Contemporary Mathematicians Gian-Carlo Rota Editor R. P. Dilworth at retirement. The Dilworth Theorems Selected Papers of Robert P. Dilworth Edited by Kenneth P. Bogart, Ralph Freese, and Joseph P. S. Kung Springer Science+Business Media, LLC 1990 Kenneth P. Bogart Department of Mathematics and Computer Science Dartmouth College Hanover, NH 03755 U.S.A. Ralph Freese Department of Mathematics University of Hawaii Honolulu, HI 96822 U.S.A. Joseph P. S. Kung Department of Mathematics University of North Texas Denton, TX 76203-5116 U.S.A. Library of Congress Cataloging-in-Publication Data The Dilworth theorems : selected papers of Robert P. Dil worth / edited by Kenneth P. Bogart, Ralph Freese, Joseph P. S. Kung. p. cm. — (Contemporary mathematicians) Includes bibliographical references. ISBN 978-1-4899-3560-1 1. Lattice theory. 2. Dilworth, Robert P. (Robert Palmer), 1914- I. Dilworth, Robert P. (Robert Palmer), 1914- . II. Bogart, Kenneth P. III. Freese, Ralph S., 1946- . IV. Kung, Joseph P. S. V. Series. QA171.5.D55 1990 511.3'3—dc20 90-31372 90-31372 Printed on acid-free paper. © Springer Science+Business Media New York 1990 Originally published by Birkhäuser Boston in 1990 Softcover reprint of the hardcover 1st edition 1990 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use, or the internal or personal use of specific clients, is granted by Springer Science+Business Media, LLC, for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $0.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress Street, Salem, MA 01970, U.S.A. Special requests should be addressed directly to Springer Science+Business Media, LLC ISBN 978-1-4899-3560-1 ISBN 978-1-4899-3558-8 (eBook) DOI 10.1007/978-1-4899-3558-8 Camera-ready text provided by the editors. 987654321 Contents List of Contributors IX Editors' Preface XI Biography XV Recollections of R. P. Dilworth Peter Crawley XIX Recollections of Professor Dilworth Phillip Chase xxi Mathematical Publications of R. P. Dilworth xxiii Doctoral Students XXVI 1 Chain Partitions in Ordered Sets Background 1 Reprinted Papers A Decomposition Theorem for Partially Ordered Sets 7 Some Combinatorial Problems on Partially Ordered Sets 13 Articles K. Bogart, C. Greene, and J. Kung The Impact of the Chain 19 Decomposition Theorem on Classical Combinatorics E. C. Milner Dilworth's Decomposition Theorem in the 30 Infinite Case H. Kierstead Effective Versions of the Chain Decomposition 36 Theorem v 2 Complementation Background 39 Reprinted Papers Lattices with Unique Complements 41 On Complemented Lattices 73 Articles M. Adams Uniquely Complemented Lattices 79 G. Kalmbach On Orthomodular Lattices 85 3 Decomposition Theory Background 89 Reprinted Papers Lattices with Unique Irreducible Decompositions 93 The Arithmetical Theory of Birkhoff Lattices 101 Ideals in Birkhoff Lattices 115 Decomposition Theory for Lattices without Chain Conditions 145 (with P. Crawley) Note on the Kurosch-Ore Theorem 167 Structure and Decomposition Theory of Lattices 173 Articles B. Jonsson Dilworth's Work on Decompositions in Semi­ 187 modular Lattices B. Monjardet The Consequences of Dilworth's Work on 192 Lattices with Unique Irreducible Decompositions J. Kung Exchange Properties for Reduced Decompositions in 201 Modular Lattices M. Stern The Impact of Dilworth's Work on Semimodular 203 Lattices on the Kurosch-Ore Theorem 4 Modular and Distributive Lattices Background 205 Reprinted Papers The Imbedding Problem for Modular Lattices (with M. Hall) 211 Proof of a Conjecture on Finite Modular Lattices Distributivity in Lattices (with J. McLaughlin) Aspects of Distributivity Articles A. Day and R. Freese The Role of Gluing Constructions in 251 Modular Lattice Theory I. Rival Dilworth's Covering Theorem for Modular Lattices 261 vi THE DILWORTH THEOREMS 5 Geometric and Semimodular Lattices Background 265 Reprinted Papers Dependence Relations in a Semi-modular Lattice 269 A Counterexample to the Generalization of Spemer's Theorem 283 (with C. Greene) Articles U. Faigle Dilworth's Completion, Submodular Functions, and 287 Combinatorial Optimiiation J. Kung Dilworth Truncations of Geometric Lattices 295 J. Griggs The Spemer Property in Geometric and Partition 298 Lattices 6 Multiplicative Lattices Background 305 Reprinted Papers Abstract Residuation over Lattices 309 Residuated Lattices (with M. Ward) 317 Non-commutative Residuated Lattices 337 Non-commutative Arithmetic 357 Abstract Commutative Ideal Theory 369 Articles D. Anderson Dilworth's Early Papers on Residuated and 387 Multiplicative Lattices E. Johnson Abstract Ideal Theory: Principals and Particulars 391 D. Anderson Representation and Embedding Theorems for 397 Noether Lattice and r-Lattices 7 Miscellaneous Papers Background 403 Reprinted Papers The Structure of Relatively Complemented Lattices 407 The Normal Completion of the Lattice of Continuous Functions 419 A Generalized Cantor Theorem (with A. Gleason) 431 Generators of Lattice Varieties (with R. Freese) 433 Contents vii Articles G. McNulty Lattice Congruences and Dilworth's Decomposition 439 of Relatively Complemented Lattices G. Gierz The Normal Completion of the Lattice of Continuous 445 Functions J. Kung Cantor Theorems for Relations 450 J. B. Nation Ideal and Filter Constructions in Lattice Varieties 451 8 Two Results from" Algebraic Theory of Lattices" Background 455 Articles J. Kung Dilworth's Proof of the Embedding Theorem 458 G. Gra.tzer On the Congruence Lattice of a Lattice 460 Permissions 465 viii THE DILWORTH THEOREMS List of Contributors M. E. Adams, Department of Mathematics, State University of New York, New Platz, New York 12561, U. S. A. Daniel D. Anderson, Department of Mathematics, University of Iowa, Iowa City, Iowa 52240, U. S. A. Kenneth P. Bogart, Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755, U. S. A. Phillip J. Chase, 8716 Oxwell Lane, Laurel, Maryland 20708, U. S. A. Peter Crawley, Department of Mathematics, Brigham Young University, Provo, Utah 84602, U. S. A. Alan Day, Department of Mathematics, Lakehead University, Thunder Bay, On­ tario, Canada P7B 5El. Ulrich Faigle, Faculty of Applied Mathematics, University of Twente, P. O. Box 217,7500 AE Enschede, the Netherlands. Ralph Freese, Department of Mathematics, University of Hawaii, Honolulu, Hawaii 96822, U. S. A. Gerhard Gierz, Department of Mathematics, University of California, Riverside, California 92521, U. S. A. George A. Gratzer, Department of Mathematics, University of Manitoba, Win­ nipeg, Manitoba, Canada R3T 2N2. Curtis Greene, Department of Mathematics, Haverford College, Haverford, Penn­ sylvania 19041, U. S. A. Jerrold R. Griggs, Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208, U. S. A. ix Eugene W. Johnson, Department of Mathematics, University of Iowa, Iowa City, Iowa 52240, U. S. A. Bjarni Jonsson, Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37235, U. S. A. Gudrun Kalmbach, Mathematische Institut, Universitat Ulm, D-7900 Ulm, Fed­ eral Republic of Germany. Henry A. Kierstead, Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208, U. S. A. Joseph P. S. Kung, Department of Mathematics, University of North Texas, Denton, Texas 76203, U. S. A. George F. McNulty, Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208, U. S. A. E. C. Milner, Department of Mathematics, University of Calgary, Calgary, Al­ berta, Canada T2N IN4. Bernard Monjardet, UniversiM Paris V and Centre d'Analyse et de MatMmatique Sociale, 54 Boulevard Raspail, 75 270 Paris Cedex 06, France. James B. Nation, Department of Mathematics, University of Hawaii, Honolulu, Hawaii 96822, U. S. A. Ivan Rival, Department of Computer Science, University of Ottawa, Ottawa, Canada KIN 6N5. Manfred Stern, Sektion Mathematik, Martin-Luther-Universitat, DDR-4010 Halle, German Democratic Republic. x THE DILWORTH THEOREMS Editors' Preface Lattice theory is one of those rare subjects to which it is possible to assign a pre­ cise birthday. This birthday occurred in 1897 with the publication of Dedekind's pa­ per [2]. Not much was done before the subject was revived in the 1930's by Birkhoff, Mac Lane, Ore, Ward, and others. At that time, lattice theory was thought of as a subject whose interest depends on other areas of mathematics and the emergence of lattice theory as a subject with ideas, theorems, and problems of its own did not really occur till the fifties. Among the many many mathematicians whose work contributed to this evolution, Robert P. Dilworth is one of the most influential. This maturing of lattice theory is best described by quoting from Dilworth's preface to the proceedings [6] of the symposium on partially ordered sets and lattice theory held at Monterey in 1959: ... on April 15, 1938, the first general symposium on lattice theory was held in Charlottesville in conjunction with a regular meeting of the Amer­ ican Mathematical Society. The three principal addresses on that occasion were entitled: Lattices and their Applications, On the Application of Struc­ ture Theory to Groups, and The Representation of Boolean Algebras. It is interesting to observe that the first and last of these titles