Orders: Description and Roles

558 Pages·1984·13.891 MB·English

by  Maurice Pouzet and Denis Richard (Eds.)

annals of
Ccnerd Editor
Peter L. HAMMER, Rutgers University, New Brunswick, NJ, U.S.A.
Advisory E ditors
C. BERGE, Universitk de Paris
M. A. HARRISON, University of California, Berkeley, CA, U.S.A.
V. KLEE, Universityofwashington, Seattle,WA. U.S.A.
J. H. VAN LINT, California Institute ofTechnology, Pasadena, CA. U.S.A.
G.-C. ROTA, Massachusetts Institute ofTechnology, Cambridge, MA, 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 on 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~teatidelaTouretteI.’ 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.~fii~lirc 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, 1942- 111. 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 Pro fesseur COROMINAS
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 interpre-
tation 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 im-
portant 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 (solva-
ble 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 ma-
thematicians 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 categori-
city 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 indica-
tor 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 possi-
ble 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 pro-
blems of consistency) and fertile valleys (lattice theory, order groups, boolean algebra,
noetherian rings, pointrset topology, problems concerning duality, representation and gene-
ration 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 rigi-
dity and boundedness of the objects there are very few general techniques available (consi-
der, for example, the RAMSEY numbers or the many famous unsettled conjectures of num-
ber 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 mathe-
matical 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 com-
puter 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 do-
mains 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 rela-
tions, 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 at 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 or 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. St 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 on the
cover. This is a reproduction of the needlepoint work he completed during the Confe-
rence.
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’occa-
sion 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 et LUMINY, 1984) et d’autres faits - comme la parution de la revue ORDER -
tout semble indiquer I’importance croissante de la thkorie des ordres et 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 et R. DEDEKIND, et
encore de la thborie des kquations et des groupes avec E. GALOIS et C. JORDAN (groupes
rksolubles, suites de JORDAN - HOLDER) et encore de la thborie des anneaux
(E. NOETHER), ceci pour ne citer que quelques exemples importants. De ces diverses ori-
gines naissent plusieurs courants tels la thkorie des treillis, les alg&bresd e Boole, la topologie,
etc..., drainant les eaux vers le fleuve en formation. Issus de I’btude de I’ensemble des sous-
groupes d’un groupe, la thborie des treillis, courant fkcond en rbsultats aujourd’hui classi-
ques (e.g. les thkordmes de reprksentation de BIRKHOFF et STONE), est ainsi, depuis plu-
sieurs dkcennies, une des pn5occupations principales des mathbmaticiens travaillant sur
I’ordre. De mSme, et ne serait-ce que dds I’abord axiomatique (Axiomes de ZERMELO,
KURATOWSKI, ZORN, ...), les courants impbtueux de la thborie des ensembles ne pou-
vaient 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 et 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 descrip-
tion des moddles de Peano fait place aux notions ordinales (extensions finales ou cofinales,
-
sections initiales et 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 logi-
que 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) et des vallkes fecondes (les grandes
structures: treillis, groupes ordonnks, algdbres de Boole, algdbre noethbrienne, topologie
X Preface
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). Cepen-
dant, ces terres sont, souvent plus difficiles i cultiver puisque I’on y substitue i la souplesse
des concepts de I’infini et 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 mathbmati-
cien pouvait narvement penser qu’elle n’btait la que pour qu’il la dbcouvre. On lui demande
a
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 forcb-
ment. C’est le cas dans les sciences sociales oh la formalisation des situations proposbes
conduit i des rbsultats et des problemes d’ordre: le paradoxe de CONDORCET est l’ori-
gine 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
a
informatique amhe 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.
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 et du rble de I’ordre en thborie des ensembles, en thborie
des treillis, en topologie, en logique (thborie des moddles, thborie des relations, arithmkti-
que), dans les groupes ordonnbs, en combinatoire, en informatique thborique et dans les
sciences sociales. Les Qditeurs ont voulu, qu’au fur et a mesure de son parcours, le lecteur
aille de situations infinitistes i des situations de plus en plus finitistes, et qu’il passe de la
a
mdme faqon du point de vue descriptif des ensembles ordonnbs leurs utilisations. Les nu-
mbros spbciaux de la revue DISCRETE MATHEMATICS contiendront les articles de recher-
che 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 btant prbcbdb d’une introduction ou d’un rbsumk exprimb dans l’autre.
Cette preface serait incomplhte si nous ne remerciions pas tous les arbitres qui ont
bien voulu nous aider, pour I’excellence de leurs critiques et de leurs suggestions. Parmi ces
arbitres, se trouvent certains des auteurs de ce livre, qui se reconnaftront ici, et 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 tout en kcoutant les confbrences.
Maurice POUZET et Denis RICHARD
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 at the UniversiG Claude
Bernard (LYON 1) and the group of French mathematicians comprising R.C.P.
698 of the C.N.R.S. in cooperation with I’Ecole Nationale Supkrieure des T616-
communications*.
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 90 papers were given at the conference
including 34 invited addresses, 15 presentations at the special sessions, 20 contri-
buted papers and 20 problems given at 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 at 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 magnifi-
cent production of Napolkon, and also to enjoy a fine concert by R. JAMISON-
WALDNER (cello) and G. BRUNS (piano).
The Scientific programme and organization of the meeting was coordina-
ted 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 assis-
tance - especially Marianne DELORME, whose efficiency and courtesy was
widely appreciated. We do 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.).