Combinatoric& and Ordered Sets Proceedings of a Summer Research Conference held August11-17,1985

AMERICAn MATHEMATICAL SOCIETY VOLUME 57

Titles in This Series

Volume 1 Markov random fields and their

applications. Ross Kindermann and J. Laurie Snell

2 Proceedings of the conference on integration, topology, and geometry in linear spaces. William H. Graves. Editor

3 The closed graph and P-closed graph properties in general topology, T. R. Hamlett and L. L. Herrington

4 Problems of elastic stability and vibrations, Vadim Komkov. Editor

5 Rational constructions of modules for simple Lie algebras. George B. Seligman

6 Umbral calculus and Hopf algebras. Robert Morris. Editor

7 Complex contour integral representation of cardinal spline functions, Walter Schempp

8 Ordered fields and real algebraic geometry, D. W. Dubois and T. Recio. Editors

9 Papers in algebra. analysis and statistics, R. Lidl. Editor

10 Operator algebras and K-theory, Ronald G. Douglas and Claude Schochet. Editors

11 Plane ellipticity and related problems. Robert P. Gilbert. Editor

12 Symposium on algebraic topology in honor of Jose Adem. Samuel Gitter. Editor

13 Algebraists' homage: Papers in ring theory and related topics, S. A. Amitsur. D. J. Saltman. and G. B. Seligman. Editors

14 Lectures on Nielsen fixed point theory, Boju Jiang

COftTEMPOIIIRY MATHEMATICS

15 Advanced analytic number theory. Part I: Ramification theoretic methods, Carlos J. Moreno

16 Complex representations of GL(2. K) for finite fields K, llya Piatetski-Shapiro

17 Nonlinear partial differential equations, Joel A. Smoller. Editor

18 Fixed points and nonexpanslve mappings, Robert C. Sine. Editor

19 Proceedings of the Northwestern homotopy theory conference. Haynes R. Miller and Stewart B. Priddy, Editors

20 Low dimensional topology, Samuel J. Lomonaco. Jr .. Editor

21 Topological methods In nonlinear functional analysis, S. P. Singh, S. Thomeier. and B. Watson. Editors

22 Factorizations of b" ± 1, b = 2, 3. 5, 6. 7,10,11.12 up to high powers, John Brillhart. D. H. Lehmer. J. L. Selfridge, Bryant Tuckerman. and S. S. Wagstaff. Jr.

23 Chapter 9 of Ramanujan's second notebook-Infinite series Identities. transformations. and evaluations. Bruce C. Berndt and Padmini T. Joshi

24 Central extensions, Galois groups, and ideal class groups of number fields. A. Frohlich

25 Value distribution theory and Its applications. Chung-Chun Yang, Editor

26 Conference In modern analysis and probability, Richard Beals. Anatole Beck. Alexandra Bellow. and Arshag Hajian. Editors

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Titles in This Series

Volume

27 Microlocal analysis. M. Salah Baouendi. Richard Beals. and Linda Preiss Rothschild. Editors

28 Fluids and plasmas: geometry and dynamics. Jerrold E. Marsden. Editor

29 Automated theorem proving, W. W. Bledsoe and Donald Loveland. Editors

30 Mathematical applications of category theory. J. W. Gray, Editor

31 Axiomatic set theory, James E. Baumgartner. Donald A. Martin. and Saharon Shelah. Editors

32 Proceedings of the conference on Banach algebras and several complex variables. F. Greenleaf and D. Gulick. Editors

33 Contributions to group theory, Kenneth I. Appel. John G. Ratcliffe. and Paul E. Schupp. Editors

34 Combinatorics and algebra, Curtis Greene. Editor

35 Four-manifold theory, Cameron Gordon and Robion Kirby. Editors

36 Group actions on manifolds. Reinhard Schultz. Editor

37 Conference on algebraic topology in honor of Peter Hilton, Renzo Piccinini and Denis Sjerve. Editors

38 Topics in complex analysis. Dorothy Browne Shaffer. Editor

39 Errett Bishop: Reflections on him and his research. Murray Rosenblatt. Editor

40 Integral bases for affine Lie algebras and their universal enveloping algebras. David Mitzman

41 Particle systems. random media and large deviations. Richard Durrett. Editor

42 Classical real analysis, Daniel Waterman. Editor

43 Group actions on rings, Susan Montgomery. Editor

44 Combinatorial methods in topology and algebraic geometry, John R. Harper and Richard Mandelbaum. Editors

45 Finite groups-coming of age. John McKay, Editor

46 Structure of the standard modules for the affine Lie algebra A~ 1 ). James Lepowsky and Mirko Prime

47 Linear algebra and its role in systems theory, Richard A. Brualdi. David H. Carlson. Biswa Nath Datta. Charles R. Johnson. and Robert J. Plemmons. Editors

48 Analytic functions of one complex variable, Chung-chun Yang and Chi-tai Chuang, Editors

49 Complex differential geometry and nonlinear differential equations, Yum-Tong Siu. Editor

50 Random matrices and their applications, Joel E. Cohen. Harry Kesten. and Charles M. Newman. Editors

51 Nonlinear problems in geometry, Dennis M. DeTurck. Editor

52 Geometry of normed linear spaces. R. G. Bartle. N. T. Peck. A. L. Peressini. and J. J. Uhl. Editors

53 The Selberg trace formula and related topics. Dennis A. Hejhal. Peter Sarnak. and Audrey Anne Terras. Editors

54 Differential analysis and infinite dimensional spaces. Kondagunta Sundaresan and Srinivasa Swaminathan. Editors

55 Applications of algebraic K-theory to algebraic geometry and number theory, Spencer J. Bloch. R. Keith Dennis. Eric M. Friedlander. and Michael R. Stein. Editors

56 Multiparameter bifurcation theory, Martin Golubitsky and John Guckenheimer. Editors

57 Combinatorics and ordered sets. Ivan Rival. Editor

Combinatoric& and Ordered Sets

Combinatoric& and Ordered Sets

I Volume 57

Proceedings of the AMS-IMS-BIAM Joint Summer Research Conference held August 11-171 19851 with support from the National Science Foundation

Ivan Rival1 Editor

AMERICAn MATHEMATICAL SOCIETY Providence • RhOda Island

EDITORIAL BOARD Irwin Kra,

managing editor

Gerald J. Janusz

Jan Mycielski

Johannes C. C. Nitsche

Carl M. Pearcy

Alan D. Weinstein

The AMS-IMS-SIAM Joint Summer Research Conference in the Mathematical Sciences on Combinatorics was held at Humboldt State University, Arcata, Cali-fornia on August 11-17, 1985, with support from the National Science Foundation, Grant DMS-8415201.

1980 Mathematics Subject Classifications. 06A10, 06A5. 06C05. 90B35. 0504. 05C20. 03015.

Library of Congress Cataloging-In-Publication Data Combinatorics and ordered sets.

(Contemporary Mathematics; v. 57} Bibliography: p. 1. Ordered sets-Congresses. 2. Combinatorial set theory-Congresses. I. Rival. Ivan.

1947- . II. American Mathematical Society. Ill. Institute of Mathematical Statistics. IV. Society for Industrial and Applied Mathematics. V. Series: Contemporary mathematics (American Mathematical Society); v. 57. QA171.48.C65 1986 511.312 86-8006 ISBN 0-8218-5051-2. ISSN 0271-4132

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Republication. systematic copying. or multiple reproduction of any material in this pub-lication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Executive Director. American Mathematical Society. P.O. Box 6248, Providence. Rhode Island 02940.

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CONTENTS

PREFACE

PARTICIPANTS

SCIENTIFIC PROGRAMME

Order-theoretic aspects of scheduling By Werner Poguntke

Radon transforms in combinatorics and lattice theory By Joseph P.S. Kung

Recursive or.dered sets By Henry A. Kierstead

Orientations and reorientations of graphs By Oliver Pretzel

Abstract convexity and meet--distributive lattices By Paul Edelman

Correlation and order By Peter M. Winkler

Retracts: graphs and ordered sets from the metric point of view By El-Mostapha Jawhari, Driss Misane and Maurice Pouzet

Antichains and cutsets By Mohamed El-Zahar and Nejib Zaguia

Stories about order and the letter N (en) By Ivan Rival

vii

ix

xili

XV

33

75

103

127

151

175

227

263

PREFACE

Ordered sets abound in mathematics, for instance in algebra, combinatorics, geometry, model theory, set theory and topology. The theory of ordered sets has applications throughout mathematics, and beyond in operations research, computer science and the physical and social sciences. The link with modern combinatorial theory is a major source of the current vitality in ordered sets itself.

This volume is a collection of the principal expository lectures presented at COMBINATORICS and ORDERED SETS a conference held at Humboldt State University, Arcata, California from August 11 to August 17, 1985. This weeklong conference was one of the 1985 Joint Summer Research Conferences in the Mathematical Sciences under the direction of the AMS-IMS-SIAM. It was supported by a grant from the National Science Foundation. The aim of this conference was to highlight several of the leading combinatorial themes in ordered sets today. As a conference featuring surveys on ordered sets it is a continuation of two recent NATO Advanced Study Institutes. The first, SYMPOSIUM on ORDERED SETS, was held in Banff, Canada, from August 28 to September 12, 1981 and was intended as an introduction to all current topics in the theory of ordered sets1• The second, GRAPHS and ORDER, was held in Banff too, from May 18 to May 31, 1984 and was to document the role of graphs in the theory of ordered sets and its applications2

The nine articles presented here cover a wide range of combinatorial themes in ordered sets. In Order-theoretic aspects of scheduling, Werner Poguntke focusses on the famous three-machine problem to illustrate order-theoretic aspects of scheduling theory. In Radon transforas in ca.binatorics and lattice thoer.y, Joseph P.S. Kung, surveys techniques he used to settle the longstanding 'matching conjecture' between the join irreducible and the meet irreducible elements in a finite modular lattice. In Recursive ordered sets, Henry Kierstead elaborates on the problem to 'effectively'

1ordered Sets (I. Rival, ed.), D. Reidel Publishing Co., Dordrecht, 1982.

2Graphs and Order (I. Rival, ed.), D. Reidel Publishing Co., Dordrecht, 1985.

ix

X PREFACE

decompose a (countable) ordered set into few chains. In Orientations and reorientations of graphs, Oliver Pretzel surveys uses of the 'pushdown' operation which reorients a diagram to produce another order with the same covering graph. In Abstract convexity and meet-distributive lattices, Paul Edelman synthesizes the many and varied occurrences of the meet-distributive property a frequently recurring concept. In Correlation and order, Peter Winkler surveys techniques that settled the conjecture that it is more likely, in a binary sorting problem, that a < b if it becomes known that a is less than some other element c. In Retracts: graphs and ordered sets fr0.111 the metric point of view, E.M. Jawhari, D. Misane and Maurice Pouzet formulate and develop an intriguing general view point for retraction, unifying ideas in ordered sets and in graphs by placing them both within a general theory of metric spaces. In Antichains and cutsets, Mohamed El-Zahar and Nejib Zaguia survey cutsets (subsets which meet every maximal chain) a theme which has seen remarkable activity in just the past few years. Finally, in Stories about order and the letterN (en), Ivan Rival considers the role played by the notion of subdiagram especially in structure and optimization themes in ordered sets.

The participation and presence of R.P. Dilworth was considerable - just as it was at the earlier SYMPOSIUM on ORDERED SETS. At that earlier meeting in 1981 there was, rather unexpectedly, one single result that seemed to play a motivating role in many of the lectures. It was the well known 'Chain Decomposition Theorem' that, in an ordered set the minimum number of chains f>'hose union is all of the set equals the JIIBXimum nl.ll6ber of pairf>'ise nonc0.111parable elements4 . At this meeting in Arcata his influence again seemed to be ubiquitous. The themes in one day's lectures alone, spanned almost two decades of his fundamental results3•4•5 • The theory of ordered sets is a lively area today and it has a rich legacy.

COMBINATORICS and ORDERED SETS was a relatively short meeting (five full working days). To avoid the prospect that some, among the many participants would perhaps, not become aware - until it was too late - of the presence of others with whom they would like to consult, we organized two opening sessions (Introductory sketches) on the first morning. these were explicitly intended as an opportunity for all participants to introduce themselves publicly. Over forty did according to these suggested guidelines: write your name on the blackboard and discuss your current research interests - all within three minutes! It seemed to work. On the basis of these introductory sketches we were able to monitor common interests and therefore, to design Special Sessions, spontaneously and following the initiatives of the participants. The value of these conferences is often in the magic and spontaneity that

3R.P. Dilworth (1940) Lattices with unique irreducible decompositions, Ann. Nath. 41, 771-777. 4R.P. Dilworth (1950) A decomposition theorem for partially ordered sets, Annals of Math. 51, 161-166. 5R.P. Dilworth (1954) Proof of a conjecture on finite modular lattices, Annals of Nath. 60, 359-364.

PREFACE

occurs between the main lectures; the lectures themselves constitute the formal structure.

These weeklong Summer Research Conferences are inspired by the American Mathematical Society. Although they are held at different locations each summer their format and overall administration is uniform and smooth. There is little doubt that we all owe much to the tireless efforts of Carole Kohanski, the Summer Research Conference Coordinator of the American Mathematical Society. Besides all of her administrative work, her comments and suggestions helped in designing an effective, yet spontaneous, scientific programme.

I am particularly grateful for the consistent patience, encouragement and enthusiasm of my wife Hetje.

Calgary, Canada, February 20, 1986 Ivan Rival

Michael Albertson (U.S.A.) Brian Alspach (Canada) Margaret Bayer (U.S.A.) Mary Bennett (U.S.A.) Gary Bloom (U.S.A.) Kenneth Bogart (U.S.A.) Graham Brightwell (England) Gerard Chang (Taiwan) Stephen Comer (U.S.A.) Julien Constantin (Canada) Maria Contessa (Italy) Bong Dang (U.S.A.) Elias David (England) Walter A. Deuber (Germany) Robert P. Dilworth (U.S.A.) Dwight Duffus (U.S.A.) Paul Edelman (U.S.A.) Paul Erdos (Hungary) Peter Frankl (France) Stephen Grantham (U.S.A.) George Gratzer (Canada) Michel Habib (France) Mark Halsey (U.S.A.) Katherine Heinrich (Canada) Roland Jegou (France) Jeffry Kahn (U.S.A.) Henry Kierstead (U.S.A.) David Klamer (U.S.A.) Daniel Kleitman (U.S.A.) V. Krishnamurthy (U.S.A)

PARTICIPANTS

Joseph Kung (U.S.A.) Renu Lasker (U.S.A.) Klaus Leeb (Germany) Ko-Wei Lih (Taiwan) Frank R. McMorris (U.S.A.) Robert Melter (U.S.A.) Shawkwei Moh (China) Joseph Neggers (U.S.A.) Heinrich Niederhausen (U.S.A.) Richard Nowakowski (Canada) Sergei Ovchinnikova (U.S.A.) Werner Poguntke (Germany) Maurice Pouzet (France) Oliver Pretzel (England) Hans Pramel (Germany) Yuyuan Qin (China) Robert Quackenbush (Canada) Ivan Rival (Canada) Vojtech ROdl (Czechoslovakia) Ivo G. Rosenberg (Canada) Michael Saks (U.S.A.) Rainer Schrader (Germany) Steven Simpson (U.S.A.) Michael Stone (Canada) Willaia Trotter (U.S.A.) Miroslaw Truszczynski (Poland) Bernd Voigt (Germany) Edward Wang (Canada) N. Zaguia (Tunisia)

xiii

SCIENTIFIC PROGRAMME

Monday, Auguet 12, 1985 Chair: I. Rival All: Introductor,ysketches W. Poguntke, Order-theoretic aspects of scheduling. (I)

All: Introductor,ysketches J. Kung, Radon transfoi'JIIS in cOIIIhinatorics and lattice theor,y. (I) H. Kierstead, Recursive cOIIIhinatorics of ordered sets. (I)

Tuesday, August 13, 1985 I. Rival, Stories about order and the letter N (en). M. Habib, Linear extensions and depth-first search. 0. Pretzel, Orientations and reorientations of graphs. P. Edelman, Abstract convexity and Jlleet-distributive lattices. J. Kung, Radon transfoi'JIIS in cOIIIhinatorics and lattice theory. (II)

Special Session 1: Chair: W. Deuber P. Erdos, Proble.s and results. M. Albertson, HOliiOIIorphiSJIIS and independence in triangle-free graphs. S. Moh, On chesSJ/IeJJ 1110ving. D. Kleitman, COIIIhinatorial enumeration.

Wednesday, August 14, 1985 M. Pouzet, Retracts: graphs and ordered sets fra. the 111etric point of view. W. Poguntke, Order-theoretic aspects of scheduling. (II)

XV

xvi SCIENTIFIC PR001WI4B

Thursday, August 15, 1985 N. Zaguia, Antichains and cutsets. W.T. Trotter, Constructiaos of linear eKtensiaos. G. Gratzer, Partition rank functiaos for general lattices.

Special Session II: Chair: R.P. Dilworth E. David, On the distributive property of quasi-ordered sets. I.G. Rosenberg, Orders admitting a JIIBjority or near-unaniRiity isotone

operation. R.W. Quackenbush, Gaps, holes and near-unani•ity functions. J. Constantin, The fixed point property in posets (and l{rBphs).

Special Session III: Chair: D. Duffus G. Brightwell, ..4 universal correlatiao inequality for finite posets. A. Rucinski, Balanced eKtensions of graphs. G. Bloom, Labelled lfraphs.

Friday, August 16, 1985 H. Kierstead, Recursive cambinatorics of ordered se~s. (II)

Special Session IV: Chair: s. Camer D. Klarner, The nUIIIber of graded posets is alternately congruent to 1 and 3

(wx/6).

Y. Qin, On a jar-1118tric principle in optiRiization.

Special Session V: Chair: B. Alspach M. Truszczynski, Natroids and jUlllp nUIIIber in tf-free posets. N. Zaguia, Interchanging chains.