MATHEMATISCHE REIHE BAND 52 LEHRBÜCHER UND MONOGRAPHIEN AUS DEM GEBIETE DER EXAKTEN WISSENSCHAFTEN G. GRÄTZER GENERAL LATTICE THEORY GENERAL LATTICE THEORY hy George Grätzer Professor of Mathematics U niversity of Manitoha Springer Basel AG CIP-Kurztitelaufnahme der Deutschen Bibliothek Grätzer, George Generallattice theory. - 1. Auf). - Basel, S~uttgart: Birkhäuser, 1978. (Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften: Math. Reihe; Bd.52) ISBN 978-3-0348-7635-3 ISBN 978-3-0348-7633-9 (eBook) DOI 10.1007/978-3-0348-7633-9 © Springer Basel AG 1978 Originally published by Birkhäuser Verlag, Basel 1978 Softcover reprint of the hardcover 1st edition 1978 All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means electronic or mechanical, including photocopy, recording, or any information storage ami retrieval system, without permission in writing from the publisher. To my lamily, Gathy, Tom, and David, and to the memory 01 my lather, J ozsel CONTENTS PREFACE AND ACKNOWLEDGEMENTS IX INTRODUCTION . . . XI I. FIRST CONCEPTS 1. Two Definitions of Lattices 1 2. How to Describe Lattices . 9 3. Some Aigebraic Concepts . 15 4. Polynomials, Identities, and Inequalities 26 5. Free Lattices . . . . . . . . 32 6. Special Elements. . . . . . . 47 Further Topics and References . 52 Problems ......... . 56 II. DISTRIBUTIVE LATTICES 1. Characterization Theorems and Representation Theorems 59 2. Polynomials and Freeness . . . . . . . . . . . . . 68 3. Congruence Relations. . . . . . . . . . . . . . . 73 4. Boolean Algebras R-generated by Distributive Lattices 86 5. Topological Representation . . . . . . . . . . 99 6. Distributive Lattices with Pseudocomplementation 111 Further Topics and References . 120 Problems ................. . 126 III. CONGRUENCES AND IDEALS 1. Weak Projectivity and Congruences 129 2. Distributive, Standard, and Neutral Elements 138 3. Distributive, Standard, and Neutral Ideals 146 4. Structure Theorems . . . . . 151 Further Topics and References . 158 Problems ......... . 159 IV. MODULAR AND SEMIMODULAR LATTICES 1. Modular Lattices. . . 161 2. Semimodular Lattices 172 3. Geometrie Lattices . . 178 VIII Contents 4. Partition Lattices . . . . . . 192 5. Complemented Modular Lattices 201 Further Topics and References . 218 Problems ......... . 224 V. EQUATIONAL CLASSES OF LATTICES 1. Characterizations of Equational Classes . . . 227 2. The Lattice of Equational Classes of Lattices 236 3. Finding Equational Bases . . . 243 4. The Amalgamation Property 252 Further Topics and References . 260 Problems ..... . 262 VI. FREE PRODUCTS 1. Free Products of Lattices 265 2. The Structure of Free Lattices . 282 3. Reduced Free Products . . . . 288 4. Hopfian Laitices . . . . . . . 298 Further Topics and References . 303 Problems ...... . 306 CONCLUDING REMARKS 311 BIBLIOGRAPHY . . . 316 TABLE OF NOTATION 362 INDEX ....... . 365 PREFACE AND ACKNOWLEDGEMENTS A book that is more than twelve years in the making has 0, long history and its final form is shaped by many. It 0,11 started in my formative years as 0, mathematician, 1955-1961, when I worked with E. T. Schmidt. We often commented upon the need for a. two- or three-volume work on lattice theory tha.t would treat the subject in depth. We feIt, however, tha.t the time was not ripefor sucha.project. Forinstance,no such work would be complete without presenting a.t lea.st one example of 0, nondistribu tive uniquely complemented la.ttice. We did not know how to do it without reproducing the a.lmost thirty pages of the famous proof of R. P. Dilworth. We also thought that much more had to be lea.rned about free lattices and equational classes of la.ttices before the project could be a.ttempted. In 1962, I wrote 0, proposal for a. volume on lattice theory that would attempt to survey the whole field in depth. Apart from doing the research necessary for the proposa.l, no writing was done on this book. M. H. Stone offered to publish 0, book on universa.l algebra in the D. Van Nostrand University Series in Higher Mathematics and I concentra.ted on that book until the end of 1967. Ma.ybe because mathematicians in general (or I, in particular) are Iike hobbits (according to J. R. R. Tolkien [1954]: "Hobbits deIighted in such things if they were accurate: they liked to have books filled with things they already knew, set out fair and square with no contradictions.") or maybe beca.use I feIt that the need for an in depth book on lattice theory had not yet been satisfied, I started in 1968 on this book. In the academic year 1968-1969, I gave a. course on lattice theory and I wrote 0, set of lecture notes. The present first two chapters are based on those notes. This material was augmented by 0, chapter on pseudocomplemented distributive lattices and published under the title "Lattice Theory: First Concepts and Distri butive Lattices" in 1971. (This book will be referred to as FC.) The Introduction of this book promised 0, compa.nion volume on generallattices. A number of research breakthroughs in the sixties now supplied me with the material (including almost 0,11 of Chapters V and VI) I needed to complete the project. But then it became apparent that a. complete revision of my plans was in order. While back in the late fifties it seemed reasonable to try to give 0, complete picture of lattice theory, this became pa.tently unfeasible in the seventies. For instance, in 1958 there was one paper on Stone algebras; by 1974 there were more than fifty papers on Stone algebras and related problems. A number of books have appeared dealing with specialized aspects of lattice theory and with various a.pplications. x Pl'eface and Acknowledgements Besides, my experience with the writing of Universal Algebra taught me not to stray too far away from my research interests. Thus it was decided that while I try to include all the basic material and research methods, the illustrations will be chosen, as far as possible, from fields in which I have some personal interest. Another change took place in the publishing field. For the second volume it became desirable to choose a publisher with a greater interest in monographs. The new ar rangement made it necessary to produce a volume that does not depend on the previous publication. That is why most of the first two chapters of the former book are repro duced here (augmented by a new section, several newexercises, with updated Further Topics and References, and wit.h a new set of Problems), thus giving the reader a self-contained book. The work on this new book started in 1972 and then continued with an advanced course on lattice theory at the University of Manitoba in 1973-1974. The lecture notes of this course form the basis of most of Chapters IH-VI. I amgrateful to mystudentswhotookthe courses in 1968-1969 or in 1973-1974 and also to my colleagues who attended for their helpful criticisms and for many simplified proofs. A corrected version of the first set of notes was read by R. Balbes, P. Bur meister, M. I. Gould, J. H. Hoffman, K. M. Koh, H. Lakser, S. M. Lee, R. Pad manabhan, P. Penner, and C. R. Platt. B. J6nsson, reading the manuscript of FC for the publisher, offered many useful suggestions. The first part was proofread by R. Anto nius, J. A. Gerhard, K. M. Koh, W. A. Lampe, R. W. Quackenbush, and I. Rival. Many readers, in particular D. D. Miller, sent me corrections to FC; this made it possible for me to improve some parts of FC that are being reproduced here. The second set of notes was distributed widely and I am grateful to all who of fered corrections, in particular, to K. A. Baker, C. C. Chen, M.1. Gould, D. Haley, K. M. Koh, V. B. Lender, G. H. Wenzel, and B. Wolk. In the proofreading of the present volume I was assisted by M. E. Adams, R. Beazer, K. A. Baker, J. Berman, B. A. Davey, J. A. Gerhard, M. 1. Gould, D. Haley, D. Kelly, C. R. Platt, and G. H. Wenzel. A great deal of organizational work was necessary in the distribution of manuscripts and the collation of corrections; this was faithfully carried out by R. Padmanabhan. M. E. Adams undertook the arduous task of getting the manuscript ready for the publisher. I received help from various individuals in specific areas, including M. Doob (matroids), I. Rival (exercises on combinatorial topics), R. Venkataraman (partially ordered vector spaces), B. Wolk (projective geometry). Thanks are due to the National Research Council of Canada for sponsoring much of the original research that has gone into this book and to Professor N. S. Mendel sohn for creating a very good environment for work. Mrs. M. McTavish did an ex cellent job of typing and retyping the manuscript. Finally, I would like to thank the members and the many visitors of my seminar who, over aperiod of eight years, have been lecturing an average of four hours a week, 52 weeks a year, in an attempt to teach me lattice theory. Without their help I could not even have tried. Despite the improvements so generously offered by so many, I am sure my original work can still be recognized: all the remaining mistakes are my own.