMISONWALDNER (cello) and G. BRUNS (piano). The Scientific programme and organization of the meeting was coordinated by M. POUZET with collaboration from R. BONNET and other members of the group, A. ACHACHE, R. ASSOUS, Ch. CHARRETTON, D. RICHARD, and the research students M. BEKKALI, M. BELHASSAN, D. MISANE, N. ZAGUIA, E. JAWHARI. Our colleagues from the logic group also gave much time and assistance - especially Marianne DELORME, whose efficiency and courtesy was widely appreciated. We d o not forget either the generous help from two other POUZET generations - Emile and Marc. Robert BONNET and Maurice POUZET

* We thank also the non-mathematical institutions that contributed to the success of the Conference especially the City of LYON and the Banque Nationale de Paris (B.N.P.).

xvi

RETOUR A LA CONFERENCE SUR LES ENSEMBLES ORDONNES ET LEURS APPLICATIONS Cette Confbrence, placbe sous les auspices du Centre National de la Recherche Scientifique (C.N.R.S.) et de la Socibtb Mathbmatique de France (S.M.F.), a Qtk organisbe par le laboratoire d’Alg8bre Ordinale de I’Universitk Claude Bernard (LYON 1)et le groupe de mathbmaticiens franqais constituant la R.C.P. 698 du C.N.R.S. avec le concours de 1’Ecole Nationale Supbrieure des Tblbcommunications*. Elle btait dbdibe au Professeur E. COROMINAS d’AlgGbre Ordinale - I’occasion de son Embritat.

- fondateur du laboratoire

Cette Confbrence rbunissait 125 participants d’une quinzaine de nationalitks. Ouverte par le Professeur I. RIVAL - organisateur du Symposium de BANFF en 1981 - elle s’est conclue par I’exposb du Professeur P. ERDOS, apr8s avoir donnb lieu a plus de 90 contributions comprenant 34 expods en sbance plbni6re, 15 exp o d s dans des sessions thbmatiques, plus de 20 contributions aux sessions de problGmes et 20 communications (Cf. Programme Scientifique). Dans le cadre du Chiteau de la Tourette, sur les hauteurs de la petite ville de I’ARBRESLE pr6s de LYON, les Dominicains ont accueilli chaleureusement les mathbmaticiens, sept jours durant, et nous les en remercions. Pendant les rares moments non mathbmatiques, certains ont eu I’occasion de dbcouvrir le Napolbon d’Abel GANCE e t tous se souviendront longtemps du concert donnb par R. JAMISON WALDNER (violoncelle) accompagnb par G. BRUNS (piano). L’organisation scientifique et rnat&rielle de la Confbrence btait coordonnbe par M. POUZET avec la participation de R. BONNET et de membres du laboratoire: A. ACHACHE, R. ASSOUS, Ch. CHARRETTON, D. RICHARD, des Qtudiants de recherche: M. BEKKALI, M. BELHASSAN, D. MISANE, N. ZAGUIA, E. JAWHARI. Nos coll6gues logiciens nous ont apportk leur aide - particuli8rement Marianne DELORME, dont tous ont apprbcib I’efficacitk e t la courtoisie mais aussi deux autres gknerations de POUZET puisqu’il faut citer a la fois Emile et Marc. Robert BONNET et Maurice POUZET

*

Nous remercions les Institutions non mathematiques qui ont apporte leur concours a cette Conference, en particulier la ville de LYON et la Banque Nationale de Paris (B.N.P.).

xvii

LISTE DES PARTICIPANTS

LIST OF PARTICIPANTS

A c h i l l e ACHACHE

DBpt. d e Math. - U n i v e r s i t C Claude B e r n a r d - 4 3 , Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e LYON 1

Roland ASSOUS

Dept. de Math. - U n i v e r s i t e C l a u d e B e r n a r d LYON 1 - 4 3 , Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

J e a n - P i e r r e BARTHELEMY

Dept. d ' I n f o r m a t i q u e - E.N.S. Teli.communic a t i o n s - 46, r u e B a r r a u l t 75634 PARIS CEDEX 13 - F r a n c e

H.

F.B. Math. T e c h n i s c h e Hoschule D a r m s t a t D 6100 DARMSTAT - R.F.A.

BAUER

Mary K a t h e r i n e BENNETT

Dept. o f Math. U n i v e r s i t y of M a s s a c h u s s e t t s AMHERST - MA 01003 - U.S.A.

Mohamed BEKKALI

Dept. d e Math. - U n i v e r s i t e Claude B e r n a r d LYON 1 - 4 3 , Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

Moulay BEL HASSAN

DBpt. de Math. - U n i v e r s i t e C l a u d e B e r n a r d LYON 1 - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

Claude BENZAKEN

I.M.A.G.

B.P. N i c o l e BLANCHARD

R o b e r t BONNET

53 X

-

U n i v e r s i t e de Grenoble - 38041 GRENOBLE CEDEX

-

France

Dept. de Math. - U n i v e r s i t e C l a u d e B e r n a r d LYON 1 - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e DBpt. d e Math. - U n i v e r s i t e C l a u d e B e r n a r d LYON 1 - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e 35

KASSEL

-

R.F.A.

B. BOSBACH

Vogelviesen,

Andre BOUCHET

1 3 , r u e Taragone -

O d i l e BOTTA

DQpt. d e Math. - U n i v e r s i t E ! Claude B e r n a r d LYON 1 - 4 3 , Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

72000

LE MANS

-

France

xviii

Gunter BRUNS

Jean-Claude CARREGA

List of participants Dept. of Mathematical S c i e n c e s Mc Master U n i v e r s i t y - HAMILTON, O n t a r i o L8S4K1 - Canada D e p t . d e Math. - U n i v e r s i t 6 Claude B e r n a r d - 4 3 , B l d du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

-

LYON 1

J o s e p h CHACRON

52, r u e C o z e t t e -

Maurice CHACRON

Dept. of Math. - C a r l e t o n U n i v e r s i t y OTTAWA, O n t a r i o K 1 5 5 ~ 6- Canada

C h r i s t i n e CHARRETTON

80000

AMIENS

-

France

DBpt. de Math. - U n i v e r s i t 6 Claude Bernard - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e LYON 1

Georges CHEVALIER

D6pt. de Math. - U n i v e r s i t g Claude B e r n a r d - 43, Bld d u 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e LYON 1

O l i v i e r COGIS

Dept. de Math e t I n f o r m a t i q u e - U n i v e r s i t 6 du Lanquedoc, P l a c e Eugene H a t a i l l o n 34060 MONTPELLIER CEDEX - F r a n c e

J u l i e n CONSTANTIN

688, r u e Vimy N . SHERBROOKE Quebec JlJ3N6 - Canada

Maria CONTESSA

Dept. de Math. - Italie

-

U n i v e r s i t e de Rome

ROME

E r n e s t COROMINAS

J e a n Louis COULON

Dept. de Math. - U n i v e r s i t e C l a u d e Bernard LYON 1 - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e Dept. de Math. - U n i v e r s i t 6 Claude B e r n a r d - 4 3 , B l d du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e LYON 1

R i c h a r d DEAN

Marianne DELORME

Dept. of Math. - CALTECH PASADENA, CA 91103 - U.S.A. DBpt. de Math. - U n i v e r s i t k C l a u d e B e r n a r d - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

LYON 1

-

Walter DEUBER

F a k u l t d t f u r Math. - U n i v e r s i t a t B i e l e f e l d D 4800 BIELEFELD - R.F.A.

Markus DICHTL

A b t e i l u n g Math. - 3 U n i v e r s i t a t Ulm 7900 ULM - R.F.A.

J e a n - P i e r r e DION

DPpt. d e Math. - U . Q . A . M . C a s e P o s t a l e 8888, SUC “ A “ - MONTREAL - P,Q. H3C3P8 Canada

Gernot DORN

-

F.B.

Math. T e c h n i s c h e Hochschule Darmstadt - R.F.A.

D 6100 DARMSTADT

Manfred DROSTE

Im Brinkmannsfeld 6 4 , D 4250 BOTTROP - R.F.A.

List ofparticipants

xix

P a u l DUBREIL

RPsidence F r k m i e t , Les Reaux 91840 SOISY-SUR-ECOLE - F r a n c e

Andre DUCAMP

14 A Dreve d e s E q u i p a g e s 1170 BRUXELLES - B e l g i q u e

Dwight DUFFUS

Dept. o f Math. and Computer S c i e n c e EMORY Univ. ATLANTA G e o r g i a 30322 - U . S . A .

I v o DUNTSCH

Inst.

f u r Math I1 Koningin-Luise s t r . 2426 33 - R.F.A.

-

1000 B E R L I N C h r i s t o p h e r EDWARDS

Dept. of E n g i n e e r i n g P r o d u c t i o n . U n i v e r s i t y o f Birmingham, PO BOX BIRMINGHAM B15 2TT Grande B r e t a g n e

P a u l ERDOS

Hungarian Academy of S c i e n c e . 1053 BUDAPEST - Hongrie

Marcel ERNE

I n s t . f u r Math. U n i v e r s i t a t Hannover W e l f e n g a r t e n l D 3000 HANNOVER - R.F.A.

Michel EYTAN

Dept. de Math. 12, r u e C u j a s -

Claude FLAMENT

L e s Blaques

Marc FORT

D6pt. de Math. - U n i v e r s i t e C l a u d e B e r n a r d LYON 1 - 43, Bld du 1 1 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

Roland FRAISSE

C a d e n e l l e - Cheverny, 122, r u e Cdt R o l l a n d 13008 MARSEILLE - F r a n c e

Claude FRASNAY

19, r u e d e l a p o s t e

J e r r o l d GRIGGS

Dept. of Mathematics and S t a t i s t i c s U n i v e r s i t y of South C a r o l i n a COLUMBIA SC 29208 - U . S . A .

-

Arnold G R U D I N

Dept. o f Math. - Denison U n i v e r s i t y GRANVILLE, Ohio 43020 - U.S.A.

-

I r h e GUESSARIAN

DBpt. de Math. - U n i v e r s i t e d e PARIS V I I T 45-55 ; 2, P l a c e J u s s i e u 75251 PARIS CEDEX 05 France

-

-

Realtanoda u .

U n i v e r s i t e Rene D e s c a r t e s 75005 PARIS - F r a n c e

04110 CERESTE -

31130

-

13-15

-

France

BALMA

- France

-

-

Marcel GUILLAUME

Math6matiques P u r e s - U n i v e r s i t e d e CLERMONT 2 Complexe S c i e n t i f i q u e d e s Cezeaux - B.P. 45 63170 AUBIERE - F r a n c e

C y p r i e n GNANVO

Departement de Math. - U n i v e r s i t e de COTONOU COTONOU - Rep. P o p u l a i r e du Benin

Michel HABIB

E c o l e d e s Mines de S a i n t E t i e n n e 158, Cours F a u r i e l 42023 SAINT ETIENNE CEDEX - F r a n c e

-

-

xx

List of participants

Geza HAHN

Dept. of Math. Quebec Canada

Mahmud HAIFAWI

Dept. of Math. ANKARA sity

Andras HAJNAL

Math. I n s t . Univ. 1088 Muzeum K r t . 6-8

Mc G i l l ,

-

-

MONTREAL

H3A2K6

- Middle E a s t T e c h n i c a l Univer- Turquie BUDAPEST

-

Honqrie

Rudolf H A L I N

Mathematisches Seminar UniversitatHamburg B u n d e s s t r . 55 D 2000 HAMBURG - R.F.A.

Georges HANSOUL

517 G r a n d ' r o u t e

E g b e r t HARZHEIM

P a l l e n b e r g s t r . 23,

R i c h a r d HENDERIKS

Econometrisch I n s t i t u t . Erasmus U n i v e r s i t e i t PO BOX 1738 - 3000 DR ROTTERDAM - Pays-Bas

W i l f r i d HODGES

Bedford C o l l e g e R e g e n t ' s Park. LONDON NWI 4NS- Grande B r e t a g n e

K a r l HOFMANN

B 4110

FLEMALLE

5 KOLN 60 -

-

Belgique

R.F.A.

Dept. of Math. Tulane U n i v e r s i t y La. 70118 - U.S.A.

NEW ORLEANS

C h a r l e s HOLLAND

Dept. of Math.

-

BOWLING GREEN Ohio

Bowling Green S t a t e U n i v e r s i t y 43403 - U.S.A.

L u i s a ITURRIOZ

DBpt. de Math. - U n i v e r s i t i . Claude B e r n a r d LYON 1 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

Michele JAMBU-GIRAUDET

32, r u e de l a Reunion

Robert JAMISON-WALDNER

Math S c i e n c e s Dept. - Clemson U n i v e r s i t y CLEMSON SC 29631 - U.S.A.

E l Mostapha JAWHARI

Dept. of Math. U n i v e r s i t e Claude B e r n a r d LYON 1 4 3 , Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

Gudrun KALMBACH

A b t . Math. 3 , OE, Univ. U l m D 7900 ULM - R.F.A.

Klaus KEIMEL

F a c h b e r e i c h Mathematik, T e c h n i s c h e Hochschule R.F.A. D a r m s t a d t , 6100 DARMSTADT

David KELLY

-

-

-

75020 PARIS

-

-

France

-

-

-

Dept. of Math.

- U n i v e r s i t y of Manitoba R3T 2 N 2 Canada

W I N N I P E G , Manitoba

Sabine KOPPELBERG

Mathematisches I n s t i t u t K o n i g i n - L u i s e - S t r . 24/26 1000

Henryk KOTLARSKI

U n i w e r s y t e t u Magellana 4m 19 02 777 VARSOVIE - Pologne

Germain KREWERAS

4 0 , r u e LacGpede

-

75005

BERLIN 33

PARIS

- France

- RFA

List ofparticipants

xxi

M a r t i n KRUSKAL

Program i n A p p l i e d Mathematics. P r i n c e t o n U n i v e r s i t - 7 - PRINCETON, N . J . 08544 - U.S.A.

Duro KUREPA

Zagrebacka 7 , 11000

Jean-Marie

LABORDE

BELGRADE

.

- Yougoslavie

I.M.A.G. Grenoble - B.P. 53 X 38041 GRENOBLE CEDEX - F r a n c e

R i c h a r d LAVER

Dept. o f Math. U n i v e r s i t y of C o l o r a d o BOULDER, C o l o r a d o -U.S.A.

Bruno LECLERC

C e n t r e de Mathematiques S o c i a l e s 5 4 , Bld R a s p a i l - 75270 PARIS CEDEX 06 France

-

- Martinsstr. - R.F.A.

K l a u s s LEEB

Informatik I n s t . 3 8520 ERLANGEN

P i e r r e LEFEBVRE

Dept. de Math. - U n i v e r s i t e Claude B e r n a r d LYON 1 - 4 3 , Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

Domenico L E N Z I

Departement de Math. Italie

L&once LESIEUR

5 , A l l e e d e s Sophoras, 92330 SCEAUX - F r a n c e

G e r a r d LOPEZ

Departement de Math. Place Victor Hugo -

Henrik MARTENS

I n s t i t u t t f o r Matematikk N-7034 NTH TRONDHEIM -

-

-

U n i v e r s i t e d e LECCE,

-

U n i v e r s i t B d e Provence 13001 MARSEILLE France

-

-

Norvege

-

M a r i e - C a t h e r i n e MAURER-VILAREM

306, A l l e e du Dragon

A r l e t t e MAYET

Dkpt. d e Math. - U n i v e r s i t k Claude B e r n a r d LYON 1 - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

Renk MAYET

Dept. de Math. - U n i v e r s i t e Claude B e r n a r d LYON 1 - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - F r a n c e

Michael MISLOVE

Dkpartement de Math. - T u l a n e U n i v e r s i t y NEW-ORLEANS - L a . 70118 - U.S.A.

E r i c MILNER

Dept. o f Math. a n d S t a t i s t i c s . U n i v e r s i t y of CALGARY - CALGARY - T2N1N4 - Canada

D r i s s MISANE

DBpt. de Math. - U n i v e r s i t e C l a u d e B e r n a r d LYON 1 - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX France

91000

-

B e r n a r d MONJARDET

Donald MONK

EVRY

France

-

C e n t r e d e Mathkmatiques S o c i a l e s 54, Bld R a s p a i l - 75270 PARIS CEDEX 06

-

U n i v e r s i t y of C o l o r a d o , Dept. of Math. BOULDER, C o l o r a d o 80309 - U.S.A.

-

- France

xxii Noel MURPHY

List ofparticipants Dept. of Math. Trent University PETERBOROUGH - Ontario K9J7BB -

Canada

Evelyn NELSON

Dept. of Math. Science. Mc Master University HAMILTON - Ontario - Canada

Bernhard NEUMA"

Dept. of Math. Inst. of Adv. Stud. Australian University POB 4 CAMBERRA ACT 2600 - Australie

Peter NEVERMANN

AG1, FB4, Technische Hochschule Darmstadt 6100 DARMSTADT - R.F.A.

Serge OVCHINNIKOV

Dept. of Math. San Francisco State University 1600 HollowayAve SANFRANCISCOCA 94132 - U.S.A.

Michel PARIGOT

UER Math. T 45.55 - Univ. PARIS VII 75221 PARIS CEDEX 05 - France

E. PICHAT

C.N.A.M. - 292, rue Saint Martin 75141 PARIS CEDEX 03 - France

Jean-Marie PLA

20, rue Elis6e Reclus France

Werner PCGUNTZE

Fachbereich Mathematik Technische Hochschule 6100 DARMSTADT - R.F.A.

Norbert POLAT

30, rue Baldeyrou -

Bruno POIZAT

Departement de Math. - Universite Pierre et Marie Curie - PARIS VI - PARIS - France

Maurice POUZET

DBpt. de Math. - UniversitB Claude Bernard Lyon 1 - 43, Bld du 1 1 novembre 1918 69622 VILLEURBANNE CEDEX - France

H.A. PRIESTLEY

St. Anne's College Grande Bretagne

Karel PRIKRY

Dept. of Math. - University of Minnesota MINNEAPOLIS Minn. 55455 - U.S.A.

Robert QUACKENBUSH

Dept. of Math. - Universitg of Manitoba WINNIPEG, Manitoba R3T2N2 - Canada

Alain QUILLOT

Dept. of Math. - University South Carolina et COLUMBIA - U.S.A. 84, rue de Nohannent 62100 CLERMONT-FERRAND - France

Norman REILLY

Dept. of Math. Simon Fraser University, BURNABY British Columbia V5A 1S6 - Canada

Serge RIBEYRE

Dept. de Math. - Universite Claude Bernard LYON 1 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - France

-

-

42000 SAINT ETIENNE

42800 RIVE DE GIER - France

OXFORD

OX2

6H5 -

-

List of participants Denis RICHARD

A.H.G. RINNOOY KAN

xxiii

D6pt. de Math. - Universite Claude Bernard LYON 1 - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - France

-

Econometric Institute, Erasmus University ROTTERDAM - Hollande

PO BOX 3000 DR

Ivan RIVAL

Dept. of Mathematics and Statistics. University of Calgary - CALGARY, Alberta T2NlN4 - Canada

Neil ROBERTSON

Dept. of Math. Ohio State University COLUMBUS, Ohio 43210 - U.S.A.

IVO G. ROSENBERG

C. Rech. Math. Appl. Universite de Montreal CP 6128 Succ "A", MONTREAL QUE. H3C 3J7 - Canada

Joseph ROSENSTEIN

Dept. of Mathematics Rutgers University NEW BRUNSWICK, New Jersey, 08903 - U.S.A.

Matatyahu RUBIN

Dept. of Math. Ben Gurion University BEER SHEVA - Israel

Bill SANDS

Dept. of Math. - University of Calgary CALGARY, Alberta T2NlN4 - Canada

James SCHMERL

Dept. of Math. University of Connecticut STORRS CT 06268 - U.S.A.

Jochanan SCHONHEIM

School of Mathematics, Tel-Aviv University TEL-AVIV - Israel

Dietmar SCHWEIGERT

FB Mathematik. University of Kaiserslautern D 675 KAISERSLAUTERN - R.F.A.

Tom TROTTER

Dept. of Mathematics and Statistics, University of South Carolina, COLUMBIA SC 29208 - U.S.A.

William TUNNICLIFFE

Faculty of Mathematics - The Open University U.K. Walton Hall - MILTONKEYNES, MK76AA

Jules VARLET

Institut de Mathematiques - Universite de Liege Avenue des Tilleuls, 15 - B400 LIEGE - Belgique

Gerard VIENNOT

Departement de MathCmaticpes - Universite de BORDEAUX - 33400 TALENCE - France

Kurt WOLFSDORF

Wilhelmstr. 2, D-1 BERLIN 61

Robert WOODROW

Dept. of Mathematics and Statistics University of Calgary CALGARY, Alberta T2N1N4 - Canada

Nejib ZAGUIA

Dept. de Math. - Universite Claude Bernard LYON 1 - 43, Bld du 11 novembre 1918 69622 VILLEURBANNE CEDEX - France

John ZELEZNIKOW

Dept. of Math. - Michigan State University EAST LANSING, Michigan 48824 - U.S.A.

-

R.F.A.

XXlV

CONFERENCE PROGRAMME

PROGRAMME SCIENTIFIQUE

_______________-__-_

Lundi 5 j u i l l e t 1982 Matinee:

..........

President de seance

E . COROMINAS

I . RIVAL ( C a l g a r y - Grenoble) , L i n e a r e x t e n s i o n s o f f i n i t e o r d e r e d sets. D. MONK ( B o u l d e r ) , Some c a r d i n a l f u n c t i o n s on Boolean a l g e b r a s . C . BENZAKEN ( G r e n o b l e ) , Decomposition Matroi'dale d'un s y s t e m e d'independance

......... R .

Aprks-midi: P r e s i d e n t d e s e a n c e

et application.

DEAN

H. PRIESTLEY ( O x f o r d ) , Ordered S e t s and d u a l i t y f o r d i s t r i b u t i v e l a t t i c e s . R. FRAISSE ( M a r s e i l l e ) , Abritement e t r a n g de non a b r i t e m e n t e n t r e c h a f n e s e t r e l a t i o n s . C . HOLLAND (Bowling G r e e n ) , C l a s s i f i c a t i o n o f l a t t i c e ordered groups.

S e s s i o n d e probl8mes:

.........

President de seance

M.

POUZET

S o i r e e : S e s s i o n s p e c i a l e q r o u p e s ordonnes: C. HOLLAND P r e s i d e n t de seance P a r t i c i p a n t s : M . JAMBU-GIRAUDET, L . LESIEUR, N . REILLY.

............

__ __

Mardi 6-- j-u _i l -l e -t -1982 --Matinee:

P r e s i d e n t d e seance

..........

B. N E U M A "

HODGES ( L o n d r e s ) , Models b u i l t on l i n e a r o r d e r i n q s . W.

E. HARZHEIM ( D u s s e l d o r f ) , On t o p o l o g i c a l p r o p e r t i e s of C a r t e s i a n p r o d u c t s o f l i n e a r l y o r d e r e d continua. C . FRASNAY ( T o u l o u s e ) , Chafnes G-compatibles, G-ranqement e t n o t i o n d e c l a s s e i n d i c a t i v e p o u r un sous-oroupe G du groupe s y m 6 t r i q u e S

m

Apr6s-midi: P r e s i d e n t d e seance J . ROSENSTEIN ( R u t g e r s )

.

..........

P. DUBREIL

,

Recursive l i n e a r orderings. R . QUACKENBUSH (Winnipeg) ,

Non modular v a r i e t i e s o f semi-modular l a t t i c e s . E . MILNER ( C a l g a r y ) ,

C o fi n a l it y

.

DUFFUS ( A t l a n t a ) , Retracts o f f i n i t e d i m e n s i o n a l l a t t i c e s . D.

xxv

Conference programme Session speciale conditions d e chaines President de seance Participants: N. ROBERTSON, M. POUZET.

....................

N.

ROBERTSON

Soiree: S e s s i o n s p e c i a l e th4orie d e s t r e i l l i s

President de seance

...........

L. LESIEUR

P a r t i c i p a n t s : H. BAUER, G . HANSOUL, K . H O F M A ” , M . M I S L O V E , J. VARLET.

__ _

-

K.

KEIMEL,

Mercredi l e t --1 -9 -8 -2 - -- - - -- 7 - j--u -i -l --

...........

Matinee: P r e s i d e n t d e s e a n c e

KELLY

D.

K.

PRIKRY

,

HAJNAL ( B u d a p e s t ) P a r t i t i on s r e 1 a t i ons . A.

(Winnipeg),

Some f r e e i n f i n i t a r y l a t t i c e s . I . GUESSARIAN

(Paris),

A p p l i c a t i o n d e la theorie d e s e n s e m b l e s o r d o n n e s d l a s e m a n t i q u e d e s schsmas d e p r o y r a m m a t i o n .

........

President de seance

AprGs-midi:

I.G.

ROSENBERG

D. KUREPA (Belgrade) On s e v e r a l k i n d of r a m i f i c a t i o n s . M.

RUBIN

(Beer Sheva)

C o n s i s t e n c y r e s u l t s on r e a l order t y p e s . J. SCHMERL ( S t o r r s )

C o m p l e x i t y and G.

,

N - c a t e g o r i c i t y o f p a r t i a l l y o r d e r e d sets.

0 BRIJNS ( H a m i l t o n )

Orthomodular l a t t i c e s . Session s p e c i a l e treillis orthomodulaires: P r e s i d e n t d e s e a n c e .................... G . BRUNS P a r t i c i p a n t s : L . I T U R R I O Z , G . KALMBACH, R . MAYET.

_______-_-__________

J e u d i 8 j u i l l e t 1982

Matinee: P r e s i d e n t d e s e a n c e J.P.

BARTHELEMY

...........

G.

KREWERAS

(Paris),

Ensembles o r d o n n e e s et t a x o n o m i e m a t h e m a t i q u e . I . ROSENBERG

(Montreal),

T h e l a t t i c e o f clones a n d t h e a s s o c i a t e d r e l a t i o n s . G.

VIENNOT

(Bordeaux),

C h a i n and a n t i c h a i n f a m i l i e s , g r i d s and Young t a b l e a u x .

- -__ __ __ __ _ -_

V e n d r e d i 9- j u i l _l e -t -1982 - --

Matinee:

P r e s i d e n t d e seance A.H.G.

RINNOOY K.’,J

.......... (Rotterdam)

P a r t i a l o r d e r e d sets i n s c h e d u l i n g t h e o r y .

B.

SANDS

xxvi

Conference programme J. GRIGGS (Columbia), The Sperner property.

M. HABIB (Saint Etienne), Une caracterisation p a r sous-qraphes exclus des qraphes sans c i r c u i t s ayant deux sauts. Aprgs-midi: President de shance

........

R . HALIN

R . JAMISON-WALDNER (Clemson), Order and convexity.

T. TROTTER (Columbia) The dimension o f the Cartesian product of p a r t i a l order.

Session speciale Combinatoire: President de seance T. TROTTER Participants: J.P. DION, P. ERDOS, J. GRIGGS, A. HAJNAL, G. KERWERAS, J.M. LABORDE, A. QUILLIOT, I. RIVAL, B. SANDS, J. SCHMERL, J. SCHONHEIM.

....................

~

~

~

~

Matin6e: President de seance

~

~

............ D. KELLY

-

A. BOUCHET (Le Mans) , Codaqes e t dimensions de r e l a t i o n s binaires. 0 . COGIS (Montpellier),

S u r l a dimension des qraphes e t des ordres.

A. DUCAMF' (Bruxelles), La bidimension e t ses applications.

D. SCHWEIGERT (Kaiserslautern), Clones of monotones functions. Apres-midi: President de seance

......... I. RIVAL

K. PRIKRY (Minneapolis) The Ramsey property and mesurable selections. E. COROMINAS (Lyon) S u r l e meilleur ordre de Nash-Williams.

Session de problemes: President de seance

.....................

D. DUFFUS

P. ERDOS (Budapest) Problems concerning the chromatic number o f f i n i t e and i n f i n i t e graphs.

*** Diner dans les jardins du Chlteau.

*

~

xxvii

SHORT COMMUNICATIONS

PROGRAMME DES COMMUNICATIONS

Yendredi-Xiuillet PrGsident: E . NELSON I. DUNTSH

PrBsident: R.W. B . MONJARDET

Axiomatique arrowienne de l a mediane dans l e s d e m i - t r e i l l i s medians.

Projective Stone A1 gebras.

S. KOPPELBERG

M.

Groups o f permutations with few fixed points. E.

PICHAT

ZAGUIA

V a r i e t y i n v a r i a n t s f o r ordered s e t s

M.

BEKKALI

S u r l e theorbme de Neumer-Fodor. une version topologique.

Deviation e t dimension de K r u l l des ensembles ordonnes. E . NELSON

EYTAN A doctrinal model for k i n s h i p .

P . NEVERMA”

Modele de donnees relationnel e t d’accbs e t ensembles ordonnes. N.

QUACKENBUSH

M.

HAIFAWI

Non archimedian a n a l y s i s and ordered sets.

Chain continuous algebras.

samedi-l2_iuillet President: R . WOODROW

President: R. HALIN C.

FRASNAY

D.

C . FLAMENT

H . KOTLARSKI

On elementary c u t s i n Models o f Ar i t h m e t ic

Comparability graphs with constraint . R.

DEAN A construction f o r large f a m i l i e s

RICHARD

Ordre nature1 e t ordre de d i v i s i b i l i t 6 dans l’arithmetigue de Peano.

Permutations b i v a r i a n t e s e t i n f i n i t ude.

.

R . LAVER

Theorems on i n f i n i t e t r e e s .

of k-element s e t s having t h e Erdos i n t e r s e c t i o n property. J. CONSTANTIN

C . CHARRETTON

Comparaison des structures engendrees p a r des chafnes.

Ordonnes escamotables e t p o i n t s fixes. M.C.

MAURER

B. P O I Z A T

S u r d e s travaux de M . Krasner.

Familles k - i n t e r s e c t a n t e s .

*****


-

PART I ORDER AND SOME SET-THEORETICAL INVARIANTS

-

PARTIEI

-

ORDRE ET INVARIANTS ENSEMBLISTES

Eric C. MILNER Recent results on the cofinality of ordered sets. . . . . . . . . . , . . . . . . . P . 1 J. Donald MONK

Cardinal functions on boolean algebras. . . . . . . . . . . . . . . . . . . . . . . . P . 9


Annals of Discrete Mathematics 23 (1984) 1-8

1

0 Elsevier Science Publishers B.V. (North-Holland)

RECENT KESULTS ON THE COFINALITY OF ORDERED SETS ERIC C. MILNEK Department of Mathematics and S t a t i s t i c s The U n i v e r s i t y of Calgary Calgary, Alberta Canada

Dedicated t o E . COROMINAS Une p a r t i e A d'un ensemble p a r t i e l l e m e n t ordonn6 P = (P, cf(A), does ( P , 5 )

contain an a n t i c h a i n o f size c f ( h )

T h i s q u e s t i o n h a s been answered p o s i t i v e l y ( [ 5 ] , [ 7 ] , [ 8 1 ) under some a d d i t i o n a l s e t - t h e o r e t i c a l assumptions. For example, t h e g e n e r a l i z e d continuum h y p o t h e s i s i m p l i e s ( 1 . 4 ) . The s t r o n g e s t r e s u l t of t h i s kind t h a t we know a t p r e s e n t i s t h e f o r m u l a t i o n g i v e n by Milner and Pouzet [ 7 ] : (1.5)

If c f ( P )

=

X

> cf(X) and

if A"

=

X for 1 5

p

=

a f o (~1 7 7

~1

-' 1!,

thm

i + . f ( l ) arid if c.orittr;ns mi R-ii2~kpxcleni subset of sine

u.

The h y p o t h e s i s IAl = X i n (3.1) may a t f i r s t a p p e a r t o o r e s t r i c t i v e . But t o r e g a i n (1.5), we simply a p p l y ( 3 . 1 ) t o a c o f i n a l s u b s e t of t h e p a r t i a l l y o r d e r e d s e t (P, f ( ~ , R )= h .

I.,

C ~ F I i:

h,

K

then there i s a tournmient

A tournament i s a r e l . a t i o n R such t h a t f o r each p a i r of d i s t i n c t e l e m e n t s x,y e x a c t l y one of t h e r e l a t i o n s (.z,p) F I? o r (:{,x)F 1'7 h o l d s , and so t h e r e i s n o t even an R-independent s e t of s i z e 2 . Note t h a t (3.1) shows t h a t t h e c o f i n a l i t y of a tournament on a s i n g u l a r c a r d i n a l X i s s t r i c t l y l e s s t h a n A .

Although we do n o t have any r e s u l t l i k e (1.5) f o r a r e l a t i o n a l s t r u c t u r e (EJ?), c ~ ( E , R )= A euf A , t h e f o l ~ o w i n gi n t e r e s t i n g r e s u l t due t o Pouzet when ( u n p u b l i s h e d ) s a y s something a b o u t t h e s i z e of independent s e t s :

1 ~ 1

(3.3) T! I c.f(x) a r k j 2" -. A .for p A nmj i~ ever3 c o f i n n l s u i i s e t o f ( E , I ~ ) co?ittr?'iis independent s p t s 0.f a r b i i ; r a q s i n c A ' c A, then (E,R) contains an indcpendent s e t of size A. Another s i t u a t i o n i n which t h e r e i s a n a t u r a l analogue t o c o f i n a l i t y i s t h e c a s e of a c l o s u r e r e l a t i o n . A elosur% r e l a t i o n on a s e t E i s a f u n c t i o n cy:P(C) - * P ( F ) such t h a t ( i ) :i sz v ( X ) , ( i i ) X C V + q ( X ) 2 cp(.y) and ( ) f$l(Q(X)) = Q(X). A (jcncrutinp s e t ( o r c o f i n a l s e t ) i s a s u b s e t X 5 E such t h a t Q ( X ) = E and we d e f i n e t h e dimension ( o r c o f i n a l i t y ! ) of ip t o he dim(@) = m i n i l x i : c r ~ ( X ) =

.

Fl

The analogue of a n a n t i - c h a i n i s a cq>-independent s e t , i . e . a s e t A 5 E such t h a t Of c o u r s e , i f (E,t having s i n g u l a r dimension f o r which t h e r e i s no i n f i n i t e (0"independent s e t . I n f a c t , i t i s c o n s i s t e n t t h a t t h e c l o s u r e r e l a t i o n ip* c o n s t ructed i n [ 7 ] i s a l g e b r a i c , i.e. it i s f i n i t e l y generated o r s a t i s f i e s (O*(X)

= II{~~>*(F F ) 2:

x

unii

IFI

.

N ~ )

An open q u e s t i o n which remains h e r e i s whether t h e r e i s a n i n f i n i t e @"-independent cf(\) and i f '4" i s 2-generated, i . e . i f s e t i f dirn(in9:) = h o , A h a s no c h a i n of t y p e cfK , and B h a s no c h a i n of t y p e K , then A*B h a s no c h a i n o f t y p e K ; but i f A h a s a c h a i n o f t y p e CfK a n d d ep th B = K , t h e n A*B h a s a c h a i n of t y p e K I f K i s i n f i n i t e and r e g u l a r , ] A 1 = K , and t h e r e i s a homomorphism from A o n t o a s u b a l g e b r a of K c o n t a i n i n g a l l s u b s e t s of power < K , t h e n t h e r e

.

exist

BA's

infinite algebras

R,C 2 A

satisfying

K + - CC

such t h a t

depth(B*AC)

2

.

K+

For any

i n a model M of GCH , t h e r e i s an e x t e n s i o n M ' of M w i t h B,C I A s u c h t h a t depth(B*AC) > max(de pthB, de pthC) ( t h i s is a result BA

A

o f S h e l a h found i n McKenzie, Monk [ 8 2 ] ) . Pro b lem 8. In ZFC i s i t t r u e t h a t f o r e v e r y i n f i n i t e B , C 2 A w i t h depth(BWAC) > max(dept hR,dept hC) ?

A

RA

there exist

Pro b lem 9 . F o r e v e r y i n f i n i t e BA A , i s t h e r e a c a r d i n a l K s u c h t h a t i f A and I B I , I C I K , then depth(BhAC) = max(de pthB, de pthC) ?

B,C

1.

2

5

d e p t hA 5 l e n g t h A 2 depthA , (H+depth)A c o i n c i d e s with t h e A ; see below. P r o b l e m s 7-9 a r e i n McKenzie, Monk [ 8 2 ]

We c l e a r l y have t i g h t n e s s of

4.

.

Incomparability

.

As w i l l incA = su p { l M l : M i s a s e t of p a i r w i s e i n c o m p a r a b l e e l e m e n t s o f A} be d e t a i l e d below, t h i s is a l a r g e f u n c t i o n . We l e t a b b r e v i a t e set o f p a i r w i s e i n c o m p a r a b l e e l e m e n t s . We f i r s t i n d i c a t e a n e q u i v a l e n t way of d e f i n i n g inc

.

tree,

Theorem 4 . 1 . T r A}

.

F o r any i n f i n i t e

BA

A

we have

No te t h a t f o r a t r e e T S A w e d o n o t i n s i s t t h a t incomparable i n T ; s e e s e c t i o n 6

.

incA = s u p { \ T I : T is a

s + t = 1

for

-

s

a nd

t

5

Proof. S i n c e any p i e is a t r i v i a l t r e e w i t h o n l y r o o t s , t h e p a r t of t h e e q u a l i t y is c l e a r . To show = , assume t h a t A h a s no p i e o f s i z e K , K r e g u l a r ; we show t h a t A h a s no t r e e of s i z e K , Suppos e t h a t T is a t r e e i n A of power K By R a u m g a r t n e r , Komjath [ S l ] , A h a s a d e n s e s u b s e t D o f power < K Now e a c h l e v e l of T i s a p i e , so h a s power < K Hence T h a s a t l e a s t K l e v e l s . L e t T' b e a s u b s e t of T of power K c o n s i s t i n g e x c l u s i v e l y o f e l e m e n t s of s u c c e s s o r L evel . For each d E D , l e t

.

Md =

It

.

ET'

.

: if

s

is t h e i mmedi at e p r e d e c e s s o r of

t

,

then

d

5 t*-s} .

17

Cardinal functions on boolean algebras

a

=udE D M d

T'

Thus

,

pie

,

d E D

so t h e r e is a

with

IM

d

I

=

.

K

But

is clearly

Md

contradiction.

.

5

2

I f A S B o r B-h'A , c l e a r l y incA incR I f A G B , t h e n inc (A xB) /A\ In f a c t , { ( a , - a ) : a E A} i s a p i e i n A X B Hence i f A i s c a r d i n a l i t y homogeneous and h a s no p i e of s i z e I A l , t h e n A is r i g i d . We ha ve inc(A*B) = m a x ( [ ~ ,\I n [ ) if \A\, 4 , s i n c e A*C ~ A X A i f Icl = 4

.

.

>

\RI

.

5

incA and \ A \ 5 Zinc* u s i n g t h e E r d o s , Rado t h e o r e m . Clearly cellA Some d e e p r e s u l t s and p r o b l e m s a r e found i n c o n n e c t i o n w i t h t r y i n g t o c o n s t r u c t a BA A w i t h no p i e of s i z e \ A \ ; w e c a l l such a BA n a r r o w . Bonnet and S h e l a h (Bo n n et

[m])

h a v e shown i n

t h a t t h e r e is a n a r r o w

ZFC

BA

of power

.

cf(Zw)

.

Bonnet [ m ] h a s shown a s s u m i n g GCH t h a t t h e r e i s a n a r r o w BA o f power K+ By a t h e o r e m o f A r h a n g e l s k i c [ 7 1 ] , i f I A l is singular strong l i m i t , then A i s n o t n arro w. B a u m g a r t n e r , Komjath and S h e l a h i n S h e l a h [ 8 3 ] ha ve shown t h a t i f A h a s no p i e of s i z e h , t h e n i t h a s a d e n s e s u s e t of s i z e < A So i f IAI i s s t r o n g l i m i t , t h e n A is not narrow. Even s t r o n g e r r e s u l t s a r e known i n which A a l s o does not have b i g c h a i n s . C a l l A c o n c e n t r a t e d i f A h a s no p i e a n d no c h a i n of s i z e \A\ S h e l a h h a s shown u n d e r GCH t h a t f o r e a c h h > w there

.

.

is a c o n c e n t r a t e d

BA

of s i z e

a A-complete

concentrated

Problem 10.

(CH)

BA

,

A+

A? o

and f o r e a c h

f

w1

t h e r e is

.

A+

of s i z e

A

with

Is t h e r e a c o n c e n t r a t e d

o-RA

w2 ?

of s i z e

Rubin [ 8 3 ] h a s shown t h a t i f B is a s u b a l g e b r a of a n i n t e r v a l a l g e b r a and is r e g u l a r , t h e n B i s not c o n c e n t r a t e d . Problem 11. I f B i s a s u b a l g e b r a o f an i n t e r v a c a n B be c o n c e n t r a t e d ?

a l g e b r a a nd

IBI

is singular,

IB[

IB I

No te by t h e a b o v e r e m a r k s t h a t t h e answer t o P rob em 11 is no f o r strong l i m i t singular. S h e l a h [ 8 0 ] and i n d e p e n d e n t l y van Wesep have shown t h a t i t is c o n s i s t e n t t o h a v e Zw a r b i t r a r i l y l a r a e and t o have a BA A of s i z e 2w w i t h c o u n t a b l e On t h e o t h e r h a n d , B a umga rtne r [801 h a s shown t h e l e n g t h and i n c o m p a r a b i l i t y . I

following consistent:

MA

+

Z w = w2

+ "every u n c o u n t a b l e

has an uncountable

BA

pie." S h e l a h h a s g e n e r a l i z e d t h i s , showi ng t h a t i t i s c o n s i s t e n t t o h a v e t h e continuum a r b i t r a r i l y l a r g e .

Is it c o n s i s t e n t t h a t every

P r o b l e m 12'.

BA

of power

w2

has a

pie

of s i z e

w2 ?

S h e l a h [831 h a s shown t h a t f o r any s i n g u l a r h w i t h c f A > w i t is c o n s i s t e n t t h a t t h e r e i s a BA A w i t h incA = A n o t a t t a i n e d . M i l n e r a nd P o u z e t ( u n p u b l i s h e d ) h a v e shown t h a t i f incA = h , w i t h w = c f h , t h e n incA is attained.

is a

BA

Problem 1 3 . cardinal K

Zw

Zw

with incomparability

not a t t a i n e d .

I s i t t r u e t h a t f o r e v e r y weakl y b u t not s t r o n g l y i n a c c e s s i b l e t h e r e i s a BA w i t h i n c o m p a r a b i l i t y K n o t a t t a i n e d ?

T o d o r r e v i t h a s shown t h a t i f t h e n A h a s a p i e of s i z e

We h a v e

i s wea kly i n a c c e s s i b l e , t h e n t h e r e

Zw

T o d o r c ' e v i i h a s shown t h a t i f

of s i z e

( H + i n c ) A = incA

cfh A

.

w f o r e v e r y homomorphic image

RA

R

Pro b lems 1 0 , 1 1 , 13 a r e ment i oned i n van Douwen, Monk, Rubin (801 r e l a t e d t o i n c o m p a r a b i l i t y has been c o n s i d e r e d by P. Nyikos: h -c o f ( A )

= min{K : e v e r y s u b s e t of

One can show t h a t

incA

5

h-cof(A)

[all has ideal.

.

w

An a l g e b r a

,

IAI = w

E

A

,

T

Clearly

Pro b lem 1 5 .

,

i s of power

CH

ow

c o n s t r u c t e d under

A

incA = w

Can one c o n s t r u c t

5

.

K)

.

well-founded)

w1

but has

by R a u m g a r t n e r , Komjath

1

( a E A : I A T a1

while

is n o t c o u n t a b l y g e n e r a t e d , s o

I

A function

and

The a l g e b r a c o n s t r u c t e d by S h e l a h [ 8 1 ] assumi ng h-cofinality

.

h a s a c o f i n a l s u b s e t of power

A

= sup{ [ T I : T

h - c o f(A)

with A ?

A of

in

ZFC

5.

a

BA

5 (d}

h-c of(A ) with

A

= I

i s a maximal

= w1

.

t o a homomorphism

C

.

C l e a r l y nB = zK To g e t nB < nA , t a k e F r ( 2 K ) + ) p ~ . One c a n s t i l l a s k , h o w e v e r , f o r what p a i r s K , A i t is t r u e t h a t e v e r y BA A o f power h a s a homomorphic image B w i t h nB = K Assuming GCH we c a n a ns w e r t h i s question fully (Corollary 5.5). B G C

A

.

Theorem 5 . 1 . f : in t a l g L- R

S uppose L i s a d e n s e l i n e a r o r d e r , Then *B 5 ID1

.

is d e n s e i n

D

.

L

,

a nd

We omit t h e e a s y p r o o f . Corollary 5 . 2 . such t h a t f o r every image

If B

Theorem 5.3. If B w i t h nB = A Proof.

K

,

cellA

.

But

.

Md

is a n

21

Cardinal functions on boolean algebras We a l s o n o t e t h a t ramA and l e n g t h A a r e , i n g e n e r a l , n o t c o m p a r a b l e . In f a c t , i n i n t a l g R t h e r e i s no r a m i f i c a t i o n s y s t e m o f power wl For, suppose

.

T is such a system. For each r E Q l e t

Let

Lr = ( t E T ' Thus

Ur E Q L r

T' =

Lr

members o f

,

c o n s i s t o f a l l nodes of

T'

: if

is t h e p r e d e c e s s o r o f

s

r E Q

so t h e r e is a n

with

T

of successor l e v e l .

r E s

t , then

Lr

\

uncountable.

a r e pairwise d i s j o i n t , contradiction.

Conversely,

t)

But t h e if

is a n

T

A r o n s z a j n t r e e , t h e n l e n g t h ( i n t a 1 g T ) = w by a t h e o r e m o f B r e n n e r , Monk [ 8 3 1 . I t would be n a t u r a l t o d e f i n e a new c a r d i n a l f u n c t i o n u s i n g t h e n o t i o n of a pseudo-tree. T h i s d o e s n o t l e a d t o a n e s s e n t i a l l y new n o t i o n , h o w e v e r . In f a c t , Kurepa 1 7 7 1 showed t h a t i f T i s a p s e u d o - t r e e o f r e g u l a r s i z e K w i t h no c h a i n s of s i z e K , t h e n T c o n t a i n s a t r e e o f s i z e K Thus

.

sup{lT\ : T s A , T a pseudo-tree} = lengthA*incA ; s u p { l B \ : B i s a p s e u d o - t r e e a l g e b r a , B S A} = 1engthA-ramA

,

The a b o v e r e s u l t o f Kurepa d o e s n o t e x t e n d t o s i n g u l a r K , , a s h e e s s e n t i a l l y On t h e o t h e r h a n d , T o d o r C e v i c o b s e r v e d t h a t by o b s e r v e d i n t h e same p a p e r . adding N Cohen r e a l s t o a model o f GCH o n e c a n g e t a AA B s u c h t h a t w.

KW

B has a pseudo-tree 1 and no c h a i n o f s i z e 1 IBI =

Finally, note t h a t

of s i z e

,

N

b u t no t r e e or

pie

of s i z e

w1

K2

(H+ram)A ; t h e p o s s i b i l i t y o f e q u a l i t y i s

spreadA

r e l a t e d t o P r o b l e m 19. (H-ram)A is always w , although it is not complete Y t r i v i a l t o see t h i s . L e t A be an a r b i t r a r y i n f i n i t e BA Then A+>B f r some BA A w i t h f i n c o w S B c w S u p p o s e R is a n u n c o u n t a b l e r a m i f i c a t i o n Now R = U n E w { x E R : n E x } , so t h e r e i s a n n E w f o r which system i n B C = { x E R : n E x } is u n c o u n t a b l e . But t h e n C is a w e l l - o r d e r e d c h a i n , c o n t r a d i c t ion.

.

B.

BA's:

.

.

Algebraic functions

We now s u r v e y c a r d i n a l f u n c t i o n s h a v i n g t o d o w i t h a l g e b r a i c a s p e c t s of s u b a l g e b r a s , a u t o m o r p h i s m s , and homomorphisms.

7. We l e t

-< , I A 1 .

Subalgebras

be t h e set o f a l l s u b a l g e b r a s o f

SubA

A

.

\A\

Clearly

w

t h e r e is a

V = L

showed a s s u m i n g

w i t h \ A 1 = ISubAl =

A

BA

that for

.

K+

In section 8

.

we n o t e t h a t i f K i s a s t r o n g l i m i t c a r d i n a l and I A l = K , t h e n l S u b A \ = Z K T h e s e two f a c t s a r e e s s e n t i a l l y a l l t h a t is known a b o u t lSubA1. In p a r t i c u l a r , t h e following q u e s t i o n s a r e open. P r o b l e m 20.

Is l S u b A \ a l w a y s a power of

Problem 21. W A l L

Can o n e p r o v e i n w ? For

\A\ =

\SubA\

A

,

then

If

and

A s; R

or

t h a t t h e r e is a

ZFC

A

RA

with

s i n g u l a r is i t c o n s i s t e n t t h a t t h e r e i s a

Problem 2 Z S . K

2 ?

K

\5

\SubA

8.

lSubR

BA

A

with

I .

Irredundance

.

A s u b s e t X o f A i s i r r e d u n d a n t i f Vx E X ( x f S g ( X \ {XI) We l e t irrA = sup{\X\ : X irredundant} T h i s is a l a r g e f u n c t i o n . S h e l a h 1 7 9 1 , g e n e r a l i z i n g Rubin [ 8 3 ] , showed a s s u m i n g V = L t h a t f o r e v e r y r e g u l a r K > w t h e r e i s a RA

.

.

A w i t h \ A \ = K + and irrA = K D e v l i n [731 showed t h a t i f K i s r e a l - v a l u e d m e a s u r a b l e , t h e n e v e r y a l g e b r a w i t h c o u n t a b l y many o p e r a t i o n s and w i t h K e l e m e n t s h a s a n i r r e d u n d a n t s u b s e t o f power K S h e l a h [ 8 3 ] showed t h a t i f nA K , then A h a s a n i r r e d u n d a n t p i e o f power K In particular, nA
(ZK)',

then

.

hs(inta1gL)

.

h s ( i n t a 1 g I R ) = {w,Zo)

Theorem 1 2 . 2 .

w Then f is deterby a convex e q u i v a l e n c e r e l a t i o n E on IR w i t h \ W E \ = \ A ) Now L ' = U { k : k is an E - c l a s s , \ k l > 1) i s B o r e l , so L" = l R \ L ' is also. Clearly

.

= \A1

IL"I

.

I A l = 2 w by t h e A l e x a n d r o f f , H a u s d o r f f t h e o r e m .

Hence

.

I f A is h e r e d i t a r i l y a t o m i c and i n f i n i t e , t h e n hA = w In fact, let [a] b e a n atom of A/IgAtA ; t h e n A 4 j A a s f i n c o K --)t f i n c o w f o r some K Also,

r

.

05 5

i t can be shown t h a t i f A i s h e r e d i t a r i l y a t o m i c , K A , then [ K , K c K ] On t h e o t h e r hand, JuhAsz, Nagy and Weiss [791 c o n s t r u c t e d u n d e r hsA f 0 V = L a BA A o f power N w l w i t h K, $? hsA van Douwen [ a ] c o n s t r u c t e d a n

.

n

.

A

h e r e d i t a r i l y a t o m i c BA the following question.

2w

of power

Problem 36. ( I n ZFC) I f A i s h e r e d i t a r i l y a t o m i c and A have a homomorphic image of s i z e K ? Problem 3 7 5 .

Con( VA(A

h e r e d i t a r i l y a t o m i c and i n f i n i t e hs :

We n o t e some o t h e r e a s y f a c t s a b o u t

5K

,

n

(5)

If

(6) (7)

I f A h a s a f r e e s u b a l g e b r a o f power hs(AxB) = hs(A*B) = hsA U hsB

w

\A\

then

hsA

[K,2K]

f

(8) I f

w

-<

K = K

w

,

does

hsA = [ w , I A l ] ) ) ?

.

0

.

.

hsA = {w,Zw}

with

,

then

such t h a t

hsA

n

f

[K,K,]

0

.

hsA = [ w , ~ ]U {2w}

.

.

I f t h e r e is a

BA

A

such t h a t

hsA = {w,w

2

1

,

then

t h e r e is a Kurepa f a m i l y .

-

P r o o f . By f a c t ( 6 ) a b o v e , A h a s no u n c o u n t a b l e i n d e p e n d e n t s u b s e t . Hence A d o e s n o t s a t i s f y c c c , Hence A h a s a homomorphic image B s u c h t h a t f i n c o w l C B E 60, C l e a r l y , s t i l l hsB = {w,w 1 I f r is any c o u n t a b l e 2 f o r b E B is a homomorphism, and hence s u b s e t of w1 , t h e n b H b

.

.

nr

{b

nr

: b E B}

is countable.

I t is c o n s i s t e n t with

CH

Thus

B

is a Kurepa f a m i l y .

t h a t t h e r e i s no Kurepa f a m i l y , hence no

BA

as i n

12.3. Problem 38". I s it c o n s i s t e n t w i t h hsA = { w , w 1 ? 2

CH

that there is a

BA

A

such t h a t

21

Cardinal functions on boolean algebras 13.

Endomorphisms

.

EndA is t h e set o f a l l e n d o m o r p h i s m s o f A Since clearly IUltAI < lEndA\ , we a r e d e a l i n g h e r e w i t h a " l a r g e " c a r d i n a l f u n c t i o n , a n d t h e most i n t e r e s t i n g q u e s t i o n is t o c o n s t r u c t B A ' s A w i t h \ E n d A \ s m a l l . Theorem 1 3 . 1 .

A

2w

,

and

Suppose

I=

IEndCintatgL)

\

Proof. -

UlttintalgA)

=

IintalgL

is a c o m p l e t e d e n s e l i n e a r o r d e r i n g o f power

L

is a d e n s e s u b s e t o f

D

o f power K a n d f u n c t i o n s from X Recall that if

I-L

1x1

= into

.

K

A

=

X

i s any i n f i n i t e c a r d i n a l ,

A

RA

A

If

.

Then

continuous

wv

w i t h a d e n s e s u b s e t of

is a n i n f i n i t e c a r d i n a l a n d

IL

\ A 1 = IEndAI = 2 "

such t h a t

A

i s minimum s u c h t h a t pv

v

and

.

C o r o l l a r y 13.2.

AK =

where

hK = A

Hence t h e r e a r e a t most as d e s i r e d .

t h e n t h e r e i s a c o m p l e t e l i n e a r o r d e r i n g of power Thus: power p

is a

,

K

is a l i n e a r l y o r d e r e d space w i t h a d e n s e subspace

X

,

of power

L

.

0

BA

.

'K

Problem 39. (GCH) For A a l i m i t c a r d i n a l o r t h e s u c c e s s o r o f a s i n g u l a r \ A \ = \EndA\ = A ? c a r d i n a l , i s t h e r e a BA A s u c h t h a t I t is e a s y t o s e e t h a t i f w

1

2w

then

IEndAI

with

\ A \ = IEndA

\

Thus t h e a s s u m p t i o n

w

=

1

P r o b l e m 40. I n ZFC c a n o n e show t h a t t h e r e a r e a r b i t r a r i l y l a r g e \A1 = \En&\ = K ? t h e r e i s a BA A w i t h Problem 41. BA A w i t h

Under a n y s e t - t h e o r e t i c a l a s s u m p t i o n s , I A l = A. a n d \EndAl = K ?

Problem 42.

Is

IEndA\

B} = sup{ilR : A

.

The p r o o f o f t h i s t h e o r e m Eollows f r o m d u a l i t y t h e o r y and J u h A s z I801 C l e a r l y spreadA hdA A l s o , hdA i n c A ; i f A- 3)B t h e n nR

5

incA

.

by s e c t i o n 5 , s o hdA incA A l s o n o t e t h a t hdA sdepthA

5

5

by 18.1. by t h e p r o o f of 15.1.

5 i n c R '
0 t h e same i s t r u e f o r t h e f r e e sequence d e f i n i t i o n . I f K is s i n g u l a r , C ~ = K w , and A has t i g h t n e s s K , then A h a s a f r e e s e q u e n c e of l e n g t h K These f a c t s a r e e a s i l y proved u s i n g t h e methods of McKenzie, Monk [82]

.

.

W e have

t(AxB) = t A U t R

and

tn;

= supi

0 t h e r e i s a BA A w i t h t i g h t n e s s f o r each K Monk [83] I t f o l l o w s from a r e s u l t o f Mal yhi n [721 t h a t

.

C l e a r l y independence, d e p t h u l t r a f i l t e r density tightness.

L

S a p i r o v s k i ? [741 showed

hdA

5

tightness.

w

On t h e o t h e r h a n d ,

s uc h t h a t

t(@A ) = K ; see

= \I] Usup. tfti 1 E ItAi. S a p i r o v s k i t [751 h a s shown t h a t

Clearly tightness

5 spread,

spreadA*(tA)+ ,

We a l s o s h o u l d m e n t i o n t h a t

.

ItAi

t A = sup{nxB : A - + B }

.

character.


depth

Possible equalities are indicated by ? . I n addition, there may b e relationships not shown; see the problems.

35

36

J.D. Monk REFERENCES

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37

Koppelherg, S. [751 Homomorphic images of o-complete Roolean algebras, Proc. AMS 51, 171-175. Koppelherg, S . (771 Boolean a l g ~ h r a s a s unions of chains of suhalgehras, Alg. llniv. 7, 195-204. Koppelherg, S . [ m ] Maximal chains in Roolean algebrds. Kunen, K. 1801 Set theory, North-Holland, 317 pp. Kurepa, D. [501 La condition de S u s l i n et une propriet; caracteristique des nombres rGels, Comp. Rendus (Paris) 231, 1113-1114. Kurepa, D. [57] Partitive sets and ordered chains, "Rad" de 1'Acad. Yougoslave 302, 197-235. Kurepa, D. [62] The Cartesian multiplication and the cellularity number, Publ. Inst. Math. (Brograd) 2, 121-139. Kurepa, D. (771 Ramified sets or pseudotrees, Publ. Inst. Math. (Beograd) 22, 149-163. Loats, J. 1771 On endomorphism semigroups of Boolean algebras and other problems, Ph.D. thesis, IJniv. of Colo., 75 pp. Malyhin, V. [72] On tightness and Souslin number in expX and i n a product of spaces, Sov. Math. Dokl. 13 (1972), 496-499. Malyhin, V., Sapirovski;, B. [731 Martin's axiom and properties of topological spaces, DAN SSSR 213, 532-535. McKenzie, R., Monk, J.D. [73] On automorphism groups of Boolean algebras, Erdjs Symposium, Colloq. Math. SOC. J. Bolyai 10, 951-988. McKenzie, R . , Monk, J.D. [82] Chains in Boolean algebras, Ann. Math. Logic 22, 137-175. Mitchell, M. [72] Aronszajn trees and the independence of the transfer property, Ann. Math. Logic 5, 21-46. Monk, J.D. 1831 Independence in Boolean algebras, Per. Math. Hungar. 14, 269-308. Rubin, M. [831 A Boolean algebra with few subalgebras, interval Boolean algebras, and retractiveness, Trans. AMS 278, 65-89. iapirovski;, 8. [74] Canonical sets and character. Density and weight in compact spaces, Sov. Math. Dokl. 1 5 , 1282-1287. iapirovskii, B. [75] On n-character and n-weight in bicompact spaces, DAN SSSR 223, 799-802. iapirovskii, B. [76] On tightness, n-weight, and related notions, Uc. Zap. Latv. Univ. 257, 88-89. iapirovskii, B. [80] On mappings onto Tychonov cubes, Ilsp. Mat. Nauk 35, 122-1 30. Shelah, S . [79] Boolean algebras with few endomorphisms, Proc. AMS 74, 135-142. Shelah, S. [SO] Remarks on Boolean algebras, Alg. Univ. 11, 77-89. Shelah, S. [80a] Independence of strong partition relation for small cardinals, and the free-subset problem, J . Symb. Logic 45, 505-509. Shelah, S . [all On uncountable Boolean algebras with no uncountable pairwise comparable o r incomparable sets of elements, Notre Dame J. Formal Logic 22, 301-308. Shelah, S. [83] Constructions of many complicated uncountable structures and Boolean algebras, Israel J . Math. 45, 100-146. Szentmiklossy, 2. (801 S-spaces and L-spaces under Martin's axiom, Colloq. Math. SOC. Janos Bolyai 23, 1139-1145. Tall, F. [80] Large cardinals f o r topologists, Surveys in general topology, Acad. Press, 445-477. Todorcevig, S . [m] Remarks on chain conditioins in products.


-

PART I1 ORDER AND LATTICES

- PARTIEII ORDRE ET TREILLIS

H.A. PRIESTLEY Ordered sets and duality f o r distributive lattices. . . . . . . . . . . . . . . . p . 39 Michael MISLOVE When are order scattered and topologically scattered the same ? . . . p . 61 Dwight DUFFUS - Maurice POUZET Representing ordered sets b y chains. . . . . . . . . . . . . . . . . . . . . . . . .

p . 81

Giinter BRUNS Orthomodular Lattices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

p . 99

George GRATZER - David KELLY The construction of some free m-lattices on posets. . . . . . . . . . . . . p . 103 I.C. ROSENBERG - D. SCHWEIGERT Compatible orderings and tolerances of lattices. . . . . . . . . . . . . . . . p . 11 9


Annals of Discrete Mathematics 23 (1984) 39-60 0 Elsevier Science Publishers B.V. (North-Holland)

39

ORDERED SETS AND DUALITY FOR DISTRIBUTIVE LATTICES

H . A. P r i e s t l e y Mathematical I n s t i t u t e U n i v e r s i t y of O x f o r d England

An a c c o u n t i s g i v e n o f t h e c a t e g o r i c a l d u a l i t y w h i c h e x i s t s b e t w e e n bounded d i s t r i b u t i v e l a t t i c e s a nd c ompa c t t o t a l l y o r d e r disconnected spaces. During t h e p a s t decade, a wide range of s t r u c t u r a l problems c o n c e r n i n g d i s t r i b u t i v e l a t t i c e s h a v e b e e n s o l v e d by t h e t o p o l o g i c a l and o r d e r t h e o r e t i c t e c h n i q u e s p r o v i d e d by d u a l i t y , and a r e p r e s e n t a t i v e s e l e c t i o n of these is presented. In addition, certain r e l a t e d d u a l i t i e s a r e b r i e f l y c o n s i d e r e d , as a r e c ompa c t t o t a l l y o r d e r d i s c o n n e c t e d s p a c e s i n t h e i r own r i g h t . The paper ends w i t h a ' d i c t i o n a r y ' of mutually d u a l p r o p e r t i e s .

ENSEMBLES O R D O N N ~ S E T

DUAL IT^

POUR LES TREILLIS

DISTRIBUTIFS

Au c o u r s d e ces d e r n i P r e s a n n g e s , l e s d u a l i t 6 s du t y p e P o n t r y a g i n o n t p r o l i f 6 r 6 . L'u n e d ' e l l e s e s t c e l l e e x i s t a n t e n t r e l a c a t e ' g o r i e des treillis d i s t r i b u t i f s e t b o r n 6 s e t la c a t 6 g o r i e d e s e s p a c e s compacts e t t o t a l e m e n t s g p a r g s pour l ' o r d r e . (Un e s p a c e t o p o l o g i q u e X a v e c une r e l a t i o n d ' o r d r e p a r t i e l g e s t t o t a l e m e n t s 6 p a r 6 p o u r l ' o r d r e s i , q u e l s q u e s o i e n t l e s p o i n t s x,y d e X a v e c LI: $ y , i l e x i s t e un ensemble d g c r o i s s a n t ( c ' e s t - & d i r e , un i d g a l p o u r l ' o r d r e ) U q u i e s t o u v e r t e t ferm 6 e t t e l s q u e z U. y E U.) Le p r 6 s e n t d 6 v e l o p p e m e n t s ' i n t 6 r e s s e 2 l a d u a l i t 6 e n t r e 1) e t P , 2 ses r e p e r c u s s i o n s e t 2 s a p l a c e d a n s l e p a y s a g e m a t h 6 m a t i q u e .

-

-

4

Au n i v e a u d e s o b j e t s , l a d u a l i t 6 p e r m e t d ' i d e n t i f i e r un t r e i l l i s L 6 D a v e c l e t r e i l l i s d e s e n s e m b l e s d g c r o i s s a n t - o u v e r t s e t f e r m 6 s d ' u n e s p a c e XL p o u r XL, o n p e u t p r e n d r e l ' e n s e m b l e d e s i de' aux p r e m i e r s , o r d o n n 6 p a r i n c l u s i o n e t convenablement t o p o l o g i s 6 . On a donc l a g 6 n 6 r a l i s a t i o n n a t u r e l l e s i m u l t a n 6 e d e l a r e p r g s e n t a t i o n d e B i r k h o f f d e s t r e i l l i s d i s t r i b u t i f s f i n i s (oh l a t o p o l o g i e e s t d i s c r P t e e t ne j o u e aucun r 6 1 e ) e t l a r e p r 6 s e n t a t i o n d e S t o n e d e s a l g P b r e s d e B o o l e (oh l ' o r d r e e s t d i s c r e t ) .

6-z;

e t P e t permet d e t r a d u i r e les concepts de Une d u a l i t 6 c a t 6 g o r i q u e p a r f a i t e u n i t l a t h 6 o r i e des t r e i l l i s dans l a langue d e s e s p a c e s topologiques ordonngs. La r e p r s s e n t a t i o n imag6e q u e ces e s p a c e s f o u r n i s s e n t ( b i e n q u e c e l l e - c i ne s o i t p l u s g u P r e c o n s t i t u 6 e q u e d e di agrammes d e Venn s o p h i s t i q u s s ) e s t 6 t o n n a m e n t p u i s s a n t e . E l l e e s t i l l u s t r 4 e p a r un mbl ange d ' a p p l i c a t i o n s e t d ' e x e m p l e s q u i m e t t e n t B nu l a s t r u c t u r e d e s t r e i l l i s e t l a s t r u c t u r e d e s v a r i e t 6 s d ' a l g P b r e s dans

u.

c-f.

En e f f e t , c ' e s t l ' o i d r e q u i d i s t i n g u e L ' o r d r e j o u e un rsle c r u c i a l e n d u d l i t 6 cette d u a l i t 6 de l a d u a l i t 6 equivalente pour m e t t a n t e n j e u les espaces spectraux. L ' a c t i o n r k c i p r o q u e e n t r e l a t o p o l o g i e e t l ' o r d r e s u r un o b j e t d e p e u t 6 t r e a s s e z s u b t i l e . A l o r s qu'on a don& une r 6 p o n s e a c e r t a i n e s q u e s t i o n s c o n c e r n a n t l ' o r d r e ( i l y a , p a r exemple, d e s d e s c r i p t i o n s d i v e r s e s d e ceux d e s beaucoup de problsmes e n c o r e ensembles ordonn6s q u i peuvent d e v e n i r o b j e t s de e n suspens semblent a u s s i a t t i r a n t s qu'6pineux !

c),

H. A . Priestle y

40

1.

INTRODUC'I'LO?:

D u r i n g t h e p a s t t e n y e a r s , d u a l i t i e s o f P o n t r y a g i n t y p e ha ve p r o l i c e r a t e d . One such d u a l i t y i s t h a t between t h e c a t e g o r y o f bounded d i s t r i b u t i v e l a t t i c e s and the category o f compact t o t a l l y o r d e r d i s c o n n e c t e d s p a c e s , w h i c h , a t t h e o b j e c t l e v e l , p r o v i d e s t h e n a t u r a l s i m u l t a n e o u s g e n c r a l i s a t i o n of B i r k h o f f ' s r e p r e s e n t a t i o n o f f i n i t e d i s t r i b u t i v e l a t t i c e s and S t o n e ' s r r p r e s e n L a t i o n o f Bo o lean a l g e b r a s , The f u l l c a t e g o r i c a l d u a l i t y whic h l i n k s a nd allows l a t t i c e t h e o r e t i c c u n c e p t s a n d p r o b l e m s t o b e t r a n s l a t e d i n t o t h e l a n g u a g e of o r d e r e d t o p o l o g i c a l s p a c e s . The f i n a l s e c t i o n of t h i s s u r v e y a s s e m b l e s f o r r e f e r e n c e 111 t h e e a r l i e r some of t h e most f r e q u e n t l y u s e d items i n t h e LLE ' d i c t i o n a r y ' . s e c t i o n s we d e s c r i b e tiow t h e d u a l i t y works and i l l u s t r a t e some of t h e ways i n which i t has b e e n u se d t o r e v e a l t h e s t r u c t u r e of l a t t i c e s i n a nd of c e r t a i n s u b c l a s s e s of In a d d i t i o n we b r i e f l y i n v e s t i g a t e t h e i n t e r a c t i o n b e t w e e n t h e t o p o l o g y and o r d e r i n R :-object .3nd, i n b a r e s t o u t l i n e , we c o n s i d e r t h e g e n e r a l s e t t i n g i n which d i s t r i b u t i v e l a t t i c e d u a l i t y s h o u l d b e p l a c e d .

2

!.

I t h a s n o t b e e n p o s s i b l e i n t h e s p a c e a v a i l a b l e t o p r o v i d e :I f u l l y c o m p r e h e n s i v e survey. I n s e l e c t i n g m a t e r i a l we h a v e t r i e d t o complement t h e a d m i r a b l e a c c o u n t o f d u a l i t y p r e s e n t e d by B . A . Uavcy and D . D u f f u s i n t h e i r p a p e r ' E x p o n e n t i a t i o n a n d d u a l i t y ' p u b l i s h e d i n t h e P r o c e e d i n g s of t h e Ba nff Symposium on O r d e r e d S e t s 1371; o u r a p p r o a c h i s somewhat more t o p o l o g i c a l and p e r h a p s l e s s i n f l u e n c e d by u n i v e r s a l a l g e b r a . We h a v e t r i e d t o make t h e b i b l i o g r a p h y c o m p l e t e i n r e s p e c t o f t h e E-E d u a l i t y a n d i t s a p p l i c a t i o n s t o p r o b l e m s c o n c e r n i n g d i s t r i b u t i v e - l n t t i r r o r d e r e d a l g e b r a s , b u t h a v e n o t s o u g h t t o g i v e full r e f e r e n c e s f o r t h e same problems t r e a t e d a l g e b r a i c a l l y .

We assu me f a m i l i a r i t y w i t h t h e r u d i m e n t s of t h e t h e o r y of d i s t r i b u t i v e l a t t i c e s a s s e t o u t i n 1 1 4 7 , 1 1 9 1 and C 5 4 1 . Our r e p r e s e n t a t i o n s p a c e s w i l l b e o r d e r e d s e t s , appropriately topologised. The n o t a t i o n ( X , C , d w i l l d e n o t e a s e t X e q u i p p e d w i t h a t o p o l o g y e a n d a p a r t i a l o r d e r ,

t Z =

iyEz'I~{>al

& P,

I-& = U [ + x I 5 E Q I . A s u b s e t Q o f P i s s a i d t o b e decreasing ( i n c r e a s i n g ) i f +Q = Q (+Q = Q). 4Q

=

UiCZ

Ia

E

Q),

In t h e l i t e r a t u r e , d e c r e a s i n g s e t s are a l s o c a l l e d o r d e r i d e a l s , h e r e d i t a r y s e t s , i n i t i a l s e g m e n t s , down s e t s , l o w e r s e t s and l o w e r e n d s . I f t' i s a n o r d e r e d s e t we d e n o t e by Pop t h e o r d e r e d s e t o b t a i n e d by r e v e r s i n g t h e o r d e r .

2.

BASIC DUALITY FOR DISTRIBUTIVE LATTICES

The c a t e g o r y h a s a s o b j e c t s t h e bounded d i s t r i b u t i v e l a t t i c e s ; t h e u n i v e r s a l b o u n d s o f a n y L, t i, a r e d e n o t e d by O,1. Morphisms i n 0, a r e t h e 0 , l - p r e s e r v i n g l a t t i c e homomorphisms. W i t h i n a r e two i m p o r t a n t f u l l s u b c a t e g o r i e s : consists of t h e f i n i t e d i s t r i b u t i v e l a t t i c e s and of t h e B o o l e a n a l g e b r a s . A s l o n g a g o a s t h e 1 9 3 0 ' s k e y r e p r e s e n t a t i o n t h e o r e m s were p r o v e d f o r t h e s e s u b c a t e g o r i e s : THEOREM 2 . 1 ( G . B i r k h o f f 1181; o r see L l 9 1 ) . Let L € L&. Then I , i s isomorpJLic to t h c latticc, of decrea::lng .:uhsct:: of t h e S L L J ( L ) o f Join-irreducible elements

of L .

Ordered sets and duality for distributive lattices

41

1

13irI f u r ; I topology the s e t s I(.c,2) .I: E A ] U l i : x 2 (c i s c l o p e n i n A w i t h i t s u s u a l t o p o l o g y } . 'The r e s u l t i n g s p a c e i s made u p o f two c o p i t l s o f A f o r m i n g two l a y e r s , w i t h Lhe d i s c r e t e t o p o l o g y i n d u c e d o n t h e u p p e r l a y e r and t h e u s u a l t o p o l o g y o n t h e l o w e r o n e . The l a y e r s a r e , o f c o u r s c , n o t t o p o l o g i c a l l y d i s j o i n t a n d i t is p o s s i b l e t o show t h a t t l i e d i s t r i b u t i v e l a t t i c e X r e p r e s e n t s h a s no ch.iin hdsc; in f a c t i t i s n o t even a s u b l a t t i c e of I' - l a t t i c e .

I

1

0

The d u a l space o f a Post a l g e b r a L h i i s t h e f o l l o w i n g p r o p e r t i e s : (7:)

(-I-)

XI, i s o r d e r e d a s a d i s j o i n t u n i o n o f maximal c h a i n s ; XI, i s t o p o l o g i s e d iis ;1 f i n i t e d i s j o i n t u n i o n of B o o l e a n s p a c e s .

h a v e b e e n c o n s i d e r e d which s h a r e t o a g r e a t c ' r B e w i l d e r i n g l y many s u b c l a s s e s o f d e g r e e t h e a l g e b r a i c p r o p e r t i e s of P o s t algcsbras (n-valued C u k a s i e w i c z a l g e b r a s , S t o n e l a t t i c e s LIE o r d e r n , ) . A L L , d u a l l y , r e t a i n e i t h e r p r o p e r t y (") o r property (-1). A s y s t e m a t i c s t u d y O F t h e i n t e r a c t i o n b e t w e e n t h e s e c l a s s e s has b e e n u n d e r t a k e n by W . G . Bowen in 1 2 0 1 .

...

From among l a t t i c e s s a t i s f y i n g ( " ) we s i n g l e o u t f o r b r i e f m e n t i o n t h e n-valued f.ukasiewicz a l g e b r a s . The du al s p a c e X I of s u c h a n a l g e b r a i. l o o k s l i k e a Post a l g e b r a d u a l s p a c e 7 x l,=.n. i n w h i c h som; o f t h e l a y e r s h a v e b e e n p i n c h e d L o g e t h c r by a map & : ( : r , L ) ++ i\.(x:), where h . : Z -+ X1 ( i = l , 2 , Y L - ~ i) s c o n t i n u o u s , s u r j e c t i v e , o r d e r - p r e k e r v i n g and s & c h t l i a i r,!] a r e i n c o m p a r a b l e i t and on1.y i f rT ( s ), ii (!y) are incomparab I e

...,

.

.

The r i g l i t , i p p r o a c h Lo c 1 , i s s i F y i n g l , i t L i c e s w i t h p r o p e r t y ( I ) seems t o b e

Lo

mike

Ordered sets and duality for distributive lattices

49

Tire d u , i l s p : i c ~ > oI 'I 1 : i t t i c . c u i t h i s t y p c i s t h e d i s j o i n t u n i o n o f t h e I!oolenn sp.Ic , a r e contained i n t h e i d e a l g e n e r a t e d by A l , A,. T h u s t h e i d e a l w h i c h r g e n e r a t e s i s c o m p a c t 111 L . One c a n e a s i l y g e n e r a l i z e t h i s a r g u m e n t t o show t h a t e v e r y f i n i t e l y g e n e r s t c d i d e a l o f R i s c o m p a c t i n L, a n d c o n v e r s e l y . S ~ n c re v e r y i d e a l i s t h e s u p o f t h e f i n i t e l y generated i d e a l s which i t conta~ns, i t follows t h a t L is a n algebraic lattice. A i l s

...,

T h i s example c a n e a s i l y b e g e n e r a l i z e d t o t h e l a t t i c e o f c o n g r u e n c e s on a n y u n i v e r s a l a l g e b r a , w h e r e i n t h e f i n i t e l y g e n e r a t e d congruences form t h e compact elements of t h e lattice. These examples a r e t h e reason t h a t s u c h l a t t i c e s a r e c a lled a Igebraic. One p r o p e r t y o f K(L) w h i c h w e j u s t u s e d l e a d s t o a c h a r a c t e r i z a t l o n o f a l g e b r a i c l a t t i c e s . Namely, i f L i s a c o m p l e t e l a t t i c e a n d k and k ' are c o m p a c t e l e m e n t s o f L, then k V k' i s a l s o compact. Indeed, l e t D c L be S i n c e k i s c o m p a c t , t h e r e 1 s some d E T) w i t h d i r e c t e d w i t h k V k ' < s u p D. k < d; likewise, since k' i s c o m p a c t , t h e r e i s some d ' 6 D w i t h k ' < d ' . 'Then, s i n c e D i s d i r e c t e d , t h e r e i s some d " E D w i t h d , d ' < d", and i t H e n c e k V k" E K ( L ) . f o l l o w s t h a t k V k ' < d".

1.3 L

LEMMA. If L i s a c o m p l e t e l a t t i c e , t h e n c o n t a i n i n g 0. [1

i s a sup-subsemilattice of

K(L)

We w a n t t o r e t r i e v e L f r o m t h e s u p - s e m i l a t t i c e K ( L ) , b u t t h i s i s n o t a l w a y s p o s s i b l e ( r e m e m b e r t h e l a t t i c e [O,l]). However, i f L i s a l g e b r a i c , t h e n t h i s is possible. N a m e l y , f o r a n y x F_ L , we h a v e x = s u p ( C x n K ( L ) ) . If S i s a s u p - s e m i l a t t i c e , a f i l t e r i n S i s a l o w e r s e t F = CF c S w h i c h i s c l o s e d Thus every point u n d e r f i n i t e suprema. T h u s , Cx n K(L) i s a f i l t e r i n K ( L ) . o f L g i v e s r i s e t o a E i l t e r i n K(L), T h i s f i l t e r i s unique, s i n c e x f y i m p l i e s t h e r e i s some c o m p a c t e l e m e n t o f L b e l o w o n e o f x a n d y , b u t n o t t h e C o n v e r s e l y , s u p p o s e t h a t F c K(L) i s a o t h e r , s o Cx n K(L) f Cy n K ( L ) . f i l t e r , a n d c o n s i d e r x = s u p F ( i n L). T h e n c l e a r l y Cx n K(L) 3 F . On t h e x. S i n c e x = s u p F and k is compact, o t h e r h a n d , s u p p o s e k E K(L) w i t h k t h e r e i s some k ' E F w i t h k < k ' , as F is d i r e c t e d . Rut, t h e n k E F as F i s a l o w e r s e t i n K ( L ) . T h i s s h o w s t h a L F = Cx n K(L). 1 . 4 PROPOSITION. F o r a n a l g e b r a i c l a t t i c e L, t h e map x + Cx n K(L) : L + F i l t K(L) f r o m L t o t h e l a t t i c e o f f i l t e r s o f K(L) i s a l a t t i c e i s o m o r p h i s m . PROOF. We h a v e a l r e a d y shown t h a t t h e map i s a b i j e c t i o n . M o r e o v e r , i f x , y E L, t h e n i t L S o b v i o u s t h a t (Cx n K ( L ) ) A (Cy n K ( L ) ) = Cxy n K ( L ) . On t h e o t h e r Conversely, i f h a n d , c l e a r l y (Cx n K ( L ) ) V (Cy n K ( L ) ) c C(xVy) n K ( L ) . z = s u p ( ( C x n K ( L ) ) V (Cy n K ( L ) ) ) , t h e n x , y C z implies that x V y C z . Since k V k ' < x V y f o r e v e r y k E Cx n K(L) a n d k ' E Cy n K ( L ) , we h a v e z C x V y . Hence, F = (Cx n K ( L ) ) V (Cy n K ( L ) ) i s a f i l t e r i n K(L) w i t h K(L) s i n c e t h e map i s s u p F = x V y , s o (Cx n K ( L ) ) V (Cy n K ( L ) ) = C(xVy) one-to-one. n On t h e o t h e r h a n d , s u p p o s e t h a t w e s t a r t w i t h a s u p - s e m i l a t t i c e S w h i c h h a s a l e a s t e l e m e n t , 0. T h e n , t h e l a t t i c e L = F i l t S o f a l l f i l t e r s o f S i s c e r t a i n l y a c o m p l e t e l a t t i c e when o r d e r e d u n d e r i n c l u s i o n . Moreover, e a c h p o i n t s o f S g i v e s r i s e t o a c o m p a c t e l e m e n t o f L, namely 4s = I t E S : t C s ) . T h i s f i l t e r i s compact. F o r a d i r e c t e d f a m i l y { F i } o f f i l t e r s o f S , s u p F, = UF,, s o , i f Cs C s u p F i , t h e n s E F i f o r some 1, a n d s o C s c F, for that 1. Moreover, i f F i s a n y f i l t e r o f S, t h e n F = s u Cs r e a l i z e s F a s t h e S

s u p o f t h e compact e l e m e n t s

Cs, s

E

F.

€8

' h i s means t h a t

L

is algebraic.

Order scattered and topologically scattered Finally,

if

F

is any compact element of

f i n i t e l y many p o i n t s ,

sl,

...,sn

L,

then

F = su S

in

F

with

6.5

F = s u p Cs,.

Cs

means t h e r e a r e

,F S i n c e

F

is a filter,

1

we h a v e s = "1 V . . . V sn r e s u l t s so f a r a s follows:

1.5

is

also i n

F,

and t h e n

F = 4s.

We s u n m a r i z e o u r

THEOREM. (a) If Id i s a n a l g e b r a i c l a t t i c e , t h e n t h e map x + Cx n K(L) : L + F i l t K(L) i s a n i s o m o r p h i s m o f I, o n t o t h e l a t t i c e of f i l t e r s of K(L), t h r s u p - s u b s e m i l a t t i c e o f c o m p a c t e l e m e n t s o f L. (h)

If S i s a s u p - s e m i l a i t i c e w i t h 0, t h e n t h e map s + Cs : S + K(L) i s a n isomurphism o f S o n t o t h e s t ~ p - s e m i l a t t i c eo f compact e l e m e n t s of L = F i l t S, t h e l a t t i c e of f i l t e r s of S.

PROOF. W e h a v e p r o v e d e v e r y t h i n g e x c e p t t h e f a c t t h a t t h e map s + J.s isomorphism o f s u p - s e m i l a t t i c e s . R u t , t h i s i s c l e a r f r o m t h e o r d e r on = Filt S. fl

is an L

So f a r we h a v e e s t a b l i s h e d a n e q u i v a l e n c e b e t w e e n a l g e b r a i c l a t t i c e s , o n t h e o n e h a n d , a n d s u p - s e m i l a t t i c e s w i t h 0, on t h e o t h e r . Our n e x t g o a l i s L O g i v e a d u a l i t y between a c a t e g o r y o f a l g e b r a i c l a t t i c e s and a c a t e g o r y o f sup-semilatt i c e s w i t h 0. To d o t h i s , w e n e e d t o i n d i c a t e w h a t m o r p h i s m s w e w i s h t o u s e i n each case. For t h e algebraic l a t t i c e s , t h e r e a r e s e v e r a l p o s s i b i l i t i e s , but f o r s u p - s e m i l a t t i c e s w i t h 0, only one p o s s i b i l i t y i s reasonable. N a m e l y , we l e t S d e n o t e t h e c a t e g o r y w h o s e o b j e c t s a r e s u p - s e m i l a t t i c e s w i t h 0, a n d w h o s e With t h e s e morphisms i n m o r p h i s m s a r e s e m i l a t t i c e m o r p h i s m s p r e s e r v i n g t h e 0. h a n d , we now d e t e r m i n e w h a t m o r p h i s m s w e s h o u l d u s e b e t w e e n a l g e b r a i c l a t t i c e s .

Let S a n d T b e s u p - s e m i l a t t i c e s w i t h 0 , a n d l e t f : S + T b e a s u p s e m i l a t t i c e m o r p h i s n p r e s e r v i n g 0. Then, L = F i l t S and M = F i l t T are a l g e b r a i c l a t t i c e s , a n d we c a n d e f i n e F i l t f : M + L by ( F i l t f)(F) = f-l(F). S i n c e F is a f i l t e r and f p r e s e r v e s f i n i t e s u p s , i t follows t h a t ( F i l t f ) ( F ) i s a f i l t e r i n S . Moreover, F i l t f obviously preserves a l l i n t e r s e c t i o n s , which a r e i n f m a i n t h e f i l t e r l a t t i c e . While suprema i n t h e f i l t e r l a t t i c e a r e n o t g e n e r a l l y u n i o n s , c e r t a i n l y d i r e c t e d suprema are, a n d F i l t f c l e a r l y preserves these as well. Thus, we have determined t h a t , €or f : S + T a sups e m i l a t t i c e morphism b e t w e e n s u p - s e m i l a t t i c e s , t h e morphism F i l t € : F i l t T + F i l t S p r e s e r v e s a l l i n f m a a n d a l l d i r e c t e d suprema, a n d c l e a r l y ( F i l t f)(T) = S, so t h a t F i l t € p r e s e r v e s t h e i d e n t i t y a s well. T h u s , w e l e t AL denote t h e c a t e g o r y of a l g e b r a i c l a t t i c e s a n d maps p r e s e r v i n g a l l i n f m a , a l l d i r e c t e d suprema, a n d t h e i d e n t i t y . Suppose t h a t L and M a r e a l g e b r a i c l a t t i c e s a n d t h a t @ : L + M pres e r v e s a l l i n f i m a , a l l d i r e c t e d s u p r e m a , a n d t h e i d e n t i t y . We w a n t t o d e f i n e t h e If k E K(M), t h e n 4k i s a f i l t e r i n M w i t h t h e m a p K ( @ ) : K(M) + K ( L ) . p r o p e r t y t h a t D n 4k f 0 i f s u p D E 4 k , f o r e v e r y d i r e c t e d s u b s e t D c M. S i n c e $r p r e s e r v e s a l l i n f i m a a n d 4 k i s c l o s e d u n d e r a l l i n f m a , i t follows t h a t k ' = A@-l(4k) E @ - l ( 4 k ) . Moreover, s i n c e @ p r e s e r v e s d i r e c t e d suprema, i t i s a l s o t r u e t h a t , i f D c L i s d i r e c t e d w i t h s u p D E @-'(4k), then D n @ - l ( 4 k ) f I?. S a i d a n o t h e r w a y , t h i s m e a n s t h a t k ' = A @ - ' ( t k ) E K ( L ) . So, we d e f i n e K ( @ ) ( k ) = A @ - l ( + k ) . T h i s i s t h e n t h e smallest compact e l e m e n t o f L w h i c h @ maps a b o v e k .

To s how t h a t K ( @ ) i s a n S m o r p h i s m , we m u s t a l s o s h o w t h a t then f i n i t e suprema. B u t , i f k l , k g E K(M),

K(@) p r e s e r v e s

M. Mislove

66

and so

given

p r e s e r v e s f i n i t e suprema.

K(@)

I t i s now a r o u t i n e e x e r c i s e t o v e r Ey t h a t t h e f o l l o w i n g d i a g r a m s c o m n u t e , f : S + T i n S a n d @ : L + M i n A1 :

lS

jL

k

I

where

is(s) = +s,

and

i ( t ) = +t.

T

I

Of c o u r s e , t h e p o i n t we a r e maki ng i s s u n m a r i z e d by t h e f o l l o w i n g :

1.6 THEOREM. L e t S d e n o t e t h e c a t e g o r y o f s u p - s e m i l a t t i c e s w i t h 0 a n d semil a t t i c e maps p r e s e r v i n g 0, a n d l e t AL d e n o t e t h e c a t e g o r y o f a l g e b r a i c l a t t i c e s a n d maps p r e s e r v i n g a l l i n f u n a , d i r e c t e d s u p r e m a , a n d t h e i d e n t i t y a r e d u a l l y e q u i v a l e n t under t h e f u n c t o r s which a s s o c i element. Then S an d a t e t o a n S E S i t s l a t t i c e o f f i l t e r s , and t o a n L E AL i t s sup-semilattice o f compact e l e m e n t s . L1.

TOPOLOGLZLNG ALGEBRAIC LATTLCES

We now h a v e f o u n d a c o m p l e t e d u a l i t y b e t w e e n a c a t e g o r y o f a l g e b r a i c l a t t i c e s a n d a c a t e g o r y o f s u p - s e m i l a t t i c e s w i t h l e a s t e l e m e n t , b u t t h e maps b e t w e e n a l g e b r a i c l a t t i c e s w h i c h a r i s e i n t h i s s e t t i n g a r e somewhat s t r a n g e , a t l e a s t f r o m a p u r e l y a l g e b r a i c v i e w p o i n t . To g i v e a n a l t e r n a t i v e f o n w l a t i o n o f t h i s d u a l i t y , we i n t r o d u c e t o p o l o g y i n t o o u r s e t t i n g . F o r a n a l g e b r a i c l a t t i c e L, we h a v e s e e n t h a t L c a n b e v i e w e d a s t h e l a t t i c e of f i l t e r s on t h e sup-semilattice K(L) o f c ompa c t e l e m e n t o f L. A n o t h e r way t o s a y t h e same t h i n g i s t o r e g a r d e a c h c ompa c t e l e m e n t k o f t h e a l g e b r a i c l a t t i c e L a s g e n e r a t i n g a m o r p h i m @k : L + 2, w h e r e 2 d e n o t e s t h e tw-point l a t t i c e {O,l}. T h i s morphisn i s simply t h e c h a r a c t e r i s t i c f u n c t i o n o f t h e u p p e r s e t o f k , 4 k . Now, t h i s map c l e a r l y p r e s e r v e s a l l i n f i m a , s i n c e 4k i s a p r i n c i p a l f i l t e r f r o m L. Moreover, @k a l s o p r e s e r v e s d i r e c t e d s u p r e m a s i n c e k i s c o mp a c t . F i n a l l y , i t is a l s o c l e a r t h a t = 1.

.

Order scattered and topologically scattered

61

We t h e r e f o r e h a v e t h e map e : L + zK(L) g i v e n by e ( x ) = ( @ k ( x ) )kEK(L), a n d t h i s map p r e s e r v e s a l l i n f i m a a n d a l l d i r e c t e d s u p r e m a . I f w e g i v e 2' the d i s c r e t e t o p o l o g y , a n d t h e n '2K(L) t h e T y c h o n o f f p r o d u c t t o p o l o g y , t h e n 2 K ( L ) i s a compact z r r o - d i m e n s i o n a l s p a c e , a n d t h e image e ( L ) i s c l o s e d i n 2 K ( L ) . R a t h e r t h a n v e r i f y t h i s l a s t d i r e c t l y , l e t u s approach t h e problem from a n o t h e r v iewpo i n t ,

We d e f i n e d t o p o l o g y o n L a 5 f o l l o w s . Let h d e n o t e t h e t o p o l o g y w h i c h a s x r a n g e s o v e r L, together h a s f o r d s u b b d s i s a l l s e t s o f t h e form L\+x, S i n c e 4 k l n 4kg = w i t h a l l s e t s o f t h e form 4 k , a s k r a n g e s o v e r K(L). n

4klvk2

f o r kl,k2

E

K(L), a t y p i c a l b a s i c open set o f L l o o k s l i k e

(4k)\(

U 1= I

fx,),

where k E K(L) d n d x 1 , . . . , x n E L. We w d n t t o d e r i v e t h e p r o p e r t i e s o f t h i s If x,y E L with x y, t o p o l o g y , a n d we b e g i n by n o t i n g t h a t i t i s H a u s d o r f f . t h e n s i n c e L i s a l g e b r a i c , t h e r e i s some c o m p a c t e l e m e n t k E L w i t h k < x but k y ( s i n c e x = s u p ( J . x (1 K ( L ) ) ) . Rut, t h e n x E 4k, which i s open, and y E L\4k, which i s a l s o open, a n d t h e s e s e t s a r e c l e a r l y d i s j o i n t . N e x t , we s h o w t h a t t h i s t o p o l o g y i s c o m p a c t , a n d t o d o t h i s , w e a p p e a l t o A l e x a n d e r ' s Theorem. T h i s m e a n s t h a t i t i s e n o u g h t o show t h a t e v e r y c o v e r o f L by s u b b d s i c o p e n s e t s So, l e t I, = ( U 4 k ) U ( U L \ 4 x ) , where H c K(L) and h a s a f i n i t e subcover.

#

4

XEA

kEH

be a cover of

A c L,

xo ( L\4x f o r any x x E 4kg f o r some k o

L

by s u b b a s i c o p e n s e t s . A, H.

E E

t h e r e m u s t b e some f i n i t e s u b s e t

u L\fxj = L\f(sup F) j=l n ( + u L \ f x j ) U f k o = L, J=l

L\+ko, s i n c e

3

ko

xo = s u

x.

Then

f o r e a c h x E A . T h u s , we m u s t h d v e x, a n d s i n c e ko i s c o m p a c t ,

F = {xl,

n

Let

XEI

s i n c e x < xo Now, xo =

k , s i n c e t k i s c l o s e d u n d e r a l l infma. This w o u l d c o n t r a d i c t t h e f a c t t h a t r E: L \ 4 k , and so we c o n c l u d e t h a t r V a = r f o r some a E A . T h i s s h o w s t h a t r i s a c o m p l e t e p r i m e . The a r g u m e n t g i v e n t o p r o v e C o r o l l a r y 3.3 now a p p l i e s t o show t h a t CSpec L o r d e r g e n e r a t e s L. To p r o v e t h a t CCoSpec L o r d e r c o - g e n e r a t e s L, we a r g u e a s f o l l o w s : y . T h e n , a s w e h a v e j u s t s e e n , t h e r e i s a complete Again, l e t x , y E L w i t h x p r i m e r w i t h y < r, b u t x r. Let s = i n f ( L \ C r ) . Then s r, since r i s a c o m p l e t e prime. Hence s < y , b u t s x. Moreover, we c l a m t h a t s is a complete ccrprime. Indeed, i f A c L w i t h s < s u p A , then A c C r cannot H e n c e we m u s t h a v e h o l d , f o r Jr i s c l o s e d u n d e r a l l s u p r e m a , a n d s $ C r . A n L \ C r # 0. But s i n c e s = i n f ( L \ $ r ) , w e h a v e t h a t t s = L\Cr. Therefore A n 4s # 0, w h i c h s h o w s t h a t s i s a c o m p l e t e c o - p r i m e . The d u a l o f t h e a r g u m e n t g i v e n t o p r o v e C o r o l l a r y 3 . 3 now a p p l i e s t o s h a w t h a t CCoSpec L o r d e r L. co-generates

4

4

1

4

n

of

I f P i s a p o s e t , t h e n I ( P ) = { A c P : A = +A}, P, is a completely d i s t r i b u t i v e a l g e b r a i c l a t t i c e .

t h e l a t t i c e o f lower ends M o r e o v e r , f o r e a c h p E P,

M. Misloue

72

t h e i d e a l Cp i s a c o m p l e t e c o - p r i m e o f I ( P ) . T h i s f o l l o w s from t h e f a c t t h a t t h e supremum o f a n y f a m i l y o f l o w e r e n d s i s j u s t t h e u n i o n , a n d t h e u n i o n c a n and hence a l s o c o n t a i n 4p i f f o n e oE t h e members o f t h e f a m i l y c o n t a i n s p , c o n t a i n s 4 p . T h u s , t h e map p -f Cp : p + CCoSpec I ( P ) i s an o r d e r i s o m o r p h i s m o f P i n t o t h e s e t o f c o m p l e t e c o - p r i m e s o f I ( P ) . R u t , i f A i s c o m p l e t e coi m p l i e s t h a t A = 4 p f o r some p E A , prime o f t(P), t h e n A = s u p { l p : p E A) a n d s o e v e r y c o m p l e t e c o-prim e o f I ( P ) i s o f t h e f o r m Cp. H e n c e , t h e map p H Cp i s a n o r d e r i s o m o r p h i s m o f P o n t o t h e s e t CCoSpec I ( P ) . Olir inrxt r e s u l t s h o w s t h a t t h e converse holds:

4.3

THEOREM. ( a ) For a n y p o s e t order-isomorphism o f P (b)

If

L

P, t h e map p -f Cp : P + CCoSpec I ( P ) o n t o CCoSpec I ( P ) .

is an

i s a c o m p l e t e l y d i s t r i b u t i v e a l g e b r a i c l a t t i c e , t h e n t h e map

x + Cx n CCoSpec 1, : L + I(CCoS pe c L )

i s a complete l a t t i c e

isomorphism. PROOF. Our com m ent s i n t h e p r e v i o u s p a r a g r a p h s e r v e t o p r o v e ( a ) . For ( b ) , l e t I, be a c o m p l e t e l y d i s t r i b u t i v e a l g e b r a i c l a t t t c e . According t o P r o p o s i t i o n 4 . 2 , t h e map x + Cx n P, w h e r e P = CCoSpec I,, i s one-to-one, s i n c e P o r d e r cog e n e r a t e s L. M o r e o v e r , i t i s e a s y t o s how t h a t t h i s map p r e s e r v e s a l l i n f i m a , s i n c e n(Cx, n P) = + ( A x i ) n P. On t h e o t h e r h a n d , we c l e a r l y h a v e U(Cx, n P) c 1

1

1

n P.

C(Vx;)

Conversely, suppose t h a t

1

p

E

C(vxl)

n P.

Th e n , s i n c e

p

is

1

c o m p l e t e l y c o - p r i m e , t h e r e m u s t b e s om e i w i t h p < xi, s o t h a t p E +xi n P c U ( J x; n P I . T h u s , t h e map a l s o p r e s e r v e s a l l s u p r e m a .

That i t

1

IS

onto

LS

obvious.

n

We c a n make t h i s i n t o a c o m p l e t e d u a l i t y i f w e t a k e f o r m o r p h i s m s b e t w e e n p o s e t s t h o s e maps w h i c h p r e s e r v e t h e o r d e r , a n d f o r morphisms b e t w e e n c o m p l e t e l y d i s t r i b u t i v e a l g e b r a i c l a t t i c e s t h o s e maps which p r e s e r v e a l l i n f i m a a n d a l l suprema. In f a c t , t h i s d u a l i t y g e n e r a l i z e s t h e d u a l i t y between complete Roolean a l g e b r a s h a v i n g enough atoms t o c l r g e n e r a t e t h e a l g e b r a , a n d s e t s , which i s i m p l e m e n t e d by t h e a s s i g n i n g t h e p o w e r s e t t o a s e t , a n d a s s i g n i n g t h e s e t o f atoms t o a Boolean a l g e b r a . The d u a l i t y w e g a v e b e t w e e n a l g e b r a i c l a t t i c e s a n d s u p - s e m ~ l a t t i c e si n S e c t i o n I i s not a g e n e r a l i z a t i o n o f t h e s e o t h e r d u a l i t i e s , s i n c e t h e f o r m e r a r e n o t P o n t r y a g i n d u a l i t i e s w h i l e t h e d u a l i t y o f S a n d AL L S a Pontryagin duality. W e c a n s p e c i a l i z e t h e S-AL a s f o l l o w s . Certainly t h e c a t e g o r y oE l a t t i c e s w i t h 0 a n d 1 a n d l a t t i c e m a p s i s a s u b c a t e g o r y o f t h e I f we r e s t r i c t t h e P o n t r y a g i n d u a l i t y b e t w e e n S a n d A L t o t h i s c a t e g o r y S. L o f S, t h e n t h e c a t e g o r y w e o b t a i n a s i t s I m a g e i n AL i s t h e subcategory c a t e g o r y AR o f a r i t h e t L c l a t t i c e s , w h e r e a n a l g e b r a i c l a t t i c e L i s u r i t h m e t i c i f K(L) is a s u b l a t t i c e of L (i.e., K(L) i s closed under i n f s as well as sups)

.

T h i s ends t h e “review” p a r t o f t h i s paper. Almost none o f what we h a v e p r e s e n t e d s o Ear i s new, b u t i s e i t h e r e x p l i c i t l y o r i m p l i c i t l y i n p r e v i o u s w o r k . Namely, t h e d u a l i t y o f a n d AL i s d e s c r i b e d c o m p l e t e l y i n [HMS], a n d s k e t c h e d i n [ C ] . The d u a l i t y b e t w e e n p o s e t s a n d c o m p l e t e l y d i s t r i b u t i v e a l g e b r a i c l a t t i c e s w h i c h we h a v e o u t l L n e d a b o v e i s i m p l i c i t l y i n [ L ] , w h e r e t h e g e n e r a l i z a t i o n i s g i v e n t o t h e c l a s s o f c o m p l e t e l y d i s t r i b u t i v e l a t t i c e s , on t h e o n e s i d e , a n d In f a c t , a l l o f t h e s t r u c t u r e t h e o r y a n d c o n t i n u o u s p o s e t s , on t h e o t h e r . s p e c t r a l theory f o r a l g e b r a i c l a t t i c e s which we have p r e s e n t e d i s t r u e for t h e more g e n e r a l c l a s s o f c o n t i n u o u s l a t t i c e s . These r e s u l t s a r e g i v e n i n d e t a i l i n [C], and are sketched f o r t h e c l a s s of continuous l a t t i c e s i n [ M l ] . The p o i n t ‘-ere h a s b e e n t o p r e s e n t t h e s e r e s u l t s i n t h e s i m p l e r s e t t i n g o f a l g e b r a i c l a t t i c e s , s o t h a t t h e more g e n e r a l t h e o r y o f c o n t i n u o u s l a t t i c e s c a n be more e a s i l y understood. T h a t we h a v e n o t d o c u m e n t e d who p r o v e d w h a t f i r s t i s m e r e l y a

s

Order scattered and topologically scattered

73

m e t h o d t o e x p e d i t e s om e wha t t h e p r e s e n t a t i o n . A very detailed list of historical remarks c a n be found a t t h e end o f e a c h s e c t i o n o f [C], and t h o s e remarks i n c l u d e t h e h i s t o r y o f t h e m a t e r i a l we h a v e j u s t p r e s e n t e d . THE CASE O F COMPLETELY DISTRLBIJTLVE A L G E B R A I C LATTICES

V.

A f t e r s o much p r e p a r a t o r y m a t e r i a l , w e r e c a l l t h e p o i n t o f t h i s w o r k . A s u b s e t A o f a p o s c t P i s order-dense i f , g i v e n a , b E A w i t h a < b, there i s some I: E A w i t h a < c < b. T h e p o s e t P i s t h e n c a l l e d order-scettered i E P h a s no o r d e r - d e n s e s u b s e t s o t h e r t h a n 4 , s i n g l e t o n s , o r a n t i - c h a i n s . A m o m e n t ' s t h o u g h t s h o w s t h a t t h i s i s e q u i v a l e n t t o t h e f a c t t h a t P h a s no o r d e r dense chains. f o r a t o p o l o g i c a l s p a c e X, we s a y X i s topologicdly A oE X h a s a t l e a s t o n e i s o l a t e d p o i n t i n t h e r e l a t i v e topology. The q u e s t i o n wc a r e a d d r e s s i n g h e r e i s t h e r e l a t i o n between t h e s e t w o c o n c e p t s i n a p o s e t L which a l s o h a s a topology. On t h e o t h e r h a n d ,

scattered i f e v e r y s u b s e t o f

DEFINITLON. T h e s p a c e X i s a Compact pOspace i f X is a compact Hausdorff s p a c e t o g e t h e r w i t h a p a r t i a l o r d e r which i s a ( t o p o l o g i c a l l y ) c l o s e d s u b s e t o f

5.1

x

x

x.

5.2 LEMMA. scattered.

A compact pospace

X

i s order-scattered i f i t is topologically

PROOF. S i n c e X i s a c o m p a c t p o s p a c e , w e know t h a t t h e t o p o l o g y o f X h a s a s u b b a s i s o f o p e n u p p e r s e t s a n d o p e n l o w e r s e t s ( s e e , e . g . , "1). This implies Now, suppose t h a t X is n o t t h a t J.x a n d 4 x a r e c l o s e d s e t s f o r e v e r y x E X. order scattered. Then X h a s a n order-dense c h a i n , C. Let Y = { x E X : x = i n f ( 4 x n C\{x})]. T h e n we h a v e Y # 0, a n d w e c l a i m Y h a s nu i s o l a t e d point. Lndeed, l e t x B Y , a n d l e t 11 c X b e a n o p e n s u b s e t oE X c o n t a i n i n g x. S i n c e X i s a c o m p a c t p o s p a c e , i t f o l l o w s t h a t e a c h f i l t e r e d s e t i n X c o n v e r g e s t o i t s i n f m u m , a n d s i n c e x E Y , we c o n c l u d e t h a t U n 4 x n C i s a n infinite set. It i s t h e n e a s y t o f i n d a p o i n t o f Y o t h e r t h a n x i n t h i s s e t , s i n c e C is order-dense. Thus Y h a s no i s o l a t e d p o i n t s , a n d s o X i s n o t topologically scattered. T h e r e s u l t f o l l o w s by c o n t r a p a s i t i o n .

n

We c a n c o n c l u d e f r o m t h i s Lemma t h a t a n y a l g e b r a i c l a t t i c e L i s o r d e r We s h o w t h a t t h e c o n v e r s e h o l d s i n scattered i f i t is topologically scattered. c a s e L i s completely d i s t r i b u t i v e .

5.3 THEOREM. A c o m p l e t e l y d i s t r i b u t i v e l a t t i c e only i f i t is topologically scattered.

L

i s order s c a t t e r e d i f and

PROOF ( P o u z e t ) . We h a v e a l r e a d y s e e n t h a t L i s o r d e r s c a t t e r e d i f L i s topologically scattered. We p r o v e t h e c o n v e r s e by c o n t r a p o s i t i o n . Suppose t h a t L is not topologically scattered. Then L c o n t a i n s a p e r f e c t s u b s p a c e ( i . e . , o n e w i t h o u t i s o l a t e d p o i n t s ) , a n d by c o n s i d e r i n g t h e c l o s u r e o f t h i s s u b s p a c e , w e h a v e a c o m p a c t p e r f e c t s u b s p a c e i n L, c a l l i t X. Now, a c c o r d i n g t o Theorem 4.3, L = I(P), w h e r e P i s t h e p o s e t o f c o m p l e t e c o - p r i m e s o f L. We recursively construct a n order-dense chain i n I(P) as follows. First, for l e t L( p) = { I E I ( P ) : p E I}, a n d l e t I ( - p ) = { I E I ( P ) : p # I}. p E P, R e c a l l t h a t J.p i s a c o m p l e t e c o - p r i m e o f I ( P ) , which means t h a t I ( p ) is a compact-open s u b s e t o f I ( P ) , as i s I(-p), f o r e v e r y p E P. Since the a n d a r e a l l o f t h e f o r m + p f o r some complete co-primes o r d e r co-generate Z(P), p E P, we c a n f i n d a p E P w i t h X n I ( p ) # 0 f X n I ( - p ) . L e t 1112 = J.p f o r t h i s p E P, a n d l e t X' = X n I ( p ) a n d X" = X n I ( - p ) . T h e n X' a n d XI' a r e a l s o compact p e r f e c t s u b s p a c e s o f I ( P ) , a n d s o t h e r e a r e p o i n t s p ' , p" E P w i t h X' n I ( p ' ) # 0 # X' n I ( - p ' ) a n d x" n I ( p " ) # 0 # X" n I ( - p " ) . We t h e n l e t 1314 = 1112 U J.p" a n d I114 = 1112 n + P I . We c a n c o n t i n u e t h i s a r g u m e n t

M . Mislove

74

recursively t o f i n d t h e desired order-dense chain i n K(P), which p r o v e s t h a t I ( P ) = L i s n o t o r d e r s c a t t e r e d . T h i s c o n c l u d e s t h e p r o o f o f t h e Theorem.

n

REMARK. T h i s proof i s due t o Maurice Pouzet. W e h a v e g e n e r a l i z e d i t a b i t by f i r s t i n v o k i n g t h e f a c t t h a t a n y c o m p l e t e l y d i s t r i b u t i v e a l g e b r a i c l a t t i c e i s oE t h e form 1 ( P ) , where P i s j u s t t h e s e t o f c o m p l e t e co-primes o f t h e l a t t i c e . I t i s a simple matter t o s e t t l e Pouzet's o r i g i n a l question ( i . e . , t o determine t h a t o r d e r s c a t t e r e d i s t h e same a s t o p o l o g i c a l l y s c a t t e r e d ) f o r a l l c o m p l e t e l y distributive l a t t i c e s . We r e c a l l t h a t a n y c o m p l e t e l y d i s t r i b u t i v e c o m p l e t e l a t t i c e L i s e m b e d d a b l e i n a p r o d u c t o f u n i t i n t e r v a l s Ix u n d e r a map p r e s e r v i n g a l l Thus, t h e n a t u r a l i n f l m a a n d a l l s u p r e m a ( t h i s i s o r i g i n a l l y d u e t o Raney [ R ] ). compact H a u s d o r f f t o p o l o g y which s u c h a l a t t i c e c a r r i e s ( a n d r e l a t i v e t o which i t i s a compact t o p o l o g i c a l l a t t i c e ) i s t h e i n h e r i t e d t o p o l o g y from Lx. The l a t t i c e L i s t h e n a l g e b r a i c p r e c i s e l y when i t c a n b e s o embedded i n a p r o d u c t ZX, a n d , t o p o l o g i c a l l y , t h i s means precisely t h a t t h e t o p o l o g y o n L i s t o t a l l y d i s c o n nected. The p o i n t i s t h a t a c o m p l e t e l y d i s t r i b u t i v e complete LaLtice which i s n o t a l g e b r a i c m u s t h a v e a n o n - d e g e n e r a t e c o n n e c t e d c o m p o n e n t , a n d we c a n t h e n i n v o k e K o c h ' s A r c Theorern ( s e e [ C ] , p . 2 9 9 ) t o p r o d u c e a n o r d e r - a r c i n t h a t c o m p o n e n t . R u t , t h i s a r c i s a n order-dense s u b s e t , a s w e l l a s a p e r f e c t s u b s p a c e o f L, and s o L i s n e i t h e r o r d e r s c a t t e r e d n o r t o p o l o g i c a l l y s c a t t e r e d ~n t h i s c a s e .

V1.

THE QUESTLON FOR ALGEBRAIC LATTICES L N G E N E R A L

We h a v e j u s t s e e n t h a t a c o m p l e t e l y d i s t r i b u t i v e a l g e b r a i c l a t t i c e i s o r d e r We now p r e s e n t a n e x a m p l e s c a t t e r e d i f and only i f i t i s t o p o l o g i c a l l y s c a t t e r e d . t o show t h i s i s n o t t h e c a s e f o r a l g e b r a i c l a t t i c e s i n g e n e r a l . To b e s u r e , Lemma 5 . 2 shows t h a t a n y a l g e b r a i c l a t t i c e w h i c h i s t o p o l o g i c a l l y s c a t t e r e d m u s t a l s o b e order scattered. T h e f o l l o w i n g e x a m p l e shows t h e c o n v e r s e i s f a l s e :

6 . 1 EXAMPLE. Let C A = {I l / n : n > I}

denote t h e u s u a l middle t h i r d Cantor s e t , and l e t { l } . We g i v e L' = C x A t h e f o l l o w i n g o r d e r : (cl,al) < ( c z , a 2 ) i f f a l , < a 2 o r a 1 = a 2 a n d c l = c 2 . We c a n t h e n d e f i n e t h e f o l l o w i n g F o r e a c h n > I , l e t R, = {((c~,l-l/n),(c2,l-l/n)) : f a m i l y o f r e l a t i o n s o n L'. ( c l - c 2 ( < 1/2n-1}. I f w e t h e n l e t R = U Rn U A ( L ' 1 , t h e n R i s a n e q u i v a l e n c e n r e l a t i o n on L', and L ' / R = L is a p a r t i a l l y ordered s e t under t h e q u o t i e n t t o L, then the resulting poset is order. I f w e a d j o i n a n i d e n t i t y e l e m e n t 1' t h e c o m p l e t e a l g e b r a i c l a t t i c e g i v e n i n t h e f o l l o w i n g diagram:

-

U

0

I'

6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

01' 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

c

0 0

L

The r e a s o n t h a t L U form [ c , a ] w i t h a

11')

have

is not a gap.

-( s i~n c e t h e j o i n o f

.

B*)

Dk

Now l e t

Ak+l

x

~

=

i s a g a p a n d , by t h e m i n i m a l i t y o f

\%+ll

=

\A(

.

-h a v~e b e e n o b t a i n e d . 1 in

Then

x

f

x

,- z n~

E

zn

~

For

~ v - z k~+

We now d e f i n e a j o i n

L.

~

.-

By ~ t h e p r o p e r t i e s of

w e can choose

An-l,

5

15 k

k ,

n - 1 ,

x ~ - ~z , ~

and t h e f a c t t h a t zk

= /A\

IDk\

Since

is again i n

n

An

i\i

=

We may a s s u m e

x

z

(Do(

Thus, and l e t

"1:.

and

Suppose t h a t

Z k c D~

,

B,

11

Let

Since

have b e e n o b t a i n e d and (A,,B) is a minimal k- 1 Apply s t a t e m e n t t o (\,B) t o o b t a i n t h e set

i t i s a minimal gap w i t h

Choose

,A.

g.

x

i n d e p e n d e n t s u b s e t of s i z e

Dk

DB

E

i s a minimal gap and

(Ai\(xk])

such t h a t

5A.

i s a s u b s e q u e n c e oT

C

p r o v i d e d by

cf(/AOI)

is minimal.

(A;),B)

and

.

IAl

and any e l e m e n t of

given i n

F

is singular.

t h e p r o p e r t i e s g u a r a n t e e d by

,

all finite

have t h e p r o p e r t i e s p r e s c r i b e d i n

Choose

\

=

(q,B)

We c h o o s e

of

m)* f o r

.

/A\

=

i\i

gap w i t h

(A,,B)

C

i s a g a p a n d , by t h e m i n i m a l i t y o f

Suppose t h a t

Then

,

]A]

F y A.

for all f i n i t e

lAl ,

]A\

and l e t

A

=

(Do,B)

with

W),

b e t h e s u b s e t of c a r d i n a l i t y

(AO,B)

Then

o f c a r d i n a l i t y the

D

regular.

A.

i A;

i A'

D

such t h a t

of c a r d i n a l i t y

A'

P r o o f of ,D. s u p p o s e t h a t

Do

,

T h i s i s o b v i o u s when o n e c o n s i d e r s t h a t

lxcllU c I A l j

,D.

2

C

P r o o f of ,C.

,

1Al

v

... v

z

.

+

z n-1

Dn-l tI Dn-1

we u s e t h e p r o p e r t i e s

..., ~ , zn

Finally, pick

% 20

t o choose

Dg

-

~

Representing ordered sets by chains

. ,. v z . I f z k 5 (z, v z1 v . . . z ~ - ~v )(zk+ ,.v z ~ D3; B, a n d xo 5 x1 2 ... - xk-1 , 'k 5 X k - l is a contradiction. Therefore, (zo.zl,. .. ,zn}

such t h a t

zo

p

z1 v z 2 v

v+

V

x.

:

1

This

has breadth a t least

L

+

n

1.

(A,B)

then because

"'

is j o i n i n d e p e n d e n t , and

1Al

Hence,

L .

Zn

n ,

is regular.

c o n t a i n s a l e f t i r r e d u c i b l e gap.

B.

Proof of

zn)

IJe c a n a c c o m p l i s h t h i s f o r a n y

c o n t r a d i c t i n g t h e f i n i t e b r e a d t h of

E.

.. .~ v

'k+ 1

1-1

1

93

Suppose t h a t

(A,B)

d o e s n o t c o n t a i n a l e f t i r r e d u c i b l e gap.

A

c o n t r a d i c t i o n i s o b t a i n e d by c o n s t r u c t i n g j o i n i n d e p e n d e n t s e t s o f a r b i t r a r y f i n i t e size. (Ao,Bo) = (A,B)

Let

and l e t

(A',B ) 0 0

(A',B ) i s n o t l e f t i r r e d u c i b l e t h e r e e x i s t

0

Co

0

that

lCol

,

=

xo t C* n Bl,( 0

(A;,B1)

and l e t

i s a gap.

(A1,B1)

A

1

Since

i s a minimal gap.

is a gap, and {xo}

.

il

(AC,Bo)

minimal,

Moreover,

y A;

and

Because lAll

that

(A,,Bk)

obtain and

Bk+l

( C k , B k + 1)

Let

A,+1

.

5 Bk

such

(Ab,Bl) and

lAOl

such t h a t

is n o t a gap. = (q\(xk])

lCkl

and

1x1, xk

Choose txk}.

1

=

Choose i s a gap,

(A1,Bl)

d e l e t e d from t h e " l e f t "

x

~

-h a v~e b e e n d e f i n e d s o

Apply

i\ t o

(Ak,Bk)

Since t h i s i s not l e f t irreducible there e x i s t

l e f t i r r e d u c i b l e subgap,

(Ai,Bk+l)

C* k

t

Since

II

w e may assume

xk

Suppose t h a t

,

E

with

lCn-ll

such t h a t

=

(A ,B ) n n

so

zn - l - # xn-*

n-1.

zk

statement zn.

V

E

Ck

Finally, pick

a s i n t h e p r o o f of

n,

{zo,zl

~ e c a u s e x n-2'

C

Once

~

An-l

zk

z o E Co

with

1

... z

+ z~k +, * ,

such t h a t

,... , z

zn

enables u s t o choose

z

x zo

~ - ~ .

h+lI

h a s t h e same p r o p e r t i e s a n d

h a s been d e f i n e d i n t h i s manner. xnTl.

,

lAn-ll

w e can again choose

15 k 5

zn

1x

i s a m i n i m a l g a p w i t h no

(A,',Bk+l)

(A,+l,Bk+l)

%

i s a g a p , and

;

Bk+l+

to

'k

=

-

b,I zn

(Ak,Bk)

c o n t a i n s no l e f t i r r e d u c i b l e gap.

(A,',Bk)

=

xo

.

(Ao,Bo)

Suppose t h a t t h e minimal gap

y Bo

B1

does not contain a l e f t i r r e d u c i b l e

(Al,Bl)

g a p as s u c h a g a p would b e a s u b g a p ( p o s s i b l y w i t h s i d e ) of

Since

is n o t a gap.

(Co,Bl)

= (A;)\(xo]) 16

A.

be a s i n statement

~ v- z k~ + l z1

V

z2

Choose and

Cn-1

z

E

n-1

5 AA-1

C

n-1

have been defined V

... v

z

V

...

z

is j o i n independent.

V

for

.

Just

D. Duffus,M. Pouzet

94

The proof of Theorem 4 . 1 i s complete. N.B. The p r o o f s of D and E a r e s i m i l a r and c o u l d be p a t c h t o g e t h e r . For a b e t t e r understanding w e prefered t h i s presentation. I t remains t o be s e e n i f a l l l a t t i c e s of f i n i t e b r e a d t h a l s o have t h e s e l e c t i o n p r o p e r t y . I n t h e n e x t s e c t i o n we p r e s e n t some consequences of t h e C-gap p r o p e r t y and t h e s e l e c t i o n p r o p e r t y .

5.

ch

REPRESENTATIONS I N

C h has a r e p r e s e n t a t i o n by i r r e d u c i b l e

Here we show t h a t every member of

ck.

c h a i n s , and t h a t t h e s e c h a i n s a r e t h e only i r r e d u c i b l e members o f For a s u b s e t

of an o r d e r e d set

X

segment g e n e r a t e d by A s above,

cf(lX\)

and

X ,

[X)

(XI

w e use

P

f o r t h e f i n a l segment g e n e r a t e d by

1x1 ,

and, a s

.

Also,

Pd

d e n o t e s t h e d u a l of

Let

let -

(A,B)

P

--

A

(iii)

C

(iv) Let

p = cf(lB1)

.

5 (C] 5 B,, =" c f ( / A ( ) ,

&z

(A,B)

such t h a t

and

l B B l < IB\

for a l l

The gap

(A,B)

Thus,

, sa

[D)

5

Sa

P

.

which p r e s e r v e s

and l e t

P

,

< Sa'

I n o t h e r words, d A @ p .

A) ,

A = U (Aala
. o r d ( r ) l

r = 2-",

n

2

ord(s)}

.

.

111

Some free m-lattices on posets

We s h a l l use

and

nl

t o denote t h e f i r s t and second p r o j e c t i o n s of

n2

J2

J . For t E J , l e t us c a l l t h e set of a E A t h a t s a t i s f y n , ( a ) t x = t l i n e i n A , and d e f i n e t h e y z t l i n e similarly. The a l t e r n a t e s l i n e s i n A. In d e f i n i t i o n s of A y i e l d e x p r e s s i o n s f o r t h e x = r and y p a r t i c u l a r , < r , r + 2- Ord(') > is t h e l a r g e s t element on t h e x = r l i n e , and <s - 2- o r d ( s ) , s> is t h e s m a l l e s t element on t h e y = s l i n e .

onto

the

Each a formulas a r e :

For example, 1>1

denoted by

has a r i g h t lower cover

a*

and a r i g h t upper cover

a*.

The

< ( r + s ) / 2 , s> < r , ( r + s ) / 2 >.

< r , s>* < r , s>, { = < r

-

2-ord(r), s>.

-ord ( s ) > S i m i l a r l y , t h e l e f t upper cover * < r , s> e x i s t s and e q u a l s < r , s + 2 when o r d ( r ) < o r d ( s ) . Observe t h a t has no l e f t (lcwer or upper) c o v e r , b u t t h a t every o t h e r element o f A h a s e x a c t l y one l e f t cover. Moreover, each cover o f an element a E A l i e s on t h e x-line through a or t h e y-line through a .

FIGURE 4.

The l a t t i c e

A

G. Gratzer, D. Kelly

I12

2 I n Figure 4 , A seenis t o be a very s p a r s e subset of J For each a = < r , s> E A, l e t us c a l l the closed i n t e r v a l [ r , s ] ( i n e i t h e r t h e r e a l s or dyadics) t h e shadow of a . We say t h a t two closed i n t e r v a l s overlap i f they have an i n t e r v a l of nonzero length i n common, but n e i t h e r i n t e r v a l contains t h e o t h e r . The " r e l a t i v e sparseness'! of A is expressed by t h e following shadow p r i n c i p l e : The shadows of two elements of A cannot overlap. Let u s i n d i c a t e an easy way t o v i s u a l i z e shadows. W e add a v e r t i c a l l i n e ( t h e shadow l i n e ) t h a t passes through t h e i n t e r s e c t i o n of the x = 1/2 and y = 1/2 lines on t h e r i g h t s i d e of A i n Figure 4 ; index t h i s l i n e i n t h e obvious way w i t h 10, 11. I f one places an opaque r i g h t angle a t a E A (forming t h e x - l i n e and t h e y-line through a ) , then t h e i n t e r s e c t i o n of t h e r i g h t angle with t h e shadow l i n e g i v e s t h e shadow of a .

.

W e found t h e shadow p r i n c i p l e t o be a very n a t u r a l t o o l i n our proofs. Let u s apply it t o "explaint1 t h e emptiness of t h e l a r g e s t open square i n Figure 4: i f t h e r e was an element a i n t h i s square, then t h e shadows of a and would overlap.

I n Figure 4 , the d i s t a n c e of each element < r , s> E A from t h e shadow l i n e is proportional t o s r , t h e length of its shadow. Consequently, f o r two incomparable elements a , b E A , we can define a t o be t o t h e l e f t of b i f f

-

has a longer shadow than b. Observe t h a t a is t o t h e l e f t of b i f f < n,(b) iff n2(a) > n 2 ( b ) . With t h i s concept, we can v i s u a l i z e t h e 1 f i n i t a r y j o i n i n A. Let a and b be incomparable elements of A , with a t o t h e l e f t of b . The j o i n of a and b is t h e l e a s t element on t h e y - l i n e through a t h a t is g r e a t e r than b. Consequently, a V b an f o r ttie l e a s t n

a

n (a)

such t h a t an > b , where an is inductively defined by: a. T h i s shows t h a t A is a l a t t i c e .

3.

THE COMPLETE LATTICE D(

a , ai+l = ( a i ) *

.

t?t)

W e build up D( m ) from t h r e e d i s j o i n t s u b l a t t i c e s : l a t t i c e A was defined i n Section 2. We d e f i n e B = { < r , s> I < s , r >

E

A , B , and C.

The

A],

a subposet of J2. Clearly, B is a l a t t i c e and its diagram is obtained by r e f l e c t i n g Figure 4 about a v e r t i c a l l i n e . I n o t h e r words, mapping < r , s> E A to <s, r > E B d e f i n e s an isomorphism of A with B. Consequently, every element has

b

of

B

has l e f t covers

a r i g h t lower cover

b*

or

a

*b

and

*b,

and i f

r i g h t upper cover

b z < 1 , 0 > , then

b

b*.

F i n a l l y , l e t I be t h e r e a l i n t e r v a l [O, 11, and r e c a l l t h a t J denotes t h e s u b s e t of I c o n s i s t i n g of dyadic r a t i o n a l s . For each t E J , we t a k e a and generators a t , a ; , pt, copy Ct of C( m ) , w i t h bounds yt and y t ,

p i , For each t E I two-element chain with

t

E

I.

Since

I

which is riot a dyadic r a t i o n a l , Ct = [y,, y i ] is t h e yt < y.; W e define C a s t h e l i n e a r sum of t h e C t ,

is complete and each

Ct

is complete,

C

is a complete

lattice. To d e f i n e C ( m ) , ( s e e Figures 5 and 6):

we must describe t h e p a r t i a l ordering on

A u b u C


FIGURE 5.

The complete l a t t i c e

FIGURE 6.

Cetails of

fi( m )

113

D(

rn)

G. Gratzer, D. Kelly

114

If

E

A,

< r , s> < r , s> < r , s> < r , s>

< t , u>

>

< t , u> E B, v E I , and < t , u> i f f s < u; < t , u> i f f r > t ; p i f f s < v holds o r s p i f f r > v holds o r r p i f f t < v holds or t p i f f u > v holds o r u

I t is e a s i l y seen t h a t

p

= = = =

Cv,

then:

v v

and

av

_c

p

hold;

and

a' 2 V

p

hold;

v

and and

pv p{

5

p

2

c

hold; hold.

E

v

is a poset.

D(m)

However, since 1 < 1 , 1 / 2 > , v was assumed t o be a dyadic pv, o r p.; Tnus, i f v is not

2.

A u B is not a subposet of J Note t h a t r a t i o n a l wherever we used t h e notation a v , a ; ,

dyadic, then i n each of the l a s t four c a s e s , only t h e f i r s t c l a u s e can apply.

I t is not d i f f i c u l t t o show t h a t C( m ) is a l a t t i c e , and t h a t each of A , B, and C is a s u b l a t t i c e of D( m ) . For < r , s> E A , < t , u> E B , v E I , and p E C v , we give t h e formulas f o r joining p a i r s : (a)

< r , s> v p

is

( i ) as v p

E

(ii) < r , s>,

(b)

<s, r>

C , where t h e j o i n is formed i n

if

r

>

v

r

or

v

w

and

(iii)

t h e l e a s t <w, s> such t h a t w > v , i f

(iv)

t h e l e a s t <w, s> such t h a t w t v , i f

p

C, i f

r r

5 5

s

5

v v

v < t , u> i s ( i ) < t , u>, i f s < u; ( i i ) < r , s>, i f t < r ; ( i i i ) the l e a s t <w, s> on t h e y s l i n e i n A such t h a t w ( i v ) t h e l e a s t < t , w> on t h e x = t l i n e i n B such t h a t w ( v ) as V ps, i f s = t , where t h e j o i n is formed i n Cs. There is an automorphism B, and interchanges a t and

E

@

pt

D( m )

of (a;

and p i )

v;

i n Cv;

5 a;

t h a t maps f o r any

p .k a; i n Cv; p 5 i n Cv;

> >

t , i f s >t; s, if s >t;

< r , s> E A t o t E J . In o t h e r

words, $ r e f l e c t s Figure 5 a b u t i t s c e n t r a l a x i s . With t h i s observation, t h e formulas f o r < t , u> v p follow from ( a ) . The pairwise meet formulas follow by duality. In order t o show t h a t D( m ) is a complete l a t t i c e , it s u f f i c e s t o f i n d X f o r any nonempty subset X of A. (The formula is s i m i l a r f o r B and we already know t h a t C is complete.) Let X, arid X2 be t h e f i r s t and t h e second p r o j e c t i o n s of X , and form u = v X1 and v = 'd X2 i n I .

v

If

u < v , then v E J , and v X is t h e l e a s t element o f y = v l i n e whose f i r s t coordinate is 2 u.

If

u

D(

m

1

v , then

-

{yo,

v yil

X

=

yu

if

uw v

ciu

if

u

is m-generated

To prove t h i s , one generates t h e four C,

-

{y,,

yi].

ao,

Po,

6

X2. , < 1 , 0>, a { , p i .

m-generators of

Then t h e four elements of

on t h e

u L X2;

and

= v and u

by

A

A u B

Co

-

{yo,

ydl

and

of o r d e r 1 a r e generated.

One then m-generates C l I 2 - {y1,2, y11/21 2 L. W e alternately m-generate Ct {yt, y;]. A t each s t e p , we increment t h e order elements of A u B and

-

115


t

V(at

yi

t.

A u B, and the order of

of elements of

t

J

and

t

Finally, for any nonzero

In the remainder of t h i s section, we discuss m). An element a = < r , s> of A i s join reducible) i n C ( m ) i f f o r d ( r ) > o r d ( s ) . (Such lower cover a*, and a, 2 x whenever a > x i n element of A i s doubly m-reducible i n D( m ) .

x

Ct,

E

x = t

6

x x

E

x

E

yt

t

E

and

E

I,

i).

n2(b) = n 2 ( y ) .

Similarly,

n,(y)

2 and

2.1.

f i n i t a r y o p e r a t i o n s on A ( i . e . maps An --tA

- =CA;F> w h e r e F is a p a i r A

so. A

for n =O,l,

0

... ) .

a set of An a l g e b r a

t e r m ( o r a l g e b r a i c ) o p e r a t i o n of

&

( o r F ) i s a n o p e r a t i o n o b t a i n e d from F v i a ( r e p e a t e d ) c o m p o s i t i o n and < < F > > s t a n d s f o r t h e s e t of t e r m o p e r a t i o n s o f F . (The c o m p o s i t i o n c o n -

s i s t s o f p r o j e c t i o n s , p e r m u t a t i o n s and f u s i o n s of v a r i a b l e s a s w e l l a s , x n ) by g ( x . , , . ,xm) p r o t h e replacement of t h e i - t h v a r i a b l e of f ( x , ,

. ..

..

d u c i n g t h e o p e r a t i o n f ( x , , . . . , X ~ - ~ , ~ ( X ~ , . . . , X ,~X+~ ~+ -, , ~, ,). . . , X ~ + ~ - ~ ) ) . For i n t e g e r s n . i and t h e n - a r y for all xl,

'

1 and a r A t h e r e a r e t h e i - t h n - a r y p r o j e c t i o n e r

c o n s t a n t cn d e f i n e d by a ct(xl, ,xn) = a ,xn) =xi, er(x and 5 d e n o t e t h e s e t s o f a l l p r o j e c t i o n s xniA. Let

,,...

...,

...

and c o n s t a n t s . The s e t

i s c a l l e d t h e set of polynomials

of fi - or F and d e n o t e d P ( A - ) o r P ( F ) . ( N o t e t h a t P ( F ) may b e r i c h e r t h a n b e c a u s e w e may u s e p r o j e c t i o n s and c o n s t a n t s ) . A c l o n e i s s u b s e t M of

0 such

that C<MUI>>

=

M ( i . e . i t i s composition c l o s e d

and c o n t a i n s a l l p r o j e c t i o n s ) . The s e t

&

o f c l o n e s , o r d e r e d by i n -

c l u s i o n , i s known t o b e a n a l g e b r a i c l a t t i c e . F o r A f i n i t e t h e l a t -

tice

5

h a s a f i n i t e number o f d u a l a t o m s , c a l l e d maximal

o r p r e c o m p l e t e ) c l o n e s and e a c h p r o p e r s u b c l o n e o f

(preprimal

0 extends

to a

maximal c l o n e . Our a p p r o a c h i s b a s e d o n t h e c o n c e p t o f p r e s e r v a t i o n of a r e l a -

2.2.

t i o n . F o r h p o s i t i v e i n t e g e r a n h-ary r e l a t i o n p i s j u s t a s u b s e t o f Ah.

I f f i s an n-ary o p e r a t i o n on A w e s a y t h a t f p r e s e r v e s

compatible with

p )

whenever a l l ( x l j

p

(or is

if

,..., x h j )

subalgebra of t h e h-th

( j =I,...,n) d i r e c t power < A ; f >h ) . Ep

(equivalently, i f P is a I n t h i s paper preserva-

t i o n w i l l be a p p l i e d almost e x c l u s i v e l y t o b i n a r y r e l a t i o n s o n l y . For example, f p r e s e r v e s an o r d e r ever x

c

yl,...,xn

2

if f(x,,

... , x n )

y n , i . e . i f f i s :-monotone.

f ( y l , . . . , y ,)

when-

Naturally

4 =

If A - Dreserves a r e f l e x i v e graph p (= symmetric b i n a r y r e l a t i o n c o n t a i n i n g o = { ( a , a ) : a C A I ) w e s a y

preserves p i f every f F F does. that

p

is a tolerance of

fi.

A t r a n s i t i v e t o l e r a n c e of

5

is a congru-

e n c e o f A. - I t i s a n e q u i v a l e n c e o n A s u c h t h a t t h e v a l u e s of e a c h

f r F s t a y w i t h i n t h e same b l o c k w h e n e v e r t h e v a r i a b l e s v a r y e a c h i n s i d e a block1.A r e f l e x i v e g r a p h p + A 2 i s l o c a l l y c e n t r a l

i f f o r each

Compatible orderings and tolerances

125

f i n i t e s u b s e t B o f A w e h a v e { a ] x B c p f o r some a C A .

I t is central

i f { a ) x A ~ from p some a F A . L e t _R d e n o t e t h e s e t of f i n i t a r y r e l a t i o n s on A . F o r p C E l e t

2.3. Pol

p

be t h e set of f C Q preserving p .

F o r R_'_R p u t P o l R := (-1

Polp.

I n t h i s p a p e r w e u s e o n l y f i n i t a r y r e l a t i o n s and t h e r e f o r e i t P C R by a c o a r s e r o n e .

i s c o n v e n i e n t t o r e p l a c e t h e c l o s u r e x ---t> on An n - a r y

f i s a l o c a l t e r m o p e r a t i o n of & - i f f o r every f i n i t e subset

B of A the r e s t r i c t i o n f t i o n g o f &.

I' B

agrees with g

rB

f o r some t e r m o p e r a -

The s e t o f l o c a l t e r m o p e r a t i o n s o f

&

i s d e n o t e d LOC

9

o r LOC F. Thus f b e l o n g s t o LOC F i f f t h e f i n i t e " p i e c e s " o f f a r e a l l among t h e f i n i t e p i e c e s o f o p e r a t i o n s f r o m < C F > > . W e s a y t h a t F

i s a local c l o n e i f L OC F = F . W e a s k t h e r e a d e r t o a c c e p t t h e f o l l o w i n g f a c t s and n o t i o n s . 2.4. some

2.5.

Proposition.

F_cO i s a l o c a l c l o n e i f and o n l y i f F = P o l R f o r

RcR.

For F s G p u t Inv F := { @ E R : f

a nonemptv f a m i l y

p r e s e r v e s o f o r a l l f F F 1 . Given

: j C. LJ) o f h . - a r v

r e l a t i o n s !P E J 3 (jiI J ) , a n i n I I j a nositive integer h, i l f . . . , i€ h1 and mans

dex set I * @ ,

{cp.

,...,

n . : 11 h . ) -I (j € J ) o u t 7 I I u :={(a(il) cx(ih)) :a € A

,...,

j EJ]

v1

J

for all

1

The s i m n l e s t and q u i t e t y p i c a l example i s t h e r e l a t i o n a l

(2.1).

product

c ~ ( n ~ ( ..., 1 ) ) a~( n j ( h . ) )€ v .

01p2

of t w o b i n a r v r e l a t i o n s d e f i n e d by

{ ( x , y ) : ( x , u ) Ecp,,

ipl 0 i p 2

=

( u , y ) E ( 0 2 f o r some u l . To see t h a t i t i s o f t h e

form ( 2 . 1 ) p u t I = { l f 2 f 3 1 , i , = 1 , i 2 = 3 n 1 ( 2 ) = 2 , n ( 1 ) = 2, n ( 2 ) 2 2 be c a l l e d a r e s o l v e n t of { L P

=

j

J = 1 1 , 2 1 and n

1

(1) = 1,

3. A r e l a t i o n o f t h e form ( 2 . 1 ) w i l l

:j E J I .

For I f i n i t e a r e s o l v e n t i s ob-

t a i n e d t h r o u g h a l o q i c a l f o r m u l a b u i l t up e x c l u s i v e l y from t h e p r e d i c a t e s c o r r e s p o n d i n a t o @ . ( jt J ) , e q u a l i t y , c o n j u n c t i o n and '1 ( t h e use of v

7

,7

a n d V i s n o t a l l o w e d ) . W e a s k t h e r e a d e r t o accept t h e

f o l l o w i n g p r o p o s i t i o n ( c f [ 2 8 1 ) which is l i s t e d h e r e f o r comDleteness s a k e and whose a p n l i c a t i o n s w i l l b e q u i t e t r a n s n a r e n t . 2.6.

P r o p o s i t i o n . L e t R g J 3 and F = Pol R .

lently,

Pol

p

2 F ) i f and o n l y i f

p

Then P E I n f F l o r e q u i v a -

i s t h e union o f a d i r e c t e d f a m i l y

of r e s o l v e n t s o f s u b s y s t e m s o f R U { A } .

I.G. Rosenberg, D.Schweigert

126

2.1.

A local clone F i s l o c a l l y m a x i m a l

i f F c j ? and F c G c O f o r no

l o c a l c l o n e G and f i n a l l y H G O i s l o c a l l y p o l y n o m i a l c o m p l e t e , i f Loc(H U C ) = 0. F o r A f i n i t e c l e a r l y LOC F = t < F > > and t h u s w e s h a l l drop t h e a d j e c t i v e "local". A polynomially complete a l g e b r a i s also c a l l e d f u n c t i o n a l l y complete, p r i m a l w i t h c o n s t a n t s o r S h e f f e r w i t h

constants.

2/3-MAJORITY OPERATIONS

63

3.1. ( )

A t e r n a r y o n e r a t i o n ( ) on A i s a 2 / 3 - m a j o r i t y

operation i f

* e 32 and (xxy) = x = (yxx)

(3.1)

h o l d s f o r a l l x , y € A . A s w e s h a l l see t h e 2 / 3 m a j o r i t y o p e r a t i o n s a r e q u i t e common. The f o l l o w i n g i s o u r s t a r t i n g p o i n t . 3.2.

Theorem.

Let

a-

=

be a f i n i t e a l g e b r a w i t h a

2/3 m a j o r i t y

polynomial. Then

i s p o l y n o m i a l l y c o m p l e t e i f a n d o n l y i f fi - is s i m p l e ,

A

A-

is

f o r no b o u n d e d o r d e r 5 a n d A - h a s no c e n t r a l t o l e r a n c e (as 2 i s s i m p l e i f it has o n l y t h e t r i v i a l congruences w and A

S-monotone

&

usual, and

C

i s b o u n d e d i f i t h a s a l e a s t and a g r e a t e s t e l e m e n t ) .

P r o o f : W e a p p l y [ 271 Thm 2 . L e t < A ; +

, O> b e a n a b e l i a n g r o u p and

l e t m d e n o t e t h e t e r n a r y o p e r a t i o n on A d e f i n e d by s e t t i n g m ( x I y I z ) =

x -y

+z

for a l l x f y l z € A . Further put

mo = I ( x l y , z I x - y + z ) : x l y l z E A I . I t s u f f i c e s t o show t h a t ( ) does n o t p r e s e r v e m0. S u o a o s e i t d o e s . L e t x , y , z € A be a r b i t r a r y . Since a l l t h r e e quadruples ~Y-Z,OfX-Y+~fX~f~Y-Zf~fzfy~f~ololzlz~

b e l o n g t o m0 and ( ) i s a 2 / 3 m a j o r i t y o p e r a t i o n , w e g e t 0

( y - z l O , z f ( x y z ) ) E m , i . e . m ( x f y I z ) = y-z-O+z = y . Now t h e c h o i c e o f x f y r z w a s a r b i t r a r y and t h e r e f o r e ( = e3. T h i s c o n t r a d i c t i o n p r o v e s 2

t h a t ( ) d o e s n o t p r e s e r v e mo. Let & - b e as i n Thm 3.2.

0

The c o n d i t i o n s f r o m t h e o r e m 3 . 2 . may p r o v i d e

a r o u g h i n f o r m a t i o n on P ( F ) . W e c o n s i d e r them i n more d e t a i l . 3.3.

L e t 5 b e a bounded p a r t i a l o r d e r and ( ) a 2 / 3 - m a j o r i t y

t i o n w h i c h i s a l s o 2-monotone.

opera-

I f a r b r c r x f y l z l ta r e e l e m e n t s o f A

127

Cornpa t ib le orderings and tolerances

such t h a t

x

yI

a

x

b

f ies

y, z

t, z

b y; z

w

x

w

t h e n w := ( a b c ) s a t i s -

t

c

t.

In particular

or

a

b

c

a

c

b - c

w h a s no c e n t r a l

1 -i

or

(yxai) = al-i

(3.1)

tolerance.

The f o l l o w i n g i s a n e x t e n s i o n of B e r g m a n ' s d o u b l e - p r o j e c t i o n t h e o r e m 111 t o s p e c i a l 2 / 3 m a j o r i t y f u n c t i o n s . F o r a n n - a r y

and k

1

=

,...,h

p u t p r k p : = { ( x 1,..., ~

S i m i l a r l y f o r 1 Y i 'j s h p u t p r i j p

-

~

prkp

relation

p

on A

..., , xxn ) I~( x,,..., + ~ xn ~)

tp}.

: = ~ ( X ~ , X ~ ) ~ ( X ~ , . E. p. }~. X The ~ )

a r e c a l l e d t h e b i n a r y r e l a t i o n s of

relations prij

see t h a t b o t h

~

p.

It is easy t o

a n d t h e b i n a r y p r o j e c t i o n s of P a r e r e s o l v e n t s o f P .

For a 2 / 3 m a j o r i t y o p e r a t i o n ( ) on A a n d a , b € A l e t Bab be t h e l e a s t s u b s e t of A such t h a t i ) a E B w h i l e t l , t 3E { b l U Bab

go B a b ) . 3.6. tion

W e have:

Proposition. Let (

(e.g.

a n d i i ) ( t , t 2 t 3E)B a b w h e n e v e r t 2 E B a b ab ( b a b ) ,( b ( b a b )( b a b ) ) and ( a ( b a b ) b ) belong

)

A-

such t h a t b €Bab

relation p

(h

2

have a 2/3 m a j o r i t y l o c a l polynomial f u n c for a l l

a,b €A.

Then P ( A - ) p r e s e r v e s an h - a r y

3 ) i f and o n l y i f i t p r e s e r v e s a l l i t s b i n a r y p r o j e c -

t i o n s . If P ( A - ) p r e s e r v e s an h - a r y h a r e equal t o A2 t h e n p = A

r e l a t i o n p whose b i n a r y p r o j e c t i o n s

.

Proof: L e t h 2 3 and l e t P ( 8 ) u r e s e r v e an h-ary r e l a t i o n

p.

Put

u . : = P r . p (i = 1 , 2 , 3 ) a n d d e f i n e 1

T:=

{(XI,

..., I Xh)

(X2'

...,Xn ) E U l

(X1r

...I ... X3r

pXn)

Ea2, (X.,,x2,X4, . . . , X n ) t o 3 1.

I. G. Rosenherg, D. Schrueigert

128 Here

i s c l e a r l y a resolvent of

T

Clearly

PST.

such t h a t a l l h-tuples

(ulx2,.

( X ~ , X ~ , W , X ~ ~ . . . , bXe~l o) n g

. . ,x n ) ,

to p.

2

and f o r i = 1 , 2

..., x n )

t

(x, ,v,x3,.

7 ,

= P.

T

There a r e u,v,w

.. , x n ) ,

By i n d u c t i o n w e Drove t h a t

( x l , t l x3 , . . . , ~ n ) E p f o r a l l t t B v x ( x l ,t 2 , X 3 , .

Suppose t h a t

We p r o v e t h a t

{al,n2,u31.

For t h e c o n v e r s e l e t ( x , ,

.

Indeed t h i s holds f o r t

.., x n )

v. (3.1)

P

. . . ,x n )

e i t h e r t i = x 2 or ( x l , t i , x 2 ,

=

c p . Then t h e r e a r e

p and r s u c h t h a t

(P, . , , x 3 , . . . , x n )

E P ,

(x1,t3,r,x4,

..., x n )

(3.2)

t p

( P u t p = u i f t 1 = x2 a n d p = x1 o t h e r w i s e and s i m i l a r l y r = w i f

t 3 = x2 and r = x 3 e l s e . ) A p p l y i n q ( ) t o t h e f i r s t h - t u p l e from (3.2 I t h e h - t u p l e from 3 . 1 a n d t h e l a s t h - t u p l e from 3 . 2 w e q e t

.

Since ( ) is a ( t l t 2 t 3 )x,3 x 3 r ) , ( x 4 x 4 x 4 ),..., ( x n x n x n ) ) r p ( x , , ( t l t 2 t 3, x)2 , . ,xn) 1 p and h a v e by i n d u c t i o n ( x l , t , x 3 , ..., x n ) C p f o r a l l t C B v x By assump((PX X I )

I

..

2 / 3 majority operation we get the required

.

t i o n w e have x C B v x

and t h e r e f o r e ( x l , .

2

. ., x n )

2 i p proving T 5 0 .

By an e a s y i n d u c t i o n w e o b t a i n t h a t p i s a r e s o l v e n t o f t h e s y s t e m

I

tpr. .P 11

1 C i < j 5 h l p r o v i n g t h e f i r s t s t a t e m e n t . The o t h e r s a r e d i -

rect consequences.

54

0

MAJORITY OPERATIONS

4.1.

A 2 / 3 m a j o r i t y o p e r a t i o n ( x y z ) s a t i s f y i n g (xyx) = x f o r a l l

x l y t A i s c a l l e d a m a j o r i t y o p e r a t i o n . For example, e v e r y l a t t i c e h a s two m a j o r i t y p o l y n o m i a l s , c a l l e d m e d i a n s ,

(xyz), = x y + x z + y z ; (xyzIu = ( x + y ) ( x + z ) ( y + z ) ( w h e r e , as u s u a l x y + x z + y z s t a n d s f o r ( x s y ) + ( x 9 z ) (

)1 a n d

(4.1)

+

(y-z)). In fact

( j U a r e t h e l e a s t a n d g r e a t e s t m a j o r i t y monotone o p e r a t i o n s

on a l a t t i c e :

4.2.

F a c t . L e t < A ; + , - > b e a l a t t i c e and

Ii t s o r d e r .

I f

(

)

is a

i f t o e a c h p r o p e r l o c a l c l o n e C Z F t h e r e i s

( - Rs u c h t h a t C s P o l

p

‘0.

The f o l l o w i n g t h e o r e m i s a r e f o r m u l a t i o n o f [ 3 8 1 Thm 3 . S i n c e i t s p r o o f i n !381 i s b u t a d i r e c t v e r i f i c a t i o n o f t h e c o n d i t i o n s o f a l o -

c a l c o m p l e t e n e s s c r i t e r i o n 1 2 9 1 w e o m i t t h e p r o o f . The s e c o n d s t a t e ment i s i n 1281 w h e r e a s t h e l a s t i s a n i m m e d i a t e c o n s e q u e n c e o f t h e first. Theorem. L e t A - = h a v e a m a j o r i t y o p e r a t i o n among i t s l o c a l p o l y n o m i a l s . L e t R c o n s i s t o f ( i ) l a t t i c e o r d e r s c o m p a t i b l e w i t h A- , 4.4.

t o l e r a n c e s o f A, - ( i i i )n o n t r i v i a l c o n g r u e n c e s o f ( i v ) t o l e r a n c e s of A - of i n f i n i t e d i a m e t e r . T h e n R i s g e n e r i c .

(ii)i l o c a l l y c e n t r a l

A-

and

Moreover, mal.

f o r p listed in

Finally

A-

( i ) - ( i i i t)h e c l o n e P o l p i s l o c a l l y m a x i -

i s l o c a l l y c o m p l e t e i f a n d o n l y i f R = @.

F o r A f i n i t e c l e a r l y l o c a l l y c e n t r a l t o l e r a n c e s are c e n t r a l a n d ( i v ) d o e s n o t a p p l y . The r e l a t i o n s l i s t e d a b o v e are l i k e l y t o p l a y a n e m i n e n t r o l e f o r a l g e b r a s w i t h a l o c a l m a j o r i t y o p e r a t i o n . The f a c t

i s n o t a c c i d e n t a l . The f o l l o w i n g P r o p o s i t i o n i s a s p e c i a l c a s e o f a more g e n e r a l t h e o r e m [ I ] .

t h a t o n l y b i n a r y r e l a t i o n s f i g u r e i n Thm 4 . 4 .

F o r c o m p l e t e n e s s s a k e w e g i v e a s h o r t p r o o f . For a n h - a r y on A and 1 < k C h , P r k p := I ( x , ,

1 5 i 5 j c h put

.. .,xk-,

,xk+., , . . . , x h )

p r . . p : = I ( x i , x . ) : (x, 13

7

resolvents of I p l 4.5.

,..., x h )

Ep).

:

(x,,

. . ., x h ) t P 1

relation P

and

I t i s e a s y t o see t h a t b o t h a r e

(2.5.).

Proposition. L e t

A-

have a m a j o r i t y

local polynomial.

Then

Lac P ( A -) = Pol R f o r a s y s t e m R o f r e f l e x i v e b i n a r y r e l a t i o n s . I f a n 2 h - a r y r e l a t i o n p i s p r e s e r v e d b y - a n d a l l p r . .p = A ( 1 C i ‘ j 5 h ) t h e n 1 7 h p = A .

The p r o o f f o l l o w s f r o m P r o n o s i t i o n 3 . 6 .


130

pn is the transitive n= 1 ( i . e . t h e least t r a n s i t i v e r e l a t i o n c o n t a i n i n g p ) . L e t p be

F o r a b i n a r y r e l a t i o n p t h e r e l a t i o n t r p := h u l l of

p

a b i n a r y r e f l e x i v e r e l a t i o n comDatible w i t h P ( & ) . Since a t h e u n i o n of a c h a i n o f r e s o l v e n t s p n o f { p l from ProD.

: = t r p

is

2.6. we i n f e r

t h a t a i s c o m p a t i b l e w i t h P ( f i ) . Now t r p i s c l e a r l y a n e q u i v a l e n c e and t h e r e f o r e a i s a c o n g r u e n c e of P ( & ) . W e h a v e : 4.6.

Proposition. Let

be a proper

A-

A-

have a m a j o r i t y l o c a l polynomial

and

let p

t o l e r a n c e o f A. - I f n e i t h e r t r p i s a proper congruence o f t h e n p n i s l o c a l l y c e n t r a l f o r some

nor p has i n f i n i t e diameter,

n >O. P r o o f . From t h e a s s u m v t i o n s w e g e t t h a t p k = A

2 f o r some k > 1 . L e t m

be t h e l e a s t i n t e g e r w i t h t h i s wrowerty and l e t n b e t h e l e a s t i n t e g e r >$n. Then a := p n i s a p r o a e r t o l e r a n c e s u c h t h a t a 2 = A 2 . : = { ( x , ,... , x h ) : ( x , , u ) € a ,..., ( x h , u ) E o f o r some u }

put Ah

.

For h ' 3 W e say

t h a t an h-ary r e l a t i o n i s t o t a l l y r e f l e x i v e i f i t c o n t a i n s each h-tup l e w i t h a t l e a s t o n e r e D e t i t i o n . From t h e s e c o n d s t a t e m e n t of P r o p .

4.5. we infer that for h

?

3 t h e r e l a t i o n Ah i s t h e s i n q l e t o t a l l y re-

f l e x i v e h - a r y r e l a t i o n from I n v P ( q ) From ci2 = A A 3 =A3.

.

2 w e o b t a i n t h a t A 3 i s t o t a l l y r e f l e x i v e of I n v P ( 4 ) . Thus

By t h e same t o k e n A 4 i s t o t a l l y r e f l e x i v e i . e .

t i n u i n g we qet Xh = A h

for all h

means t h a t a i s l o c a l l y c e n t r a l . I f P(A) c o n t a i n s t h e o n e r a t i o n s

h4 =A

4

.

Con-

3. Accordinq t o t h e d e f i n i t i o n t h i s

2

0

+

and

-

o f a l a t t i c e w e can r e s t r i c t

R t o t h e n a t u r a l o r d e r 2 o f t h e l a t t i c e and t h e t o l e r a n c e s of P ( & ) .

F o r t h e f i n i t e c a s e t h i s was p r o v e d i n [ 311 and i n g e n e r a l i n [ 2 1 . Our p r o o f f o l l o w s t h e i d e a s o f [ 2 1 . L e t A f l e x i v e c o m p a t i b l e b i n a r y r e l a t i o n s of r a n c e s o f &. 4.7.

(5) d e n o t e t h e s e t o f reg t h e s e t o f tole-

and To1

Theorem. L e t L_ - b e a l a t t i c e and 5 i t s o r d e r . Then = P o l ( 5 U T o 1 & ) , i e . an o p e r a t i o n on A i s l o c a l p o l y n o m i a l

LOC ),P.I(of

&

.

i f and o n l y i f f i s %'-monotone and p r e s e r v e s e a c h t o l e r a n c e o f

Proof. L e t L = < A ; + , - >

and l e t 6 € A ( & ) .

C l e a r l y 6 + and 6 ) a r e t o l e r a n c e s o f

&.

Set

+-

P u t p : = { ( x , y ) :x6+u6 y

for

&.

Compatible orderings and tolerances

some x

iu 1 y l .

Clearly p is a resolvent of

131

and t h e t w o t o l e r a n c e s

2

6 + and 6 + . I t r e n a i n s t o show t h a t p = 6 .

1. L e t ( x , y )

C=

6.

Setting u

(u,y) = (x+y,y+y) t 6 .

=

x+y w e h a v e ( x , u ) = ( x + x , x + y ) t 6 a n d

S i n c e x :u w e h a v e ( x , u ) C 6 + a n d from ( x , u ) F 6 +

and ( u , u ) t 6 + a l s o ( x , u ) C 6 + . I n a s i m i l a r f a s h i o n ( u , y ) E 6 + a n d therefore (x,y) E p .

2 . L e t ( x , y ) F p . Then t h e r e e x i s t s x r u " y r v ( x , v ) t6+, ( u , v ) C 6 + ,

(u,w) F 6 - ,

(y,w) t 6 - .

?

x + u and w Luy s u c h t h a t

Now

' x , u ) = ( x u , v u ) € 6 and

( u , y ) = (u+y,w+y) t & . F i n a l l y ( x , y ) = ( x u , u y ) E 6 which c o n c l u d e s t h e proofon 4.8.

Remark. L e t

r ( L- ) b e t h e s e t o f c o m p a t i b l e e x t e n s i o n s o f

h (L ) t h e set of compatible s u b r e l a t i o n s of s .

e.g.

$5

2,

Con

&

t h e t h r e e f i r s t are i s o m o r p h i c .

I n v i e w o f Thm 4 . 5 . s t u d i e d i n $96

and

I n ( 2 1 and [ 3 5 1 t h e r e

a r e s e v e r a l i n t e r e s t i n g i n t e r a c t i o n s b e t w e e n r(A),A(L),Tol a n d A(&),

~1

t h e set T o 1

&

i s of p r i m e i m p o r t a n c e . I t w i l l b e

- 8.

COMPATIBLE ORDERS I N ALGEBRAS W I T H LOCAL M A J O R I T Y POLYNOMIALS, SEMILATTICES AND LATTICES

5.1.

I n t h i s s e c t i o n w e c o n s i d e r t h e c o m p a t i b l e d i r e c t e d orders i n

a l g e b r a s w i t h a l o c a l m a j o r i t y p o l y n o m i a l ( ) . They t u r n o u t t o b e completely determined by ( ) .

For b e t t e r r e s u l t s we look a t algebras

h a v i n g l a t t i c e o p e r a t i o n s among i t s l o c a l p o l y n o m i a l s . A c c o r d i n g t o Thm 4 . 7 .

t h e orders compatible with a lattice

&

distinct f r o m its

o r d e r are n o t e s s e n t i a l ( i n t h e s e n s e t h a t t h e y d o n o t r e s t r i c t LOC P ( & ) i . e . may b e r e p l a c e d by t o l e r a n c e s ) . S t i l l i t may b e i n t e -

r e s t i n g t o know t h e s t r u c t u r e o f c o m p a t i b l e d i r e c t e d a n d d o w n - d i r e c t e d o r d e r s of a l a t t i c e

&.

T h i s h a s been completely s o l v e d i n [ 9 1

w h e r e t h e y a r e shown t o b e i n 1-1 c o r r e s p o n d e n c e w i t h d i r e c t decompo-

s i t o n s of & ( a n d t h u s f o r bounded l a t t i c e s w i t h t h e c e n t e r o f & ) i . e . t h e y are d e t e r m i n e d b y s p e c i a l c o n g r u e n c e s o n l y . Q u i t e s u r p r i s i n g l y , s i m i l a r a n d more g e n e r a l r e s u l t h o l d s f o r c o m p a t i b l e o r d e r s o f s e m i lattices [22]. F i r s t we study compatible orders f o r majority operat i o n s (5.2.

-

5.7.)

and t h e n f o r semilattices (5.8.

-

5.23).

The f o l l o w i n g i s i m p l i c i t i n [ 3 8 ] a n d [ 2 ] . W e s a y t h a t 5 i s t h e o r d e r of a p a r t i a l

u p p e r s e m i l a t t i c e < A ; v > i f a l e a s t u p p e r bound xvy e x i s t s

I. G. Rosen berg, D.Schweigert

I32

f o r e a c h p a i r x , y € A h a v i n g a common u p p e r bound. 5.2.

Lemma. L e t

o p e r a t i o n on A.

be a n o r d e r o n A a n d Then

i

Proof. L e t x

monotone m a j o r i t y

i s the o r d e r o f a p a r t i a l

x 5 z,y 5 z

a n d

) a

(

=$

upper semilattice

(xyz) = (xzy) = (zxy) =xvy

S e t x v y = (xyz).Then x = (xyx)

z and y 5 z .

and s i m i l a r l y y = ( x y y ) 5 x v y . Moreover, i f x

(5.1)

(xyu) = x v y

t and y - t , t h e n

x v y = (xyz) z ( t t z ) = t proving t h a t x v y = ( x y z ) i s t h e least upper bound of x and y . An a r b i t r a r y e x c h a n g e o f v a r i a b l e s i n ( ) l e a d s t o a .--monotone m a j o r i t y o p e r a t i o n ( 1 . From t h e a l r e a d y p r o v e d x v y = (xyz) we q e t (xzy) = (zxy) = x v y . 5.3.

COrOllarY. I f 2 i n L e m m a 5 . 2 .

o f a n u p p e r s e m i l a t t i c e

CI

i s directed,

then

i s the o r d e r

(5.1.) and

satisfying

(L1) ((xyu)zu) = (x(yzu)u) whenever

u is

an upper bound

o f

X,y a n d

Z.

(L, ) i s t h e a s s o c i a t i v e l a w .

Proof.

I n view o f Thm 4 . 4 .

w e a r e i n t e r e s t e d i n b o t h d i r e c t e d and down d i -

rected orders. 5.4.

C o r o l l a r y . L e t 2 be a d i r e c t e d a n d d o w n - d i r e c t e d o r d e r on A a n d

( ) a 6-monotone

lattice

i s a n A-enl a t t i c e . A p r o g r e s s i o n o f a c o m p l e t e l a t t i c e K_ domorphism G s u c h t h a t b (ii)o ( v x

i=

xx)

=

v

XFX

~ ( x €or ) every d i r e c t e d 0 e X 5 B . We have [ 3 1 :

P r o p o s i t i o n . For P

6.9.

a ( b ) f o r a l l b F B and

To1 L_ - and I C I ( L -) p u t p * ( I )

f o r some i C I ) . T h e n P + o *

a b i j e c t i o n from To1 _L - onto the s e t 0 El(&). -

is

S k e t c h o f t h e p r o o f : One h a s v e r i f y t h a t f o r p* ( I )

{x C A :xpi

n

s o f I__ (_L ) such t h a t sp) = p) i f

o f a l l proqressions

set

:=

(-1( L ) a n d t h a t I

-%

P* (I)

L- a n d I E I ( & ) t h e

P ETol

i s A-nreservinq,

satisfies I sp*(I)

and o r e s e r v e s d i r e c t e d f a m i l i e s . Conversely f o r s C I l p u t s*

(where ( a ] = ( x : x

'

:= { ( a , b :

a f : s ( b ] ,b t s ( a 1 )

a ) ) . By a d i r e c t c h e c k s* F T o l

somewhat t e d i o u s v e r i f i c a t i o n o f

** p

= p

and

** s

&.

I t remains t h e

= s for all p ETol

-

and s E n . 6.10.

T h i s a u o r o a c h i s e x p a n d e d i n [ 3 1 t o show t h e d i s t r i b u t i v i t y

o f To1

& f o r d i s t r i b u t i v e l a t t i c e s and t o p r o v e a d e s c r i p t i o n o f to-

l e r a n c e s o f f i n i t e l a t t i c e s by s p e c i a l + - e n d o m o r o h i s m s . W e a d d r e s s o u r s e l v e s t o s u c h a d e s c r i p t i o n f o r s l i g h t l y more g e n e r a l l a t t i c e s . A regression of a lattice L = H0

and

&

a di-

be t h e union o f t h e s q u a r e s o f i n t e r v a l s


144

w h i c h a r e c l o s e d u n d e r j o i n s a n d meets of f a m i l i e s of c a r d i n a l i t i e s O f o r j = i + l , rn. 1 1 From r ( a . ) a . w e q e t r ( a . ) = a . a n d r ( 1 ) = r ( a ) + ...+r ( a ) = a1+1 . ran. 7 1 1 1 1 By t h e same t o k e n r ( a l + . . + a . ) = O . By C o r . 7 . 5 . t h e r e i s a ? r ( 1 ) = Proof. L e t

p

...

...

=...

- -.

.

= ai+.,+.

. .+an s u c h

show a ' 1

i.e.

p

that r(a) =O.

Then a

= A2 i s t r i v i a l .

.

.

a l + . .+a. and a 1 a i + l + . .+an

0

Note t h a t a d i s t r i b u t i v e l a t t i c e w i t h more t h a n 2 e l e m e n t s i s n o t simple and hence does n o t p o s s e s s t h e O I P . For modular l a t t i c e s wehave

8.6.

Theorem. L e t L b e a m o d u l a r l a t t i c e w i t h a l e a s t o n e c o m p l e m e n -

ted interval. i f L i s simple

Then L h a s the o r d e r i n t e r p o l a t i o n p r o p e r t y i f and o n l y and r e l a t i v e l y complemented.

The c o n c l u s i o n h o l d s i f L h a s a t l e a s t o n e p r i m e i n t e r v a l ( i . e . nonempty c o v e r i n g r e l a t i o n ) . P r o o f . N e c e s s i t y . L e t p be t h e u n i o n o f t h e s q u a r e s o f complemented inervals of tolerance of

&. U s i n g P r o p . 6 . 7 . i t 2. By a s s u m p t i o n p + w

i s shown i n [ 2 1 Thm 3 t h a t hence P = A 2 , i . e .

p

is a

each i n t e r v a l of

L- i s complemented. S u f f i c i e n c y . By Lemma 6 . 8 .

the lattice

&

is tolerance t r i v i a l .

F o r modular l a t t i c e s o f f i n i t e l e n g t h w e h a v e t h e f o l l o w i n g remarka b l e r e s u l t [ 371 [ 321: 8.7.

Theorem. L e t L_ - be a m o d u l a r l a t t i c e of f i n i t e l e n g t h .

T h e n L_ - has

t h e o r d e r i n t e r p o l a t i o n p r o p e r t y i f a n d o n l y i f L_ - i s a d i r e c t l y indecomposable p r o j e c t i v e

geometry.


148

P r o o f . N e c e s s i t y . The t o l e r a n c e s z f r o m E x . 6 . 1 2 .

1'

=O.

T h u s b y [ 51 Thm I V . 6 .

the lattice

&

i s t r i v i a l Droving

i s t h e n a complemented

s i m p l e l a t t i c e and hence a n indecomposable p r o j e c t i v e qeometry. S u f f i c i e n c y . C l e a r l y 1 is t h e j o i n o f a t o m s , h e n c e b y L . 8 . 5 .

tice

4

is loaf-complete.

the lat-

S i n c e i t i s a l s o s i m p l e i t h a s t h e OIP.

0

REFERENCES [

11 Baker K.A. and P i x l e y A.F., Polynomial i n t e r p o l a t i o n and t h e C h i n e s e r e m a i n d e r t h e o r e m f o r a l q e b r a i c s y s t e m s , Math. Z . , 1 4 3 ( 1 9 7 5 ) , 165-174.

[ 21 B a n d e l t H. J . , L o c a l p o l y n o m i a l f u n c t i o n s o n l a t t i c e s , H o u s t o n J. M a t h . 7 ( 1 9 8 1 ) , 3 1 7 - 3 2 5 . [

31 B a n d e l t H . J . , T o l e r a n c e r e l a t i o n s o n l a t t i c e s , B u l l . A u s t r . M a t h . SOC. 2 3 ( 1 9 8 1 ) , 3 6 7 - 3 8 1 .

[ 41 B a n d e l t H. print.

J.,

[ 51 B i r k h o f f G . ,

Toleranzrelationen als Galoisverbindungen, pre-

L a t t i c e Theory 3rd e d . A m e r .

Math. SOC. 1967.

[ 6 1 C h a j d a I., R e c e n t r e s u l t s a n d t r e n d s i n t o l e r a n c e s o n a l q e b r a s and v a r i e t i e s , P r o c . Colloq. F i n i t e a l g e b r a M u l t i p l e - v a l u e l o q i c ( S z e g e d 1979) C o l l . Math. SOC. I . B o l y a i 28, N o r t h H o l l a n d 1 9 8 1 , p p 69-95. [

71 C h a j d a I . , T h e o r y o f t o l e r a n c e s o n a l g e b r a i c s y s t e m s , b i b l i o g r a p h y , 1981- 19 8 3 .

[

81 C h a j d a I., D i s t r i b u t i v i t y a n d m o d u l a r i t y o f l a t t i c e o f t o l e r a n c e r e l a t i o n s , A l g e b r a U n i v e r s a l i s 1 2 ( 1 9 8 1 ) , 247-255.

[ 91 C z e d l i G . , Huhn A . P . , S z a b 6 , L . , On Compatible o r d e r i n g o f l a t t i c e s , C o l l . Math. SOC. I . B o l y a i 33, N o r t h H o l l a n d 1 9 8 0 , 87-99.

[ l o ] CzCdli G . , K l u k o v i t s L . , A n o t e on t o l e r a n c e s o f i d e m p o t e n t alg e b r a s , G l a s n i k M a t e m . 1 8 ( 3 8 ) ( 1 9 8 3 ) 35-38. 1111 C z C d l i G . , F a c t o r l a t t i c e s b y t o l e r a n c e s , A c t a S c i . M a t h . g e d ) 4 4 1-2 ( 1 9 8 2 ) 35-42.

(Sze-

[ I 2 1 Davey B . A . , D u f f u s D . , Q u a c k e n b u s h R.W. a n d R i v a l I . , E x p o n e n t s o f f i n i t e s i m p l e l a t t i c e s , J. London M a t h . S O C . , 1 7 ( 1 9 7 8 ) , 203-21 1 . [ 1 3 1 Davey B . A . a n d R i v a l I . , E x p o n e n t s of l a t t i c e - o r d e r e d a l g e b r a s , A l g e b r a U n i v e r s a l i s 1 4 ( 1 9 8 2 ) 87-98. [141 D o r n i n g e r , D., Nobauer W . , L o c a l p o l y n o m i a l f u n c t i o n s on l a t t i c e s a n d u n i v e r s a l a l g e b r a s , C o l l o q . M a t h . 42 ( 1 9 8 0 ) 8 3 - 9 3 . [ I 5 1 D o r n i n g e r D . , E i g e n t h a l e r G . , On c o m p a t i b l e a n d o r d e r - p r e s e r v i n g f u n c t i o n s o n l a t t i c e s , B a n a c h C e n t e r P u b l i c a t i o n s 9 , Warsaw 1 9 8 2 , 97- 1 0 4 . [ 1 6 1 G r a t z e r G., B o o l e a n f u n c t i o n s o n d i s t r i b u t i v e l a t t i c e s , A c t a M a t h . Acad. S c i . H u n g a r . 1 5 ( 1 9 6 4 ) 9 5 - 2 0 . [I71 Gratzer G., Lattice theory: F i r s t concepts anddistributive latt i c e s , Freeman ( S a n F r a n c i s c o ) 1971.

149

Compatible orderings and tolerances G e n e r a l l a t t i c e t h e o r y , (Akademie V e r l a q B e r l i n

I18

Gratzer G.,

i 19

Herrmann C . , S . - v e r k l e b t e ( 1 9 7 3 ) 255-274.

[ 20

H e d l i k o v h J . , B e t w e e n e s s i s o m o r p h i s m s of m o d u l a r l a t t i c e s A r c h . Math 37 ( 1 9 8 1 ) 1 5 4 - 1 6 2 .

Summen von V e r b a n d e n , Math. Z . ,

1978. 130

Kindermann M . , u b e r d i e X q u i v a l e n z von Ordnungspolynomvollstand i g k e i t und T o l e r a n z e i n f a c h h e i t e n d l i c h e r V e r b a n d e , C o n t r i b u t i o n s t o G e n e r a l Algebra ( e d . H. K a u t s c h i t s c h e t a l . ) , 1 9 7 9 , 145-149. K o l i b i a r M . , Compatible o r d e r i n g s i n s e m i l a t t i c e s , C o n t r i b u t i o n t o General Alqebra 2 , Proceedings o f t h e Klagenfurt Conference 1 982, (Teubner, S t u t t g a r t ) 215-220. K o l i b i a r M . , S e m i l a t t i c e s w i t h i s o m o r p h i c q r a p h s , C o l l o q . Math. SOC. J . B o l y a i 2 9 , Esztergom 1977 473-481. K o l i b i a r M . , I n t e r v a l s , c o n v e x s u b l a t t i c e s a n d s u b d i r e c t repres e n t a t i o n s of l a t t i c e s , U n i v e r s a l A l g e b r a a n d A p p l i c a t i o n s B a n a c h C e n t e r P u b l . v o l 9 , Warsaw 1 9 8 2 , 3 3 5 - 3 3 9 . N i e d e r l e J . , R e l a t i v e bicomplements and t o l e r a n c e e x t e n s i o n prop e r t y i n d i s t r i b u t i v e l a t t i c e s , C a s o p i s P e s t . Mat., 1 0 3 ( 1 9 7 8 ) , 2 50-25 4 . Rosenberg I . G . , F u n c t i o n a l l y c o m p l e t e a l g e b r a s i n congruence d i s t r i b u t i v e v a r i e t i e s , A c t a S c i . M a t h . ( S z e g e d ) 4 3 , 3-4 ( 1 9 8 1 ) 34 7- 3 5 2 . Rosenberg I . G . , F u n c t i o n a l completeness o f s i n g l e g e n e r a t e d o r s u r j e c t i v e a l g e b r a s , Proc. Colloq. F i n i t e Algebra Multiple-val u e d L o g i c , S z e g e d ( 1 9 7 9 ) , C o l l . Math. SOC. J . B o l y a i 2 8 , N o r t h H o l l a n d 1 9 8 1 , pp. 635-652. R o s e n b e r g I.G., S c h w e i q e r t D . , L o c a l l y maximal c l o n e s , E l e c t r o n i s c h e Informationsverarbeitung u . K y b e r n e t i k 1 8 ( 1 9 8 2 ) 7 / 8 p p . 389-401. Rosenberg I . G . , Szab6 L . , bra Universalis.

L o c a l c o m p l e t e n e s s I , t o a p p e a r Alge-

S c h w e i g e r t D . , Uber e n d l i c h e , ordnungspolynomvollstandige V e r b a n d e , Monatsh. Math. 7 8 ( 1 9 7 4 ) , 6 8 - 7 6 . S c h w e i g e r t D . , Some r e m a r k s o n p o l a r i t y l a t t i c e s a n d on o r t h o l a t t i c e s , P r o c e e d i n g s of t h e L a t t i c e T h e o r y C o n f e r e n c e , U l m ( 1 9 7 5 ) 254-256. S c h w e i g e r t D . , C o m p a t i b l e r e l a t i o n s of m o d u l a r a n d o r t h o m o d u l a r l a t t i c e s , P r o c e e d i n g s of t h e AMS, 81 ( 1 9 8 1 ) , 4 6 2 - 4 6 4 . S c h w e i g e r t D. , P o l y n o m v o l l s t a n d i g e a t o m i s t i s c h e P o l a r i t a t s v e r b a n d e , A b h a n d l u n g e n Math. Sem. U n i v . Hamburg 5 1 ( 1 9 8 1 ) 5 0 - 5 9 . Schweigert D., A f f i n e complete o r t h o l a t t i c e , Proceedings Amer. Math. SOC. 6 7 , 2 ( 1 9 7 7 ) 1 9 8 - 2 0 0 . S c h w e i g e r t D . , C e n t r a l r e l a t i o n s on l a t t i c e s , J . A u s t r a l . Math. SOC. ( S e r i e s A ) 35 ( 1 9 8 3 ) . S c h w e i g e r t D . , Szymanska M . , P o l y n o m i a l f u n c t i o n s o f c o r r e l a t i o n l a t t i c e s , A l g e b r a U n i v e r s a l i s 16 ( 1 9 8 3 ) 3 5 5 - 3 5 9 . S c h w e i g e r t D . , Szymanska M . , On c e n t r a l r e l a t i o n s o f c o m p l e t e l a t t i c e s , U n i v e r s i t a t K a i s e r s l a u t e r n , P r e p r i n t no. 6 4 , 1 9 8 3 . Szabb L . , T o l e r a n c e - f r e e a l g e b r a s w i t h a m a j o r i t y f u n c t i o n , Prep r i n t , Bolyai I n s t i t u t e Szeged.

I.G. Rosenberg, D. Schweigert

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[ 3 9 ] Szabb L . , C h a r a c t e r i z a t i o n o f compatible q u a s i o r d e r i n g s of l a t t i c e ordered a l g e b r a s , p r e p r i n t Bolyai I n s t i t u t e Szeqed. [ 4 0 ] W i l l e R.,

H a l b k o m p l e m e n t a r e V e r b a n d e , Math.

2.

94 ( 1 9 6 6 ) 1 - 3 1 .

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Eine C h a r a k t e r i s i e r u n g e n d l i c h e r , ordnungspolynomvolls t a n d i g e r V e r b a n d e , A r c h . Math. 2 8 ( 1 9 7 7 ) , 557-560.

[ 4 2 ] Wille R . , A n o t e o n a l g e b r a i c o p e r a t i o n s a n d a l g e b r a i c f u n c t i o n s of f i n i t e l a t t i c e s , H o u s t o n J . Math. 3 ( 1 9 7 7 ) 593-597.

[ 4 3 ] W i l l e , R . , uber e n d l i c h e , ordnungsaffinvollstandige V e r b a n d e , Math. 2. , 1 5 5 ( 1 9 7 7 ) 103-107.

- PART I11 ORDER AND ORDERED GROUPS -

PARTIEIII ORDRE ET GROUPES ORDONNES

W. Charles HOLLAND

Classification o f lattice ordered groups.

. . . . . . . . . . . . . . . . . . . . . p . 151

A.M.W. GLASS The isomorphism problem and undecidable properties f o r finitely presented lattice-ordered groups. . . . . . . . . . . . . . . . . . . . . . . . . . . p . 157



151

W . Clrirles llnlland

Depurtricnt 01' !kithemutics tind S t s t i s t i e s Ecowlinrf Green S t a t e University t i i v ~ l l n r : Green. Ohio U .S . A .

IJn ~:roupc r6ti,.!u16 est. un p o u p e dont les 6lCnients c o n s t i t u e n t un t r * e i l l i s cornpntible avec l ' o p 6 r a t i o n du Troupe. Nous d i s c u t e r o n s t r o I s iii6thodes d e classific!at,ion d e ces s t r u c t u r e s . Is premiPre est. c e l l e de:! v w i 6 t c s . q u i sont. !es c l a s s e s d 6 f i n i e s par les Pqii" I.ioiis. La c o l l e c t i o n tles variEt6r: c s t un t r e i l l i s coniplet, 4 61Pment:: qui a un plus p e t i t Plecieiit n o n - t r i v i a l e t un p l u s ::rcind Element n o n - t r i v i a l . IA deuxiPme mEthode est, c e l l e des c l a s s e s de t o r s i o n . J.es c l a s s e s d e t o r s i o n c o n s t i t u e n t iin t r e i i l i s cmnplet, mais w r i t t.rop nombr*euses pour c o n s t i - t u e r un ensemble. T.a t r o i s i t i n e m6thode est c e l l e d e s c l a s s e s Cl&nent.aires, qui s o n t d6firile:; par des enonces d u l a i c a f e dii premier o r d r e des groupes

PO

r6ticii16s. e i e n t l u ' i l n'y a i t qiie classes ~ ~ e m e n t n i r e s , besucoulj d e croupes r 6 t i cul6s bien connus peuvent etre caract e r i s d s dans l a classe d e s groupes d'aulomorphismes d e chnines p a r un s e u l 6noncE du l a r i p g e d u premier o r d r e . 0.

INI'HODUC'I'ION

:I Kroup whose elements a l s o have a I . a t t i c e s t r u c t u r e compatible with t h e croup t r a n s l a t i o n s , so t h a t x(y v z)w = (xyw) V (xzw) and d u a l l y . From t h e standpoint of ordered s e t s , e-groups m y be viewed a s l a t t i c e s endowed with a group s t r u c t u r e which f o r c e s them t o have n i c e l a t t i c e p r o p e r t i e s . For exarnple, they must be homogeneous (with r e s p e c t t o l a t t i c e outomorphisiris) and d i s t r i b u t i v e . There is a n o t h e r c l o s e r e l a t i o n s h i p between P-groups and ordered sets. The automorphism group of every t o t a l l y ordered set is an P-group under t h e point-wise o r d e r i w . Moreover, every P-group is n subgroup and s u b l a t t i c e of such an automorphism group [ 8 ] ,

A httict' drdereJ p w4 p (L-group) i s

Much of t h e 11 t e r a t u r e on !-groups i s concerned with s t r u c t u r e and r e p r e s e n t a t i o n theory. Good r e f e r e n c e s f o r backl:round m a t e r i a l a r e 111, 123, [ A ] , and [ > I . The emphasis i n t h i s paper, however, is on c l a s s i f ' i c a t i o n of 1-groups by various schemes, i n parLicular ( i ) e q u a t i o n a l c l a s s e s , o r v a r i e t i e s , ( i i ) t o r s i o n c l a s s e s , and ( i i i ) elementary c l a s s e s . The ninin i n t e r e s t h e r e l i e s not s o much i n t h e s t r u c t u r e of p a r t i c u l a r !-groups b u t i n t h e r e l a t i o n s h i p between d i f f e r e n t c l a s s e s of P-groups. For example t h e set of a l l equational c l a s s e s of !-groups i t s e l f forms a l a t t i c e with some remarkable p r o p e r t i e s . In t h i s paper, which corresponds roughly with a one-hour address e i v e n a t t h e Conference on Ordered S e t s a t t h e Chdtenu de l a Tourette i n 1982, I w i l l not t r y t o cover t h e s u b j e c t of f-groups i n g r e a t d e t a i l , nor even g i v e a g e n e r a l overview, but r a t h e r , I w i l l c o n c e n t r a t e on a few d e t a i l s of the c l a s s i f i c a t i o n problem with t h e hope of whettin(! t h e r e a d e r ' s i n t e r e s t , and I w i l l provide a guide t o f u r t h e r i n v e s t i f a t i o n f o r those i n t e r e s t e d . 1.

VAR1F:TIES

A w r i e t $ (of P-groups) is a c l a s s defined by any set of u n i v e r s a l l y q u a n t i f i e d

W.C. Holland

152

e q u a t i o n s , which may i n v o l v e t h e group or l a t t i c e o p e r a t i o n s or b o t h . For example, t h e c l a s s A of a b e l i a n !-groups d e f i n e d by t h e e q u a t i o n xy = yx i s a v a r i e t y . A more t y p i c a l example, because i t i n v o l v e s b o t h t h e group and l a t , t i c e o p e r a t i o n s , i s t h e representable v a r i e t y R d e f i n e d by e , where e i s t h e i d e n t i t y element, of t h e group. (x-l(y V e ) x ) A (y-l V e ) A c l a s s i c a l r e s u l t of Lorenzen [ l 4 ] shows t h a t t h e r e p r e s e n t a b l e L-groups a r e Just. t h o s e which a r e s u b d i r e c t p r o d u c t s of t o t a l l y o r d e r e d groups. Another v a r i e t y of s p e c i a l i n t e r e s t i s t h e norma2 vulued v a r i e t y N d e f i n e d by t h e law xy < y?x:‘ i f x , y > e . I n t h i s form, t h e law i s n o t a n e q u a t i o n , b u t i t i s e a s i l y s e e n t n 2 be e q u i v a l e n t t o t h e e q u a t i o n ( x v e ) ( y i/ e ) ( x V e ) - ( y v e ) - * V e - e . The c o l l e c t i o n of a l l v a r i e t i e s of L-groups i s i t s e l f a complete l a t t i c e o r d e r e d by containment w i t h l a r g e s t member i: ( = a l l [-groups, d e f i n e d by x = x) and s m a l l e s t member E ( = a l l one-element L-groups, defined by x = e ) . The f i r s t remarkable r e s u l t about t h e l a t t i c e of a l l v a r i e t i e s of L-groups was d i s c o v e r e d by Weinberg [26] who showed t h a t t h e a b e l i a n v a r i e t y A i s t h e o n l y cover of E; t h a t i s , any v a r i e t y which c o n t a i n s a n !-group w i t h more t h a n one element must c o n t a i n a l l a b e l i a n !-groups. Another dramatic way t o e x p r e s s t h i s r e s u l t i s t h a t i f a n f-group s a t i s f i e s xy = yx t h e n i t s a t i s f i e s every e q u a t i o n ( e x c e p t t h o s e which would f o r c e t h e group t o c o n t a i n o n l y e ) . Much of t h e e a r l y work i n v a r i e t i e s of L-groups was done by Martinez ( [151, [ 1 6 ] , [17]). He observed t h a t t h e normal valued v a r i e t y N i s v e r y l a r g e . I l a t e r showed [9] t h a t N i s t h e unique l a r g e s t proper v a r i e t y . That i s , i f an L-group s a t i s f i e s any n o n - t r i v i a l e q u a t i o n , i t must a l s o s a t i s f y xy < ~ 2 x 2 for x , y 2 e . I n a s e n s e , t h e law i n t h e p r e v i o u s s e n t e n c e i s t h e weakest e q u a t i o n a l c o n d i t i o n t h a t can b e r e q u i r e d of an !-group, w h i l e xy = yx i s t h e s t r o n g e s t .

I t f o l l o w s from u n i v e r s a l a l g e b r a i c c o n s i d e r a t i o n s t h a t s i n c e A i s d e f i n e d by one e q u a t i o n , A cannot b e a n o n - t r i v i a l i n t e r s e c t i o n of a tower of v a r i e t i e s . T h i s makes i t i n t e r e s t i n g t o look f o r covers of A . I n [ 2 3 ] , Scrimger d i s c o v e r e d a n i n f i n i t e c o l l e c t i o n of c o v e r s of A , one f o r each prime number p . Like most small v a r i e t i e s , each o f t h e s e i s g e n e r a t e d by a s i n g l e k-group

i s t h e L-group g e n e r a t e d by abp Let all

=

a.

a,b

>

e

such t h a t

a

A

abi = e

S

1< i

if

P’

And

e w i t h an < ab for -I + n , and l e t fl b e the v a r i e t y g e n e r a t e d by M . S i m i l a r l y , !I i s defined

an < a b - l . F i n a l l y , Ill i s t h e v a r i e t y g e n e r a t e d by t h e f r e e n i l p o t e n t i j k c l a s s 2 group on a , b , o r d e r e d l e x i c o g r a p h i c a l l y on a b [ a , b ] Just before t h i s c o n f e r e n c e , G. Bergmann, f o l l o w i n g a s u g g e s t i o n of T . F e i l , found a n o t h e r r e p r e s e n t a b l e cover. These a r e a l l o f t h e p r e s e n t l y known c o v e r s o f A . The d i f f i c u l t y i n f i n d i n g more l e a d s one t o c o n j e c t u r e t h e r e may n o t b e many more.

with

.

F i n a l l y , c o n s i d e r t h e q u e s t i o n of t h e number and d i s t r i b u t i o n o f v a r i e t i e s . How many v a r i e t i e s a r e t h e r e ? How broad and how t a l l i s t h e l a t t i c e of v a r i e t i e s ? S i n c e a l l e q u a t i o n s may be c o n s i d e r e d t o be w r i t t e n i n a c o u n t a b l e a l p h a b e t , t h e r e a r e o n l y countably many e q u a t i o n s , t h u s

2

no

s e t s of e q u a t i o n s , and hence no more

t h a n 2’0 v a r i e t i e s . Kopytov and Medvedev [13] showed t h e r e a r e indeed ZHJ v a r i e t i e s . R e i l l y [22] gave a g e n e r a l method t o c o n s t r u c t v a r i e t i e s which produced a n uncountable c o l l e c t i o n o f mutually incomparable v a r i e t i e s , showing t h a t t h e l a t t i c e i s v e r y broad. hnd F e i l [ 3 ] c o n s t r u c t e d a tower of v a r i e t i e s whose o r d e r t y p e i s t h a t o f a r e a l i n t e r v a l , showing t h a t t h e l a t t i c e i s a l s o very tall.

Classificationo f lattice ordered groups

153

We have n o t d i s c u s s e d h e r e t h e very i n t e r e s t i n & a r i t h m e t i c s t r u c t u r e of t h e l s t t i - e of v a r i e t i e s . ? h e r e a d e r is r e f e r r e d t o t h e f a i r l y comprehensive paper

[6l. 2. 'I'!)K,'TON

CLSIXE:'

Mmy #:if the c l a s s e s o f i n t e r e s t i n f-groups a r e n o t e q u a t i o n a l l y d e f i n a b l e . Tn p a r t i c r i l x r , t h e r e a r e rimy c l a s s e s of i n t e r e s t a l r e a d y c o n t a i n e d i n t h e a b e l i a n c l a s s . F o r example, Archimedean L-iwoups must lie a b e l i a n (where Archimedean n:eans t h e r e a r e no n o n - t r i v i a l bounded c y c l i c s u b g r o u p s ) . Tn o r d e r t o make a Yiiier d i s t i r l c t i o n , t h e n , Martine:: [181, [l'3] i n t r o d u c e d t h e n o t i o n of a t o r s i o n c l a s s of k-t:roups: :I c l a s s c l o s e d under L-homomorphic images and under j o i n s of convex L-suljgrorips. I t i s c l e a r t h a t v a r i e t i e s a r e c l o s e d under L-homomorphic image". I t i s n o t so c l e a r t h a t v a r i e t i e s a r e c l o s e d under j o i n s o f convex f-suh$:roups. N e v e r t h e l e s s , i t i s t r u e , a s I showed i n [lo]. Other examples o f t o r s i o n c l a s s e s a r e the h,yperarchiirieiiean L-groups--those f o r which e v e r y k?-homomorphic image i s Archimedean, and t h e h y p e r i n i e g r a l L-groups--those for which e v e r y L-honmnorphic image i s a s u b d i r e c t p r o d u c t of c o p i e s of t h e L-group o f integers.

I t i s e a s i l y s e e n t h a t t h e i n t e r s e c t i o n of any c o l l e c t i o n of t o r s i o n c l a s s e s i s a t o r s i o n ~13:;s. Thus, l i k e v a r i e t i e s , the c o l l e c t i o n of a l l t o r s i o n c l a s s e s forms a complete l a t t i c e . Iicivever, t h e previous. s t a t e m e n t i s riot e n t i r e l y t r u e i n t h e usual s e n s e , for t h e " l a t t i c e " of t o r s i o n c l a s s e s i s a p r o p e r c l a s s , n o t a s e t . There simply a r e t o o many t o r s i o n c l a s s e s . Some f e e l i n g f o r t h e q u a n t i t y o f t o r s i o n c l a s s e s may be had by c o n s i d e r i n g t h e way i n which t h e l a t , t i c e of v a r i e t i e s i s eiiibedded i n t h e l a t , t i c e of t o r s i o n c l a s s e s . I t i s n o t a s u b l a t t i c e (Smith [24]) . Moreover, a s Martinez and I showed [ZO], no v a r i e t y c o v e r s a t o r s i o n c l a s s , s o e v e r y v a r i e t y i s the j o i n of a l l t o r s i o n c l a s s e s it, p r o p e r l y c o n t a i n s . B u t no n o n - t r i v i a l v a r i e t y i s t h e j o i n o f any s e t of t o r s i o n c l a s s e s i t p r o p e r l y c o n t a i n s . The irnport,ant word i n t h e p r e v i o u s s e n t e n c e i s " s e t . " T am n o t aware of any o t h e r n a t u r a l c o n t e x t i n which i t is i n t e r e s t i n g t o make a d i s t i n c t i o n between "complete" arid " s e t complete." That i s , a l a t t i c e may have t h e p r o p e r t y t h a t e v e r y s e t of i t s elements h a s an upper bound, y e t n o t be complete.

Many s i m i l a r phenomena o c c u r i n r a d i c a l c l a s s e s of groups ( w i t h o u t o r d e r ) a s s t u d i e d by Vovsi 1251.

3. bLEMENlARY CLASSES I n many ways the most n a t u r a l c l a s s i f i c a t i o n o f L-groups, o r any a l g e b r a i c s t r u c t u r e , i s by elementary c l a s s e s , t h a t is, c l a s s e s d e f i n e d by f i r s t o r d e r s t a t e m e n t s i n t h e language a p p r o p r i a t e t o t h e s t r u c t u r e . V a r i e t i e s a r e e l e m e n t a r y c l a s s e s , t o r s i o n c l a s s e s i n & e n e r a 1 a r e n o t . Other i n t e r e s t i n g c l a s s e s are ( i ) t h e c l a s s d e f i n e d by t h e s t a t e m e n t : i f x 2 = y 7 t h e n x - y , and ( i i )t h e c l a s s of d i v j s i b l e L-groiips, d e f i n e d by the i n f i n i t e s e t o f s t a t e m e n t s ( f o r each p o s i t i v e integer

n):

for a l l

x

there exists

y

such t h a t

y"

-

x.

The f i r s t o r d e r lanr:uage of o r d e r e d s e t s does n o t mnke very f i n e d i s t i n c t i o n s . I t is w e l l known, for example, t h a t any two dense t o t a l l y o r d e r e d s e t s w i t h o u t e n d p o i n t s a r e i n d i s t i n g u i s h a b l e i n t h e f i r s t o r d e r language. Because e v e r y L-group i s c o n t a i n e d i n an L-group A ( U ) of nutnniorphisms of a t o t a l l y o r d e r e d s e t $2, i t i s i n t e r e s t i n p t o c o n s i d e r the elementary t h e o r y o f such t-jiroups A(0). In 171, Gurevich and I i n v e s t i i 7 a t e d t h e q u e s t i o n of which e-groups a r e e l e m e n t w i l y e q u i v a l e n t t o t h e L-group A( B) of automorphisms of t h e r e a l l i n e . A t f i r s t g l a n c e , i t might h e expected t h a t s i i i c e 1 and Q, t h e r a t i o n a l s , a r e e q u i v a l e n t i n t h e f i r s t order lmipiage of o r d e r e d s e t s , perhaps A ( 1 ) and A ( Q )

W.C. Holland

154

are equivalent. Yhis i s fur from beinc the case, however, 9s con be seen f r m I.he following considerations. Iln element f E R ( Q ) i s said Lo be n ~?on:,es cyclrc if for some u € R, a < crf and i f R E Q and i3f # 13 then f o r :some intelTer n,

UP< P < a?’’.

Roughly speakinc, r i l l the points moved by

f

l i e In the

convexification or the o r b i t f n f n l of one point. This description of convex cycles i s not even i n t h e lanjp3,qe of‘ L-croups, l e t alone t h e f i r f i t order language. However, i t is seen t.0 be equivalent t o the followiw f i r s t order statement: f > e arid i f f = g V h and g A 11 e then E = e or h = e. I 1 can be shown [8] (and i s easy t o believe) thnt m y two convex cycles i n A ( 1 ) :ire condugate. ‘ihat i s , i f f and f ’ a r e convex cycles then there exisLs k such -1 t h a t f’ = k f’k. This f i r s t order stnten.ent is f a l s e for A ( Q ) bemuse a cmivex cycle whose interval of moved points hnu end points i n Q crmnot be con./u:nt.e to one whose i n t e r v a l of moved points has i r r a t i o n a l end pui!itn.

-

In 1’71 we showed the surprising r e s u l t t h a t there is R sentence + i n the first, order l w u a g e of L-groups such that A ( 1 ) satisfies $, and i f A ( R ) i s t r a n s i t i v e on R ( a technical requirement) arid s a t i s f i e s ti8 then 11 .is isoriiorphic t o P. That is, B i s completely characterized by a f i r s t order ststemen% concernint: i t s autoniorphism L-group. I t is not possi.ble t o chnracterixe Q i n t h i s way, since the L-eroup R(Q) is L-isomorphic t o A ( I I ), IT the i r r a t i o n n l s . B u t again i n [7] we showed there is u f i r s t order statement $1 i r i the 1aq:uabre of L-groups such thut A ( Q ) and A ( I I ) sntlvfy $ arid i f A ( R ) is t r a n s i l i v e arid s a t i s f i e s JI then R i s isomorphic tCJ e i t h e r Q o r IT. The sentences 4 and JI of the precedinf: pnrograph can be taken t o be purely i n the language of groups. And a s shown by .JRmbu-Ciraudet 1121 they cnn also be taken t o be purely i n the languwe of l a t t i c e s . The paper [l?] contains many more and deeper r e s u l t s along these l i n e s . REFERENCES Dieard, A., Keimel, K. and Wolfenstein, S., Groupee e t Anneaux RBticul6s (Lecture Notes i n Mathematics 608, Springer, 1077). Birkhoff, C., l a t t i c e Theory (Amer. Math. Soc. Colloquium Pub., vol. 25, 1969 1. Peil, T., An uncountable tower of L-group v a r i e t i e s , Algebra Universalis 11, 129-1 31.

( 1981)

Puchs, L., P a r t i a l l y rJrdered Algebraic Systems (Addison Wesley, New York, 19631. Class, A. M. W., Ordered Permutation Groups (London Math. Soc. Lecture Notes 55, Cambridge U. Press, 19H1). Class, A. M . W., Ilolland, W. C. und McCleary, Tr. H., v a r i e t i e s , Algebra Universalin 10 (1980), 1-20. Curevich, 7 . and Holland, W. C., hhth. SOC. 265 (1381), 527-534.

l?ie s t r u c t u r e of k-eroup

Hecoenixing the r e a l l i n e , Trans. Amer.

Hollnnd, W. C., The l a t t i c e ordered group of automorphisme of an ordered s e t , Mchlfzan Math. .T. 10 (1963), 399-408. Hollrind, W. C., The l a r g e s t proper variety of l a t t i c e ordered groups, Prvc. Amer. Math. Soo. 57 (19761, 25-28.

Classification of lattice ordered groups

155

I L+J I151

M a r t i n e z , .T., 'Torsion theor:/ Ti.r l a t t i c e ordered i:roups, ( 1.W5 ), ? g 4 - P ? .

Czech. Math.

.r. 25

Mart.iner,, :. , The fundarcental t,heorenl on torsion classes o f l a t t i c e o r d e r e d ):youps, '['rims. h e r . IMzith. :'oc . (19~?0),j11-317. M a r t i n e z , ,T. and Tiolland, iz'. f'. , A c c e s s i b i l i t y of' t . o r s i o n c l a s s e s , A lpe lirn U n i v e r s a l i s 9 197') ) , 19q-?Oh. Medvedev, N . ,Ta., 'The l a t t i c e s of v a r i e t i e s of l a t t i c e o r d e r e d p r o u p s and Lie a l g e b r a s , A1h;ehra and L n c i c 16 (1977), 27-30.

R e i l l y , N . R . , A s u b s e r ' i i l a t t . i c e tnf' t h e l a t t i c e of v a r i e t i e s of l a t t i c e o r d e r e d groups, Canad. ,T. Math. 33 (1981 ), 1109-1318.

Scrimger, E. 2,., A large class of small v a r i e t i e s of l a t , t i c e ordered groups, €'roc. h e r . Math. S o c . 51 (1975), 301-106. Smith, t T . , The l a t t i c e of !-group ( 11801, 347-?57.

v a r i e t i e s , Trans. h e r . hlath. S o c . 257

Vovsi, S. M . , i l n r a d i c a l and c o r a d i c a l classes of e-groilps, Algebra Universalis ( t o appear).

[26] Weinberg, E. C., F r e e l a t t i c e o r d e r e d groups, Math. Ann. 151 (1961), 187-18'4.


Annals of Discrete Mathematics 23 (1984) 157-170 0 Elsevier Science Publishers B.V.(North-Holland)

THE ISOMORPHISM

157

PROBLEM AND UNDECIDABLE P R O P E R T I E S

FOR F I N I 'TELY PRESENTED L A T T I C E - ORDERED GROUPS A. M.

W. Glass

1

Department of Mathematics Bowling Green State University Bowling Green, Ohio 43403, U.S.A.

Resume

Nous donnons une version abstraite d'une technique gbnkrale permettant de montrer, s o u s des hypotheses convenables, que la plupart des problemes algorithmiques concernant l'ensemble rPcursif des algebres finiment prhsent6es ( d'une variete d'algkbres rkcursivement axiomatisee ) ne sont pas resolubles recursivement quand i l existe line algkbre finiment presentee avec un problhe des mots non rksoluble. Nous Gtablissons un lemme technique ( The messuage Lemma ) pour les groupes rkticulks, qui utilise Les groupes d'automorphisrnes de chaines, e t nous l'utilisons pour montrer, par exemple, qu'il n'y a pas d'algorithme pour determiner si oui ou non un groupe r6ticul6 finiiient present6 quelconque a un seul elbment, ou est abelien, o u est totalement ordonne.

A Zattice-ordered group G i s a group and a l a t t i c e such t h a t f ( g V h)k

=

{xi: i E I}

-1

,

under I f each

and

f(g A h)k

fgk A fhk

=

for a l l

{xi: i E 1) be a non-empty s e t of symbols.

Let in

fgk V fhk

is t h e s m a l l e s t s e t containing

the l a t t i c e operations V

r . ( x ) i s a word i n J

i s c a l l e d a presentation.

and A ,

{xi: i E I},

f,g,h,k E G.

The set of uords

{xi: i E I } closed and t h e group o p e r a t i o n .

then

(xi; r j ( s ) ) i E T

In t h e s p e c i a l case t h a t

I

,jcJ

and

J

are

EJ,

we form t h e

f i n i t e , w e o b t a i n a finite presentation. Given a f i n i t e p r e s e n t a t i o n

n

=

(xi; r j ( x ) ) i

finitely presented lattice-ordered group t h e f r e e l a t t i c e - o r d e r e d group

F

on

Gn

by taking the q u o t i e n t of

{xi: i E I} by t h e k e r n e l (convex

normal s u b l a t t i c e subgroup) generated by t h e subset F.

So

r.(z) J

=

e

in

G

n

for a l l

j E J.

{ r . ( L ) : j E J} J

(Throughout we w i l l use

f o r t h e i d e n t i t y element of an a r b i t r a r y l a t t i c e - o r d e r e d group. ) 'Research supported i n p a r t by a Bowling Green S t a t e University Faculty Research Grant.

of e

A.M. W. Glass

158

THEOREM (THE ISOMORPHILSM PROBLEM).

There i s no recursive algorithm

t o determine whether o r not two a r b i t r a r y f i n i t e l y presented l a t t i c r ordered groups are isomorphic; i . e . , t o determine whether or not thio arbitrary f i n i t e presentations give r i s e t o isomorphic f i n i t e l ? / presented Zattice-ordered groups. Let

P

be a property of ( f i n i t e l y p r e s e n t e d ) l a t t i c e - o r d e r e d groups

t h a t i s preserved under isomorphisms.

nl, the f i n i t e l y presented

( i ) f o r some f i n i t e p r e s e n t a t i o n l a t t i c e - o r d e r e d group

i s c a l l e d a Markov property i f

P

enjoys

G,

P,

and

1 (ii)

there i s a f i r r t e presentation presented l a t t i c e - o r d e r e d group

n2

such t h a t t h e f i n i t e l y cannot be er?,bedded i n

G,,

2 any f i n i t e l y presented l a t t i c e - o r d e r e d group which enjoys

THEOREM. Let

P

be a Markov property.

Then t h e r e i s no recursive

algorithm which determines, f o r every f i n i t e presentation n ,

not

G

P.

whether or

enjoys P.

COROLLARY. There i s no recursive algorithm which determines, f o r every f i n i t e presentation

TT,

whether o r not

G

TI

enjoys

P,

is any o f the fozlowing:

( i ) abelian

(ii) Archimedean (iii) t r i v i a 2 (ivl

free

(v) f r e e abelian (vil

l i n e a r l y ordered

( v i i ) l i n e a r l y orderable ( p r e s e m i n g the group s t m c t u r e l (viiil

representable

( i x ) normal valued

(21 unique e x t r a c t i o n o f a l l roots ( x i ) soluble word problem.

where

P

Isomorphism problem and undecidable properties

159

Clearly examples l i k e t h e s e can be m u l t i p l i e d till t h e cows come We leave it t o the reader t o a d j o i n h i s own f a v o l i r i t e s .

home.

Most of t h e above p r o p e r t i e s a r e h e r e d i t a r y - - i . e . ,

a finitely

presented l a t t i c e - o r d e r e d proup which can be enhedded i n one possessing

P a l s o enjoys

P.

However, unlike groups, being a f r e e l a t t i c e - o r d e r e d

group i s not a h e r e d i t a r y property t h o w h i t i s Markov as we w i l l s e e . Every f i n i t e l y presented l a t t i c e - o r d e r e d group i s e f f e c t i v e l y isomorphic t o one defined by a s i n p l e r e l a t i o n - - I f

n

T j ( 5 ) l i E I ,jcJ

(xi;

=

with

r ( 2 ) E V { l r . ( & ) l : j E J} where Then

G,

7

J G,

,

since

IyI

>e

and

I

IyI

(ii

=

{0,1,2,... 1 ,

f i n i t e and

y Vy

& Iy/ = e

only consider f i n i t e p r e s e n t a t i o n s subset of

=

J

(xi;

-1, l e t

only if y

= =

e.

(xi;

r(x))iEI.

Hence we need

r ( z ) ) i E I . Regarding

I

as a

we can e f f e c t i v e l y ( r e c u r s i v e l y ) a t t a c h a

n a t u r a l number t o each f i n i t e p r e s e n t a t i o n i n a one-to-one onto way.

If

denotes t h i s number, then t h e theorem, s t a t e d p r e c i s e l y , says t h a t if

P

i s a Markov property, then

recursive s e t .

{ # ( n ) : Gn

er.jojis P}

i s not a

However, s i n c e t h e i n t u i t i o n i s more informative and can

e a s i l y be converted t o formally c o r r e c t statements, we w i l l use t h e i n t u i t i v e words from now on and leave t h e conversion t o t h e f o r m a l i s t . Whereas t h e proofs of t h e corresponding r e s u l t s i n group theory r e l y heavily on f r e e products with amalgamation or t h e Higman-Neumann-Neumann theorem ( s e e [6] and

“71 ),

t h e s e c o n s t r u c t i o n s a r e not v a l i d f o r l a t t i c e -

ordered groups ( [l, Theorem lac] and [ 2 ] ) and we must r e s o r t t o .&permutation groups and r e s u l t s (proved using 8-permutation groups) whose group-theoretic analogues n a t u r a l l y belong t o combinatorial group theory.

The isomorphism problem and ( i ) , ( T i ) , (iv), ( v ) a n d ( x i ) of

t h e c o r o l l a r y can be proved q u i t e e a s i l y and we g i v e a s e p a r a t e s o l u t i o n f o r t h i s p a r t ; t h e main theorem r e q u i r e s more d e l i c a t e techniques which beside

giving a l l p a r t s of t h e c o r o l l a r y a l s o y i e l d

problem by ( i i i )of t h e c o r o l l a r y

( L e t nn

=

t h e isomorphism

(xo; xo).

I f we could

solve t h e isomorphism problem, we could r e c u r s i v e l y determine f o r an

A.M. W. Glass

160

a r b i t r a r y f i n i t e presentation whether or not

G,,

P

{el.

whether or not

TI,

G

*

E

G

i.e.,

;

no

This c o n t r a d i c t s ( i i i )of t h e c o r o l l a r y ) .

The key t o a l l t h e proofs i s t h e existence o f a f i n i t e l y presented (non-abelian) l a t t i c e - o r d e r e d group with i n s o l u b l e word problem ([l, Chapter 131 or, more f u l l y , i n

[&I).

We show t h e non-existence of

r e c u r s i v e algorithms by coding i n t h e word problem for t h i s l a t t i c e ordered group.

Most of t h e proofs a r e general a l g e b r a ; t h e exceptions

a r e t h e proofs f o r t h e Isomorphism Problem Theorem and t h e Messuage The Messuage Lemma i s p i v o t a l and i s proved by using l a t t i c e -

Lemma below.

ordered groups of automorphisms of ordered s e t r . To prove t h e isomorphism problem we use t h e followinp r e s u l t of Keith R . Pierce [l, Theorem 10B] which i s proved by using b?-perinutation groups ( i. e . , l a t t i c e - o r d e r e d groups of order-preserving permutations of l i n e a r l y ordered s e t s )

Lam 0. -

.

Every lattice-ordered group can be embedded in one in

which any two strictly positive Now l e t n o

I>

el elements are conjugate.

be t h e f i n i t e p r e s e n t a t i o n of t h e f i n i t e l y presented

non-abelian l a t t i c e - o r d e r e d group with i n s o l u b l e word problem; say

no

= ( x l , . , .,xn;

Let

= (xO,xl , . . . , x n ;

Ti

0,w

r(E),

G

-1

( x o ~ w l - ~ x ~ )](x i ( ) ) . BY Lema 0 , i=l

can be embedded i n a l a t t i c e - o r d e r e d group

G

. . , xn .

r ( 5 ) ) . Let w be an a r b i t r a r y word i n x l , .

H

(wI

i n which

and

,O n

v

\xi]

a r e conjugate ( i f

i=l

h-'/w(h

=

v

(xi( i=l

by t h e image of of

w # e

in

G

).

h E H be such t h a t

Let

n

and

G,,

Ho be t h e s u b l a t t i c e subgroup of together with

0 hold i n Ho i f

G,

xo

h.

image of

h,

G and t h e diagram

G,

generated

Since a l l t h e defining r e l a t i o n s

i s replaced by

0,w

H

Ho i s a homomorphic

-G noc

0,w

/

=o,w

HO i s commutative under t h e mapping

xi

M

xi (1< i

< n)

of

into

Gn

0


.

G n 0,w

If

,

e # g 6 Gn

then t h e image of

e.

0

Since t h e diagram i s commutative, t h e image of Hence t h e homomorphism of in

Ho i s not

in

g

161

.

G,,

into

G

n0 However, i f

w

e

=

in

in

i s not e . 0,W i s an embedding if w # e g

Gn

0,w G n O , then

w

=

G,,

e

i n any l s t t i c e -

0

r(2) = e .

ordered group i n which in

, xo-1Iw(xo

G,,

v

=

0,w IyI

e

=

only i f

y

=

e).

group of i n t e g e r s , Z ElZ

(l,-l); note t h a t

I-+

( x i ( ; hence

x

i=O

l a t t i c e - o r d e r e d group on

xo

In p a r t i c u l a r ,

w

=

e

in

Thus

xo

i

=

But,

.

cn

n

0,w ( / y (2 e

e (1 4 i S n )

with

is j u s t the free (abelian) 0,w ( I t i s isomorphic t o Z m Z where Z i s t h e Gn

i s ordered by:

(m,n) 2 ( 0 , O ) i f

m,n 2 0,

and

( 1 , O )= (l,-l)v (0,O)& ( 0 , l ) = -(l,-l) V (0,O)).

Now t h e f r e e ( a b e l i a n ) l a t t i c e - o r d e r e d group on one g e n e r a t o r has s o l u b l e word problem [l, Theorem 1 1 . 5 1 .

I f t h e r e were a r e c u r s i v e algorithm

which determined whether t h e f i n i t e l y presented l a t t i c e - o r d e r e d groups given by two a r b i t r a r y f i n i t e p r e s e n t a t i o n s were isomorphic or n o t , we would be a b l e t o decide whether or not t h e f i n i t e l y presented l a t t i c e ordered group given by an a r b i t r a r y f i n i t e p r e s e n t a t i o n were isomorphic t o t h e f r e e ( a b e l i a n ) l a t t i c e - o r d e r e d group on one g e n e r a t o r . could determine f o r an a r b i t r a r y word not

w

in

xl, ..., xn,

So w e

whether o r

were t h e f r e e ( a b e l i a n ) l a t t i c e - o r d e r e d group on one 0,w i s non-abelian and has i n s o l u b l e word problem, g e n e r a t o r . Since G

Gn G

GTI

i s non-abelian and has i n s o l u b l e word problem whenever

w # e

in

0,w

.

(An algorithm f o r t h e word problem f o r

would imply t h e

0,W e x i s t e n c e of an algorithm f o r t h e word problem for G,,

.) 0

Hence

Gn

0,w

i s isomorphic t o t h e f r e e ( a b e l i a n ) l a t t i c e - o r d e r e d group on one generator i f and only i f

w = e

in

GTT 0

.

Thus t h e r e c u r s i v e algorithm f o r t h e

ismorphism problem l e a d s t o a r e c u r s i v e algorithm f o r t h e word problem in

.

Gn

This l a s t i s a c o n t r a d i c t i o n and e s t a b l i s h e s t h e Isomorphism

0 Problem Theorem.

Since any f i n i t e l y generated f r e e l a t t i c e - o r d e r e d group

162

A.M. W. Glass

has s o l u b l e word problem [l, Theorem 1 1 . 5 1 and any Archimedean l a t t i c e ordered group i s a b e l i an,

i s a b e l i a n (Archimedean, f r e e , f r e e 0,w . a b e l i a n , has soluble word problem) i f and only i f w = e i n G

Gn

no

//

This e s t a b l i s h e s ( i ) , ( T i ) , ( i v ) , ( v ) and ( x i ) of t h e c o r o l l a r y .

The above proof a l s o e s t a b l i s h e s t h a t " f r e e " i s indeed a Markov property. I n o r d e r t o prove t h e Markov Property Theorem (and hence t h e e n t i r e c o r o l l a r y ) , we need t o do some work.

The proof hinges on t h e Messuage

Lemma which we now g i v e . Let word i n

d x ) ) i E I be

IT = (xi;

{xi: i

E I}. T I ( w )

f i n i t e p r e s e n t a t i o n such t h a t ( i i ) GTI can be embedded i n

THE MESSUAGE -

LEMMA.

a f i n i t e p r e s e n t a t i o n and

i s c a l l e d a w-rnessuage' of

Gn(w) i f

w # e

TI

w = e

{e} i f

( i )G n ( w )

in

be a

w

i f it i s a in

GTI and

G,.

There i s a recursive algorithm which, given an

a r b i t r a r y f i n i t e presentation

TI =

(xi; r l ~ ) l i f l and a word w i n

{xi: i E I}, constructs ( e f f e c t i v e l y ) a w-messuage of

K.

We temporarily postpone t h e proof of the Messuage Lemma (which a l s o appears as Theorem

D i n [ 3 ] where i t i s used t o prove t h a t every countable

l a t t i c e - o r d e r e d group can be embedded i n a 7 generator L-simple l a t t i c e ordered group) and show how t o deduce t h e Markov Property Theorem from i t .

x: Let

no

be t h e f i n i t e p r e s e n t a t i o n such t h a t

n1 and n 2

has

KO

i n s o l u b l e word problem.

no,

G

Let

nl, n 2 be f i n i t e p r e s e n t a t i o n s such t h a t

a r e d i s j o i n t s e t s of symbols,

enjoys

G,

P

and

1 G,

can not be embedded i n any f i n i t e l y presented l a t t i c e - o r d e r e d

2

'Messuage was " o r i g i n a l l y t h e p o r t i o n of land intended t o be occupied, o r a c t u a l l y occupied, a s a s i t e f o r a dwelling-house and i t s The word seems p a r t i c u l a r l y appurtenances" [Oxford English Dictionary] a p t f o r t h e property we a r e describing; " t e s t " was used i n [ 7 ] .

.


163

group that enjoys P. Let G be the free product (as lattice-ordered groups) of G

,

and G

TI

.

2 the union of those of n o and

Then G

=

G,

where

has as "generators"

0

where r(2) and s(y-)

and "relation" Ir(,x)l V I s ( x ) l

TI2,

are the "relations" of no

is effectively obtained from no

Note that

and n2

and n2.

respectively.

Since G

TI

0 has insoluble word problem, so does G,

can be (even effectively)

(Sn

0

embedded in GTI) .

L,et w be a word in the "generators" of

and G(w) be free product (as lattice-ordered

be the w-messuage of n g r o u p s ) of

G

,

and G

1

n(w).

Now if w

=

e

in G , , G(w)

and hence enjoys P. If w # e in G ,

isomorphic to G,

TI~W)

TI,

is

,

G

be embedded in G

2

n(w)

and so in G ( w ) ;

does not enjoy P.

thus G(w)

Consequently, G(w) enjoys P if and only if w G,

can

TI

1

=

e in GTI. Since

has insoluble word problem, the Markov Property Theorem follows. // We next note some consequences of the proof. A property P

of finitely presented lattice-ordered groups that is

preserved under isomorphisms is said to be incompatible x i t h f r e e ( d i r e c t ) productsiffor some finite presentation

(i )

TI

1

enjoys P,

the finitely presented lattice-ordered group G n 1

and (Ti)

the free (direct) product (as lattice-ordered groups ) of G 1 and any non-trivial finitely presented lattice-ordered group always fails to enjoy P.

THEOREM. I f

P

i s a property t h a t i s incompatible w i t h f r e e

( d i r e c t ) products, then t h e r e is no recursive algorithm t o determine, f o r every f i n i t e presentation

n,

whether or not

G n enjoys

P.

In

particuZar, there i s no recursive aZgorithm t o determine whether or not

a f i n i t e l y presented lattice-ordered group can be w i t t e n as t h e f r e e ( d i r e c t ) product ( a s Zattice-ordered groups) of two non-trivia2 l a t t i c e ordered groups.

A.M. W. Glass

164

prooS: Let n 1 be the guaranteed presentation and no as in the previous proof. Again we assume that the generators of no and are disjoint. For an arbitrary word

TI

1

w in the generators of n o ,

G(w) be the free (direct) product of G

);

and G

let

the presenta-

n1

tion n for G(w)

can be effectively given from that of

In either case, G(w)

enjoys P if and only if w

=

TI^

and n,(w)

.

e in G TI

This

0

proves the first part of the theorem. Let n1

=

(xl: x1 2 e);

f o r the “in particular. I’

so

G TI,

3

Z

has the desired requirements

//

Finally, we note that the above proofs have nothing to do with lattice-ordered groups. Hence: THEOREM (essentially Rabin [7]) ,

Assume a c l a s s of algebras has

f i n i t e signature and a one element t r i v i a l algebra contained i n a l l algebras.

Further suppose t h a t f r e e algebras e x i s t i n t h i s c l a s s , t h e

Messuage Lema h02ds and there e x i s t s a f i n i t e Z y presented algebra i n t h e c l a s s with insoluble word problem.

If, moreover, the c l a s s admits

f r e e ( d i r e c t ) products and every f i n i t e l y presented algebra can be embedded i n i t s f r e e ( d i r e c t ) product with any other f i n i t e l y presented algebra (and t h e d i r e c t product of t v o f i n i t e l y presented algebras can be given a f i n i t e presentation which i s e f f e c t i v e l y obtainable from that of t h e two swnmands), then t h e Markov Property Theorem holds a s does

t h e previous theorem.

In direct contrast to the above result, the proof of the Messuage Lemma that we give is very concretely rooted in lattice-ordered groups it is quite unlike Rabin‘s proof for groups (or the easiest proof given in [6,Section IV 4 1 ) and is clearly the pivotal result in this paper. Let

TI =

(xi; (r(z))ieIr

Assume, for ease of notation, that

xG,x1,x2,x3,xh j! {xi: i E I} and let w be a word in {xi: i E 11.


First, we assume that w

=

.

e in G ,

Then since r(5) = e in

-1

But xo lwlxo = V { l x i / : i E TI; hence -1 A lx41 = e; xi e for all i E I. Sin1.e w = e, lxcl = \x4 1 = ~wx/,~l -1 so x4 = e. Now x2 Ix Ix ixOI; thus x : e. By the last relation 4 2 6 i n the presentation of G x1 = e . Since 1x21 V ix3i Q xl, it (w)' f o l l o w s that x Therefore a11 generators of are 2 - x3 = T I (W)

w

G,(W)'

=

e

in

G n ( W ) '

equal to e in Gn(w);

i.e.,

G n L w ) 5 {el.

To prove the other part of the Messuage Lemma (viz.: in G,

, then G, can be embedded in G

if w # e

we need the following (w)' facts from [I]. The first is the Cayley-Holland Theorem (Appendix I), ll

the second comes from the proof of Lemma 2.2.1 and the third comes from the proof of Theorem 2E ( see the remark on page 58 ).

Lemmas 2 and 3

can be viewed as showing the considerable degree of "local" freedom in the choice of roots and conjugators. If R

for the lattice-ordered group of order-preserving permutations

i\(R)

of

is a linearly ordered set ( o r chain for short), we write

R

(where f Q g

. a

< al

(i

=

& Bo

O,l),

B = U ( F i ~ F i I i E11 if I is finite. But if X is equipped with the Tietze-topology then for arbitrary I (finite or not) the boundary of B is the union of all faces of B, that means Fiu Fi(iE I ] ,

x(

x

u(

Topological properties of Cartesian products

177

4. THE GENERALIZATION OF BROUWER’S FIXED POINT THEOREM FOR PRODUCTS OF LINEARLY ORDERED CONTINUA 4 ] proved a fixed point theorem for chainable In 1956 E.Dyer [ continua which contained among others a generalization of Brouwer‘s fixed point theorem to n-continua. Here we shall give a new proof of this latter result which uses only elementary methods in addition to Brouwer‘s fixed point theorem. The idea of proof is to translate the theorem to an equivalent statement for which the euclidean situation is easier to transfer. By the way, we extend Dyer’s fixed point theorem for n-continua to products of arbitrarily many factors.

The following statement is equivalent to Brouwer‘s fixed point theorem (see e.g. [6J1p.161,Satz V.4.10 and also the proof of theorem 4.4 given below): STATEMENT 4.1. Let C = LO,l]” be the unit cube of Rn. For every n) let T, be a closed subset of C which separates in C V€{l, the opposite faces Fv and F; of C. Then IT,,/3 E { l , n)) # 0.

...,

...,

n

Now we transfer this statement to n-continua:

...

x [an,bn] be a box THEOREM 4 . 2 . Let B = [al,bllx . x Cn. For each V E { 1,. ,nt let Tv nuum X = C1x subset of B which separates in B the left v -face F, v-face F.; of B. Then we have n{T,,lVE i1,...,ntt

..

..

of an n-contibe a closed and the right # 0.

PROOF. We assume the contrary. Then T1nr\{TVIV = 2,...,n) = 0. Since the sets T9 are compact we can find finitely many open boxes of X such that their union V contains T1 but the union of their closures is disjoint to 1 T, I Y = 2,. ,n) Then also there exist 1 B2,.. 1 finitely many closed boxes B1, ,Bil which are subsets of B

n

.

.. .

such that their union V1 contains T1 and satisfies (1) V1n T2n

...

n

Tn

=

0.

...

Now T2 and V 1 n T 3 n T 4 n n Tn are closed, bounded and disjoint because of (1). Then there exist analogously finitely many closed 2 2 boxes B1, B 2 , . , , B I which are subsets of B , such that their union 2 V2 satisfies V 2 2 T 2 and V2n(VlnT3nT4n nTn) = 0.

.

...

So (1) remains valid if T2 is replaced by V2.

So

continuing we ob-

E. Harzheim

178

tain after n steps sets V,,=TV (2)

n{v,lv

=

~,...~nt =

for V E { ll...,nl with

PI,

9 where each Vv ,V = 1,...,n, is a union of finitely many boxes B1,.. ,Byv which are subsets of B. Let & = {B1,. ,Bk) be the set of all boxes B Y v = 1,. ,n,U , = 1,. , iy in an arbitrary ordering. P' For v = 0 we define Bo := B. Of course also Vv 2 TV separates the faces FV,F$ in B. And since 2& is a finite set we can accomplish our proof by a method of translating the situation into the euclidean space Rn. Let B, = 1 1 2 2 n n Lav,by] x Lav,] , b x x [a,, ] , b for v = 0 , k. Then we ascribe to these boxes By certain boxes

..

..

..

..

.. .

. ..

of IR", such that the coordinate relations between the boxes B, , V = O,...,k, are the same as those between the boxes B:,v= O,..,k. More precisely: For each,urz { I , n\ let Dpbe the set containing bgl ay, br,, ,ac, b t Then Dp is a finite suball elements set of C p . For each p~ { 1,. ,n] let D* be a set of real numbers

q,

...

.. ..

...,

.

P

,*!a b[*, , for 0 < o L < I. Be-

x

\.

z.... -

cause of the compactness of B there follows (6) E : = n { A , , , I O

< = < I)

0.

...

E. Harzheim

180

But now we have (7) E

G fXTiJiEI\.

For if x E E then for every i E I we have ~ ( x ) E -(Ti) for a l l a with i < oc < I , which gives x E Ti for all i E I since the Ti are closed, (6) and (7) now complete the proof. Using 4 . 3 we now derive the general fixed point theorem for boxes with arbitrarily many factors: THEOREM 4 . 4 . Let B be a box in a space are linearly ordered continua, f: B + B fixed point.

x {Cili€ I\ where

the Ci continuous. Then f has a

It can easily be seen that the sets Ti, i E I , are closed. Also we have for i E I: (8) Ti separates Fi and Fi in B.

Indeed, let i E I and K C B be a continuum satisfying K n Fi # $ K n Fi. If K n Ti would be void we would have K = K'u K" where K' : = \ x € Klxi < fi(x)i and Kff : = { x c K(xi > fi(x)l are closed sets and nonempty because of $ # K n Fi E; K' , $ # K n Fi E K". But this yields a contradiction to K being a continuum. So (8) is valid for all i E I. BY 4 . 3 the set n { T i l i E 11 of fixed points of f is nonempty.

#

A well-known theorem of Brouwer states that the fixed-point property of the euclidean unit n-ball Bn = x E IRn I 1x1 f 11 is equivalent to the statement that its boundary Sn-1 is not a retract of Bnl that means: There is no continuous mapping f: Bn Sn-l for which f(x) = x holds for all x E Sn-l. This theorem was generalized by van Dalen ([20] ,p.91-95) to boxes of n-continua. In his proof he applied a method of Dyer [4]. Using 4 . 3 we shall give a new (and shorter) proof and at the same time an extension of this theorem to boxes of arbitrarily many factors.

+

THEOREM 4.5. Let Ci, i g I , be linearly ordered continua. For i E I


181

let cai,bi] be a closed interval of Ci. We put B : =

x {[ai,bi] IiE I}. The union u { Fiu Fi li€ It. Then there with f(x)

=

x for all x E

of all faces of the box B is U : = is no continuous mapping f: B + U

U.

PROOF. To construct a contradiction we assume that such a mapping f does exist. Then for each i E I we choose an element e i E Ci satisfying ai < e. < bi. We define for i E I Gi {Eji jG I] 1 where Ei : = [ai,ei) and E . * - C . for the indices j # i from I. J '- J If we replace here Ei = [ai,ei) by Ei = (ei,bi] , we obtain the definition for Gi. -1 -1 Further for i € I we put Oi : = f (Ci) and 0 ; := f ( G I ) . Of course, Oi and 0: are disjoint, and we have for i E I:

:=x

(9) Oi is open in B, Oi 2 Fi, O i n Fi

=

$

(9') 0 ; is open in B, O! 2 F i , 0; n Fi

=

$.

and

For i E I now Ti : = B \ (Oi u 0 ; ) is non-empty, because B is connected. Also T. is closed, and Ti separates Fi and FI in B. In1 deed, if K is a continuum -C B meeting Fi and F I we have K n Oi # 0 # K n 0 ; . Now if K n T i would be void we would have K G 0 i. u 0i' and thus K = (KnOi)d ( K n O I ) contradicting to the fact that K is connected. Hence, according to 4.3, D : = n { T i l i € 11 is non-empty. Now let z be an element of D. Then there exists an index j € I with f(z) E FjuF; 5 G . u G ! Thus z s f-l(Cj)u f-l(G!) = 0 u O! = B \ T J J' J J J j which contradicts to z € T . J'

In connection with 4.5 we mention the following: If p is any point of the interior of the unit n-ball Bn Wn, then its boundary Sn-l is a retract of Bn\ { p] Van Dalen ( [20] ,p.91-104) shows that an analogous statement is valid in certain product spaces X x X if X is a closed interval of a linearly ordered continuum satisfying a certain homogeneity condition. But in the general case such a statement is no longer valid. Van Dalen gives a counter-example in

.

[ZO], p.106, Theorem 10. There he proves: Let X be a segment [a,b] of a linearly ordered continuum, such that the corresponding open interval (a,b) is coinitial with -* and cofinal with , where L X , ~are regular initial ordinals with Let R be the boundary of X X X and p € X x X \ R , then R is cY>p not a retract of x x x \ { p}

.

p

.

In connection with 4.4 there arises a natural question, namely

182

E. Harzheim

whether the statement of 4.4 remains valid if the set )( 1 CiliG I) is equipped with the Tietze-topology (see 3 . 2 ! ) . But here the answer is negative in all cases where I is infinite. This can easily be shown using a theorem of Knight GO]: THEOREM 4.6. If I is an infinite set, Ci a linearly ordered continuum for i E I, [ai,bi] a closed interval of Ci, then the set B : = 1 [ai,bi] IiE I} , equipped with the Tietze-topology, is not connected, Then there exists a continuous mapping (with respect to the Tietze-topology) f: B + B which has no fixed points.

x

PROOF. The first part is a special case of Theorem 5 . 1 of [lo] according to which every two points of B which differ in infinitely many components, are in different connectivity components of B. Here one needs also that the spaces, where the components differ, are regular T1-spaces (see 3 . 2 ! ) . So there exists a decomposition B = B'LIB", where B' and B" are non-empty disjoint open sets of B. We choose points x'E B', E B" and put f(x) := X I if x E BIl and f(x) := XI! if x E B'. Then f has no fixed point but is continuous in the Tietze-topology. 5 . CLOSED CONNECTED SETS IN n-CONTINUA

In this section we shall investigate closed connected sets in n-continua. Here we have the opportunity to illustrate another technique of proving theorems concerning n-continua. In this context it turns out to be useful to generalize the notion of manifold to n-continua. In the euclidean case one defines an n-dimensional manifold to be a separable metric space, which is locally euclidean, that means: Each point has a neighborhood which is homeomorphic to Rn. Of course, we have to modify this concept now: DEFINITION 5 . 1 . Let n be a positive integer, C an n-continuum. We call a set Q S C a box union, if there exist finitely many closed boxes B1,...,Bm in C, such that the following condition is satisf ied: (10) For each two indices i,j E { l , . . , , m ) is void o r a box of C.

the intersection B i n B

j

If p e C \ Q we denote by R (Q) the boundary of the connectiP vity component of p in C \ Q. Evidently (as in the euclidean case)


183

R ( Q ) is a subset of the union of the boundaries of the sets B1,.. P , Bm. Now a subset M C C is called a manifold, if M is connected and if there exists a box union Q and a point p E C \ Q such that M = R (Q). P What we have defined here corresponds roughly to the concept of a closed connected piecewise linear manifold of dimension n-1 which is embeddable into Rn.

..

Since there was only used a finite number of boxes to define a manifold and since this is required to be connected one can easily check the validity of the following theorem: THEOREM 5.2. If M is a manifold in an n-continuum C, then the set C \ M has exactly two connectivity components. One of them is bounded, the other not. The first one is called the inner region of M and the second one the outer region. We denote them by I(M) resp. A(M). We omit the proof of 5.2 which could be constructed analogously to the corresponding statement in the euclidean case or by reducing the above situation to the euclidean case (using the same Iltranslation principlettas in the proof of 4.2).But we give a sketch how the proof for the case C = Rn can be arranged: First the following analogon to the theorem of the Jordan arc is to be proved:

1) If M is a manifold (in the sense of 5.1) of IR", and if W is a subset of M which is isometric to an (n-1)-dimensional cube, then the set M \ W does not separate IR". 2) Let p be a point of IRn\ M. Then all rays starting at p and satisfying a certain condition of general position have an even (resp. odd) number of intersection points with M. In the first case p is called an even point, in the second an odd one. 3) If p~ IRn\ M is even (resp. odd) and if S is a segment containing p and avoiding M then all points of S are even (resp. odd). Thus all points of a connectivity component of Rn\ M are even or all are odd. 4) If p,q are in Rn\ M and one of these points is even and the other one odd then by 3) p and q are in different components of IRn\ M,

E. Harzheim

184

5 ) There exist points p,g E IRn \ M such that one of them is

even and the other one odd. So by 4 ) IRn\ tivity components.

M has at least two connec-

6) Let W be a subset of M which is isometric to an (n-l)-dimensional cube, c the center of W. Then for each p E IRn \ M there exists (because of 1)) a polygonal arc lT which links p with c P such that TT n M = 1 c) , Then TT arrives at c at one of the two siP P des of W. All points p J whose 7 arrives at the same side of W P then can be joined by a polygonal arc avoiding the whole set M. Thus IRn \ M has at most two connectivity components.

Next we need a lemma concerning the separation of disjoint closed sets by manifolds: LEMMA 5.3. If T is a closed bounded subset of an n-continuum C and if T is not connected then there exists a manifold M such that T n M = 0 , TnA(M) # 0 # T n I ( M ) holds. By assumption there exist disjoint nonempty closed sets T1, = T1u T2. For each x E T1 there exists an open box Bx containing x such that its closure Fx is disjoint to T2. There are finitely many open boxes of that kind such that their union covers TIJ say BlJ...,Bm. We can assume that their closures B I J . . .'Bm have pairwise an intersection which is void or a box. We divide the set of boxes B1,...,B, into classes such that two B , B , belong to the same class if they are subsets of the P same connectivity component of B 1 u . . u Bm' Let & be such a class. Then V := {ilB E& 1 is a box union. We choose a point p E T2. Then one can verify that M : = R p ( V ) (see 5.1!) is a manifold for which p and T1n V are situated in different connectivity components of C \ M. PROOF.

T2 with T

u

.

Now we generalize Janiszewski's theorem on continua ([8]) and We cannot with this the theorem of Sierpinski on continua ([IS]). take over Janiszewski's proof in which a continuum is constructed

185


as limes superior of a sequence of point sets which are and wheref tends to 0. First we need a simple lemma:

-chains

LEMMA 5 . 4 . Let B be a box of an n-continuum, a a limit ordinal, and ( P A < A a transfinite sequence of points of B. Then there exists a point h E B such that for each ordinal o C < and each open box Q containing h there exists an index e with OC < e < such that p E

e

Q. Shortly spoken: In each neighborhood of h are cofinally many

points of (P,,),, 0 and if for all ordinals (F O , or y = I and x > O > is open in X but fails to have an injective hull, since condition 3.7(3) fails for x = ( 1 , l ) . (This shows that Corollary 4 in [B-771 is false.) Notice that Y is locally quasicompact and sober.

By contrast one should mention a result of R.-E.Hoffmann's according to which a dZo;cu subspace of a To-space with an injective hull has itself an injective hull. (Indeed, if Y L X S A X with injective A X and Y closed in X, then there is a Scott closed subset S of A X with Y = X n S . If S E S, then s = sup(+snX) since X is sup-dense in AX. But S is a lower set, whence J s n X c S n X = Y so that s = sup(JsnY). Set e = sup S and A = S u {el. Then A is a subalgebra of + e , hence is a continuous lattice, in which Y is sup-dense. The Scott topology 01 A is that induced from the Scott topology of A X , whence Y is topologically embedded into A . Hence Y has an injective hull by 3.11.)

4.2 EXAMPLE. belongs to X (a)

x

(b) (c)

1/2 < x

x

= =

Let C be the following subset of the square L: A point (x,y) if at least one of the following conditions is satisfied:

0 or

y

or

= 1/2

y

=

0,

1/2

K+

whenever

K

has uncountable c o f i n a l i t y .

The

f u l l e s t i n f o r m a t i o n a v a i l a b l e on w e l l - o r d e r i n g numbers 6 ( ~ i) s i n Theorem V I I . 5 . 5 of S h e l a h 1551 ( S h e l a h w r i t e s 6 ( ~ , 1 ) f o r 6 ( ~ ) ) .

S e t t h e o r i s t s know some o t h e r ways of f i n d i n g EM f u n c t o r s . For example, i f cardinality
i.

For example, l e t A be t h e symmetric group S3, l e t a and b be the cycles ( I x-2yx2 = y .

2 ) and ( I 2 3 ) r e s p e c t i v e l y , and l e t $ ( x , y ) be t h e e q u a t i o n Then t h e h y p o t h e s i s of Lemma 9 h o l d s .

From S h e l a h ' s theorem

we i n f e r t h a t e v e r y v a r i e t y of groups which c o n t a i n s S3 c o n t a i n s ZK p a i r w i s e non-isomorphic g r o u p s i n e a c h u n c o u n t a b l e c a r d i n a l i t y

K.

Baldwin and

McKenzie [ 3 1 u s e t h e same method t o show among o t h e r t h i n g s t h a t i f V i s a congruence-modular non-abelian v a r i e t y , t h e n i n e v e r y u n c o u n t a b l e c a r d i n a l i t y K,

V c o n t a i n s ZK p a i r w i s e non-isomorphic a l g e b r a s .

(Hodges [ 3 0 a ] s t r e n g t h e n s

this last result.) What happens i f w e want t h e 2'

s t r u c t u r e s i n Theorem 8 t o be

e q u i v a l e n t t o A i n some i n f i n i t a r y language?

The answer i s t h a t we u s e

e x a c t l y t h e same argument a s f o r Theorem 8 , b u t w i t h E x i s t e n c e Theorem 6 i n p l a c e of E x i s t e n c e Theorem 5.

The h y p o t h e s i s i n Theorem 8 h a s t o be

strengthened t o say t h a t t h e chain enough o r d e r - t y p e . an i n f i n i t a r y s e n s e ;

n is

well-ordered

by $ ( x , y ) i n some l o n g

(Such c h a i n s e x i s t i f t h e t h e o r y of A i s ' u n s t a b l e ' i n see Shelah 1511.)

M a c i n t y r e and S h e l a h [381 used t h e i n f i n i t a r y v e r s i o n of Theorem 8 t o answer a q u e s t i o n of Kegel and W e h r f r i t z [331 a b o u t t h e number of non-isomorphic u n i v e r s a l l o c a l l y f i n i t e g r o u p s i n a n u n c o u n t a b l e c a r d i n a l i t y (A group i s l o c a l l y f i n i t e i f f e v e r y f i n i t e l y g e n e r a t e d subgroup i s f i n i t e ;

K.

W.Hodges

220

i t i s u n i v e r s a l l o c a l l y f i n i t e i f f i t i s l o c a l l y f i n i t e , i t c o n t a i n s a copy

of every f i n i t e group, and e v e r y isomorphism between f i n i t e subgroups

e x t e n d s t o a n i n n e r automorphism of t h e whole g r o u p . ) They showed t h a t t h e number i s always Z K .

T h i s r e s u l t has somehow gained a

q u i t e undeserved r e p u t a t i o n f o r b e i n g o b s c u r e ( p r o b a b l y b e c a u s e t h e a u t h o r s r e f e r t o [511,

which i s a r e d h e r r i n g ) .

In f a c t t h e proof f a l l s s t r a i g h t o u t L e t me s k e t c h i t .

of r e s u l t s a l r e a d y used above, t o g e t h e r w i t h f a c t s i n [331.

The f i r s t s t e p i s t o check t h a t t h e c l a s s of u n i v e r s a l l o c a l l y f i n i t e groups i s a x i o m a t i s e d by a s i n g l e s e n t e n c e $ of L immediate from t h e d e f i n i t i o n .

f i n i t e and hence e v e r y power of i t i s l o c a l l y f i n i t e . ordinal

.

This i s I Next we n o t e t h a t t h e symmetric group S3 i s

(Iw s u f f i c e s , given t h e value

6(w)

= w

1

1

w w

L e t a b e some l a r g e

i n 13).

By Lemma 9 t h e

l o c a l l y f i n i t e group ( S )a c o n t a i n s a c h a i n rj of e l e m e n t s o r d e r e d i n t y p e a 3 by a q u a n t i f i e r - f r e e f o r m u l a .

Now (S )a c a n b e extended t o a u n i v e r s a l 3

l o c a l l y f i n i t e group G (by Theorem 6.5 i n [ 3 3 1 ) .

S i n c e G i s a model o f @ and

r~ is o r d e r e d by t h e same q u a n t i f i e r - f r e e formula i n G a s i n (S )a, E x i s t e n c e 3

Theorem 6 f i n d s a n EM f u n c t o r F j u s t a s r e q u i r e d i n t h e proof of Theorem 8 above.

The ' o u t p u t s t e p ' of t h a t proof g i v e s u s ZK p a i r w i s e non-isomorphic

models of $ i n e a c h u n c o u n t a b l e c a r d i n a l i t y

K.

A c t u a l l y most of t h i s proof h a s n o t h i n g t o do w i t h g r o u p s . A g e n e r a l s t a t e m e n t of t h e r e s u l t i s a s f o l l o w s .

L e t V be a v a r i e t y

such t h a t ( i ) V c o n t a i n s a f i n i t e a l g e b r a s a t i s f y i n g t h e h y p o t h e s i s of Lemma 9 f o r some e q u a t i o n J l ( x , y ) , and ( i i ) f o r any embeddings f : A g: A

-+

-+

B and

C between f i n i t e a l g e b r a s i n V, t h e r e i s a f i n i t e a l g e b r a D i n V

w i t h embeddings f ' : B + D and g ' : C uncountable c a r d i n a l i t y

K

+ .

D such t h a t f ' f = g ' g .

Then i n e v e r y

t h e r e a r e ZK p a i r w i s e non-isomorphic e x i s t e n t i a l l y

c l o s e d l o c a l l y f i n i t e V-algebras. Can w e e x t e n d Theorem 8 by making t h e ZK s t r u c t u r e s non-isomorphic i n some s t r o n g e r s e n s e ?

For example, c a n they b e chosen t o have o n l y a

small overlap with each o t h e r ?

Models built o n linear orderings For a l l u n c o u n t a b l e r e g u l a r t h e f a m i l y of Z K models of c a r d i n a l i t y embeddable i n any o t h e r .

S h e l a h [551 was a b l e t o choose

K,

K

22 1

s o t h a t none was e l e m e n t a r i l y

I n any r e g u l a r non-weakly-compact

cardinality

K,

Hodges [301 found two s t r u c t u r e s B, C e l e m e n t a r i l y e q u i v a l e n t t o any g i v e n i n f i n i t e A , such t h a t i f D i s e l e m e n t a r i l y embeddable i n b o t h B and C t h e n D c o n t a i n s no c h a i n of c a r d i n a l i t y

(This e x t e n d s E h r e n f e u c h t [ 1 7 1 . )

K

which i s o r d e r e d by a f i r s t - o r d e r f o r m u l a .

For some t h e o r i e s s u c h a s ZFC, one c a n

c o n s t r u c t p a i r s of a r b i t r a r i l y l a r g e models B , C such t h a t a n y t h i n g e l e m e n t a r i l y embeddable i n b o t h B and C i s c o u n t a b l e (Hodges [ 2 9 1 ) . used EM f u n c t o r s .

A l l these constructions

R e c e n t l y C h a r r e t t o n and Pouzet [ I 2 1 showed:

THEOREM 10.

(GCH) L e t T be a f i r s t - o r d e r t h e o r y whose models

a r e p a r t i a l l y o r d e r e d by a formula $, and suppose some model of T c o n t a i n s a n i n f i n i t e c h a i n which i s o r d e r e d by I$. Then i n e v e r y s u c c e s s o r c a r d i n a l i t y K,

T h a s a f a m i l y of 2 K models such t h a t any s t r u c t u r e e l e m e n t a r i l y

embeddable i n two of them h a s c a r d i n a l i t y
w i s s i m i l a r b u t more c o m p l i c a t e d .

S h e l a h 1521 proved some s t r e n g t h e n i n g s of Theorem 13. example he b u i l t s t r u c t u r e s B w i t h IP

B

I

=

A,

I QB I

= K

For

u

and ( R B I =

whenever

A ' K , L J L W .

7.

Sporadic a p p l i c a t i o n s

( I ) Automorphisms.

E h r e n f e u c h t and Mostowski [ I 9 1 a d v e r t i s e d

t h e i r c o n s t r u c t i o n a s a way of f i n d i n g models w i t h many automorphisms. Ebbinghaus [ I 6 1 combined EM f u n c t o r s w i t h some o t h e r t r i c k s t o g e t s t r u c t u r e s which have many automorphisms and a r e models of s e n t e n c e s w i t h q u a n t i f i e r s "There a r e a t l e a s t A e l e m e n t s such t h a t (2) R e a l i s i n g few t y p e s .

. . . ' I .

of a n element b of a

The 0-

s t r u c t u r e A i s t h e s e t of f o r m u l a s $ ( X I E 0 s u c h t h a t A

1$(b).

We s a y t h a t

A r e a l i s e s t h e @-type p i f f p i s t h e @-type of some element of A .

Suppose F i s a n EM f u n c t o r , rl i s a c h a i n and t ( x ) i s a term. Then Lemma I s a y s t h a t f o r any two i n c r e a s i n g n - t u p l e s e l e m e n t s t F ( r l ) ( a ) and t

F(rl)

a,

b

from n , t h e

( b ) have t h e same q u a n t i f i e r - f r e e t y p e .

We a r e

s t i l l assuming t h a t t h e s i m i l a r i t y t y p e of F ( n ) i s c o u n t a b l e , so t h a t t h e r e a r e j u s t c o u n t a b l y many d i f f e r e n t terms t

THEOREM 1 4 .

of c a r d i n a l i t y

K.

(x).

Hence:

L e t F b e a n EM f u n c t o r and rl a n i n f i n i t e c h a i n

Then:

(a)

F ( n ) r e a l i s e s o n l y c o u n t a b l y many q u a n t i f i e r - f r e e

(b)

For e v e r y 1-ary r e l a t i o n symbol P , t h e number of e l e m e n t s of F(n) s a t i s f y i n g Px i s e i t h e r

zw

or =

K.

types.

W.Hodges

228

When Th(F) c o n t a i n s t h e a p p r o p r i a t e Skolem t h e o r y Z a s i n Lemma 4 , we can replace 'quantifier-free' formula Ji (x)

by ' f i r s t - o r d e r ' ,

and ' P x ' by any f i r s t - o r d e r

. E h r e n f e u c h t [I81 n o t i c e d t h a t p a r t ( a ) of t h i s theorem i s

r e l e v a n t t o t h e c a t e g o r i c i t y problem f o r f i r s t - o r d e r t h e o r i e s .

In f a c t

Morley's improvements of t h i s r e s u l t played a v i t a l p a r t i n h i s s o l u t i o n of Lo6's problem [ 4 0 1 , and they l a i d t h e f o u n d a t i o n s of s t a b i l i t y t h e o r y . Rowbottom [48] adapted p a r t (b) of t h e theorem t o show t h a t i f t h e r e i s a measurable c a r d i n a l , t h e n f o r e v e r y c a r d i n a l X c o n s t r u c t i b l e s u b s e t s of A .

(3)

Heirs.

and P o i z a t [361.

t h e r e a r e a t most X

T h i s preceded S i l v e r ' s r e s u l t , Theorem 1 1 above

T h i s a p p l i c a t i o n p r e s u p p o s e s knowledge o f L a s c a r

These a u t h o r s i n t r o d u c e t h e n o t i o n o f t h e

They show (Theorem 4 . 9 of p a i r of models M -4 over N.

'> w

heir of

a type.

[361) t h a t i f T i s a s t a b l e t h e o r y t h e n f o r e v e r y

N and e v e r y t y p e p over M ,

t h e r e i s a u n i q u e h e i r of p

The f o l l o w i n g argument p r o v e s t h e c o n v e r s e :

i f h e i r s a r e unique

then T i s s t a b l e .

Assume T i s u n s t a b l e , so t h a t some formula $ ( x , y ) l i n e a r l y o r d e r s a n i n f i n i t e c h a i n of n - t u p l e s i n some model. be 1 ( t h e proof a d a p t s t o t h e c a s e n > I ) . EM f u n c t o r F w i t h T

u

{$(xl,x2)

Skolem t h e o r y from Lemma 4 .

of t h e Skolem t h e o r y 2,

A

For s i m p l i c i t y l e t n

By E x i s t e n c e Theorem 5 t h e r e i s a n

z

- I $ ( X ~ , X ~u ) } 5 Th(F), where C i s t h e

L e t q be t h e c h a i n w+{a}+{b}+{c}+w*.

* i n c l u s i o n F(w+w )

the

+

F(n) i s elementary.

Because

Let p

*

be t h e type of b o v e r F(w+w ) , and l e t q , r be r e s p e c t i v e l y t h e t y p e s of a , c over F (w+{b)+w*). F(q)

F(q)

b X(c,t(7,b,46*))

Then q , r a r e b o t h h e i r s of p: t h e n by s l i d i n g , F ( n )

x(b,t(7,&3,46*)) by a second s l i d e .

s i n c e Jl(x,b) i s i n q b u t n o t i n r .

f o r example i f

1 x(c,t(7,8,46*)),

and s o

However, q and r a r e d i s t i n c t

229

Models built o n linear orderings A s i m i l a r proof shows t h a t i f

t h e formula

has the order

@

p r o p e r t y then t h e r e a r e u n d e f i n a b l e $-types o v e r models. r e s u l t of S h e l a h - s e e f o r example Theorem 1 1 . 2 . 2 i n [ 5 5 ] .

T h i s i s a well-known But t h e proof

by EM f u n c t o r s has one d i s t i n c t a e s t h e t i c a d v a n t a g e over t h e p r o o f s i n p r i n t , namely t h i s .

The p u b l i s h e d p r o o f s u s e c a l c u l a t i o n s w i t h u n c o u n t a b l e

c a r d i n a l s i n o r d e r t o prove a s t a t e m e n t a b o u t c o u n t a b l e t h e o r i e s ;

t h e proof

by EM f u n c t o r s u s e s o n l y e l e m e n t a r y arguments and t h e compactness theorem f o r countable logic.

(4) P e r m u t a t i o n g r o u p s . the s e t X.

L e t G be a group of p e r m u t a t i o n s of

We s a y t h a t G i s k - s e t - t r a n s i t i v e

i f f f o r any two k-element

s u b s e t s of X, t h e r e i s a n element o f G which t a k e s t h e one s u b s e t t o t h e o t h e r .

THEOREM 15.

Let m

2

5, and l e t t h e p e r m u t a t i o n group G on t h e

c o u n t a b l e s e t X be k - s e t - t r a n s i t i v e

f o r some k 2 2 m - 2 ,

b u t not m - t r a n s i t i v e .

Then t h e r e i s a l i n e a r o r d e r i n g < of X s u c h t h a t e v e r y element of G p r e s e r v e s t h e r e l a t i o n “ e x a c t l y one of a , c i s i n t h e i n t e r v a l from b t o d i n t h e ,v,,

sont localement v -isomorphes, alors m

ces d e u x r e l a t i o n s s o n t i s o m o r p h e s " . 0.2.

L a s e c o n d e p a r t i e de l'expos6, c o n s a c r E e 2 la t h 6 o r i e d e s

c h a 7 n e s p e r m u t 6 e s e t a u x g h o u p e 6 d ' a u t o m o k p h i n m e n l!ocaux d e s relations e n c h a y n a b l e s , v a c o n d u i r e p r o g r e s s i v e m e n t a u c a l c u l d e s nombres v

m

.

a) E n p r e m i e r l i e u , n o u s f a i s o n s i n t e r v e n i r des r e l a t i o n s m-aires e n c h a P n a b l e s p a r t i c u l i s r e s a p p e l 6 e s G-kaRgtrnentb (pour la s i m p l e r a i s o n qu'une

t e l l e r e l a t i o n e s t s u s c i t 6 e p a r u n e c h a y n e t et r e l i 6 e

5 un s o u s - g r o u p e G d u g r o u p e s y m s t r i q u e S ) , a i n s i q u e l e u r s d g f o r m m a t i o n s a p p e l E e s pbeud0-G-hangernent6. P a r e x e m p l e , e n n o t a n t b r i s v e -

Relations enchaina bles, rangements et pseudo-rangements m e n t (abc)=

{(a,b,c),

(b,c,a),

239

(c,a,b)j, o n peut d 6 f i n i r d e u x r e l a -

tions t e r n a i r e s r et s de s u p p o r t { O ,

I ,

2 , 3 1 par leurs e n s e m b l e s

c a r a c t c r i s t i q u e s (123) u (023) I: ( 0 1 3 ) u ( 0 1 2 ) et (321) u (023)

L)

(310) ~ ( 0 1 2 ) :

d2s lors (pour lc s o u s - g r o u p e i n d i c a t i f T 3 de S ) r e s t u n T 3 3 + r a n g e m e n t et s e s t u n p s e u d o - T 3 - r a n g e m e n t . La chai'ne u s u e l l e o (dont le s u p p o r t est l'ensemble w d e s n a t u r e l s ) s u s c i t e des G - r a n g e m e n t s " n a t u r e l s " G dont la v a l e n c e peut s t r e n o t 6 e s i m p l e m e n t v(G).

En

1 9 7 2 - 7 3 , l e s t r a v a u x d e D . CLARK et P . KRAUSS d'une p a r t , d e M . P O U Z E T d'autre p a r t , d o n n a n t u n r a l e m a j e u r 2 l a s e u l e e x i s t e n c e (maximum des v a l e n c e s de t 0 n f e . s les r e l a t i o n s m-aires m enchaTnables), n o u s c o n d u i s e n t 5 r e v o i r e n c e s e n s l a t e c h n i q u e d e

des nombres v

hecaeYemehif d e s G - r a n g e m e n t s c o m p a t i b l e s

( o u de

G - 4 e c o L l e m e n t des

c h a r n e s G - c u r n p U t i b l ? c h q u i l e s suscitent) G l a b o r c e e n 1 9 6 3 - 6 4 ,

et B

est a u s s i la v a l e n c e m a x i m u m d e s A c u e b r a n g e m e n t s m m-aires. L a r 6 u s s i t e d e c e t t e t e c h n i q u e t i e n t a u f a i t q u e (pour tout

montrer que v sous-groupe

que

G

de Sm) : "la c l a s s e d e s G - r a n g e m e n t s e s t u n i v e r s e l l e "

plus pr&cisZment,

v(G)

est le plus petit des naturels n tels

"si t o u t e s l e s n - r e s t r i c t i o n s d'une r e l a t i o n m - a i r e r

:

> n ) sont des G-rangements, alors r est un

(de c a r d i n a l i t 6 G-rangement".

A s s e z c u r i e u s e m e n t , c e t t e m u l t i t u d e de c l a s s e s uni-

v e r s e l l e s n o n t r i v i a l e s (li6e a u x g r o u p e s de p e r m u t a t i o n s ) est r e s t 6 e m c c o n n u e jusqu'en

1976 - p a r c o n t r e , e n u t i l i s a n t le d e g r 6

d e m o n o m o r p h i e m - a i r e , M. J E A N a v a i t d E m o n t r 4 d 2 s 1967 m que : "la classe d e s r e l a t i o n s m-aires m o n o m o r p h e s et l a classe

optimal d

d e s r e l a t i o n s m - a i r e s enchai'nables s o n t d e u x c l a s s e s u n i v e r -

s e 11 es" b)

.

D6signons par

support w )

rm

l'ensemble

(fini) d e s r e l a t i o n s m - a i r e s (de

libre-interpr6tables par la charne usuelle

p o u r toute r e l a t i o n r e

t i o n (C(n))ne

r m'

d . En

1964-65,

nous avions envisag6 la suite de d i e a t a -

d e r d a n s l a q u e l l e G(n)

d 6 s i g n e le s o u s - g r o u p e d e

Sm+n c o n s t i t u 6 p a r les a u t o m o r p h i s m e s d e la r e s t r i c t i o n rl (m+n) support { O , I , 2 ,

. . . , m+n-l]).

r = 6 p o u r l e q u e l G(O)=G

(de

D a n s le c a s p a r t i c u l i e r d u G - r a n g e m e n t

(et e n u t i l i s a n t l e s u p . d e m i - t r e i l l i s d e s

s o u s - g r o u p e s i n d i c a t i f s d e Sm' c e q u i a t t r i b u e 1 G une d i c h e i n d i c a t i u i ? H d e t y p e S , I p t q , J p , T , D t e l l e q u e Hm s o i t l e g r o u p e i n d i catif maximum contenu dans G ) ,

c l a o b e i n d i c a t i v e y(G)

nous avions mis e n Svidence une

a u s e n s s u i v a n t : le g h o u p e d i P a t P G(n)

n o n i n d i c a t i f p o u r n , y ( G )

2 40

C. Frasnay

(de sorte que y(G)=O

caractgrise G comme groupe indicatif).

dehors d u groupe indicatif .J:

En

(bien connu comme groupe d'invariance

+ x )=2(x 0 x 1 + x 2 x 3 ) , 3 par exemple sur le corps Q ) pour lequel v(Ji)=6, tous l e s s o u s de la relation quaternaire harmonique (x + x )(x 0

1

2

groupes G de Sm vsrifient l'inegalitt? : v(G)cm+y(G)+l.

La classe

fournit ainsi la majoration :

indicative m-aire maximum y

m v <m+l+y et cette technique nous avait permis d'obtenir en m m' 1 1964-65, grzce B une inggalitd y 3) un sous-groupe G de Sm verifiant y(G)=m-3 tO,I,

et v(G)=2m-2,

. . . , m-11

fixant

o

:

h savoir le groupe S 1 des permutations de m et m-1. Connaissant les valeurs y , = y 2 = y 3 = 0 ,

y 4 = 2 et v l = l , v 2= 3 , v 3 = 4 , v 4 = 6 , l'int?galitt? y ' m <m-3 pour m>5 (de HODGES-LACHLAN-SHELAH) combinse avec notre inggalitg vm<m+l+y et m I 1 nos deux ggalitgs y ( S )=m-3, v(S )=2m-2 pour m23, conduit aux deux m m rssultats dEfinitifs : y =m-3 pour tout m25, et v =2m-2 pour tout m m m23. c) Nous achevons la seconde partie de l'expost? e n justifiant un lemme (de la thdorie des chafnes permutdes) qui apporte u n argument simplificateur dans la ddmonstration par J.P. CAMERON (1976) d'un thborsme sur les groupes h a m o g P n c 4 de permutations d'un ensemble infini. Nous mentionnons ggalement l'extension B toute relation

r e T m par

G.

HIGMAN (1976) d u resultat que nous avions obtenu e n

1964 pour l e s G-rangements r=G : toute relation r c T une cJ!?adAe i n d i c a t i w c 6(r) locaux Aut(rl(m+n))

admet aussi m telle que le groupe d'automorphismes

soit un groupe indicatif pour n>S(r).

Bien

pour la classe indicative maximum 6 m (mais les valeurs 6 n e semblent pas connues en m m dehors des valeurs triviales S,=S,=O).

entendu y(G)=G(G),

et y, ,

[rl=

:

( $(r))

'L

d 6 f i n i e par

21

de.s r e l a t i o n s i b o m u k p h e . 4 B r , La C L a b b e d ' i b o m a k p h i e d e r e s t

: y e O ( E ) I e t , pour t o u t e r e l a t i o n r ' k r r l , on r'.

%

S i r'= y(r),

E'= y<E>, p'= y
, la bijection

e s t un ibomokphibme e n t r e r=(m,E,p) e t r ' = ( m , E ' , P ' ) , 9 q u ' o n n o t e e n c o r e : r 'L r ' . E

-f

ce

Pour rERm, s 6 R m , l o r s q u e r e s t isomorphe 2 une r e s t r i c t i o n

1.2.3 r'

E'

d e s , o n d i t q u e r b'abki-te danb s , o u que s a b h i t e r , e t l ' o n

n o t e : rss. Dans l e c a s c o n t r a i r e r q s , o n p e u t d i r e q u e s P v i Z e r et

( p l u s g6n6ralement)

lorsque s

Pvite chacune des r e l a t i o n s appar-

d e @Lm, o n p e u t d i r e q u e s B v i t e

t e n a n t 1 une s o u s - c l a s s e

L a formule rss d 6 f i n i t dans chacune d e s c l a s s e s k

m

d'abfiitement m-aire.

l e pkfio4dhe

21.

un p r 6 o r d r e :

On a p p e l l e z g e d e s l a c l a s s e B d e s

-3 e s t

r e l a t io n s r de c a r dinalit6 f i n i e v 6 r i f i a n t r<s. S i

l'ensern-

b l e d e s p a r t i e s f i n i e s du s u p p o r t de s , a l o r s E e s t l a r e u n i o n d e s classes d'isomorphie [SIX]

l'on note s(n)

(pour n e w )

n-restrictions

de s (grossisrement

-

on d 6 f i n i t une s u i t e s

: w

.

A cet egard, s i 9 s l e nombre d e s c l a s s e s d ' i s o m o r p h i e d e s

lorsque X parcourt

-f

w

-

: s ( n ) < 2 (nm) pour

appel6e

pkodid d e

tout

sE Q L ~ ) ,

s

Une r e l a t i o n s t e l l e q u e s ( n ) < l e s t d i t e n - t t t o n o m o t p h e ( c e q u i r e v i e n t 1 d i r e q u e , p o u r t o u t c o u p l e (X,Y) d e n - p a r t i e s de s : SIX dit

'L

sly).

L o r s q u e s e s t n-monomorphe

( b r i s v e m e n t ) q u e s e s t manomo4phe. S i l ' o n n o t e

des r e l a t i o n s m-aires d'exemple)

que

monomorphes, on p e u t rernarquer

]\t2c o n t i e n t

du s u p p o r t

pour t o u t n e w ,

]\^em

(5 t i t r e

l a classe des relations binaires

(ou ChaintA).

d'ordre

total

1.2.4

E t a n t donne une r e l a t i o n r = ( k , E , P ) , t o u t isomorphisme

rlX

'L

local

on

la classe

r l Y e n t r e d e u x r e s t r i c t i o n s d e r e s t a p p e l 6 un i o o m o 4 p h i ~ m e d e r. P o u r d e u x r e l a t i o n s

lorsque tout

et

s e R S md e mEme s u p p o r t E ,

isomorphisme l o c a l d e r e s t un isomorphisme l o c a l d e s ,

on d i t que s e s t Libfie-intehpkZtabLe p a r r .

Relations enchaina bles, rongements et pseudo-rangements a)

l o r s q u e r est u n e

Par e x e m p l e ,

~ € ( 8E )

libre-interprgtablr

un

IXI=lYl,

l'injection

croissante

isomorphisme e n t r e S I X e t sly". p r o p o s i t i o n 5.1.)

( [ I l l ,

les couples

pour b)

r e s t d i t e C ~ i c h u i x a b ~ (Cp l u s

par

2 d i r e : " P o u r t o u t c o u p l e (X,Y) d e p a r t i e s f i n i e s d e

ce13 revient que

c h a f n e d e s u p p o r t E, u n e r e l a t i o n

C ~ I C ~ I C ( T I I G pC a r r e t o n m o n t r e f a c i l e m e n t q u e

pr@cis&rnent, e l l e est

E tel

(X,Y)

enchaTnables,

n a b l e est m o n o m o r p h e ) .

dSmontr6 "Pour

18,

§

il rGsulte En

lY64,

r)

de vsrifier

cette condition

de E).

rm

la classe des

:

3

t m ]I[,

relations

(toute r e l a t i o n enchaf-

l'instigation

99 - v o i r a u s s i I l l 1 ,

p.

de X s u r Y est

I E [ > m + l , on p e u t p r o u v e r

partielle

de

R. F R A T S S E ( q u i

: "Toute

de cardinalit6 i n f i n i e est enchafnable"

monomorphe annonci.

suffit

1954 une r g c i p r o q u e

dGs

a v a i t demontre

qu'il

(mod.

Lorsque

de m-parties

s i l'on n o t e

De c c s r e m a r q u c s ,

m-aires

243

proposition

(r41,

relation r6sultat

I l.2),

le tti6orGme s u i v a n t ( n o t e [ I 0 1 e t t h s s e 1 1 1 1 , t h .

tout

m e o, i l e x i s t e u n c o u p l e ( n , p ) e

r e l a t i o n m-aire

n-monomorphe

UXU

pour

nous avons 12.1.1.):

lequel

toute

2p est encha4nable

de cardinalit6

(donc

le plus p e t i t des n a t u r e l s n apparaism ( n , p ) , nous disons que d est le d@ghc m < ~ l ~ t i m c cde e m o n o m o r p h i e m - a i r e . P l u s p r S c i s G m e n t , on p e u t m o n t r e r

S i l'on n o t e d

monomorphe)". sant dans de

qu'il

couples

e x i s t e deux s u i t e s de n a t u r e l s

: "Pour

que

tels

t o u t n a t u r e 1 n2d

de c a r d i n a l i t 6

enctiafnable)"

>pn (resp.

(P,),,~

m' de c a r d i n a l i t 6 si p

(bien entendu,

sont

: d =0, d = I , I

c)

tout

n a t u r e 1 m22,

Pour

phes

d

2

('n'n2d

m-aire

telles

m

n-monomorphe

>q ) e s t monomorphe

e t qn s o n t p r i s

male : pn. m + l

on

244

C. Frasnay

1.3

Si

&.

est u n e s o u s - c l a s s e de

(autrement dit : (vu)('dr)

c l o s e par a b r i t e m e n t

((uCr et rs?t)

=j

--

o u encore :

(ue-A)),

est une r 6 u n i o n de c l a s s e s d ' i s o m o r p h i e t e l l e q u e , pour t o u t e r e l a t i o n r de s u p p o r t E et t o u t X c E :(re&)

.h

d i s o n s , pour a b r g g e r , q u e

(rlXc2k ) ) , nous

est u n e c l a s s e

i n 4 t i U 8 t

DSs l o r s , n a t a n t X c Y l'inclusion s t r i c t e ( X & Y

n o u s a p p e l o n s p J ~ u d ~ - P l ? $ m e fdlet

ft. ( o u

"bu4ne" de

de r e l a t i o n s et X # Y ) ,

.k ,

a u s e n s de

R. F R A ' L S S E ) t u u t e r e l a t i o n m-aire s d e s u p p o r t fini Y t e l l e qut? ( s + . f t ) et

(1x1

:

( s l x t ~ ) .

X I Y Bien entendu, si

s

s'

et s i s est p s e u d o - 6 l C m e n t d e

a l o r s s'est a u s s i p s e u d o - 6 1 6 m e n t de

h .

S i b(h)

s u p 6 r i e u r e d e s c a r d i n a l i t s s d e s p s e u d o - 6 l g m e n t s de b(h) n , a l o r s i l e x i s t e

X d e E pour l a q u e l l e les cha'ines r e s t r e i n t e s S I X ,

tlX s o n t i d e n t i q u e s o u o p p o s 6 e s " . b) A l o r s q u e l a f o r m u l e v mE O ( e x p r i m a n t le c a r a c t s r e f i n i - b o r n 6 d e t o u t e r e l a t i o n m-aire encha?nable) n ' o c c u p a i t d a n s n o t r e t h s s e d e 1965qu'une p o s i t i o n m a r g i n a l e , u n r e n v e r s e m e n t d e t e n d a n c e ( d e p u i s 1970) a r e n f o r c 6 le r c l e de c e t t e p r o p r i E t 6 : c e t t e n o u v e l l e

o r i e n t a t i o n est a p p a r u e d a n s les t r a v a u x d e D. C L A R K et P .

KRAUSS

d'une p a r t , R. F R A T S S i et M. P O U Z E T ( C 8 1 C 2 0 1 ) d'autre p a r t

(C21C191)

( t r a v a u x d o n t t i e n t c o m p t e la n o t e C 1 3 1 s u r les v a l e n c e s m - a i r e s m a x i m a l e s v ) . A i n s i e n 1972, d a n s u n e a p p r o c h e p u r e m e n t q u a l i t a t i v e m (note r 2 0 1 , s a n s i n c i d e n c e s u r les v a l e u r s v m ) , M. P O U Z E T a d g f i n i (pour tout m s o) la c l a s s e

firn d e s

b e P P ~ 0h e t a t s o n 0 r n - u L h e i ,

X420

de telle sorte que : ( I )

P o u r tout

m g IW,

s i re

(2) P o u r tout ( k , m ) t o u o , - i n t e r p r & t a b l e par u , Comme

(2) :

q)2

rm,/Rm

bm,a l o r s si

~

6

r est fini-born6e. est libreet % s i~ re'% m

alors r t 6 m .

c o n t i e n t la c l a s s e d e s cha'ines, il r 6 s u l t e d e ( I )

et

et toute r e l a t i o n encha'inable est f i n i - b o r n 6 e .

c) L'6tude q u a n t i t a t i v e e f f e c t u 6 e e n 1963-65 d a n s l e s n o t e s [ 9 l [ l O l et la thSse [ I l l u t i l i s a i t la " t h 6 o r i e d e s cha'ines p e r m u t 6 e s " et l e

.

I' des sous-groupes indicatifs de S Elle fournissait m m (pour le d e g r 6 o p t i m a l d m d e m o n o m o r p h i e m-aire et l a v a l e n c e m-aire

demi-treillis

m a x i m a l e v ) les i n 6 g a l i t 6 s : d m < v m (pour tout m e w ) et m+l,2). L ' e s p o i r d'une m a j o r a t i o n d e l a f o r m e v m < c x m + B s e r 6 a l i s e r a e n 1 9 7 6 - 7 7 , par a p p l i c a t i o n d'un n o u v e a u r e s u l t a t s u r I' m dCi 1 W. H O D G E S , A.H. L A C H L A N et S. S H E L A H

1161.

E t a n t d o n n 6 ( m , n ) ~W X W , o n a p p e l l e o p P 4 a t c u t L i b f i e

1.3.2

les a r i t s s m , n )

une application f

r o l a t i o n m-aire r d e s u p p o r t E ) , ( I )

f(r)

:

Rm +%

(pour

v 6 r i f i a n t (pour t o u t e

les t r o i s c o n d i t i o n s s u i v a n t e s :

est u n e r e l a t i o n n - a i r e d e s u p p o r t E.

246

C. Frasnay

(2) P o u r tout (3)

P o u r tout

X E

P(E)

Y=@(E)

: f ( r J ~ )= f(r)

IX

%

f(j(r))

:

=

y(f(r))

a) Etant d o n n e u n tel o p 6 r a t e u r l i b r e f (pour les a r i t 6 s m,n), s o i t

h

la c l a s s e des r e l a t i o n s r=(m.E, p)

k

On dit q u e

t e l l e s q u e f(r)=(n,k.,E").

est la c e u 5 5 e a n c u e 4 6 e e f c (au s e n s de A .

h e p h P n e n t P e par l ' o p e r a t e u r libre f . Le d c g r l g de

petit d e s n a t u r e l s n tels q u e libre f :

fLm

h

e s t le plus

s o i t r e p r 6 s e n t a b l e par un o p G r a t e u r

(Par e x e m p l e , la c l a s s e des c h a i n e s est u n i v e r -

!?L

+

h

T A R S K I 1231)

s e l l e de d e g r E 3 ) . T o u t e c l a s s e u n i v e r s e l l e est i n i t i a l e (close par abritement),

et on peut m o n t r e r q u e la c o n c e p t i o n de A .

TARSKI

e q u i v a u t 2 d e u x a u t r e s c o n c e p t i o n s q u i n e n e c e s s i t e n t pas de f a i r e appel 2 la n o t i o n d'opgrateur l i b r e : ( I )

de

Au s e n s e q u i v a l e n t d e R . L .

Rm est

V A U G H T 1241, une s o u s - c l a s s e

u n i v e r s e l l e s i e l l e est i n i t i a l e et s'il e x i s t e n t

(I

v E r i f i a n t l a c o n d i t i o n s u i v a n t e : " L o r s q u e r est une r e l a t i o n m - a i r e d o n t t o u t e s les r e s t r i c t i o n s d e c a r d i n a l i t 6 alors r appartient 2

.k

"

v

sont indsformables". Bien entendu, puisque m les r e l a t i o n s m - a i r e s m o n o m o r p h e s n o n e n c h a P n a b l e s o n t u n e c a r d i n a l i t 6 m a x i m u m f i n i e k m , il e n r 6 s u l t e q u e "toute r e l a t i o n m o n o m o r p h e

est f i n i - b o r n 6 e " e t e n c o r e : " P o u r t o u t m e w ,

il e x i s t e p c w

lequel toute r e l a t i o n m-aire m o n o m o r p h e d e c a r d i n a l i t 6 f i n i e

pour >p

es t indEf o r m a b le" .

1.4

Etant donn6 u n ensemble I et une famille

u=(mi)itI

r e l s , o n a p p e l l e b t ' Z u c - t u h ( ! h d u , t i u n n t L L e u - u i h e de

[muLtihei!ution

ou

de natu-

Auppohf E

k-tclatiu~ p-aire d e s u p p o r t E s i I est f i n i ,

/ I / = k ) t o u t e f a m i l l e (ri)ie I t e l l e q u e , p o u r tout i t I , r i

soit

u n e r e l a t i o n d'arit6 m i et d e s u p p o r t E. S i n e w e s t t e l q u e mi$" pour t o u t i r I , o n dit a u s s i q u e l'arit6 p

e s t b v t t n a e et q u e

(ri)ie I est u n e s t r u c t u r e r e l a t i o n n e l l e u u p l u h n - u i h e .

1.4.1

T o u t e l a t e r m i n o l o g i e i n t r o d u i t e B p r o p o s d e s r e l a t i o n s (en

1.2 et 1.3) s'Btend a i s 6 m e n t a u x s t r u c t u r e s r e l a t i o n n e l l e s . P a r

248

C. Frasnay et tout y c Q(E),

exemple, pour tout X t P(E)

chaque structure rela-

de support E admet une restriction i it1 % 'L L'isoet une image isomorphique y(R)=(y(ri))ik I.

tionnelle p-aire R=(r ) RI X=(ri 1 X)

E I

morphisme explicite (R

?,

R') ou implicite ( R

'L

R'),

l'abritement

l'zge R , de m t m e que les propri6tGs

R3).

G du groupe sym6trique

2.

L A T H ~ O R I ED E S CHAINES

PERMUTEIES.

Un groupe de permutations s'associe tout naturellement 1 une relation r=(m,E,p) (1)

dans deux circonstances :

Si S E d6signe le groupe des permutations de E , o n note Aut(r)

(et o n appelle g h o u p e d t n u u t o m o h p h i n r n t A

y~

S E telles que

%

(p(r)=r.

de r) le groupe des permu-

P l u s ggngralement, lorsque X parcourt

les g h o u p e n d'automokphiomen e a c a u x de r sont les groupes

P(E),

Aut (rlX) associ6s aux restrictions r ( X de r. Par exemple, une relation r de support w admet (parrni ses groupes (Gn)npw Si

B:E

+

est un isomorphisme entre deux ?elations r et r'

E'

de supports respectifs E et E ' ) , tion Aut(r)

locaux) une suite

de groupes G n € En d6finie par : G =Aut(rl n).

9-0

o

Y

o

en Aut(r').

0-I)

i l lui correspond (par transmuta-

un isomorphisme de S E sur S E l qui transforme

250

C. Frasnay

(2) L e s p e r m u t a t i o n s o ez S m c o m p o s d e s a v e c les m-uples x c E d e s m-uples x o x

0

m

e E

0

0 6

donnent

( p l u s e x p l i c i t e m e n t , p o u r x=(xO,x l , . . . , ~ m - , ) : )).

=(XO(o)’xa(l)’.”’xo(m-l)

permutations

m

S i Inv(r)

S m t e l l e s q u e : (\dx)(x&p

d6signe l e groupe des

-y x o

(?

C

o), o n d i t

q u e I n v ( r ) ~ Zm est le gkaupe d ’ i i i u a k i a n c e m a x i m u m d e la r e l a t i o n r=(m,E,p).

Tout isomorphisme r

.?,

r ’ i m p l i q u e : Inv(r)

=

Inv(r’).

P l u s g B n g r a l e m e n t , o n peut a s s o c i e r B tout g r o u p e G 6 C m et 2 toute relation re&, l a r e l a t i o n G x r( ddfinie comme suit :

km

si r=(m,E,p),

alors G

x

r = (m,E,p’)

avec p ’ = { x o o : x ~ pet

0 6 GI.

P u i s q u e r et G x r o n t m 8 m e a r i t 6 m , m 8 m e s u p p o r t E , et q u e l e u r s e n s e m b l e s caract6ristique.s p . G

x

p’

vdrifient

p g p ’ ,

o n peut d i r e que

r e s t u n e e x p u n h i O n de r : plus p r B c i s E m e n t , G x r est la

G - e x p a n b i o n d e r. B i e n e n t e n d u (si E

m d s s i g n e la p e r m u t a t i o n i d e n t i q u e , B l g m e n t n e u t r e de S ) , le g r o u p e t r i v i a l I = { E ~ } est u n a g e n t m m n e u t r e : I~ x r = r (pour t o u t r e P l u s g B n B r a l e m e n t , p o u r G E Cm ’

Rm).

la condition G x r=r Bquivaut 1 la condition G

C_

Inv(r)

et (dans ce

cas) o n dit q u e l a r e l a t i o n r est G - i n u a k i a n t e , o u e n c o r e q u e G est u n g h a u p e d’inuakiance de r. L e s r e l a t i o n s m-aires hymEtkiqucA s o n t les r e l a t i o n s r E @(E),

Rm telles

q u e Inv(r)

=

sm .

p o u r rt(R,(~),

X ~ ZP(E),

les f o r m u l e s :

G x r E (?Lm(E),

G

X

(r(X)

que la G-expansion r

I--+

=

(G

G

X

X

r)(X, G x

y% (r)=

%

~ ( G x r ) , montrent

r d 6 f i n i t u n o p 6 r a t e u r l i b r e de

(k m

d a n s lui-m8me. E n f i n , p o u r tout G t C m , o n p r o u v e a i s s m e n t q u e l a c l a s s e d e s r e l a t i o n s m-aires G - i n v a r i a n t e s e s t u n e c l a s s e u n i v e r selle. N o u s a l l o n s a d a p t e r ces c o n s i d 6 r a t i o n s g 6 n B r a l e s ( p o r t a n t s u r

les r e l a t i o n s et l e s permutations) a u x c h a f n e s et a u x r e l a t i o n s enchafnables,

2.1

+ E t a n t d o n n 6 u n n a t u r e 1 m (support d e l a c h a f n e m) et une c h a f n e

t de s u p p o r t E , l ’ e n s e m b l e p d e s i n j e c t i o n s c r o i s s a n t e s de

m={O,l,

. . . ,m - 1 )

+ (ordonn6 p a r m) d a n s E ( o r d o n n 6 par t ) d s f i n i t u n e

r e l a t i o n m-aire tCm)=(m,E,p) uples x=(xo,xI, (mod. t). t(m)

. . . ,xm- 1 )

d e s u p p o r t E , s a t i s f a i t e par les m-

d ’ B l 6 m e n t s d e E t e l s q u e : x < x C...<x 0

1

e s t le h u n g e m e n t m - a i k e t k i u i a k ? s u s c i t B (dans E) par l a

c h a f n e t.

m- I

P a r a l l u s i o n a u g r o u p e t r i v i a l I m = { ~ m } , o n peut d i r e q u e

Relations enchainables, rangements et pseudo-rangementr

25 1

Le premier acte de la thgorie "des chafnes permut6es"

2.1.1

consiste 2 introduire, pour chaque groupe de permutations G E C le G - h a n g c m e n t t G tG =

=

est donc la G-expansion du rangement trivial t(m)

(m,E,D')

s o n ensemble caractgristique

('a(o)

9

UGG

'n(l)

*

1

e t x <x 0

m'

G X ~ ' ~ suscit6 ) (dans E) par l a chafne t :

1

'Xo(m-~)) C...<x

p'

e Em

vsrifiant les deux conditions :

(mod. t).

m- I

et

est l'ensemble des m-uples

Bien entendu, si / E l C m , les

G-rangements suscit6s dans E coincident avec la relation m-aire "vide" : t G = (m,E,d) (qui admet S comme groupe d'invariance). Par m contre, si IEl a m , les G-rangements suscit6s dans E sont des relations m-aires t G " n o n vides" dont l e groupe d'invariance maximum est prgcis6ment

: Inv (t

G

)=G.

Lorsqu'un G-rangement t G abrite une relation m-aire

2.1.2

alors s est a u s s i u n G-rangement

s,

(la preuve s'effectue d'abord dans

le cas trivial G = I m , puis dans le cas gEn6ral G E E ) . Autrement m la classe des G-rangements est une m' classe initiale de relations m-aires". Si l'on appelle p h e u d u - G -

d i t : "Pour tout groupe G e i :

h a n g e m e n f tout pseudo-616ment de cette classe initiale, et si l'on tient compte des deux remarques suivantes : ( I )

tout ensemble E supporte u n G-rangement,

( 2 ) deux G-rangements tG,th de m8me cardinalit6 finie sont

implique t'=%(t G Y

isomorphes (puisque t'=;(t)

alors i l r6sulte de la terminologie g6n6rale ( 1 . 3 )

G) ) .

qu'un pseudo-

G-rangement est une relation m-aire r de cardinalits finie n+l verifiant l'une o u l'autre des d e u x conditions suivantes : (I)

r n'est pas un G-rangement, mais toute n-restriction de r est un G-rangement.

(2)

r est une dsformation d'un G-rangement.

Par ailleurs, tout G-rangement t

G

est enchalnable ( p l u s pr6ci-

sEment : t G est libre-interprgtable par la chafne t qui le suscite) et tout pseudo-G-rangement 2.2

est monomorphe.

Etant donn6 u n groupe G c C

respectifs A,B, on dit que

s

m

et deux chafnes s,t de supports

et t sont G - c o m p a t i b L Q d d S s que les

G-rangements s G et t G ont la mEme restriction de support A n B : s

G/ A n B

= t

G

[ A n B. Cette condition est trivialement r6alis6e

C. Frasnay

252

lorsque ( A n B ( c m e t , dans le cas ggn6ra1, elle revient 5 dire que les deux chafnes s ( A n B , t l A n B (de support commun A n B )

suscitent

l e mgme G-rangement. Si (s,t) est une bichafne de support E , la G-compatibilitG

2.2.1

des chaPnes s,t peut s'exprimer plus directement en associant 1 toute m-partie X de E une permutation

x

=

0

e

X ~ , . . . , X ~ - ~x} ,< x 2 r , J: d i r e c t (d'ordre 2(r!) b)

2

)

: J:

=

. . . , m-q-I.

Pour r22,

e s t le g r o u p e p r o d u i t

J A I:'r.

On p e u t c o n s t a t e r q u e t o u t g r o u p e i n d i c a t i f H

m

# Sm appar-

t i e n t 2 u n e s u i t e i n d i c a t i v e u n i q u e et q u e 1' e s t un s u p . d e m i m il t r e i l l i s (si G ' e t G " s o n t d e u x s o u s - g r o u p e s i n d i c a t i f s de S m' e n e s t d e m t m e p o u r le g r o u p e G ' B G " , t a n d i s q u e G' A G " = G ' n G" n'est pas n 6 c e s s a i r e m e n t un g r o u p e indicatif).

Tout groupe G e C

m

contient donc un groupe indicatif maximum H E C ' qui dgtermine sans m m

C. Frasnay

254 ambiguit6 la diche

indicative H de G .

Les dilatss successifs d'un groupe indicatif sont des groupes indicatifs de msme fiche. Notamment, p o u r les groupes symgtriques (de fiche S )

:

(Sm)(n)=Sm+n.

Pour toute suite indicative H P S , de

rang h , o n d6montre sgalement : (Hm)(n)

Hm+n p o u r tout m>h.

=

sont indicatifs : c e m (pour le degr6 I ) , S 2 et 1;" (pour

c) Pour m 3 ) un g r o u p e G c Z m ,

(d'oii i l r 6 s u l t e que l a majoration

par l e s auteurs prCcit6s 6 t a i t

l a meilleure

possible). P o u r m=2p o u m = 2 p + l , e t n > l , n o t o n s Sn l e s o u s - g r o u p e

m

d 6 f i n i de l a faqon s u i v a n t e ( I )

Si

I < n < p , S:

est

de S

m

:

l e g r o u p e (non i n d i c a t i f )

des permutations

256

C. Frasnay

conservant chacun des Elfments O , l , 2 , (au t o t a l

2n GlEments s u r m,

. . . , n-I

e t m-n,

m-n+l,

. . . ,m - 1

c e q u i l a i s s e s u b s i s t e r au moins

2 o u 3 ClSments i n t e r c a l a i r e s ) . S i nDp,

(2)

S t (d'ordre

S:

=

11"

I

=

m

(groupe i n d i c a t i f

et

l a fiche triviale II'I,

f a c i l e (s:)(')

=

Sn+ 1 m+l

'

I

trivial).

s i 2n)m)

i r i t e h c a t a i k e i d e Sm

appelEs l e s hOuh-ghoupe4 ve e s t

m

s i 2n&m, e t d ' o r d r e

(m-2n)!

Les groupcs

peuvent g t r e

: leur

fiche indicati-

ils s e d i l a t e n t s u i v a n t l a r 6 g l e

DBs l o r s , portons n o t r e a t t e n t i o n s u r l e (pour m>3)

premier groupe i n t e r c a l a i r e S'

m

:

a)

S i m = 3 , a l o r s S 1 = 1;" (groupe i n d i c a t i f t r i v i a l ) e t 3 1 S i m=4, a l o r s S 4 n ' e s t p a s i n d i c a t i f , m a i s s o n p r e m i e r y(S3)=0. I

dilatE (Si)(l)=S:

=

b ) P l u s gGnGralement, dilatEs successifs sont

(S;)(n)

=

I donc y ( S ) = I

est indicatif,

1;"

p o u r m 2 5 , S;

4

(mais y 4 = 2 ) .

n ' e s t pas i n d i c a t i f e t s e s

:

( S m1 ) (n) = S m - 3 2m-4

Sn+'. A i n s i , p o u r n = m - 4 , m+n

(laissant

s u b s i s t e r 2 ElEments i n t e r c a l a i r e s ) n ' e s t i n d i c a t i f . P a r c o n t r e , p o u r n=m-3 indicatif et

(finalement)

I

pas encore un groupe Sm-2 I,1 est : (SA)(n) = 2m-3 - '2m-3

y(Sm)=m-3.

En t e n a n t c o m p t e d e c e g r o u p e p a r t i c u l i e r t i o n obtenue dans [16],

S 1 e t de l a majora-

m

nous o b t e n o n s donc b i e n

: y,=m-3

comme

c l a s s e i n d i c a t i v e maximum p o u r t o u t m 2 5 .

2.4

De 1 9 6 3 1 1 9 7 6 , u n a u t r e r E s u l t a t - e l 6

c h a i n e s permutEes G-recollement

(prEsentC i n i t i a l e m e n t

des ordres totaux")

de l a t h 6 o r i e des

1 9 1 comme " t h E o r B m e d e

e s t apparu sous d i v e r s e s formes

S q u i v a l e n t e s . La f o r m e l a p l u s r E c e n t e de l a SociEtE f r a n q a i s e de Logique,

(annoncEe a u Colloque

S c i e n c e s , mais d o n t l a p u b l i c a t i o n a Ct6 r e t a r d s e problBmes d ' E d i t i o n ) G&Cm,

e s t a u s s i l a plus brPve

l a c l a s s e d e s G-rangements

1976

Philosophie e t MEthodologie des

1141 par des

: "Pour

est universelle".

tout

groupe

Le d e g r 6 d e

c e t t e c l a s s e ( c a r d i n a l i t 6 maximum d e s p s e u d o - G - r a n g e m e n t s )

e s t Egal

1 l a v a l e n c e v(G) du G-rangement n a t u r e 1 ( s u s c i t C p a r l a c h a ? n e -+

u s u e l l e a) : p a r a b u s d ' 6 c r i t u r e e t d e l a n g a g e , n o u s n o t e r o n s s i m -

Relations enchahables, rangements et pseudo-rangements

p l e m e n t v(G) y4OUpC

2.4.1

251

cette valence et nous l'appellerons la w u ! e ~ c e du

G. E x p l i q u o n s c o m m e n t c e t t e v a l e n c e v(G)

i n t e r v i e n t d a n s le

thGorSme de G-recollement. E t a n t d o n n 6 u n n a t u r e 1 n et u n e n s e m b l e E , o n dit qu'une f a m i l l e (Ai)icI d ' e n s e m b l e s e s t u n ~ - 4 C C 0 u w h e m e n t d e E l o r s q u e , il e x i s t e u n i n d i c e i t 1 tel q u e X & A i

p o u r tout X,Pn(E),

n o t i o n u s u e l l e de r e c o u v r e m e n t c o r r e s p o n d a u c a s n=l).

DSs

la c o n j o n c t i o n d e s d i f f 6 r e n t s t r a v a u x e f f e c t u g s d e 1 9 6 3 - 6 5

1 1972-73

(121C131)

f a i t a p p a r a i t r e l a v a l e n c e v(G)

(la lors,

(C911111)

c o m m e G t a n t le

plus p e t i t d e s n a t u r e l s n v g r i f i a n t la c o n d i t i o n s u i v a n t e : "Pour tout e n s e m b l e E d e c a r d i n a l 1El2n et p o u r t o u t e f a m i l l e d e c h a l n e s d e u x 1 d e u x G - c o m p a t i b l e s d o n t les s u p p o r t s f o r m e n t u n n-recouvrement de E ,

il e x i s t e u n e c h a f n e s d e s u p p o r t E q u i e s t

G - c o m p a t i b l e a v e c c h a c u n e d e s c h a f n e s t . ( p o u r i eI)". S i deux relations m-aires r , r '

d e s u p p o r t r e s p e c t i f s A,A',

sont d i t e s c o r n p a t i 6 L e h d S s q u e rl ( A n A')=r'l ( A n A'),

o n obtient une

v a r i a n t e ( t r s s proche) d e l a c o n d i t i o n p r 6 c G d e n t e e n p r e n a n t u n e f a m i l l e (ri)it I d e G - r a n g e m e n t s d e u x 1 d e u x c o m p a t i b l e s , c e q u i p e r m e t d ' i n t r o d u i r e u n e r e l a t i o n r l e s a d m e t t a n t c o m m e restrictions: r

i

rlAi.

=

D S s lors, l a valence v ( G )

est le p l u s petit des naturels

n t e l s q u e : "Si les n - r e s t r i c t i o n s d'une r e l a t i o n m-aire r (de c a r d i n a l i t s )n) 2.4.2

s o n t d e s G - r a n g e m e n t s , a l o r s r e s t u n G-rangement".

L e s t r a v a u x c i t e s prec6dernment ( n o t a m m e n t C 1 1 1 1 1 3 1 )

rnontrent a u s s i q u e la v a l e n c e m - a i r e m a x i m a l e v m ( c a r d i n a l i t s m a x i m u m d e s r e l a t i o n s m-aires "finies" e n c h a l n a b l e s et d g f o r m a b l e s ) e s t l a p l u s g r a n d e d e s v a l e n c e v(G)

l o r s q u e G p a r c o u r t Cm. Autrement

d i t , p a r m i l e s r e l a t i o n s m-aires e n c h a l n a b l e s et d s f o r m a b l e s d e cardinalits finie maximum v o n est assurg de trouver certains m' G - r a n g e m e n t s . N o u s d i r o n s qu'un g r o u p e G c 1m e s t u n g4Oupe C 4 i t i g u e s'il e x i s t e u n p s e u d o - G - r a n g e m e n t d e c a r d i n a l i t s m a x i m u m v m (autrement d i t : s i v(G) = vm). a) Nos c a l c u l s 1111 et c e u x d e P. J U L L I E N [ I 8 1 o n t m o n t r 6 q u e l a v a l e n c e d e s g r o u p e s i n d i c a t i f s G c C m' e s t e n g e n 6 r a l v(G)=m+l. S e u l s f o n t e x c e p t i o n les g r o u p e s s y m s t r i q u e s S m et l e g r o u p e 2

J 2 : v(S )=m et v(J4)=6. 4 m

C Frasnay

258 b) L o r s q u ' u n g r o u p e G c E

m rnontrg [ I l l que s a v a l e n c e v(G)

n ' e s t pas i n d i c a t i f , n o u s a v o n s e s t li6e 2 s a c l a s s e i n d i c a t i v e

p a r l'insgalitg : v(G)<m+y(G)tI.

C e t t e i n 6 g a l i t E (valable 2 e n c o r e p o u r les g r o u p e s i n d i c a t i f s a u t r e s q u e J 4 ) , j o i n t e B la y(G)

m a j o r a t i o n d e y m (HODGES tion : v

m

3 , L e p4emiefi gficrupe i n t e . f i c a g h o u p ~c f i i t i q u e . de. v a l e n c e v m - 2 m - 2 " . O n s a i t (2.3.3) q u e S mI e s t un g r o u p e d e f i c h e

Preuve :

,

I

d e c l a s s e i n d i c a t i v e y ( S )=m-3 ( m a x i m u m s i m2,5), d e v a l e n c e m I I m+y(Sm)tl = 2m-2. P u i s q u e v(S ) v 6 2m-2 ( 2 . 4 . 2 ) , i l m m

4 , 1'inE-

\< m+y(G)+I

g a l i t 6 v(G)

e t 1 ' 6 g a l i t E v = 2 m - 2 m o n t r e n t qu'un g r o u p e m c r i t i q u e G n o n i n d i c a t i f doit a v o i r la c l a s s e i n d i c a t i v e m a x i m u m y(G)=m-3,

m a i s c e t t e c o n d i t i o n n k c e s s a i r e n'est p a s s u f f i s a n t e

(par e x e m p l e , p o u r le g r o u p e G v(G)=5).

=

{r,,

1

m

)=2m-2,

de d e g r E m=4:y(G)=l

et

I

- y ( S )=m-3 n ' i m p l i q u a i t pas m une d s m o n s t r a t i o n d u t h G o r E m e 2 . 5

Pour cette raison, puisque

n g c e s s a i r e m e n t v(S

(O,3)}

s ' i m p o s a i t !.. D e m z m e , l e s d e u x e x e m p l e s d e H O D G E S - L A C H L A N - S H E L A H [I61

(montrant q u e l e u r m a j o r a t i o n y m 6 m - 3 e s t "pointue", s o i t -clans I 1 d o n n e n t seulement:v(A ),, m - 3 ?

pour

tout m 3 5.

A-t-on

y ni d o n n e s e u l e (ou non)

:

Relations enchaina bles, rangernents et pseudo-rangernents

267

B I B L LOGRAPH I E

I I 1 CAMERON, ordered

I:!I C L A R K ,

I).

EKDOS, try,

sets,

P.,

and

R.,

relations,

I 5 I FRATSSE, formule

R.,

des

par

R.,

(1971),

1 9 i FRASNAY,

Alger,

et

e t

1974),

M.,

POUZET,

C.R.

Acad.

I'OUZET,

Sc.

Pf.,

Paris,

Sur une

C.R.

de permutations

finies

C.,

Groupes

de cornpatihilit;

thGorie

A-259

Quelques

totaux et

(Grenoble,

1 1 2 I PRASNAY, C . , tntaux

(1964),

les

problcmes

1624-1627.

relations

Sc.

e t

Paris,

faniilles

relations,

de deux o r d r e s

d'orcires

Acad.

C.R.

r e l a t i o n m-aire,

combinatoires

d e G-recollcnient

: Permutations

in

Villars

1974),

mcnt d ' u n 865-868.

(1971),

Acad.

monomorphes,

; application aux relations

C.,

d'une relation

Sc.

totaux C.R.

;

Acad.

concernant

Ann.

Tnst.

les

Fourier,

1965) 415-524.

multirelations,

I I 3 I FRASNAY,

Gauthier-

3910-3913.

relations

'I'h6orGme

(Paris,

des

d'une

a p p l i c a t i o n a u x n-morphismes

15-2

A-272

f i n i de bornes,

2944-2947.

C.,

en th6o-

1972),

275-278.

(1963),

ordres

et

102-113.

relations

(Paris,

c l a s s e de

A-257

I I I FRASNAY,

(Relation

1971),

Interpr6tabilitC

Paris,

Paris,

t. I

35-182.

211-228.

; a p p l i c a t i o n '2 l a

Sc.

,

les

: Permutations

in

systsmes d e

(1954),

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***

3 9 1 a n d Indag. M a t h . ,


269

H,-CATEGORICAL PARTIALLY ORDERED SETS James H. Schmerl Department of Mathematics U n i v e r s i t y of C o n n e c t i c u t S t o r r s , CT 06268, U . S . A .

To Professor Emzest Corominas RCsumC: Un ensemble dBnombrable p a r t i e l l e m e n t ordonnB (A; < ) e s t .-cat&orique l o r s q u e , pour t o u t ensemble g dcnombrable q u i e s t p a r t i e l l e m e n t ordonne e t q u i p o s s l d e l a mcme t h B o r i e du p r e m i e r o r d r e que A ( i . e . Th(A) = T h ( B ) ) , A 8 . Nous c o n s i d g r o n s l a q u e s t i o n g 6 n 6 r a l e s u i v a n t e :

A

=

Q u e l l e complexit; d o i t a v o i r un ensemble A k o - c a t 6 g o r i q u e p a r t i e l l e m e n t ordonng pour que Th(A) s o i t complexe? Les deux emplois du terme "complexe" d a n s l a q u e s t i o n d o i v e n t e t r e vagues e t d i f f 6 r e n t s . L e p r e m i e r "complexe" f a i t r'efgrence aux p r o p r i g t 6 s s t r u c t u r e l l e s de A t e l l e s que l a h a u t e u r , l a l a r g e u r , l a dimension ou l e plongement d e c e r t a i n e s c o n f i g u r a t i o n s . Le second emploi du mot f a i t r g f g r e n c e aux a t t r i b u t s l o g i q u e s t e l s que l a d B d c i d a b i l i t 6 ou l ' a x i o m a t i s a t i o n f i n i e . Cet a r t i c l e , q u i s ' a d r e s s e au l e c t e u r a y a n t peu d e c o n n a i s s a n c e s dans l e domaine d e l a l o g i q u e , p r & e n t e un risum; d e s r g s u l t a t s e t l e s t e c h n i q u e s p e r m e t t a n t d e les o b t e n i r , c e q u i a p p o r t e une rgponse l a q u e s t i o n posge e t aux q u e s t i o n s a d j a c e n t e s . Un exemple t y p i q u e d e r ' e s u l t a t partiellement e s t c e l u i d e t o u t ensemble H .-cat;gorique ordonng de l a r g e u r f i n i e d o n t l a t h B o r i e n ' a q u ' u n nombre f i n i d'axiomes. Quelques thBorlmes nouveaux, complets e t accompagnBs de p r e u v e s , s o n t e'galement p r g s e n t b s . Par exemple, chaque ensemble K .-cat&gorique p a r t i e l l e m e n t ordonn; de dimension 2 e t de h a u t e u r f i n i e p o s s l d e une t h g o r i e d g c i d a b l e . Par c o n t r e , il e x i s t e d e s ensembles H .-cat&orique p a r t i e l l e m e n t ordonnBs, de dimension 3 e t de h a u t e u r 2. L ' a r t i c l e c o n t i e n t Bgalement d e s q u e s t i o n s o u v e r t e s .

e

1.

INTRODUCTION

Probably t h e e a r l i e s t theorem on K . - c a t e g o r i c a l

p a r t i a l l y o r d e r e d sets ( o r itself, i s C a n t o r ' s theorem c h a r a c t e r i z i n g t h e o r d e r t y p e of t h e r a t i o n a l numbers.

p o s e t s ) , proved l o n g b e f o r e t h e i s o l a t i o n of t h e n o t i o n of H . - c a t e g o r i c i t y THEOREM 1.1. (A; R ) is order-isomorphic t o t h e r a t i o n u l s foZlowing two conditions hold:

endpoints.

I&;

Ibl

.

l e a s t a prime d i v i s o r of

m

for

a

(mod A) ; I'ensemble des x tels que a < x < b est un intervalle re-

latif admettant la borne {a,b}, entre autres. Si maintenant i l existe x,y de l'intervalle D consid6r6, et t de E - D ,

avec a < x

3

un cyclordre sur

...

suivis pour terminer de 0, I , 2, ..., p - I .

616ments, D une partie stricte de la base E. Prenons

u , v distincts appartenant 1 D : La transposition (u,v) est tin automorphisme l o c a l

de C, non extensible, &me

pas par l'idrntit6 sur u n singleton dans E-D. Donc un

tel cyclordre n'admct pour intervalles relatifs q u e le vide, la base et l e s singlctons. Par contre, l a consgcutivit6 des entirrs naturels, qui n'admet c o m e intervalles absolus que le vide, l a base et les singletdns (voir l . l ) , admet bien d'autres intervalles relatifs : par exemple l a paire { O , l } en prenant c o m e borne l'ensemhle

> 3.

des entiers

La base entisre est l e s e u l intervalle qui admette c o m e borne l'ensemble

vide (consgquence de l a condition de maximalitg). Soit D et D' 3 D deux A-intervalles relatifs ; si F est une borne de D , ni F ni aucun ensemble incluant F n'est une borne de D ' (autre consgquence de I s maximalit&). Si n est I'aritE de A, i l suffit, c o m e pour l'intervalle absolu, de prendre un automorphisme local arbitraire de A/D' sur un domaine de p S n-l

616ments

de D', et d'exiger qu'il s o i t extensible par I'identitE sur tout (n-p)-emble inclus dans F ; un intervalle relatif D 6tant encore un D' maximal par inclusion.

3.1. Etant donng

une

relation A et une partie F de sa base, l a d u n i o n

des (A,F)-intervalles est E-F. En effet, pour chaque 6lEment u de E-F, le singleton de u est un ensemble D' au sens du paragraphe 3 : il existe donc au moins un intervall e relatif qui inclut ce singleton.

Dans certains cas particuliers, c o m e le cas d'une relation binaire r6flexive A , les (A,F)-intervalles sont mutuellement disjoints. En effet ici n-l

= I

i l suffit de considgrer les automorphismes locaux dgfinis sur un unique ElEment, c'est 5 dire toutes l e s transformations d'un 6lGment de E-F en un autre, et d'exiger qu'une telle transformation soit extensible par I'identitE sur chaque Slsment de F. Autrement dit, si a dEsigne u n 616rnent de F, et x un Elsment quelconque de l'intervalle, ledit intervalle est dEfini par les valeurs A(a,x) et A(x,a)

qui

doivent rester les mEmes lorsque x varie dans le (A,F)-intervalle. 11 en rEsulte qu'un msme x ne peut appartenir 1 deux (A,F)-intervalles distincts. Passons au c a s u n peu plus gEnEral d'une relation ou multirelation bim i r e (arit6 maximum 2 ) A et d'un ensemble F. En ce cas r6partissons l e s 6lEment-s de l a base en classes

d'isomorphie des singletons (voir 2.3).

Cela donne au plus

2 classes pour une relation (A(x,x) = + ou - ) , un nombre fini de classes pour une

;

323

L'intervalle en theorie des relations

multirelation. Chaque classe s e subdivise en sous-classes, en mettant x et y dans une mzme sous-classe lorsque l'automnrphismr local de A q u i transforme x en y , est extensible par 1'identitP sur F : donc chaque sous-classe est dSfinie par les valeurs A(a,x) et A ( x , a ) qui pour a 616ment de F, doivent rester constantes lorsque x varie dans une soi~s-classe.Alors chaque (A,F)-intervalle est obtenu en prenant dans chaque classe non vide, une de s e s sous-classes non vides, p u i s la r6union des sous-classes ainsi choisies. En consgquence, Stant donnt? line relation binaire A et une partie F de sa base ; si deux (A,F)-intervalles ont un BlPment commun dans chaque classe d'isomorphie des sinyletons, ces intervalles sont identiques. ProblPme. Contre-exemple de l'Pnonc6 ci-dessus avec une relation ternaire. 3.2. Soit A une relation, F,G deux parties de sa base avec G C F ; alors c h a g u e (A,F)-intervalle est inclus dans un (A,G)-intervalle.

Dans le cas d'une relation binaire A avec G

c F,

chaque (A,G)-interval-

le non vide inclut un (A,F)-intervalle non vide. 0

Rgpartissons les ElPments du (A,G)-intervalle considErE, en classes d'isomorphie

d e s singletons (voir 2.3).

Cela donne au plus 2 classes si A est une relation, un

nombre fini de classes dans le cas gEn6ral d'une multirelation. Prenons un 61Ement t reprgsentant de chaque classe. Pour assurer la maximalit6 qui fait partie de la

dEfinition de l'intervalle relatif (voir paragraphe 3), joignons 2 chaque reprgsentant t les El6ments t' de la m@me classe, pour lesquels la transformation de t en t', qui est d6j2 extensible par tit6 sur

1'identitE sur G, est de plus extensible par l'iden-

F.0 Problsme. Existence d'une relation ternaire A avec un F et G

c F,

et un

(A,G)-intervalle qui n'inclut aucun (A,F)-intervalle. 3.3.

L'intersection de deux intervalles relatifs distincts mais ayant

une borne commune F, ne peut Stre un intervalle relatif de borne F, en raison de la maximalite. Soit A une relation binaire ; consid6ron.s un ensemble de A-intervalles relatifs Di, chacun admettant entre autres une borne Fi. Alors l'intersection des

D. est incluse dans un A-intervalle relatif ayant p o u r borne la reunion des F.. D6j2 pour une relation binaire, l'intersection de deux intervalles re-

latifs n'est p a s forcement un intervalle relatif. Prenons une base E de 6 El6ments a,b,c,c',d,e avec A(x,x) x

=

a,b,d,e, et

A(e,a)

=

- ,

tions (b,e),

-

pour x=c,c'. Orientons les paires suivantes : A(a,e)

= +

pour

= +

et

ce que nous nommons l'orientation (a,e) ; prenons de mgme les orienta(e,d),

(c,d), (d,c') ; les autres paires prenant par exemple la valeur

+ d a m les deux sens. L'ensemble A = {a,c,c') est un intervalle admettant la

borne F

=

{b,e} : on ne peut lui ajouter d en raison des orientation (a,e) et (e,d).

L'ensemble B

=

{b,c,c'} est un intervalle admettant la borne G

=

{a,e] : on ne peut

lui ajouter d en raison d e s orientation (b,e) et (e,d) : la maximalit6 est bien

R. Fraisse'

324 satisfaite. L'intersection A n B

=

{c,~'} n'est pas un intervalle.

En effet la borne ne peut pas comprendre d, en raison des orientations (c,d) et (d,c'). Donc cette borne Bventuelle est incluse dans {a,b,e). Alors la maximalit6 n'est pas satisfaite, puisqu'5 c et c' n o u s pourrons toujours ajouter d (de valeur A (d,d)

= +). 0

3 . 4 . Soit A une r e l a t i o n binaire d e base E , e t s o i t D un A-intervalle r e l a t i f ; a l o r s la reunion des bornes de D e s t une borne de D. 0

Soit u,v deux Qlgments de D, avec isomorphie des resteictions de A aux

singletons de u et de v. Alors pour chaque 616ment c de la rBunion des bornes, l'automorphisme local transformant u en v est extensible par l'identitg sur c. De plus D est maximal par inclusion, parmi les ensembles qui vBrifient ce qui prBcPde. ProblZmes. Contre-exemple P ce qui pr&cZde, dans le cas d'une relation ternaire. Existence d'une relation ternaire A, d'un A-intervalle D de borne F , d'un A-intervalle D' de borne F' avec un automorphisme local de AID' n D' inextensible par l'identit6 sur F U F'.

3.5. Contrairement 5 l'intervalle absolu (voir 1.7),

l'intervalle re-

latif n'est pas prCservC par restriction : si D est un sous-ensemble de la base de A, et si F est un A-intervalle relatif, l ' i n t e r s e c t i o n D n F n ' e s t p a s forcement un (AID)-intervalle r e l a t i f .

lorsque y

Prenons la cons6cutivit6

=

x+l. Alors l'ensemble des entiers > 2 est un intervalle, admettant

A

siir les entiers naturels : A(x,y)

= +

0

pour borne le singleton de 0. Supprimons l'ClCment 0 : nous obtenons la consscutivit6 sur les entiers positifs, pour laquelle l'ensemble des entiers un intervalle.

>2

n'est plus

0

3 . 6 . Etant donne une r e l a t i o n binaire A e t une p a r t i e f i n i e F de sa base, l e s (A,F)-intervalles sont en nombre f i n i . 0

Disons que deux ClCments u,v de I A I - F sont equivalents lorsque la

transformation de u en v est un automorphisme local de A, extensible par l'identit6 sur F. Puisque F est fini, les classes de cette 6quivalence sont en nombre fini,

ce qui entraPne l'bnonc6. L'Bnonc6 prCc6dent ne s'Ctend pas au cas ternaire. Par exemple prenons pour A un cyclordre infini ; pour prgciser, le cyclordre des entiers naturels, et prenons pour F le singleton de 0 . Chaque singleton autre que 0 , est un (A,F)-intervalle ; en effet la maximalit6 est vCrifibe, puisque l'addition d'un second entier v B l'entier u qui constitue notre intervalle, autoriserait 5 considsrer la transposition (u,v) qui est un automorphisme local non extensible par l'identite sur 0. ProblSme, suggerd par 1 . 3 . Etant donn6 un ensemble, filtrant par inclusion, d'intervalles relatifs, leur rdunion est-elle t o u j o u r s un intervalle relatif.

0

325

L'interualle en thiorie des relations 4- LE F I N I V A L L E , UNE N O T I O N BOOLEENNE

Soit A une relation, D une partie de sa base. Pour chaque entier posirt v i ( i = I , ...,p) dans D sont e q u i v a l e n t s i lorsque la transformation qui change chaque u . en v. est bijective

tif p, disons que deux p-uplrs u modulo ( A , D ) ,

et est un automorphisme local de A extensible par l'identit6 sur I A I -D. L'ensemble D est d i t un A - f i n i v a l l e

lorsque, pour chaque p, i l n'existe qu'un nombre

fini de classes d'6quivalence mod(A,D) entre p-uples. Si n est l'arit6 de A (le maximum de l'arit6, pour une multirelation), il suffit de limiter la longueur d e s suites aux p

< n-I,

et de limiter l'extensi

bilit6 1 n-p 616ments de I A I - D . L'6nonc6 1 . 7 reste valable pour les finivalles : e t a n t d o n n e u n e r e l a t i o n A et un sous-ensemble

D d e s a b a s e , t o u t A - f i n i v a l l e , intersecte p a r D

donne un

(AID)-finivalle. 4.1 . T o u t e r e u n i o n f i n i e d ' i n t e r v a l l e s d i s j o i n t s e s t u n f i n i v a l l e (la

condition de disjonction sera inutile aprSs le 4 . 4 ci-dessous). 0

Deux p-uples sont Cquivalents s i la transformation d e l'un en l'autre

est un automorphisme local, et si de plus les premiers termes sont dans un mzme intervalle, les deuxismes termes sont dans un mGme intervalle, et ainsi de suite. En particulier, pour toute relation, t o u t e n s e m b l e f i n i e s t u n f i n i v a l l e , puisque r6union finie de singletons, qui sont des intervalles ; cela se voit

aussi immgdiatement sur la dgfinition. P o u r u n e c h a i n e , l e s f i n i v a l l e s sont e x a c t e m e n t l e s r e u n i o n s f i n i e s d ' i n t e r v a l 1es

.

A chaque chaine A nous avons associ6 la relation ternaire de c y c l o r d r e

( o u ordre cyclique) voir paragraphe 3 . Nous avons vu alors que les seuls interval-

les (absolus ou relatifs) du cyclordre sont le vide, la base et les singletons. Soit alors A la chaine des entiers relatifs, C son cyclordre engendrG : l'ensemble D des entiers positifs n'est ni un C-intervalle ni un extervalle ni une r6union ou intersection finie d'entre eux ; mais D est un C-finivalle, deux p-uples 6tant Gquivalents mod(C,D) lorsque la transformation d'un en l'autre pr6serve l'ordre A . 4 . 2 . L e c o m p l e m e n t a i r e d ' u n f i n i v a l l e est u n f i n i v a l l e . 0

Soit A la relation, n s o n aritc, E sa base, D un finivalle.

Pour chaque entier positif p, il n'existe qu'un nombre fini d e classes d'Gquiva-

lence mod(A,D).

Pour chaque p-uple (p

< n-1)

dans D, prenons un unique reprGsen-

tant appartenant P la mzme classe, et appelons H la partie finie de D, rEunion de ces

reprcsentants. Pour chaque entier positif r. disons que deux r-uples dans E-D

sont Gquivalents, lorsque la transformation de l'un e n l'autre est un automorphisme local, extensible par l'identit6 sur H. Puisque H

est fini, i l n'existe, pour

chaque r, qu'un nombre fini de classes pour cette Equivalence.

R. Fra'isse

326

Pour voir que E-D est un finivalle, i l suffit maintenant de

prouver que, si g

est un automorphisme local de A/(E-D), extensible par 1'identitE sur H, alors g est encore extensible par l'identitE sur D. I1 suffit mcme de prouver que, s i g est extensible par l'identit6 sur H, alors g est extensible par l'identit6 sur toute partie F de D, de cardinal

< n-1.

Par hypothsse, il existe une partie F ' de H

et un automorphismc local f de domaine F et codomaine F'. extensible par l'identi-

t6 sur E-D, donc aussi par l'identit6 IG sur G

=

Dom g. Donc f U I et g G

u

IF'

(06 I F ' est l'identitE sur F') sont deux automorphismes locaux de A ; il en est de

mGme de f U g par composition. Puisque f est extensible par l'identit6 s u r E-D,

il l'est aussi par I G l qui est l'identit6 sur G '

=

-1

Cod g ; donc f

U I G l est u n

automorphisme local de A , et par composition I u g est un automorphisme local. 0 F En particulier, pour toute relation, l e complementaire d'une p a r t i s f i n i e quelconque de l a base e s t un f i n i v a l l e . 4.3. (1)

S o i t A une r e l a t i o n , B une autre de msme base, libre-interpr4-

table en A ; alors t o u t A - f i n i v a l l e e s t un B - f i n i v a l l e . 0

Prenons deux suites finies de mzme longueur et telles que la transfor-

mation de l'une en l'autre soit un automorphisme local de A , donc encore de B. Ces suites 6tant dans une partie D de la base, si la transformation est extensible, pour A, par l'identitE hors de D, elle est extensible de mzme pour B .

0

En particulier, s ' i l e x i s t e une m u l t i r e l a t i o n unaire en l a q u e l l e A s o i t l i b r e - i n t e r p r e t a b l e , a l o r s t o u t e p a r t i e de l a base e s t un A - f i n i v a l l e . ( 2 ) Etant donne une m u l t i r e l a t i o n B e t un B - f i n i v a l l e D , i l e x i s t e

une m u l t i r e l a t i o n A de m6me base, en l a q u e l l e B s o i t l i b r e - i n t e r p r e t a b l e e t t e l l e gue D s o i t un A-intervalle

(POUZET, 1975, non publi6).

Soit n l'arit6 de B . Pour chaque p

< n,

considgrons 1'6quivalence dE-

finie en mettant deux p-uples extraits de D dans une mzme classe lorsque la transformation de l'un en l'autre est un automorphisme local de B , extensible par l'identitb sur 1 B 1 - D. Pour chaque classe U, prenons la relation p-aire encore appel6e U, valant + pour les p-uples qui appartiennent 1 la classe et - pour tous autres p-uples dans la base. Nos classes Gtant en nombre fini, la suite formge de B et des U pour p = I ,

...,n, constitue une multirelation A

en laquelle B est

libre-interprgtable. De plus si deux p-uples dans D sont transformgs l'un en l'autre par un automorphisme local de A , ils appartiennent 1 une mzme classe, donc ladite transformation est extensible par l'identitg hors de D, pour A c o m e pour B.n ( 3 ) En consEquence, e t a n t donne une r e l a t i o n B e t une p a r t i e D d e

sa base, D e s t un B - f i n i v a l l e s i e t seulement s'il e x i s t e une A en l a q u e l l e B s o i t l i b r e - i n t e r p r e t a b l e , avec l a condition que D s o i t un A-intervalle.

Notons qu'en g6n6ra1, B gtant donnbe, il n'existe pas de multirelation A en laquelle B soit libre-interprgtable, avec la condition que t o u t B-finivalle soit un A-intervalle. En effet, tout ensemble fini 6tant un B-finivalle, il faudrait que tout ensemble fini, donc toute partie de la base, soit un A-intervalle (voir 1 . 8 ) .

327

L'interualle en thiorie des relations I1 faudrait donc que A , donc B, soit librr-interprdtable en une multirelation

unaire (voir 2.5).

4.4. (I)

Tout finivalle, augmente ou diminue d'un nombre fini d'elements

de l a base, donne un finivalle. 0

A chacun des El6ments ajoutEs, associons sa relation unaire singleton,

puis ajoutons ces relations singletons 5 la multirelation A de l'dnoncE ( 3 ) cidessus. On passe ensuite au cas d e 1 3 suppression d'un nomhre fini d'616ments, par passage au complEmentaire.

(2)

La

0

reunion et 1 'intersection de deux finivalles quelconques

c s t un finivalle (POUZET,

1976, non publiE ; ut. ax. choix dgnombrable ; ZF suf-

fisant lorsque la base est dEnombrahle). 0

Prouvons l'i.nonc6 pour la r6union ; on passera ensuite au compl6men-

taire. Rrtisonnons par l'absurde

en

supposant q u e U et V sont deux finivalles, mais

non leur rEunion. Soit p le plus petit entier pour lequel existe une infinit6 de classes de p-uples, deux p-uples dans U U V Etant dits Equivalents lorsque la transformation de l'un en I'autre est un automorphisme local extensible par l'identit6 hors de U U V. Prenons une o-suite de p-uples mutuellement non 6quivalents u les termes appartenant 5 U , et non

(ax. choix d 6 n . ) . Pour chaque p-uple, notons

1 V ; notons v l e s termes appartenant 5 V et non 1 U ; enfin w les termes appartenant 1 ]'intersection U

n V. Puisqu'il y a une infinit6 de p-uples en considEra-

tion, n o u s pouvons supposer que, pour chaque indice r

< p,

l e r-6me terme est

toujours un u , ou toujours un v, o u toujours un w. En restant dans le cas gEn6ra1, nuus pouvons supposer p

=

3 avec chaque suite formEe d'un u , un v et un w : pour

chaque entier i, nous avons donc la suite d e s trois termes u.,v.,w.. 1

1

1

Puisque U est un finivalle, et que chaque suite u i wi est formCe de deux ElEments de U, ces suites se r6partissent en un nombre fini de classes, pour 1 ' 6 quivalence d6finie par un automorphisme local extensible par l'identits hors de U. Nous pouvons donc supposer que toutes Les suites u . wi appartiennent i une m&ne

classe. Donc pour chaque i, la transformation d e u

vi wi en uo vi wo est un auto-

morphisme local extensible par Z'identitE hors d e U. Par ailleurs puisque V est un finivalle, i l en est d e &me

de V diminus d e 1'61Ement wo : voir le ( 1 ) pr6c6dent.

Les termes v appartenant Z ce nouveau finivalle, et les termes uo et wo 6tant en dehors, nous pouvons supposer que, pour chaque paire d'entiers i , j , la transformation de u

vi wo en uo vj wo es t un automorphisme local extensible par l'identi-

ti3 hors de V. Finalement pour tots i,j, nous avons un automorplisme local tralsformant u . v. w. en u 1

1

1

vi wo, puis en u

O

v. wo, puis en u J

vj v.,

avec 1 chaque 6tape

l'extersitilit6 par l'identitg h o r s de U U V : contradiction avec la d6finition de la suite infinie d e p -uple non Cquivalents 4.5.

.

0

Soit A une relation, E s a b e , D une partie de E . A l o r s pour que D

soit un finivalle, il faut et suffit qu'il existe une partie finie F du complementaire E-D,

telle que chaque automorphisme local de A I D , ou bien n'est pas extensible

R. Fra'isse

328

p a r l ' i d e n t i t k s u r F . o u b i e n e s t e x t e n s i b l e p a r l ' i d c n t i t d sur E-D t o u t e n t i e r (6nonc6 communiqu6 p a r POUZET, 1 9 7 8 ) . 0

S u p p o s o m d ' a b r d que F e x i s t e . Alors pour chaque e n t i e r p . r 6 p a r t i s -

s o m l e s p - u p l f f i d a m D e n un n o m k e f i n i d e c l a s s ~ s ,deux p ? ~ p I p s 6 t a n t d i t s

6 q u i v a l e n t s l o r s q u e l a t r a m f o r m a t i o n d e l'un e n l ' a u t r e , r 6 u n i e 3 l ' i d e n t i t s s u r F , donne un a u t o m a r h i s m e l o c a l d e A . La c o n d i t i o n d e l ' h o n c e c o n c e r n a n t F

e n t r a ' i n e a l o r s que D s o i t un f i n i v a l l e . I n v e r s e m e n t s o i t D un f i n i v a l l e ; n o t o m n l'arit6 d e A . Pour c h a q u e e n t i e r p o s i t i f p < n , r 6 p a r t i s s o f f i Iffi p i p l r s e n leiits c l a s s e s d ' P q u i v a l e n c e mod(A,D) : v o i r d 6 t u t d u p r k e n t p a r a g r a p h e 4 ; ces c l a s s -

s o n t e n nombre f i n i .

# C i , p r e n o r s un p j Gl&ments d e E - D , t e l que la

P o u r c l a q u e p e t c h a q u e p a i r e d e c l a s s e s d e p - u p l ~C ~. e t C

d ' a u p l u s n-I

u p l e dam c h a c u n e , e t u n e r s e m b l e Fi

c h o i s i pour C ne s o i t p"s e x i ' e n 'j j' L a r 6 u n i o n f i n i e F des F p o u r t o e les p < n , ij* ij v 6 r i f i e l ' 6 n o n c 6 . En e f f e t 6 t a n t donn6 deux p l l p l f f i u ' . d e C e t u' d e C la i j j' t r a m f o r m a t i o n d e u i e n u ' est p a r h y p o t h k e e x t e n s i b l e p a r l ' i d e n t i t 6 s u r E-D ; t r a n s f o r m a t i o n du p - u p l e u

i

chois i p o u r C

teffiihle par l ' i d e n t i t 6 s u r F

i

d e mEme l a traffi f o r m a t i o n d e

11

j

en u '

j'

Donc s i l a t r a m f o r m a t i o n d e u! e n u ! 6 t a i t J

e x t e n s i b l e p a r l ' i d e n t i t 6 s u r F , i l e n s e r a i t d e mZme d e l a t r a n s f o r m a t i o n de u i en u . : c o n t r a d i c t i o n . F i n a l e m e n t F v 6 r i f i e 1 ' 6 n o n c 6 p o u r l e e n t i e r s p < n , e t i l

J en r f f i u l t e que F v E r i f i e e n c o r e l ' 6 n o n c 6 p o u r u n automorphisme l o c a l q u e l c o n q u e d e AJD.

4 . 6 . En v u e des 6noncffi s u i v a n ' s , ciaprk

l e l e c t e u r r e t r o u v e r a les t r o i s lcmmffi

( d o n t l e d e r n i e r remonte 1 LOPEZ, 1 9 6 9 ) :

(I)

S o i t A une r e l a t i o n n a i r e , u , v deux 616ments d e l a h e ; s i l a

t r a n s p o s i t i o n ( u , v ) m o d i f i e A ( i . e . c e t t e trampos i t i o n , 6 t e n d u e p a r l ' i d e n t i t 6

s u r 1 s 616ments a u t r e s que u , v , n ' f f i t pas u n a u t o m o r p h k m e d e A ) , a l o r s i l e x i s t e u n e n s e m b l e H d ' a u p l u s n+l e l e m e n t s , p a r m i l e s g u e l s u e t v , t e l q u e ( u , v ) m o d i f i e A/H. ( 2 ) S o i t E un e m e m t l e , f une p e r m u t a t i o n d e E e t F une p a r t i e f i n i e d e

E ; i l e x i s t e u n e s u i t e s a n s r k p e t i t i o n d ' P l 6 m e n t s u I , ...,u,,

d e F, t e l l e que p o u r

c h a q u e Plernent x d e F , l e t r a n s f o r m e f ( x ) s o i t i d e n t i q u e a u t r a n s f o r m 4 d e x p a r l a composee d s s t r a n s p o s i t i o n s s u c c e s s i v e s (u

, f ( u I ) ) , . . . , (y,,f (r,) ) .

( 3 ) Soit A une r e l a t i o n d e b a s e E, et f une p e r m u t a t i o n d r E g u i m o d i f i e A ; a l o r s i l existe un e l e m e n t u d e E t e l g u e l a t r a n s p o s i t i o n ( u , f ( u ) ) m o d i f i e A .

4.7.

(I)

S o i t E l ' e m e m t l e dffi e n t i e r s n a t u r e l s , N l a c h a i n e u s u e l l e s u r

ces e n t i e r s . S i A e s t l i b r e i n t e r p r e t a b l e en N e t s i , p o u r c h a q u e p r o g r e s s i o n arithmetique D de raison

> 2,

i l e x i s t e d e u x e l e m e n t s x e t y > x d e D t e l s q u e la

t r a n s p o s i t i o n (X,y) s o i t u n a u t o m o r p h i s m e l o c a l d e A, e x t e n s i b l e p a r 1 ' i d e n t i t 8

sur E-D,

a l o r s t o u t e p e r m u t a t i o n d e E e s t u n a u t o r n o r p h i s m e d e A (6nonc6s ( I )

et

( 2 ) dus 1 POUZET e t YASUHARA). 0

S u p p o s o n s qu'il e x i s r e une p e r m u t a t i o n d e E qui m o d i f i e A . P a r

lffi

lemmes p r G c 6 -

329

L'interualle en theorie des relations

d e n t s , i l e x i s t e un e n s e m b l e f i n i F d ' e n t i e r s e t d e u x e l 6 m e n t s a , l a t r a n s p o s i t i o n ( a , t) m o d i f i e A / F .

Fixiins

b de F tels que

t > a . Soit u =

1 6 idges en posant

Max(2, b a ) ; s o i t v l e n o m l r e d e s 616ments < a e t w c e l u i d e s 616me nts Puisque A ffit l i b - e - i n t e r p r 6 t a b l e

rn N,

>

k da ns F .

t o u t e i n j e c t i o n c r o i s s a n t e d e doma ine F

> x+u,

e s t un a u t o m o r p h i s m e l o c a l d e A . Donc p o u r t o u s e n t i e r s x 2 a e t y

l a trans

-

p o s i t i o n ( x , y ) m o d i f i e A . P l us p r G c i s 6 m e n t e l l e m o d i f i e l a r e s t r i c t i o n d e A 1 t o u t e n s e m t l e q u i comprend a u moins v e l e m e n t s

<x

et

wi.16ments > y

e t u-l

616ments

e n t r e x e t y . En p a r t i c u l i e r G t a n t donne l a p r o g r e s s i o n a r i t h m 6 t i q u e D = { a , a+u, a+2u,.

. . 1,

p o u r d e u x 6l Gmrnt s q u e l c o n q u e s x e t y

>x

de D, l a trans -

p o s i t i o n ( x , y ) e t e n d u e p a r l ' i d e n t i t e s u r E-D, n ' e s t p a s un a u t o m o r p h i s m e l o c a l de A. 0

( 2 ) S o i t D un e r s t . m t l e i n f i n i d ' e n t i e r s , d o n t l e c o m p l g m e n t a i r e E - D e s t i n f i n i . S i t o u t e p a r t i e d e D est un A - f i n i v a l l e , a l o r s il e x i s t e d e u x elements x et y

>x

d e D,

t e l s q u e l a t r a n s p o s i t i o n (x,y) soit un automorphisme

local d e A, e x t e n s i b l e p a r l ' i d e n t i t e s u r E - D . Co rsCq u e n c e de ( I )

e t (2) : s i A e s t l i b r e - i n t e r p r e t a b l e

en N e t s i t o u t e n s e m b l e

d ' e n t i e r s est u n A - f i n i v a l l e , a l o r s t o u t e p e r m u t a t i o n d e E est u n a u t o m o r p h i s m e d e A. 0

P r e u v e du ( 2 ) . A c h a q u e p a i r e d ' 6 1 6 m e n t s x , y d e D a s s o c i o n s l e c o u p l e o r d o n n d

(x,y

> x),

c e q u i nous p e r m e t d e r E p a r t i r l e s p a i r e s e n un n o m I r e f i n i d e c l a s s e s

d ' g q u i v a l e n c e mod (A,D). P a r l e t h e o r s m e d e RAMSEY, i l e x k t e u n e p a r t i e i n f i n i e D* d e D d a n s l a q u e l l e t o u t e les p a i r e s a p p a r t i e n n e n t 1 une m t m e c l a s s e . Donc p o u r tous x < y

e t x' < y '

616ments do D*, l a t r a n s f o r m a t i o n d e x e n x ' e t y e n y ' e s t

un a u t o m o r p h i s m e l o c a l d e A , e x t e r s i t l e p a r l ' i d e n t i t g s u r E - D . P r e n o n s d a n s D* les 6lGments d e r a n g s p a i r s : s o i t D*. P u k q u e ce D* e s t un A - f i n i v a l l e , p r e n o n s P P u n e p a r t i e i n f i n i e D** d e D* d a m l a q u r l l e l e s s i n g l e t o n s s o i e n t t o u s g q u i v a l e n t s

P

avec x < y < z , l a mod ( A , D * ) . Donc d t a n t d o n n b x , z d a n s D** e t y d a m D*-D* P P t r a n s f o r m a t i o n d e x e n z e t d e y e n l u i m z m e , e s t un a u t o m o r p h i s m e l o c a l d e A ,

e x t e n s i b l e p a r l ' i d e n t i t 6 s u r E-D.

P u i s q u e x , y , z a p p a r t i e n n e n t 1 D*,

p a r composi-

t i o n l a t r a n s p o s i t i o n (x,y) e s t un a u t o m o r p h i s m e l o c a l d e A , e x t e n s i k l e p a r l ' i d e n t i t 6 s u r E-D. 4.8.

0

S o i t A u n e r e l a t i o n , E s a t a s r . Supposom q u e t o u t e p a r t i e d e E

s o i t un A i i n i v a l l e ; a l o r s ( c o m e en 2 . 3 ( 2 ) e t e n 2 . 5 ( I )

(I) fini

) :

L e s c l a s s e s d ' e q u i v a l e n c e p a r t r a n s p o s i t i o n (mod A ) sont en n o m b r e

;

(2) I 1 e x i s t e u n e m u l t i r e l a t i o n u n a i r e en l a q u e l l e A e s t l i b r e - i n t e r p r e t a b l e ( u t i l i s e l ' a x i o m e d e c h o i x d G n o m l r a t l e , ZF s u f f k a n t l o r s q u e l a h e E e s t

d6 n o m b ra b l e ; GnoncCs dus 5 POUZET). P a r c o n t r e i l n ' e x i s t e pas f o r c E m e n t u n e m u l t i r e l a t i o n u n a i r e

u

telle

que A e t U s o i e n t chacune l i t r e i n t e r p r 6 t a U e e n l ' a u t r e . Prenons e n e f f e t pour A une r e l a t i o n

t i n a i r e d ' g q u i v a l e n c e B un n o m t r e f i n i d e c l a s s e s . P o u r c h a q u e e n t i e r

R. Fra'issB

330

pour n ' i m p o r t e q u e l l e p a r t i e D d e l a

p, d e u x p - u p l e s s o n t 6 q u i v a l e n t s mod(A.D) bse qui

lffi

c o n t i e n t , p o u r v u q u ' i l s s o i r n t t r a n s f o r m & p a r une E j e c t i o n e t quc

d e u x termes de &me

r a n g d a m les d e u x p - u p l e s a p p a r t i e n n e n t 2 u n e m6me c l a s s e .

A i n s i t o u t e p a r t i e de l a Lase e s t un f i n i v a l l r ; mais une m u t t i r e l a t i o n U e n l a q u e l l e A s e r a i t l i l r e i n t e r p r g t a t l e , c o m p r e n d r a i t p a r exemple une r e l a t i o n u n a i r e valant

+ s u r une c l a s s e d e

A et

-ailleurs,

e t c e l a p o u r c h a q u e c l a s s e . Nous voyons

q u ' a l o r s U ne s e r a i t pas l i h - e - i n t r r p r 6 t d b l e 0

en A .

( I ) R e p a r t i s s o n s l e s 616ments d e E e n c l a s s e s d ' f q u i v a l e n c e p a r t r a n s p o s i t i o n

(mod A) : v o i r 2 . 2 .

Supposons q u ' i l e x i s t e une i n f i n i t 6 d e c e s c l a s s e s . P a r

l ' a x i o m e d e c h o i x d 6 n o m k r a U e , p r e n o n s un e n s e m t l e d 6 n o m l r a ' d e D d'6lGments d e l a Lase, m u t u e l l e m e n t non g q u i v a l e n t s . C o n s t r u i s o n s s u r l a Lase D u n e isomorphe N d e l a cha'ine des e n t i e r s n a t u r e l s . P a r l e thilorsme d e RAMSEY, now pouvons t o u j o u r s s u p p o s e r que l a r f f i t r i c t i o n A/D e s t l i h - e - i n t e r p r E t a b l e

e n N . Plus p r S c i s g m e n t ,

m e t t o n s dans une msme c l a s s e d e u x p a i r e s d ' E l 6 m e n t s x , y e t x ' x D n .

u , v d e u x glements d i s t i n c t s q u e l c o n q u e s d e D

.

Par c e q u i prscsde,

5 n+l

l a t r a n s p o s i t i o n ( u , v ) m o d i f i e A . P a r 4 . 6 , i l e x i s t e une p a r t i e H d e

d e E , parmi l e s q u e l s u e t v , t e l l e q u e ( u , v ) m o d i f i e A / H . un e n t i e r h ( I 5 h a u c u n des

5

n-l

5 n)

t e l que l a d i f f e r e n c e e n s e m b l i s t e

Glements d e H

- Iu,v}.

6lEnents

Donc i l e x i s t e a u moins

4,

- 4,

n e comprenne

Donc c e t t e d i f f B r e n c e s t d k j o i n t e d e H .

P a r l ' G n o n c 6 4 . 7 , l a t r a n s p o s i t i o n ( u , v ) e s t un a u t o m o r p h i s m e d e A / D ,

puisque toute

p a r t i e d e D e t un (A/D) X i n i v a l l e

:

donc ( u , v ) automorphisme d e A / D h . Mais ( u , v )

n ' e s t p a s un automorphisme de A / ( E

-

(Dh-I-Dh)),

Autrement d i t , B t a n t donnE u , v d e D n , phisme l o c a l d e A / D h ,

non e x t e n s i b l e p a r 1 ' i d e n t i t B s u r E

D ' a u t r e p a r t B t a n t donnB u , v , u ' , v ' la t r a m formation de u en u' e t v en v '

phisme l o c a l d e N / 9 , donc d e A / \ r a i s o n d e l a d g f i n i t i o n msme d e u,v,u',v'

de D

d u f a i t que H C E

avec u < v e t u '

,

-

(Dh-,-Dh).

i l e x i s t e u n h t e l q u e ( u , v ) s o i t un a u t o m o r -

ffi

-?I - 1 '

d e Dn a v e c u < v e t u ' < v '

t pour chaque h

(1

< h 5 n)

(mod N ) ,

un a u t o m o r -

e x t e n s i b l e p a r l ' i d e n t i t s s u r E - Dh -1 ' en p a r c o m p o s i t i o n , q u ' E t a n t donne

4.11 en r b u l t e , > v'

(mod N ) ,

i l e x i s t e un h t e l que l a t r a n s f o r -

33 1

L'intetvalle en theorie des relations

, v ' s o i t un automorphisme l o c a l d e A/Dh, non

mation de u,v respectivement en u ' extens g

H e par l'identiti. s u r E -

t un a u t o m o r p h i s m e l o c a l d e A / D

4,

n'

A pl16

f o r t e raison, c e t t e transformation

lion e x t r n s i

F i n a l e m e n t , 6 t a n t donn6 u , v , w d e D

Ue par l ' i d e n t i t e s u r E - D

avec u

< w

F U F' ;

G , toute relation 6lQment de UG, une fois restreinte B F ,

donne un Qldment de UF ; alors il existe une relation R de l'arit6 commune donnQe, bas6e sur la reunion des F , et telle que pour chaque F , la restriction R/F EUF.

Ce lenime 6quivaut 2 l'axiome de l'ultrafiltre. dans le cas gQn6ral ; mais ZF suffit lorsque les F sont finis et leur rsunion dsnombrable. Note 2. La notion usuelle d'intervalles contigus

se

g6n6ralise de manisre

naturelle : deux A-intervalles sont dits c o n t i g u s lorsque leur r6union est un A-intervalle. Le lecteur d6finira ainsi 1'616ment adhPrent cl6ture topologique d'un A-intervalle.

D un A-intervalle, donc la

34 1

Istvan FOLDES, 1973

-

R e l a t i o n s denses et d i s p e r s e e s

;

extension d ' m t h b o r e m e d e H a u s d o r f f .

C o m p t e s r e n d u s A c a d . S c i . P a r i s , vol 2 7 7 A , p . 2 6 9 - 2 7 1 .

Roland FRAYSSA, 1971-72Cours d e l o q i q u e mathematique, ( 1 ) R e l a t i o n et f o r m u l e l o g i q u e , 197 p . : ( 2 ) T h e o r i e d e s m o d e l e s , 1 7 7 p . : P a r i s ( G a u t h i e r - V i l l a r s ) : trad.

1973-74-

C o u r s e of m a t h e m a t i c a l l o g i c , D o r d r e c h t ( R e i d e l ) .

1977- P r e s e n t p r o b l e m s a b o u t i n t e r v a l s i n r e l a t i o n - t h e o r y

and l o g i c .

Non c l a s s i c a l l o g i c s , m o d e l t h e o r y a n d c o m p u t a b i l i t y , vol 8 9 d e S t u d i e s

i n L o g i c , Amsterdam (North-llolland).

David GILLIAM, 1978 A

-

concrete r e p r e s e n t a t i o n t h e o r e m f o r i n t e r v a l s o f m u l t i r e l a t i o n s .

Z e i t s c h r i f t M a t h . L o g i k vol 2 4 p . 4 6 3 - 4 6 6 . 1 9 7 9 - I n t e r v a l s o f b i n a r y r e l a t i o n s . I b i d e m vol 2 5 p . 5 7 - 6 0 .

Gerard L O P E Z , 1978 L ' i n d e f o r m a b i l i t e d e s r e l a t i o n s et m u l t i r e l a t i o n s b i n a i r e s . Z e i t s c h r i f t M a t h . L o g i k vol 2 4 p . 3 0 3 - 3 1 7 .

Maurice POUZET, 1979

-

R e l a t i o n m i n i m a l e p o u r son d g e . Z e i t s c h r i f t M a t h . L o g i k vol 2 5 p . 3 1 5 - 3 4 4 . 1981- R e l a t i o n s i n p a r t i b l e s . D i s s e r t a t i o n e s Mathematica

, vol 1 9 3 , p . 1 - 4 3 .


-

PART VI ORDER AND CHAIN CONDITION ON CLASSES OF STRUCTURES

- PARTIE VI ORDRE ET CONDITIONS DE CHAINE SUR DES CLASSES DE STRUCTURES

P.D. SEYMOUR

-

Neil ROBERTSON

Some new results on the well-quasi ordering of graphs.

.... . ... .

p . 343


Annals of Discrete Mathematics 23 (1984) 343-354 0 Elsevier Science PublishersB.V. (North-Holland)

343

SOME NEW RESULTS ON THE WELL-QUASI-ORDERING OF GRAPHS

P.D. SEYMOUR* and Neil ROBERTSON Department of Mathematics The Ohio State University 231 West 18th Avenue Columbus, Ohio 43210 U.S.A.

D e d i c a t e d t o Professor E . COROMINAS.

RESUME.-

Quelques resultats nouveaux sur la relation de be1 ordre des graphes finis.

Designons par 9 l'ensemble des graphes finis, et, lorsque G, H € 9 , Bcrivons H E G si H est isomorphe a un mineur de G, c'est-8-dire a une contraction d'un sous-graphe partiel de G . Dans les annees 1960, K. WAGNER a fait la conjecture (non publiee) que 9 est belordonne par cette relation d'inclusion. Pour des graphes finis, cela signifie que dans tout sousensemble infini un certain graphe sera isomorphe a un mineur donne d'un autre graphe. Recemment, les auteurs de cet article ont demontre des theorsmes qui laissent fortement presager que la conjecture de WAGNER est correcte. Notre methode consiste ?I montrer que l'ensemble des graphes G qui ne contiennent pas un certain graphe fixe H comme mineur a une structure definie, et de montrer alors, qu'etant donnee cette structure, l'ensemble des graphes qui la possedent est belordonnd. Si on savait le faire pour tout graphe H, il en decoulerait que ( 3 5 ) est lui-meme belordonnd, donc que la conjecture de WAGNER est vraie. La structure de graphes mentionnde p l u s haut est construite en trois Btapes: (1) plonger un graphe dans une surface S ; ( 2 ) adjoindre au plus n sommets, qui peuvent etre adjacents a n'importe quels des autres sommets, et (3) joindre les unes aux autres des copies disjointes de tels graphes en une structure d'arbre. Ici, S et n sont determines par le graphe H exclu, et les graphes adjacents dans la structure d'arbre formant G sont joints en identifiant des sousensembles de leurs sommets d'une maniere univoque. De tels sous-ensembles ne peuvent intersecter la surface de plongement S q u e trivialement, c'estA-dire au p l u s en trois sommets contenus dans la frontisre d'une face du plongement. Ce programme a maintenant BtB execute pour des classes de graphes G qui ne contiennent pas c o m e mineur un graphe planaire fixe H; c'est un cas oh il

n'est pas necessaire d'utiliser la structure de la surface. Pour tout graphe planaire H, il existe un entier w tel que tout graphe G qui ne

*

Partially supported by National Science Foundation grant number MCS 8103440

.

P.D. Seymour, N. Robertson

344

contient pas H comme mineur peut &tre construit en joignant en une structure d'arbre des graphes avec au plus w sommets. De plus, on montre que les classes de graphes G qui peuvent etre construites de cette mani6re sont belordonnees; dans cette demonstration on utilise une generalisation du th6orhe de KRUSKAL sur la structure d'arbre, que l'on prouve pour des structures d'arbre de certains hypergraphes structures (familles d'ensembles) de dimension bornee. Les consequences de ce travail sont:

(1) que toutes les anti-chaines (ensembles d'elements non relies deux B deux) dans ( 9 , C_ ) qui contiennent au moins un graphe planaire H sont finies, et ( 2 ) qu'il existe un algorithme pour determiner la relation

H

5G

(H est isomorphe B un mineur de GI lorsque H est un graphe planaire fixe et G E 9 , avec temps de calcul born6 par des polyndmes de la dimension de G . Nous pensons que nos methodes nous permettront de demontrer la conjecture generale structurelle ainsi qu'un theoreme de be1 ordre, avec les deux memes consequences qu'auparavant sans la condition que H soit planaire, c'est-5dire de demontrer la conjecture de WAGNER et d'obtenir un bon algorithm pour determiner la relation H 5 G pour tout G E 9 . ABSTRACT.It has been conjectured by K. WAGNER that finite graphs are wellquasi-ordered by minor inclusion, i.e. being isomorphic to a contraction of a subgraph. A method is reported on here that shows promise of settling this conjecture. We have proved (1) that all graphs G not including a fixed planar graph H as a minor can be constructed by piecing together graphs on a bounded number of vertices in a tree-structure, and (2) by elaborating the KRUSKAL tree theorem that the class of graphs formed by piecing together graphs of bounded size in treestructures is well-quasi-ordered. It follows from this that no infinite antichain of finite graphs can include even one planar graph and that there is a "good" algorithm for testing the presence of a fixed planar graph as a minor.

Background. Let Q be a quasi-ordered set, i.e. a set on which a reflexive and transitive relation < is defined. If x , y E Q and x < y , we say that y includes x . We write x < y if x < y % x . A subset S of Q is (i) an antichain if there do not exist distinct elements, x, y E S such that x < y , (ii) an upper order ideal if y E S whenever x < y for some x E S , (iii) a lower order ideal or lower ideal if y E S whenever y < x f o r some x E S . An upper order ideal S is finitely generated if it has a finite subset F such that S = {x E Q : f < x for some f E F} . It is an easy exercise to prove that the following conditions on a quasi-ordered set Q are equivalent: (a) all upper order ideals of I) are finitely generated, (b) in Q there is neither an infinite antichain nor an infinite descending chain (i.e. an infinite ), sequence x2, such that x > x2 > x3 > (c) for every infinite 1.

'

sequence and

XI'

xi < x.

. ..

x*,

.

...

...

1

of elements of

Q

there exist

i, j

such that

i< j

A quasi-ordered set satisfying any of the equivalent conditions

1 (a), (b), (c) is said to be well-quasi-ordered.

Well-quasi ordering of graphs

345

2.

Graph I n c l u s i o n R e l a t i o n s . Examples o f w e l l - q u a s i - o r d e r e d s e t s a r i s e n a t r i r a l l y i n a l g e b r a , g r a p h t h e o r y and l o g i c . S i n c e t h e 1940's, a b r a n c h of o r d e r e d s e t t h e o r y h a s developed from t h e s t u d y of w e l l - q u a s i - o r d e r i n t h e s e f i e l d s . The a u t h o r s of t h i s p a p e r have r e c e n t l y made p r o g r e s s on t h e open g r a p h t h e o r e t i c a l problems and r e p o r t h e r e on t h e r e s u l t s t h e y have o b t a i n e d . Denote by 9 t h e c l a s s o f f i n i t e g r a p h s . Under subgraph c o n t a i n m e n t 9 i s n o t w e l l - q u a s i - o r d e r e d , a s i s shown by t h e i n f i n i t e a n t i c h a i n o f f i n i t e c i r c u i t s . Most o f t h e t h e o r y o f g r a p h w e l l - q u a s i - o r d e r i n g c o n c e r n s t h r e e c l o s e l y r e l a t e d q u a s i - o r d e r s on 9 o f a t o p o l o g i c a l n a t u r e . To d e f i n e t h e s e r e l a t i o n s w e i n t r o d u c e some n o t a t i o n . Suppose G E 3 and X 5 V ( G ) A simple path i n G i s a p a t h w i t h a t l e a s t one edge and no r e p e a t e d v e r t i c e s , e x c e p t p o s s i b l y i t s e n d v e r t i c e s . The g r a p h of a s i m p l e p a t h ( i . e . t h e subgraph o f G d e t e r m i n e d by i t s v e r t i c e s and e d g e s ) w i t h i t s e n d v e r t i c e s i n X and no o t h e r v e r t i c e s i n X i s c a l l e d a n X-join, and i s a n x - l i n k o r an X-loop i f t i s endvertices a r e , r e s p e c t i v e l y , d i s t i n c t or e q u a l . Then P(G, X ) d e n o t e s t h e g r a p h w i t h vertex-set X , e d g e - s e t t h e s e t o f X - j o i n s o f G , and i n which e a c h X-join i s i n c i d e n t with i t s e n d v e r t i c e s i n X

.

.

X-joins i n G t h e n a c h o i c e f u n c t i o n f o r P i s a mapping s u c h t h a t f ( P ) E E ( P ) f o r a l l P E !J' Given such a c h o i c e f : f' + E ( G ) w i t h e n d v e r t e x x E X , d e n o t e by P f ( x ) t h e segment o f f u n c t i o n and P E 9) If

P

from

9' i s a s e t o f

x

.

t o t h e f i r s t vertex of

may g e n e r a t e t o t h e s i n g l e v e r t e x

i n c i d e n t with

P

f(P)

.

Note t h a t

Pf(x)

.

x

The g r a p h P ( G , X) o f X-joins can be used t o def i ne t h e i ncl usi on r e l a t i o n s f o r w e l l - q u a s i - o r d e r i n g i n a u n i f i e d manner. H ' 5 P(G, X ) f o r any X 5 V ( G ) i s embedded i n G when t h e X-joins o f E ( H ' ) meet p a i r w i s e o n l y a t common e n d v e r t i c e s . Then a g r a p h H i s c o n t a i n e d i n G by embedding i n c l u s i o n , w r i t t e n H <e G

A graph

,

when H i s i s o m o r p h i c t o a g r a p h H' embedded i n G. A graph

H'

5 P(G,

X)

f o r any

X

5 V(G)

i s immersed i n G when t h e

X-joins of E ( H ' ) have p a i r w i s e no common e d g e s . Then a g r a p h H i s c o n t a i n e d i n G by immersion i n c l u s i o n , w r i t t e n H a 3 , a 3 < a4,. 1 i s s a i d t o s a t i s f y

..

i=O and from t h i s , t h e 2n-element c?om C2n = { a l < b,, b , > a 2 , a , < b ,

pi:]

,...,

L ( F * i ) L(Fn'n-1-2i )

an < bn, bn

>

a 11 s a t i s f i e s

There i s l i t t l e hope f o r a u n i f y i n g formula e x p r e s s i n g L ( P ) i n terms of some s t r u c t u r a l i n v a r i a n t s o f an o r d e r e d s e t P. N e v e r t h e l e s s , in r e c e n t y e a r s , t h e r e have appeared s e v e r a l a r t i c l e s which d a r e t o c o n f r o n t t h i s c o u n t i n g problem fairly directly.

359

Linear extensions o f finite ordered sets

Let P he an ordered set. For elements 2 , y of P such that y $ z let P ( z < y ) stand for the transitive closure of P with the additional comparability 3: < y , that is, u < u in P ( z < 9) if u < u in P or if u < z in P and y < v in P. How likely is it that a "typical" extension L of P contains z < y? One way to find out is by a "simple majority" criterion. Write zBy if

L(P(z < Y)) > L(P(y < z)). Observe that L(P(x < y ) ) = L ( P ) if 3: < y in P and in this case just put L ( P ( y < 2 ) ) = 0. For instance, suppose the elements of P represent candidates for some elective office and the order u < U in P stands for v is unanimously preferred to u. Now suppose each elector ranks all of the candidates in a total order. If there are L ( P ) electors each selecting a different linear extension then the binary relation ii summarizes the outcome according to simple majority rule. Notice that R need not impose a total ordering on P: if P is an antichain then L ( P ( z < y ) ) = L ( P ( y < 2)) for each z # y s o D leaves P totally unordered. What is surprising though is that R need not even he a transitive order on P. A n example to illustrate this (see Figure 3) was first') put forward by FISHBURN [ 1 9 7 4 ] . This 31-element ordered set P has a three-element "6-cycle": L ( P ) = 6270; L ( P ( z iy ) ) = 3218 so zBy; L ( P ( y < z ) ) = 3192 so yBz: L ( P ( z < x)) = 3150 so Z E X . (Recently both M. AIGNER [1982] and P.M. WINKLER [1982] have remarked that 6 is a transitive ordering of P if and only if li decomposes P into a linear sum of antichains.) P.C.

A more encouraging direction is proposed in the series of articles R.L. GRAHAM, A . C . YAO and F.F. YAO [1980], L.A. SHEPP [1980], [1982], P.M. WINKLER [1982a], [1982b]. At this writing the most important result is this so-called "transitivity inequality": f o r elements a, b, c of G f i n i t e ordered set P

X

Figure

L P ( G < b ) ) . L ( P ( a < c)) 5 L ( P ( a < b , a < c)).

L(P)

(L.A. SHEPP [1982]). Why is this "transitivity inequality" interesting? One motivation is an algorithmic one. It is concerned with the problem of sorting numbers a , b , c,... by binary comparisons " a : b" to construct successively "finer" orders P on { a , b , c, ) until a total order is finally established. It is said that a quantity fundamental in deciding the expected efficiency of such algorithms is

....

the "probability" that the result of a : b is a c b , assuming that all linear extensions of P are equally likely. (In this model P is thought to be some intermediate state of order (disorder) of { a , b , c, . . . I and a , b are not yet + as in Figure 4 , then sorted.) For instance, if P

22 2

1) The well known "voting paradox" example of CONDORCET [1785] will not do. Suppose there are three candidates x, y, z and three voters ranking the z < x. Evidently y is "preferred" candidates by z < y < z, z < z < y and y to x, z is "preferred" to y , but x is "preferred" to z . Nevertheless, if a l l of the linear extensions of a three-element antichain are considered then, by symmetry, B still leaves all pairs noncomparable.

360

I. Rival

'1 1:

Q

P

while

Pr(d < b(P)

=

0.

Moreover, "conditional probabilities" can also be considered. For instance

C

Pr(a c blP(a < d ) )

9(P) Figure 4

=

L(P(a < b , a < d)) L(P(a < d ) )

=

2 5

and

Pr(a

4 implies IPI = 1 this decomposition must be a trivial one. But G (K) And thus we have: G = H

1 ) When K is irreducible and

.

2) When K is a stable, therefore G(K) is also a stable. Then using the transitivity we have:

*

*

If \ X 3 \ # 0 then 1X 1 = 1 and IX I = IX 1 = 0 and so 3 1 2 reduced to a single vertex and the result is trivial. If

IX

have:

3

I

=

0

G = H

,

then whether

G (K)

IX

2

I

=

0 or

IX

1

I=

3 ) When K is a complete graph. If I P / = 1 the result is obtained, let us now suppose:

0

IPI

G(K)

and thus we

>

2

.

is

M.Habib

316

As K is a prime component of Comp(G), Ktx is not a prime component of Comp(G). Thus there exists y E H - a - x such that [x, 21 is an edge of Comp(H), and [y, 21 is not an edge of Comp(H) for some z E K . /PI = 2 and using transitivity there exists necessarily some vertex E G(K) connected in G with y , and thus using the substitution decomposition of Comp(G), we can see that K+x is a prime component of Comp(G) , which is excluded. and G = H G (K) And thus ( P I = 1

As z'

Since (KI = 2 and K irreducible implies that K is a stable or a complete graph, we have considered all the cases. From this result we can easily deduce the very useful following result. COROLLARY

-

Let G, H, K E ^ e

and

G = H

K

with

a E H

.

If Comp(K)

is a prime component of Comp(G) , then for any G' E % G'

H'

-

-

, , there exist H' and K' E % , a' E H' such that: K' (where a' corresponds H , K' K , and G' = H' a ,

G

-

to a in the isomorphism from Comp(H') to Comp(H)

)

.

In other words if K is a prime component of an undirected comparability graph G, then every transitive orientation $(GI admit a substitution decomposition "around" $(K) Of course this result is still valid for any generalized prime component of Comp(G) .

.

Now we can express the main result of this paper. THEOREM 3 -

Let f : % + R (resp. ? + R ) a graph-invariant then f is a comparability Z-invariant (resp. ? -invariant) if and' only if f satisfies the following conditions (O), (i) and (ii):

*

.

t ,

(Where T and K2 denote respectively 2 the total order and the symmetric complete graph on two vertices).

( 0 ) f (T2 = f(K2)

(i) For every G E Z (ii) For every H, H' E^G For every K, K' E t

(resp. 3'

)

(resp. 3 (resp. 9

) )

f(G) such that such that

=

H K

.

f(G-)

--

H' K'

, ,

For a E H and a ' its image . by -the isomorphism from Comp(H) to Comp(H') , the condition f(H) = f(H') and f(K) = f(K') implies K K' f(H a) = f ( H ' a,)

.

pof:

These conditions are obviously necessary. Let us consider their sufficiency, first in the case of comparability %-invariants.

The proof goes by induction on the number of vertices, and since there exists only one simple directed graph having one vertex, the result is obvious in this case. Let u s suppose for every have: f ( G ' ) = f (G")

.

W e consider now

G',

GI, G2 E ?f,

G" E %

such that

GI

-

G"

and

(GI(

< n , we

two transitive orientations of the same comparabi-

Comparability invariants

lity graph having n vertices

(i.e. Comp(G

)

1

Comp(G2)

=

Comp(G1) ,

Let us take K a prime component of

377

)

.

IKl > 1

with

,

(there exists

at least one such prime component, Comp(G1) itself). Then using theorem 2, we have: Gl(K) and

rily we have: 1") If

GI a(K)

Gl(K)

K # Comp(G

-

G2(K)

,

)

1

and

l

H1

H

N

IG1(K)I < n

then

2") If K

H2 G2 (K) where a2 associated with K. Necessa-

,

=

f(H2)

=

Comp(G)

2 '

.

By induction hypothesis we know that

f (H 1

G2 =

and 1 G2(K) are the subgraphs of resp. G1, G2 G1 = H

f (GI(K)) = f (G2(K)) and

hence we conclude with the condition

,

H1 = ({a,}, p )

then

H2

fi:

,

(ii).

I

and

/G1(K)

=

n

.

a) If K is irreducible and IKI > 3 using theorem 1 and condition (:)

,

b) If K is a stable, then G1 and

are also isomorphic stables.

Therefore G

1

G

M

and

2

G2

.

then the result is obtained by

f(G1) = f(G2)

since f is a graph-

invariant. c) If K is a complete graph. We notice that any transitive orientation of Comp(G1) can be obtained by substituting complete graphs in a total order.

cl, Thus

G1

- O1

xl,

..., c ..., xP

D1, and

is a total order on p vertices

...,

D

. . . I

Yq

G2 = 0

2 Y1'

(resp. q ) , and

,

C . D. I' I'

where

1 < i

O1 (resp. 0 2 )

p(G2 )

Let us suppose these paths are

If

h

p(G2)

If

h > p(G2)

=

, ,

P(G2) is obvious as we can

)

a

Let us consider now the other inequality. Let

G2

.

by a path of a path-partition of G

S P(G2)

partition of G. There exist

1

a

we have finished we contract in

P

{Cl,

=

...,

2'

Ckj

a path-

paths of this partition P that cross cl,

. . . I

.

'1'

' * . I

.

h'

the vertices of

'p(G2)

G

2

to o n e

vertex (for example the first one of the path in G2). Thus we obtain CI1

, ...,

Sp (G2) C'

PtG2)

paths

Of

Furthermore, we delete from possible as C'

p(G2)+1 ,

Thus Since

...,

Cp(G2)+1

. , ...,

Ck

the vertices of

G

2

(thus is

and we obtain

new paths.

Cth

S

...,

IPI = IP' I

,

=

a

GI was supposed to be transitive),

iC'l,

P'

G1

Ctk}

is a path-partition of

G1

P(G-) 2 a

we have the desired result. m

Thus by proposition 1, it is not very hard to notice that we can apply the corollary 2 of theorem 3, with mapping @ equal to the second projection mapping. So the path-partition number is a comparability 3 -invariant. An

exactly argument shows that the path-covering number is a comparability

6 -invariant.

M . Habib

380

Note -

I n t h e f o l l o w i n g examples, we s h a l l now o n l y c o n s i d e r p a r t i a l o r d e r s and hence c o m p a r a b i l i t y 9 - i n v a r i a n t s .

DEFINITION:

L e t P be a p a r t i a l o r d e r , we d e n o t e by C ( P ) t h e s e t of a l l t o t a l ord ers g reat er than P ( a l s o c a l l e d t h e s e t of a l l l i n e a r extensions of P )

.

P a r t i a l o r d e r dimension

W e r e c a l l t h a t t h e dimension of a p a r t i a l o r d e r P , d e n o t e d by d e f i n e d by DUSHNIK, MILLER i n [ 7 ] , a s t h e minimum i n t e g e r h ,

n

P =

with

T~

Ti

d i m ( P ) , was such t h a t :

c(P)

E

1 6 i c h For a r e c e n t s u r v e y on p a r t i a l o r d e r dimension t h e o r y , s e e KELLY, TROTTER [141 W e j u s t r e c a l l t h e p r o o f s made by A R D I T T I [ l ] and TROTTER, MOORE, SUMMER 1191 a s t h e y i n s e r t t h e m s e l v e s v e r y w e l l i n o u r frame.

.

HIRAGUCHI f i r s t noticed i n t 1 3 1 :

,

V P1, P2 E .’P

V a

E P1

dim(P p 2 ) l a

:

.

= max { d i m ( P 1 ) , d i m ( P 2 ) )

With t h i s formula and t h e p r e v i o u s c o r o l l a r y 1 o f t h e theorem 3 , w e can t r i v i a l l y show t h a t t h e dimension i s a c o m p a r a b i l i t y 9 - i n v a r i a n t , by i d e n t i f y i n g $ w i t h t h e a p p l i c a t i o n max. Number of l i n e a r e x t e n s i o n s of a p a r t i a l o r d e r

I

1

.

L e t u s adopt t h e following n o t a t i o n L(P) = f (P) From GOLUMBIC [ 101, ?-invariant. w e know t h a t STANLEY h a s proved t h a t L ( P ) i s a c o m p a r a b i l i t y S i n c e it i s very e a s y t o be c o n v i n c e d of t h e f o l l o w i n g formula P V P1, P2 E 9 and V a E P1 , L ( P 1 a 2 ) = L ( P 2 ) . L(P1

”,2

f o r any

o2 E

c

(P2)

,.

t h e n w e c a n p r o v e , a s i n c o r o l l a r y 2 of theorem 3 , t h a t t h i s formula i m p l i e s t h e c o n d i t i o n ( i i ) o f theorem 3 . Here t h e argument g o e s by i n d u c t i o n on t h e number of uncomparable p a i r s of v e r t i c e s of t h e p a r t i a l o r d e r . Hence w e have a n o t h e r p r o o f of STANLEY’S r e s u l t .

E ~ t g-

T h i s r e s u l t c a n n o t b e g e n e r a l i z e d t o t h e number of t o t a l p r e o r d e r s g r e a t e r than a given p r e o r d e r .

D) J u m p number

For

P E9

(x., x .

and

1

1

, u

and o n l y i f t h e r e e x i s t s

u(.rl, P

that

)

= a(P1)

(P) =

T~

,

=

1

a x a a

1 2 3 sup(a ) < x

1

and f u r t h e r -

0(P2)

u(P ) + o(P2)

-

1

f (P1)

E

,

x'

.

(x.y)

and

(x', y')

iff

x = x'

5

). We put a n

and

y > y'

or

T h e n u m b e r of v e r t i c e s of t h i s g r a p h i s t h e n u m b e r of

such that

i

<j

n u m b e r of t h e p e r m u t a t i o n

and 0

.

o ( i ) 3 O(j) , i . e . t h e s o - c a l l e d i n v e r s i o n s

Such d i a g r a m s have been i n t r o d u c e d by Rothe

[8l] i n o r d e r t o give a n i c e ( g e o m e t r i c ) proof of t h e f a c t t h a t t h e i n v e r s i o n

435

Chain and antichain families

n u m b e r i s invariant under the t r a n s f o r m

0

=;b

u

-1

.

In t h e i r r e c e n t w o r k r e l a t i n g c o m b i n a t o r i c s (Young t a b l e a u x , p l a c t i c m o noid, S c h u r f u n c t i o n s , . . . ) a n d a l g e b r a i c g e o m e t r y (flag m a n i f o l d s ,

. . .)

[611,

1651, r66], L a s c o u x and S c h u t z e n b e r g e r showed that t h e s e " i n v e r s i o n p a i r s " g r a p h s play t h e s a m e r o l e f o r S c h u b e r t f u n c t i o n s a s F e r r e r s d i a g r a m s f o r S c h u r f u n c t i o n s . T h e y c a l l e d t h e s e g r a p h s Riguet d i a g r a m s .

F i g u r e 12

-

T h e i n v e r s i o n p a i r s g r a p h (on Riguet d i a g r a m ) of a p e r m u t a t i o n .

A c l a s s i c a l notion i n t h e t h e o r y of d i r e c t e d g r a p h s i s t h e k e r n e l of a g r a p h , that i s a s e t of points s u c h t h a t e v e r y v e r t e x of the g r a p h i s t h e s o u r c e of a n edge ending i n the k e r n e l , a n d e v e r y edge having i t s s o u r c e i n t h e k e r n e l h a s i t s end not i n t h e k e r n e l ( s e e B e r g e [5]). Such a s e t i s unique if i t e x i s t s . It i s a l s o t h e s e t of "winning p o s i t i o n s " of a " N i m g a m e " played on t h e g r a p h . F r o m [lo91 t h e s k e l e t o n

S(0)

of t h e p e r m u t a t i o n

CJ

is e x a c t l y the

G. Viennot

436

k e r n e l of i t s i n v e r s i o n p a i r s g r a p h . Writing t h i s p r o p e r t y f o r all s k e l e t o n s S.(U) leads to Foata's characterization. F r o m F r e d m a n [24] a n d Viennot [lo91 t h e c o n s t r u c t i o n of the l o n g e s t s u b -

s e q u e n c e of a p e r m u t a t i o n

u

n e e d s the c o n s t r u c t i o n of the s k e l e t o n . T h u s ,

t h e c o n s t r u c t i o n of the s k e l e t o n c a n be done by a n o p t i m a l a l g o r i t h m ( f o r t h e w o r s t c a s e i n "on-line" r e a d i n g ) i n

nlogn-nloglogn

comparisons, For more

u n d e r s t a n d i n g about t h e s e t h e o r e t i c a l c o m p u t e r s c i e n c e c o n c e p t s , s e e f o r e x a m p l e Knuth [52] o r Aho, H o p c r o f t , Ullman [2]

6

6

-

.

Plactic monoid. We have s e e n t h a t t h e Robinson-Schensted c o r r e s p o n d e n c e i s r e l a t e d t o

t h e r e p r e s e n t a t i o n t h e o r y of t h e s y m m e t r i c g r o u p . T h e c o m b i n a t o r i c s of Young t a b l e a u x i s a l s o i n t i m a t e l y c o n n e c t e d with t h e t h e o r y of s y m m e t r i c f u n c tions. A b a s i s of s y m m e t r i c f u n c t i o n s h a s b e e n i n t r o d u c e d by J a c o b i [48] a s a quotient of a n t i s y m m e t r i c functions ( e x p r e s s e d a s d e t e r m i n a n t s ) . Following F r o b e n i u s [25), S c h u r [9O] d i s c o v e r e d t h e i r r e l e v a n c e t o t h e r e p r e s e n t a t i o n t h e o r y of t h e s y m m e t r i c g r o u p s a n d t h e g e n e r a l l i n e a r g r o u p s . T h e s e f u n c t i o n s h a v e b e e n c a l l e d S c h u r f u n c t i o n s ( o r S - f u n c t i o n s ) by Littlewood and R i c h a r d s o n [TO),

A l s o they a r e r e l a t e d t o s o m e w o r k i n i t i a t e d by P i e r i i n 1873, f o l l o w e d by

S c h u b e r t , G i a m b e l l i a n d o t h e r s , a n d which i s nowadays n a m e d a "cohomology r i n g of G r a s s m a n n v a r i e t i e s " . We s h a l l not t o u c h t h i s d e e p a n d huge s u b j e c t . We s h a l l only give t h e w e l l known c o m b i n a t o r i a l definition of t h e S c h u r f u n c t i o n s ( s e e f o r e x a m p l e Littlewood [ 6 9 ] ).

X

Let

be a p a r t i t i o n a n d

be a Young t a b l e a u of s h a p e

Y

X

(with

e n t r i e s s t r i c t l y i n c r e a s i n g i n c o l u m n s a n d w e a k l y i n c r e a s i n g i n r o w s ) . We d e i xp" , w h e r e i.!, i s t h e n u m b e r of e n note by m(Y) t h e m o n o m i a l

4'. . .

' trie s equal to

.!,

in the tableau

s u m of all m o n o m i a l s

m(Y)

and e n t r i e s taken among Example

-

s ( 2 , 1) (xl,

X2'

1,2,

Y

. T h e S c h u r function

Sx (xl,

. . . ,xn)

e x t e n d e d o v e r all Young t a b l e a u x with s h a p e

is t h e

X

. . ., n .

x 3 ) = x12 x 2 + x 2 x3+ x3 2 xl+ x12 x3+ x 2 XI+ x 2 3 x2 + 2x1 x2 x 3 *

Chain and ontichain families

437

c o r r e s p o n d i n g t o the e i g h t Young t a b l e a u x d i s p l a y e d i n t h e following f i g u r e .

F i g u r e 13 -

A Schur function.

L a s c o u x a n d S c h u t z e n b e r g e r i n t r o d u c e d the p l a c t i c monoid a s a n o n - c o m m u t a t i v e c a l c u l u s , unifying a n d extending p r e v i o u s p r o p e r t i e s of the R o b i n s o n S c h e n s t e d c o r r e s p o n d e n c e a n d of the t h e o r y of s y m m e t r i c f u n c t i o n s [62]. F o r example, the r a t h e r m y s t e r i o u s , s o - c a l l e d "Littlewood-Richardson rule" (giving t h e p r o d u c t of two S c h u r f u n c t i o n s a s s u m of S c h u r f u n c t i o n s ) i s m a d e clear. T h e c o m b i n a t o r i a l p a r t of t h i s t h e o r y is b a s e d on t h e "jeu d e taquin", [62] ,[97].

Y

+

d(Y)

T h i s r u l e f o r moving l a b e l s i n d i a g r a m s g e n e r a l i z e s t h e o p e r a t i o n m a d e i n t h e c o n s t r u c t i o n of t h e d u a l

YJ

of a t a b l e a u

Y

.

We s h a l l only g i v e h e r e a f e w h i n t s of t h i s beautiful t h e o r y . T h e i n t e r e s t e d r e a d e r w i l l s e e t h e p a p e r s of L a s c o u x a n d S c h t i t z e n b e r g e r [58] ,[59], [60], [62], [63], [95], [97]

a n d of T h o m a s [103], [104], (105], [lob], [107], t h e e x t e n s i o n to

t h e n i l p l a c t i c monoid [66] a n d t h e s u r v e y p a p e r of C a r t i e r a t t h e " S k m i n a i r e B o u r b a k i [7]. F o r t h e c l a s s i c a l t h e o r y of s y m m e t r i c f u n c t i o n s , s e e t h e book of Macdonald [72]

o r S t a n l e y 6983, F o u l k e s [l8]

,

J e u de t a q u i n . Let

X

contained i n

and

Fx

.

U

be two p a r t i t i o n s s u c h t h a t t h e F e r r e r s d i a g r a m

The diagram

g r a m s , is a n i n t e r v a l of

TI

F

cr

is

Fx\FcI , t h e d i f f e r e n c e of t h e t w o F e r r e r s d i a -

= Z x Z

A s k e w Young t a b l e a u of s h a p e

( o r d e r e d by t h e p r o d u c t o r d e r ( 2 ) ) . X\P

is a l a b e l i n g of

FA\Fcr with i n t e -

g e r s s u c h t h a t they a r e weakly i n c r e a s i n g i n t h e r o w s ( f r o m l e f t t o r i g h t ) a n d s t r i c t l y i n c r e a s i n g i n t h e c o l u m n s (down-up, i n t h e " F r e n c h notation").

G. Viennot

438

p = (5,5,2) = (8,7,5,5,2) 0

comer cell of F

r

F i g u r e 14 - A s k e w Young t a b l e a u .

T h e lljeu de taquin" i s d e s c r i b e d as f o l l o w s . F i r s t we c h o o s e one of t h e c o r n e r c e l l s (defined i n

9 4) of the F e r r e r s d i a g r a m F . We t a k e the m i n i m u m CI

of the t w o v a l u e s that a r e l o c a t e d at t h e North a n d E a s t , a n d move t h i s v a l u e into the c o r n e r c e l l . T h i s c r e a t e s a "hole" i n t h e s k e w Young t a b l e a u . In a sim i l a r way t o t h e c o n s t r u c t i o n of the d u a l i n

5 4, we

hole h a s moved t o the N o r t h - E a s t o u t s i d e of

Fx

r e p e a t t h e p r o c e s s until t h e

. In c a s e the

two v a l u e s a t t h e

N o r t h o r the E a s t of a hole a r e e q u a l , we move i n the hole the value l o c a t e d a t t h e N o r t h (thus p r e s e r v i n g the condition t o have a s k e w Young t a b l e a u ) . A l s o note t h a t if the hole i s a t t h e " b o r d e r " of

Fx , at m o s t one of the N o r t h a n d E a s t

c e l l s i s n o n - e m p t y . We move a l s o t h i s s i n g l e v a l u e i n t h e h o l e . T h e p r o c e s s t e r m i n a t e s when both t h e N o r t h a n d E a s t c e l l of t h e hole a r e e m p t y (that i s i n f a c t when the hole i s a c o r n e r c e l l of

Fx

1.

T h e t e r m i n o l o g y b n d t h e t h e o r y !) lljeu de taquin" is due t o S c h u t z e n b e r g e r . T h e lljeu de taquin" is a g a m e played with a n x n b o a r d . I n s i d e the b o a r d , 2 t h e r e a r e n -1 e l e m e n t a r y c e l l s . On e a c h c e l l i s p r i n t e d a l e t t e r . At e a c h s t e p of the g a m e , one c a n move one of t h e n e a r e s t c e l l s to t h e unique e m p t y p o s i t i o n . At e a c h s t e p t h i s e m p t y position t a k e s the p l a c e of the c e l l which h a s m o v e d . T h e r u l e i s t o obtain some w o r d s ( o r l e t t e r s i n a l p h a b e t i c o r d e r ) .

439


F i g u r e 15

-

T h e lljeu de taquin"

We o b t a i n a n o t h e r s k e w Young t a b l e a u with s h a p e ( r e s p . FuI ) i s obtained f r o m

X'\bi',

Fx

1

( r e s p . F@) by r e m o v i n g a c o r n e r c e l l .

Fx , Now, if we r e p e a t t h e p r o c e s s going f r o m a t a b l e a u

t o a t a b l e a u with s h a p e

where

Y

with s h a p e

I\@

F is e m p t y , we o b t a i n a U Young t a b l e a u , c a l l e d t h e " r e d r e s s d " of t h e s k e w Young t a b l e a u Y u n d e r t h e A'\P'

until t h e d i a g r a m

s u c c e s s i v e c h o i c e s of t h e "jeu de taquin". A r e m a r k a b l e f a c t i s t h e i n v a r i a n c e of the " r e d r e s s C " u n d e r t h e "jeu de taquin",

i . e . the " r e d r e s s d " i s independant

of t h e s u c c e s s i v e c h o i c e s of t h e c o r n e r c e l l s of e a c h d i a g r a m

F

u

. This inva-

r i a n c e h a s b e e n p r o v e d by S c h a t z e n b e r g e r [ 9 7 ] a n d T h o m a s [lo51 ( s e e a n i d e a of t h e proof b e l o w ) . T h e " r e d r e s s C " of the s k e w Young t a b l e a u

Y

is d e n o t e d by

R(Y). We follow a g a i n S c h u t z e n b e r g e r ' s t e r m i n o l o g y ; " r e d r e s s 6 " i s b a d l y t r a n s l a t e d by s a y i n g t h a t t h e South-West b o r d e r of s k e w Young t a b l e a u

Y

is n o

m o r e a zig-zag line. F o r example, the tableau where

a

11

d(Y)

defined i n

8

4 is nothing but

R ( Y \ all)

d e n o t e s t h e ( s k e w ) Young t a b l e a u o b t a i n e d by d e l e t i n g t h e v a l u e

Y \ all (1,l) c e l l f r o m the Young t a b l e a u

i n the

Y

.

In the e x a m p l e d i s p l a y e d i n f i g u r e 16, we h a v e s h o r t e n e d t h e c o n s t r u c t i o n by giving only the t a b l e a u x o b t a i n e d o n c e t h e "hole" i s o u t s i d e of the d i a g r a m

Fx

. We h a v e

c i r c l e d t h e v a l u e s which m o v e f o r e a c h c h o i c e of t h e c o r n e r c e l l s

of t h e s u c c e s s i v e d i a g r a m s

F

U

.

,w

G. Viennot

440

&+

Y =

3 8 9

choice of the

comer cell

0

F i g u r e 16

-

The "redressk"

v a l u e w i n g in the " j e u d e taquin"

R ( Y ) of a s k e w Young t a b l e a u

Y

.

6 . We denote by Y(o) the s k e w Young n t a b l e a u obtained by w r i t t i n g t h e s u c c e s s i v e v a l u e s U(l), ,u(n) f r o m N o r t h Let

u

be a p e r m u t a t i o n of

...

West t o S o u t h - E a s t ( s e e f i g u r e 17). A f u n d a m e n t a l p r o p e r t y of t h e lljeu de taquin", a n d giving a n o t h e r definition of t h e R o b i n s o n - S c h e n s t e d c o r r e s p o n d e n c e is t h e following

PROPOSITION 8 .

(Schiitzenberger)

-

F o r any permutation

" r e d r e s s k " of t h e s k e w Young t a b l e a u P-symbol

P(u)

Y(U)

coding

u

u

of

en ,

the

i s identical to t h e

o b t a i n e d by t h e Robinson-Schensted c o r r e s p o n d e n c e .

Define a s t r i p t o be a d i a g r a m

Fcl a r e disjoint and s u c h t h a t

Fx\Fu

Fx\ Fu

F

s u c h t h a t t h e b o r d e r s of

d o e s not c o n t a i n a n y s q u a r e

x

F

(292)

and of

Chain and antichain families f o u r c e l l s . Such d i a g r a m s with

n

44 1

c e l l s a r e coded by a w o r d of l e n g t h

two l e t t e r s . T h e r e e x i s t s a t r i v i a l b i j e c t i o n between p e r m u t a t i o n s of s k e w Young t a b l e a u x with d i s t i n c t e n t r i e s

1, 2,

. . .,n

6,

n-1

on

and

a n d with a s t r i p s h a p e ,

F r o m s u c h t a b l e a u x , we a s s o c i a t e a p e r m u t a t i o n by r e a d i n g t h e e n t r i e s f r o m N o r t h - W e s t t o S o u t h - E a s t following t h e s t r i p . C o n v e r s e l y , f r o m a p e r m u t a t i o n 0

we define a s k e w Young t a b l e a u

the c e l l containing c e l l containing

iff

having a s t r i p s h a p e a n d s u c h t h a t

i s c o n s e c u t i v e a n d a t t h e E a s t ( r e s p . South) of t h e

a(it1)

O(i)

Str(0)

0(i) < o(it1)

( r e s p . 0 ( i ) 3 o ( i t 1 ) ) . In o t h e r w o r d s ,

t h e s t r i p i s nothing but a coding of t h e up-down s e q u e n c e of t h e p e r m u t a t i o n

8

4

Y h)

-

defined i n

.

k-, 3 9

F i g u r e 17

-

T h e s k e w Young t a b l e a u x Y ( 0 ) a n d Str(ff) coding a p e r m u t a t i o n 0 .

Obviously, the t a b l e a u

Str(0)

c a n be o b t a i n e d f r o m

Y(0)

by applying

t h e "jeu de taquin", a n d t h u s , by S c h f i t z e n b e r g e r ' s t h e o r e m s ( i n v a r i a n c e of t h e " r e d r e s s k " u n d e r t h e "jeu de taquin" a n d p r o p o s i t i o n 8 ) , t h e " r e d r e s s k " R(Str(0))

i s the P-sy mb o l

P(0) of t h e p e r m u t a t i o n

U

.

T h e r e a d e r will

c h e c k t h i s f u n d a m e n t a l fact with f i g u r e 16 f o r t h e g e n e r i c p e r m u t a t i o n a p p e a r i n g i n all the e x a m p l e s of t h i s p a p e r . T h e i n v a r i a n c e of t h e " r e d r e s s k " u n d e r t h e lljeu de taquin" i s p r o v e d by introducing t h e following f u n c t i o n s , b a s e d on G r e e n e ' s i n t e r p r e t a t i o n of the s h a pe d the t a b l e a u x

P(0) a n d

Q(0)

(we s h a l l a s s u m e t h a t t h e e n t r i e s a r e d i s -

t i n c t , t h e g e n e r a l c a s e c a n be d e d u c e d f r o m t h a t ) .

G. Viennot

442 Let

Y

be a s k e w Young t a b l e a u (with p o s s i b l y a "hole" i n s i d e ) . We d e -

f i n e a n i n c r e a s i n g s u b s e q u e n c e of

< xk

such that f o r e v er y

E a s t of t h e c e l l w h e r e we define

i , 1G i

x.

to be a s e q u e n c e of i n t e g e r s k , x.

1t1

. ..

x

1 i s i n a c e l l which i s a t t h e S o u t h -

i s l o c a t e d . T h e n , f o r any i n t e g e r s

k

a2

and

1,

Y

to be the m a x i m u m c a r d i n a l i t y of s u b s e t s of e n t r i e s of

Gk,,(Y)

t h a t a r e unions of

Y

k

.

i n c r e a s i n g s u b s e q u e n c e s with v a l u e s

i s i n v a r i a n t u n d e r the e l e m e n t a r y 'k, sliding of t h e 'ljeu de taquin". T h e f a c t mentioned above i s p r o v e d f r o m the r e It c a n be shown t h a t t h e function

m a r k t h a t a s t a n d a r d Young t a b l e a u is c h a r a c t e r i z e d by the function

G

k,

.

u + (R(0) , R( 0-l))

T h e n one c a n s h o w d i r e c t l y that the c o r r e s p o n d e n c e

i s a b i j e c t i o n . T h e "bumping p r o c e s s " c a n be s e e n as p a r t i c u l a r c a s e of t h e s l i d i n g s of the lljeu de taquin", a n d t h u s we h a v e p r o p o s i t i o n 8 . F r o m t h e r e , G r e e n e ' s i n t e r p r e t a t i o n and p r o p e r t i e s of Knuth's t r a n s p o s i t i o n s c a n be d e d u c e d . Other p r o p e r t i e s mentioned in

5

4 a r e a l s o d e d u c e d . F o r e x a m p l e , the p r o p e r t y

r e l a t i n g t h e up-down s e q u e n c e and t h e "line of r o u t e ' ' i s e a s i l y shown a s a p r o p e r t y i n v a r i a n t u n d e r the "jeu de taquin". A l s o , taking t h e dual of a t a b l e a u c o r r e s p o n d s to r e v e r s i n g the I'jeu de taquin", t h a t is r e v e r s e the o r d e r b e t w e e n i n t e g e r s and move t h e e n t r i e s t o t h e N o r t h - E a s t ( i n s t e a d of t h e S o u t h - W e s t ) .

P r o d u c t s of Young t a b l e a u x . T h e above c o m b i n a t o r i a l c o n s i d e r a t i o n s c a n be e x p r e s s e d in a m o r e a l g e b r a i c way by introducing the plactic m o n o i d . Let

Y

and

Z

be two Y o u n g t a b l e a u x . Denote by

Young t a b l e a u obtained by putting

Z

the skew

Y

a s shown o n

a t t h e S o u t h - E a s t of

f i g u r e 18. T h e p r o d u c t of the two Young t a b l e a u x the " r e d r e s s 6 "

Y. Z

Y

Z

and

i s defined to be

R(Y. Z ) . By t h e i n v a r i a n c e of t h e " r e d r e s s & " u n d e r t h e lljeu de

taquin", t h i s p r o d u c t i s a s s o c i a t i v e . T h e e m p t y t a b l e a u i s t h e identity e l e m e n t . A , and with the

T h e s e t of Young t a b l e a u x with e n t r i e s i n the s e t of i n t e g e r s p r o d u c t j u s t defined, i s t h e p l a c t i c monoid g e n e r a t e d by

A

.

443


=B 1

1

2

Y.2

-@ F i g u r e 18 -

=%

n Young t a b l e a u x r e d u c e d t o one c e l l

insertion process

I(T,x)

with the s i n g l e - c e l l

x

R] 2

2

4

T h e p r o d u c t of two Young t a b l e a u x .

P r o p o s i t i o n 8 s a y s nothing but t h a t t h e P - s y m b o l p r o d u c t of the

m =

R(Y*Z)

5

defined in

P(0)

is the p l a c t i c

O(l), 0 ( 2 ) ,

. . . ,~

( n .) T h e

3 i s in f a c t t h e p l a c t i c p r o d u c t of

T

placed a t the South-East.

A n o t h e r way t o define t h e p l a c t i c monoid c o m e s f r o m K n u t h ' s t r a n s p o s i be a t o t a l l y o r d e r e d a l p h a b e t . L e t

tions . Let

A

n e r a t e d by

A , t h a t i s the s e t of w o r d s

w = wl.. .w

a n d with p r o d u c t the c o n c a t e n a t i o n p r o d u c t : f o r

be the f r e e monoid g e with l e t t e r s

n

of

x,y,z

A

x y z s y x z

(6) f o r any l e t t e r s

x,y

of

A

x y x s y x x

such that and

x c z

in

A

i a t ~ i r ce J n c t m i u n t I e s LThainc.s, sst-ii Si n o i , cmrn-nt peut-on mesui=i;r 1 ' e f - f e c t i o i t P

Un exemple de th6orsme combinatoire vrai effectivement est le r6sultat de Cantor selon lequel toute chaine dsnombrable est isomorphe 2 un sous-ensemble des nombres rationnels. On peut modifier la d6monstration standard de ce r6sultat pour dBcrire un algorithme qui produit le th6orCme suivant:

Touts chainc rdcursivc est d c u r s i v e m e n t isornorplie a' un sousenscmbie rz'zursif des nnmbres rationneis. Remarquez que tout emploi du mot "recursif" doit Ctre prScis6ment dgfini; ainsi par exemple, la phrase "chaine recursive" entraine l'existence d'un algorithme au moyen duquel on peut determiner si a < b ou non, et la phrase "r6cursivement isomorphe" entraine l'existence d'un algorithme au moyen duquel on peut calculer les valeurs de la fonction. Dans l'exemple que j'ai cit6 ci-dessus, la d6monstration traditionnelle peut Ctre effectivisge; il y a des exemples oij cela n'est pas possible. Dansces cas-li on doit ou fabriquer un nouvel algorithme ou demontrer qu'il n'existe pas d'algorithme convenable. Par exemple, la dgmonstration standard quetoutensemble bien ordonn6 a une extension lin6aire bien ordonn6e ne peut pas s'effectiviser. Cependant, un autre algorithme, que j'ai dgcrit avec H. Kierstead, demontre que:

Tout ensemb7,e bier1 o r r h n 8 qui est re'cursif i unp extension line'aire qui est bien ordonne'e ct re'cursive. La question de savoir si tout ensemble ordonne qui est dispers6 et rdcursif a une extension lingaire qui est dispers6e et rgcursive reste ouverte. 11 n'y a pas toujours une manisre unique d'effectiviser une formulation combinatoire. Par exemple, une autre manisre d'approcher l'exemple mentionnd cidessus est de demander si tout ensemble ordonne qui est r6cursif et recursivement bien ordonn6 a une extension lin6aire qui est rgcursive et recursivement bien ordonnke. ("Recursivementbien ordonn6" signifie que l'ensemble ordonn6 n'a aucune suite infinie r6cursive qui soit dgcroissante). Cette approche mPne plutst 1 une solution nggative, due h moi-m&ne et h R. Statman: I 1 existe un ensemble ordonni q u i est re'rursif e t re'cursivement bien ordonne', mais qui n'a pas d'extension 7in6aire qui soit re'cursivc et re'cursivement bien ordonnie. AprSs avoir introduit la higrarchie arithm6tique de Kleene, nous rgpondons B l'autre moitig de la question g6nCrale: Etant donn6 un ensemble ordonn6 qui est recursif et r6cursivement bien ordonn6, quelle complexit6 l'extension lin6aire d6sir6e doit-elle poss6der ?

466

J.G. Rosenstein Tout ensemble ordonnd q u i e s t r e c u r s i f e s t re'cursivement ordonnd a m e extension Lindaire qui e s t rdcursivement bien ordonneeet e s t s i t u d e au niveau h 2 dans La hidrarchie arithmdtique de KZeene.

La dgmonstration de ce thSorPme e s t un exemple de l ' u t i l i s a t i o n des "argumentat i o n s d i a g o n a l e s " dans l a t h g o r i e de l a r g c u r s i v i t g .

Les v e r s i o n s e f f e c t i v e s de quelques a u t r e s r g s u l t a t s combinatoires s o n t d i s c u t g e s dans l ' a r t i c l e y compris l e thLorOme de Dushnik e t M i l l e r s e l o n l e q u e l t o u t e c h a i n e dgnombrable p e u t s ' a b r i t e r dans un sous-ensemble p r o p r e d ' e l l e - m i k e . Ces s u j e t s s o n t d i s c u t g s 3 fond dans mon l i v r e L i n e a r Orderings (Academic P r e s s , 1982) dans l e c h a p i t r e i n t i t u l 6 , comme on p e u t s ' y a t t e n d r e , L i n e a r Orderings and Recursion Theory.

*** The framework f o r t h i s survey a r t i c l e about r e c u r s i v e l i n e a r order i n g s i s provided by t h e f o l l o w i n g Reneral q u e s t i o n s : Given a c o m b i n a t o r i a l theorem about l i n e a r o r d e r i n g s , i s i t t r u e e f f e c t i v e l y ? I f n o t , how can t h e e f f e c t i v e n e s s of t h e r e s u l t be measured ? Among t h e s p e c i f i c c o m b i n a t o r i a l theorems considered i n d e t a i l are ( a ) Every c o u n t a b l e l i n e a r o r d e r i n g i s order-isomorphic t o a s u b s e t of t h e r a t i o n a l s ( C a n t o r ) , ( b ) Every well-founded ( r e s p . , s c a t t e r e d ) p a r t i a l o r d e r i n g h a s a well-founded ( r e s p ; , s c a t t e r e d ) l i n e a r e x t e n s i o n , and ( c ) Every c o u n t a b l e l i n e a r o r d e r i n g can be embedded i n t o a p r o p e r s u b s e t of i t s e l f (Dushnik and M i l l e r ) . I n t h i s b r i e f t a l k about r e c u r s i v e l i n e a r o r d e r i n g s , I would l i k e t o convey t o you some s e n s e of t h e q u e s t i o n s that a r e r a i s e d and some of t h e f l a v o r of t h e I w i l l not methods and t e c h n i q u e s t h a t a r e used i n answering t h e s e q u e s t i o n s . t r y t o be comprehensive. The framework f o r t h i s t a l k i s provided by t h e f o l l o w i n g g e n e r a l q u e s t i o n :

Given a combinatorial theorem about linear ordsrings, i s it true e f f e c t i v e l y ? If not, how can the effectiveness of the result be measured? I u s e t h e phrase " l i n e a r o r d e r i n g s " t h e way o t h e r s u s e t h e p h r a s e s " t o t a l o r d e r i n g s " o r "chains." Information about l i n e a r o r d e r i n g s can be found i n my r e c e n t book [ S ] on t h e s u b j e c t ; t h e c h a p t e r on r e c u r s i v e l i n e a r o r d e r i n g s also c o n t a i n s t h e a p p r o p r i a t e material on r e c u r s i o n t h e o r y . A l l l i n e a r o r d e r i n g s t h a t I d i s c u s s w i l l be c o u n t a b l e , and t h e r e f o r e w e need o n l y look at s u b s e t s of Q ( t h e r a t i o n a l numbers.) This i s c o r r e c t by t h e f o l l o w i n g theorem, due t o Cantor:

E v e v countable linear ordering i s (order) isomorphic t o a subset of Q . Is t h i s c o m b i n a t o r i a l theorem t r u e e f f e c t i v e l y ? Before answering t h i s q u e s t i o n . we must f i r s t f o r m u l a t e and e x p l a i n t h e e f f e c t i v e v e r s i o n of C a n t o r ' s theorem. Loosely speaking, t h e way t o a r r i v e a t a n e f f e c t i v e v e r s i o n of a c o m b i n a t o r i a l r e s u l t is t o add t h e word " r e c u r s i v e " a t a l l a p p r o p r i a t e l o c a t i o n s . Using t h i s h e u r i s t i c , we a r r i v e at t h e f o l l o w i n g statement:

E v e y recursive linear ordering is recureiveZy isomorphic t o a recursive subset of Q.

46 7

Recursive linear orderings

Each of the three usages of the term "recursive" requires an explanation. Let us think of a countable linear ordering as a structure where N is the set of natural numbers and R is a binary relation on N which defines a linear ordering of N. Then we define the linear ordering to be recursive if there is an algorithm which, given a and b, will determine whether aRb or bRa is correct. Similarly, a subset A C- Q is recursive if there is an algorithm which, given a E Q, will determine whether or not a E A. Also, a function f:N + Q is recursive if there is an algorithm which, given a E N, will produce f(a) E Q; thus to say that is recursively isomorphic to a subset A C_ Q means that there is a recursive function f:N + A which defines an order isomorphism between and A c o n t r a d i c t i n g t h e assumption Indeed, s i n c e c j + l Wk9 X

hL, = 1 0 0 ) . E n f i n , p o u r x , y E X ,

p a r t i t i o n g r o s s i s r e d e X.

(Dans l ' e x e m p l e

i l e x i s t e un X minimum t e l q u e x e t y s o i e n t d a n s

u n e c l a s s e d e IT*. P a r e x e m p l e , X = 38 p o u r 1 e t 4 . On e s t a i n s i c o n d u i t h l a not i o n d e R tc h a 9 n e d e p a r t i t i o n s d e X , c ' e s t - 2 - d i r e d ' u n e a p p l i c a t i o n i s o t o n e d e IR" d a n s l e t r e i l l i s TIX d e s p a r t i t i o n s d e X t e l l e que F-'(TI ) = b e t q u e p o u r t o u t x # y i l e x i s t e X minimum a v e c F(X) 3 { x , y } . L ' i mage d ' u n e t e l l e a p p l i c a t i o n F c o n s t i t u e u n e c h a i n e d e I I X , a l l a n t d e l a p a r t i t i o n l a p l u s f i n e n o , $ l a moins f i n e une chaEne Ctendue d e JIX).

(i.e.

T[

Une m a n i s r e d e s e d o n n e r l ' a p p l i c a t i o n F e s t

d ' b c r i r e c e t t e c h a f n e d e p a r t i t i o n s i n d i c e e p a r l e s valeurs F - ' ( n ) . p l e , on o b t i e n t n e u f p a r t i t i o n s d i s t i n c t e s :

TI^

= {{x],x

E

I

x
f ' ( h o ) l A [(V h E L - I h 1) €(A) = f'(h)l. Les t r o i s t r e i l l i s n l ' , I s o ( l , , n ) e t r c s ( l , , n ) o n t l a m e m e g r a d u a t i o n = g ( f ( h ) ) e t s o n t s e m i - m o d u l a i r e s s u p & r i e u r e m e n t (Moniez 1 9 8 2 ) , AEL [,cur lonpueur commune e s t I + ( I L I - 1 ) ( 1 x 1 - 1 ) . Le t r c i l l i s f i n i U Gt:int g(f)

sous-suo-denii-trei

I 1 i s d e L p 2 ( X ) c u n t e n a n t s o n 6lGment n u l . Y i n d u i t

un

L I ~ C ouvc'r-

Ensembles ordonnes et taxonomie mathematique

535

L'application lilt e s t idempotente, isotone, anti-extensive [ult(d) 5 d] image e s t u . L'ultrambtrique ult(d) -0cii.e d a de nombreuses appellations traditionnelles : ultramtitrique inftrieure maximale (ultimax), s o u s dominante, distance du moindrc saut, fernirture transitive floue de d (par dualiti.) ... Elle correspond h I'une des methodes de classification les p l u s connues dont on rappellc quelques propri6tii.sau paragraphe 4 . 3 . ) . e t son

L'ouverture ultram6trique esiste encore lorsque L n'est p a s fini. Pour l a construire, on utilise le fait que L

est un treillis d'applications dbfinies P2 (X) sur un ensemble fini, en posant, pour d t L , L~ = Imd(82(x)) et '2(x)

e, (x)

ult ( d ) , oil u l t d cst I'ouverture ultrambtrique dans I. . De facon d d analogue, on montre que le treillis U vhrifiela condition de semi-modularitb d e Dubreil-Jacotin, Lesieur et Croisot (1953) (Cf. Leclerc 1979). ult(d)

=

Une Ctude plus combinntoire des ultram6triques et de leur treillis a b t b faite par Leclerc (1979, 1981). Reprenant une idbe d e Hubert ( 1 9 7 7 1 , on y gbnbralise la coatomicitb de n par l'hcriture de t o u t e ultrambtrique comme supremum d'au plus n-1 ultrametriquessup-irrbductibles. Si L a un hlbment universel, il y a une ghn6ralisation analogue de 1'atomicitP de n . Finalement, les proprietbs ordinales des treillis d'ultrambtriques apparaissent trks proches de celles d e s treillis des partitions. Pour L fini, le dbnombrement d e s L- ultram6triques de msme graduation serait intbressant pour la statistique des classifications. Cn donne plus loin une facon de dhnombrer toutes les L-ultrambtriques. 3.3.ULTRAPREORDONNANCES ET HIERARCHIES : DEMI-TREILLIS A MEDIANES. 3 . 3 . 1 . Demi-treillis b mhdianes.

Un demi-treillis 5 mhdianes est un inf-demi-treillis vhrifiant les deux conditions suivantes (Sholander 1954 ; cf. Bandelt et Hedlikova 1983) : (Ml) Tout idbal principal de M est un treillis distributif. (M2) Pour tout triplet x,y,z de M tel q u e x v y , y v z , z v x existent, x v y v z existe. On dbfinit dualement un sup-demi-treillis 5 mbdianes. Sholander montre notamment qu'un demi-treillis B medianes M admet un plongement avec de bonnes propribths dans un treillis distributif M'. I1 suffit en fait de prendre pour M' le treillis d e s parties commencantes des irrbductibles de M : M est une partie commencante de M' et tout element de M' est supremum d'une partie finie d e M. I1 en resulte que le plongement est isom6trique. Le fait pour un ensemble d'avoir cette structure peut donc a priori faciliter l'obtention d e solutions aux problkmes d'ajustement et d'agrhgation. Des exemples en seront donnbs aux paragraphes 4 . 3 . et 4.6. 11s utilisent notamment le fait que tout triplet x,y,z E M posshde une mediane unique (x A y) v (y A z ) v ( z A x). Pour M fini et k impair, tout k-uple a egalement une mbdiane. De plus, celle-ci conserve les propribtbs mbtriques &tablies par Barbut (1961) pour l a mbdiane dans les treillis distributifs (Bandelt et Barthblemy 1983).

Les demi-treillis 5 mbdianes qui vont Stre considerhs ici sont des demitreillis de charpentes (Bandelt 1982). Soit E un ensemble ordonnh fini, Min E(Max E ) l'ensemble de s e s Clbments minimaux (maximaux). Une partie C de E une charpente ("frame") de E ssi les deux conditions suivantes sont verifihes

est

J,-P.Barthelemy e t al.

536 par C : (Cl) (Min E) U (Max E ) 5 C ( V x E Min E ) ( V y E Max E ) (C2)

(z

E C/x 5 z 5 y) est totnlement ordonnh.

L'ensemble e ( E ) des charpentes de E, ordonn6 par l'inclusion, est un demitreillis 5 m6dianes (avec pour Cl6ment nu1 Co = (Min E) U (Max E)). On dira de deux charpentes C et C ' qu'elles sont presque disjointes lorsque C n C ' = Co. L'exemple le plus imm6diat de demi-treillis de charpentes est don& par l'ensemble des chafnes Ctendues (c'est-$-dire contenant le maximum et le minimum) d'un treillis fini. Si E est un treillis distributif, et F l'ensemble ordonn6 des bl6ments sup-irrbductibles de E, l'inf-demi-treillis des chaines btendues (charpentes) de E est dualement isomorphe au sup-demi-treillis des pr6ordres totaux compatibles avec F. Celui-ciest donc h medianes. En particulier l e supdemi-treillis des preordres totaux sur un ensemble fini S est h mtdianes. 3.3.2. Ultrapreordonnances. Une prbordonnance (dissimilarity relation, PR) sur X est un pr4ordre total P sur P z ( X ) . Soient S 1 , ...,Sk 5 f'z (XI les classes de P indicCes dans 1 'ordre induit par P, C en posant Ro

=

P

= (Ro,

8, Ri

=

Rl,

..., %)

la chafne Ctendue de

S 1 , Rz = S i U S z , .

..R.1

= S.U 1

associbe

ff(82(X))

Ri-l,...,\

=

6)2(X).

i P

On

reprend la demarche du paragraphe 3.2.2. en s'intbressant aux prbordonnances P pour lesquelles les blbments de la chaine Cp ont de bonnes propribtfs, et plus prtcisbment au cas oij ce sont des equivalences sur X. Proposition 3.6. Les deux conditions suivantes sont equivalentes pour une priordonnance D sur X : (1)

La cha?ne CD est une chaine de

nX'

(2) Pour tous x,y,z E X, on a : ({x,y},{x,z})ED

ou ((x,y),(y,z1)€

D.

Une pr4ordonnance vbrifiant ( 1 ) et (2) est une ~rltruprdordonnance (ultradissimilarity relation, UPR) sur X. Les conditions de la proposition 3.6. correspondent respectivement h celles de la proposition 3.5. : le prbordre D(u) induit sur 6 ' 2 ( X ) par une ultrambtrique u est une ultrapreordonnance sur X tandis que l'image de L par laL-chaine fU est la chaine des partitions C (privbe de D(u) R O si fu(0) # no). L'ensemble

d3X

des UPR sur X est un sup-demi-treillis B mbdianes, dualement

isomorphe au demi-treillis des charpentes (chaines) de T l . De plus, d'aprgs l a condition (2) ci-dessus, c'est une partie finissante du demi-treillis px des prbordonnances sur X et, donc, un sous-sup-demi-treillis absorbant (Schader 1980). On dispose dc formules de recurrences pour le c a l c u l du nombre D des -n,k ultraprbordonnances B k classes (de mgme graduation) et du nombre total D des -n ultraprbordonnances sur X (avec n = 1x1). S dCsignant les nombres de Stirling de seconde espece -9P Proposition 3 . 7 .

Les nombres D -n

et

vCrifient :

n- 1

D -n

=

=

i=l

sn,; D i

(Lerman 1970)

n- 1

D -n,k

=

E S D i=1 -n,i -i,k-1

(Schader 1980)

Ensembles ordonnis et taxonomie mathematique

avcc

O n iin t l e d u i t , p o u r I. f i n i , I c nombre U -11,. " = IL :

531

des L - u l t r a m ~ t r i q u c s s u r

1x1

soit,

U

-n,V

i=l 154 u l t r a m k t r i q u e s sur 1 1 , 2 , 3 , 4 3 h v a l e u r s d a n s

I 1 y a , pCir exemple, {O, 1,2,3 t.

3.3.3.

lli6rnrcliies.

P o u r iine u l t r n p r G o r d o n n a n c e D s u r X e t p o u r l a cii;i?ne & t e n d u e CD = ( T ~ , T T ~ , . . . , ~ a~ s)s o c i g e , on p e u t c o n s i d b r e r I ' e n s e m b l e H = HD d e s c l a s s e s d e s C q u i v a l e n c c s d e CD : i l e s t n a t u r e 1 en c l a s s i f i c a t i o n d e s ' i n t h r e s s e r a u x c l a s s e s .

H vPrifie lcs propribtes : X E H b ?! f l (H3) ( V x E X ) [xi E H (H4) ( V h , h ' E H) h n (H1)

(H2)

Une note

11'

E {h,h',@!

hiirarchie s u r X e s t uii h l g m c n t d e P ( p (X)) v k r i f i a n t ( H l ) 5 (H4). On

VX l ' e n s e r n b l e

des h i e r a r c h i e s s u r X.

Pour H

EgXe t

Y

5 X,

l'ensemble

{ h F H/Y 5 h l est totalement ordonnt p a r l ' i n c l u s i o n , c e q u i a deux consequences : e s t I'ensemble dcs charpentes de

p(X) - i01

d e m i - t r e i l l i s arborescent, c'est-;-dire h l v 11: c 111 v h3 ou h l V ti2 5 hz minimale i n i q u e d e H c o n t e n a n t h n h ' . Soit alors D

H

V

; t o u t e h i t r a r c h i e H e s t un s u p -

v t r i f i a n t : (V h , , h , , h i € H) h ; . Pour h , h ' E H , h v h ' e s t l a c l a s s e

l e preordre p a r t i e l d e f i n i s u r ( { x , y j , { z , t } ) E DH

Q

x v y

5

P?(X) p a r

:

z v t.

I,e p r b o r d r e DH v e r i f i e l a c o n d i t i o n ( 2 ) d e l a p r o p o s i t i o n 3 . 7 . , mais n ' e s t p a s t o t a l . P a r c o n t r e , t o u t e p r k o r d o n n a n c e s u r X c o n t e n a n t DH e s t u n e u l t r a p r t o r d o n n a n c e s u r X . 011 a s s o c i e a i n s i 2 t o u t e hierarchic u n e c l a s s e d'UPR : S o i e n t kI u n e h i t r a r c h i e s u r X e t D un p r t o r d r e t o t a l s u r $?(X) Proposition 3.8. c o n t e n a n t DH e t m i n i m a l avec c e t t e p r o p r i g t i . . A l o r s D e s t u n e u 1 t r a p r ; o r d o n n a n c e s u r X e t HD = H .

Les h i g r a c h i e s s o n t d o n c e x a c t e m e n t l e s e n s e m b l e s d e c l a s s e s d e f i n i s p a r l e s u l t r a p r e o r d o n n a n c e s nu l e s u l t r a m e t r i q u e s s u r X ( c e s o n t les f a m i l l e s d e b o u l e s de c e s d e r n i k r e s ) ; d'oh l e s a p p e l l a t i o n s c l a s s i q u e s , B l a s u i t e de Benzecri, de h i G r a r c h i e s s t r a t i f i g e s pour les u l t r a p r 6 o r d o n n a n c e s , e t d e h i e r a r c h i e s i n d i c h e s p a r les u l t r a m e t r i q u e s ( r e s p e c t i v e m e n t " o r d e r e d t r e e s " e t "valued trees" chez Boorman e t O l i v i e r ) . Une e t u d e d e s d e m i - t r e i l l i s d e h i e r a r c h i e s , o r i e n t g e v e r s l a t y p o l o g i e e t l a c o m p a r a i s o n d e c e l l e s - c i , a e t b f a i t e p a r L e c l e r c ( 1 9 8 2 ) . On y met notamment l e demi-treillis e n r e l a t i o n avec l ' o r d r e rnodulaire d e s p a r t a g e s d e I ' e n t i e r n - 1 e n e x h i b a n t un m o r p h i s m e d ' o r d r e , p r e s e r v a n t l a c o u v e r t u r e , d u p r e m i e r s u r l e s e c o n d . Le p r o b l P m e d u dbnombrement d e s h i e r a r c h i e s r e m o n t e B S c h r i j d e r ( 1 8 7 0 ) . Comtet ( 1 9 7 0 ) a dCnombr6 l e s h i e r a r c h i e s d e m&ne g r a d u a t i o n p a r l e s n o m b r e s d e S t i r l i n g 2 - a s s o c i e s d e s e c o n d e e s p t c e , e t e n a d b d u i t un c a l c u l p a r r b c u r r e n c e e t un G q u i v a l e n t a s y m p t o t i q u e d e s n o m b r e s ( d u q u a t r i k m e p r o b l k m e ) d e S c h r o d e r .

xX

Les h i e r a r c h i e s maximales ( d i t e s gbneralement h i e r a r c h i e s b i n a i r e s ) o n t 2n-1 b l b m e n t s . P o u r l a s t a t i s t i q u e d e s h i e r a r c h i e s , l a r e p o n s e B c e r t a i n e s q u e s -

J.-P. Barthelerny et al.

5 38

tions surl'intersection des hiCrarchies serait intCressantc. Plus gbneralement, les problemes classiques d'intersertion de ch,iPnes peuvent stre posGs pour l e s intersections de chdrpentes. Par exemple, pour un ensemble ordonn6 F fini, on pc'iit chercher l e nombre maximum de ses charpentes maximales presque disjointes deux 1' deux. Si E a un plus petit et un plus grand flfment c'est le nombre maximum d c chaPnes maximales presque disjointes de E. Une rCponse est fournie p a r l e th6ot-Cme de Menger (1927, cf. Berge 1 9 7 0 ) . 4 - COMPARAISONS, AJUSTEMENTS, AGREGATION : METHODES ORDINALLES.

4.1. Situations. Nous nous intkressons, ici, d u s mCthodes de classification qui conduisent construire une fonction 9 : E + F, E et F Ctant deux ensembles ordonnes (dans l a pratique, il s'agira, bien sSr, des ensembles ordonnbs, treillis ou demi-treillis dCcrits au paragraphe 3 ) . Une telle fonction sera appelCe Line prucd&irt di,

classification. Deux cas se rencontrent essentiellement : 2 F. On dira que 9 est un ajustement. Un exemple classique est : E = L "(') (ensemble des indices de dissimilarit4 bur X h valeurs dans I.) et F = L,X (ensemble des ultramktriques sur X, 2 vnleurs dans L). Un autre serait E = 8 ( 8 , ( X ) ) (ensemble des relations reflexives et sym6triques sur X) et F = r'' X (ensemble des kquivalences, ou des partitions, de X). Dans le premier cas on "approche" une dissimilarit6 par une distance ultrametrique de manihre B construire une hi6rarchie indicke ; dans le second on "construit" une partition 5 partir d'une "relation de ressemblance" quelconque. (1) E

u

(2) E = U FV. On dira alors que 9 est une agre'gaiation. Ce c a s se rencontre, par exemple,"dans la situation suivante : l'ensemble X est dkcrit par v variables, chacune d e ces variables induit sur X une partition : deux blkments pour lesquels elle prend la mOme valeur sont dans l a m@me classe. I1 s'agira, des lors de rksumer, en une seule, ces v partitions (on a donc, ici, F = I$,,). Bien scr la fonction 9 ne sera pas n'importe quai. On exigera qu'elle vCrifie certaines "contraintes" (au sens le plus large d e c e terme). On peut regrouper ces contraintes en trois rubriques (d'ailleurs largement empibtantes). a) .Rhgles On indique explicitement l'art et la manihre de . . . . . . .de . . . construction. ........... construire la fonction 9. On ohtient ainsi les proce'dures a2gorithrnique.s (par exemple, la construction d'une hierarchic indicbe, ou stratifiee, en fusionnant, h chaque niveau, des classes, de maniere B optimiser un critgre). On se restreint aux fonctions 0 verifiant des b) ContrainteS-"axjomatiques". conditions jugees indispensables du point de vue oh l'on se place. On dCbouche alors sur des problemes d'existence et le cas &heant, d'unicite o u sur la caract6risation de toutes les "solutions". ___________S__-T_____-------

On s'intkresse aux fonctions 9 qui permettent c) Contraintes_dloetjmjsatjon_. --'-r-------"--T-r--n------Par exemple, lorsque est un ajustement, d 6tant d'optimiser un critere une distance sur E, on demande que 0(r) soit une solution de Min d(r,s).

.

sEF

Cette r6partition en (a), (b), (c) appelle un certain nombre de remarques : Remarque 1. Les ensembles ordonn6s interviennent fort diversement dans ces trois cas. Si les constructions algorithmiques (cas (a)) ne sont qu'exceptionnellement fond6es sur des crithres ordinaux (la methode dite du "lien simple" que nous rencontrerons un peu plus loin constitue une telle exception), les "axiomes" (cas (b)) le plus souvent rencontr6s sont fortement libs aux ordres de E et F. Enfin, pour le cas (c), les criteres qu'utilise le taxonomiste sont parfois lies B ces structures ordonn6es.

Ensembles ordonnes et taxonomie mathematigue

5 39

Remarque 2 . 1.a r u b r i q u e ( c ) ( e t d a n s une c e r t a i n e m e s u r e ( a ) ) i m p o s e l ' e x t e n s i o n de l a notion de procedure de c l a s s i f i c a t i o n . I 1 n ' y a , en e f f e t aucune r a i s o n de p r c s u p p o s e r 1 ' r i n i c i t c d ' u n h l c m e n t d e F q u i o p t i m i s e un c r i t t r e . On e s t a i n s i conclriit 311x < ,< l -L i icati:or;, q u i s o n t d e s a p p l i c a t i o n s d e E d a n s 1 ' e n s e m b l e cles p a r t i e s d c F . I'

Keni;irqur. 3 . L ' c x e n i p l e q u ' o n R d o n n e p o u r ( c ) m o n t r e q u ' i l p e u t 8 t r e n G c e s s a i r e d ' i i t i l i s e r une d i s t a n c e (ou un i n d i c e d e d i s t a n c e ) s u r E . Indgpendamment d c t o u t s o u c i d ' a j u s t e m e n t , d e t e l l e s m c t r i q u e s s o n t f o r t u t i l e s pour comparer d e s c l a s s i f i c a t i o n s . A c e p r n p o s , nous r e t r o u v o n s nos t r o i s r u b r i q u e s . Les r t g l e s d e c o n s t r u c t i o n p c rme t t e t i t d e c a 1c u 1 e r r f f e c t ivemen t un e d i s t a n c e Le s c o n t r a i n t e s a x i o n i a t i q u e s c u n d u i s e n t 5 ne c u n s i d e r e r q u e c e l l e s q u i v 6 r i f i e n t c e r t a i n e s c o n d i t i o n s , j u ~ : G e s " n a t u r e l l e s " clans l e c o n t e x t e oh l'on t r a v a i l l e . E n f i n , on p e u t t'xiger q u ' u n c r i t G r e s o i t o p t i m i s r ( e n t r a v a i l l a n t , p a r exemple, a v e c d e s " d i s t a n c e s d u p l u s c o u r t rhemin")

.

.

Reniarque 4 . Cornme s o t i v e n t , p l u t 8 t q u e 1 ' C t u d c d e ('11, ( b ) , ( c ) p r i s i s u l e m e n t , un p o i n t t o u t 11 f a i t e s s e n t i e l s e r a l a d G t e r r n i n a t i o n d e r a p p o r t s e n t r e c e s t r o i s c a s . P a r e x e m p l e , Line p r o c e d u r e v e r i f i a n t u n e c o n t r a i n t e d ' o p t i m i s a t i o n s e p e u t c l l e c o n s t r u i r e I: l ' , l i d e d ' u n n l g o r i t h m e ? q u e l l e s s o n t s e s p r o p r i G t 6 s ? ( c o n f r o n t a t i o n aux a x i o m a t i q u c s d ' a u t r c s p r o c G d u r e s : q u e c o n s e r v e - t - o n , q u ' a b a n d o n n e t-on ? ) . P e u t - o n n l l e r j u s q u ' 2 lcas c a r a c t 6 r i s e r a x i o m a t i q u e m e n t ? D;ins c c t t e qlJatrii.nie p a r t i e , n o u s G t u d i e r o n s s u c c e s s i v e m e n t l e s c o m p a r a i s o n s , a j u s t e m e n t s e t a,qrGgntions d e c l ~ s s i f i c a t i o n s , : I l ' a i d e d ' e x e m p l e s , sous l ' 6 c l a i rage des t r o i s rubriques ( a ) , ( b ) , ( c ) . 4 . 2 . Compnraisuns de c l , i s s i f i c a t i o n s . 1.e s ~ 1 6 < ~ ~ & s iS~U~I !E< ce o: n s t i t u e n t d e s m G t r i q u e s f o r t l i 6 c . s h sa s t r u c t u r e ordonnee. L k t e l l e s d i s t a n c e s s ~ n td c s " d i s t a n c e s du p l u s c o u r t chemin" d n n s l e g r a p h e ( n o n o r i e n t h ) d e c u u v e r t u r e d e E , u n e a r S t e uv d e c e g r a p h e e t a n t 6valui:e p a r I . b ( u ) - $ ( v ) I , 9 G t a n t une f o n c t i o n , 5 v a l e u r s r e e l l e s , s t r i c t e m e n t c r o i s s i i n t c siir E . O n t r o u v e r a , clans l ' n r t i c l e d e s y n t h s s e d c M o n j a r d e t ( 1 9 8 1 ) , l e s d i v e r s e s p r o p r i 6 t i . s a c t u e l lenient c o n n u e s d e c e s d i s t a n c e s a v e c l e s i n d i c a t i o n s b i b l i o g r a p h i q u e s c o r r e s p o n d a n t e s . En p a r t i c u l i e r , u n e h y p o t h t s e s u r $ ( v a l u a t i o n i n f e r i e u r e ou s u p 6 r i e u r e ) f o u r n i r a une c o n d i t i o n n g c e s s a i r e e t s u f f i s a n t e p o u r q u e l a d i s t a n c e g G o d 6 s i q u e dd, ( d o n c d e f i n i e p a r une c o n t r a i n t e d ' o p t i m i s a t i o n ) p u i s s e S t r e c a l c r i l t e p a r l a "formule" :

(1)

d (r,s)

(1")

d (r,s)

=

cb(r) + g ( s ) - 2 + ( r

=

? , b ( r v s ) - , + ( r ) - $ ( s ) , s i E e s t un sup-derni-treillis.,

A

s ) , s i E e s t un i n f - d e m i - t r e i l l i s .

,(I

'b

P a r a i l l e u r s , m @ m e l o r s q u e (1 n ' c s t p a s m e v a l u a t i o n ( i n f b r i e u r e o u s u p e r i e u r e ) , l e s f o r m u l e s ( 1 ) e t ( 1 * ) p e u v e n t C t r e u t i l i s 6 e s p o u r c a l c u l e r un i n d i c e de d i s t a n c e . Celui-ci s e r a p e r t i n e n t en " t h t o r i e de I'Information". Lorsque 9 r e p r t s e n t e une m e s u r e d ' i n f o r m a t i o n (on s e p l a c e d a n s I e c a s E = II ) , ( 1 ) e t ( 1 9 0 X r e p r G s e n t e n t u n c summe d ' i n f o r m n t i o n s c o n d i t i o n n e l l e s ( p o u r q u e l q u e s p r 6 c i s i o n s c f . Kampe d e F e r i e t 1 9 7 4 , L o s f e l d 1 9 7 4 ) . I1 e s t d ' a i l l e u r s r e m a r q u a b l e q u e l e s mesures, c l 3 s s i q u e s , d'informnt ion conduisent des "valuations". I':encc s u p p l 6 m e n t a i r e : & ( d ) e s t une u l t r a m G t r i q u e ) . A c h a q u e s r u i l X E L c o r r c s p o n d I n r e l a t i o n s y m C t r i q u e f d ( X ) = t I x , y l t e l q u e c l ( x , y ) 5 A , } . On demandc q u e f ~ ; , ( ~ i ) ( I ) ( i . c . " @ ( d ) a u s e u i l A " ) ne d6pcnde que de f d ( A ) ( i . e . tie " d a11 s e u i l A"). Te c h n iq u e me n t : i c h a q u e d i ssi ni i I n r i t e d c o r r c s p o n d I J I.-clinine f d E R e s( L, 8 ( p,(X)) ( 3 . 1 . 2 ) . On c i i t a l o r s clue I n p r o c 6 d u r e d e c l n s s i i c a t i o n '11 : I, i p 2 ( X ) + I . p 3 2 ( X ) e s t L: scL4i/ s ' i l e x i s t e t e l l e q u e , pour t o u t e L - d i s s i m i l n r i t & d : a l o r s npye1i.e u n e j b n i r t i o n

line

y

o

a p 7 l i c a t i o n y : x'(P,(x))+ fd = 1,a [ u n c t i o n y

P( P , ( x ) ) rst

Dks l o r s s e p o s e n t deux q u e s t i o n s (les " f l a t proble ms " tle J n n o w i t z 1 9 7 8 ~ ) . 1) c a r a c t t r i s e r I e s p r o c P d u r e s h s e u i l , 2) c a r a c t b r i s e r l e s f o n c t i o n s s e u i l . J a n o w i t z l e s t r a n s f o r m e e n p r o h l 6 m e s plus g b n e r a u x e t pure me nt o r d i n n u x e n r e m p l a c a n t L p a r un s u p - d c m i - t r e i l l i s ;aver u n p l u s p e t i t 616nrent e t ff( B~(x)) p a r deiix e n s e m b l e s o r d o n n 6 s q u e l c o n q u e s M e t N , a v e c un p l u s g r a n d 6 l b m e n t . Dans c e c a d r e , u n e f o n c t i o n s ' 1 s e r a une a p p l i c a t i o telle q u c , pour t o u t g F Re s( L,M ) , y o g E R e s ( L , N ) . De rnSnie ;inyC. BERGE and V. CHVATAL 1984 xiv + 370 pages Vol. 22: Trees and Hills: Methodology for Maximizing Functions of Systems of Linear Relations Rick G R E E R 1984 xiv + 352 pages


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