http://dx.doi.org/10.1090/pspum/002

LATTICE THEORY PROCEEDINGS OF THE SECOND SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY

Held at the U.S. Naval Postgraduate School Monterey, California April 16-18, 1959

WITH THE FINANCIAL SUPPORT OF THE NATIONAL SCIENCE FOUNDATION

R. P. Dilworth EDITOR PROCEEDINGS OP SYMPOSIA IN PURE MATHEMATICS

VOLUME II

LATTICE THEORY

AMERICAN MATHEMATICAL SOCIETY

190 HOPE STREET, PROVIDENCE, RHODE ISLAND

1961 Library of Congress Catalog Number 50—1183

Prepared by the American Mathematical Society under Grant No. NSF-G8292 with the National Science Foundation

Copyright © 1960 by the American Mathematical Society. Printed in the United States of America. All rights reserved except those granted to the United States Government. Otherwise, this book, or parts thereof, may not be reproduced in any form without permission of the publishers. CONTENTS

PREFACE ...... vii

PART I. LATTICE STRUCTURE THEORY Structure and decomposition theory of lattices . . . 3 By R. P. DILWORTH

Status of word problems for lattices ...... 17 By P. M. WHITMAN

Generalized partitions and lattice embedding theorems . . .22 By JURIS HARTMANIS

Sublattices of free lattices ...... 31 By R. A. DEAN

Prime ideal characterization of generalized Post algebras . . 43 By C. C. CHANG and ALFRED HORN

PART II. COMPLEMENTED MODULAR LATTICES Complemented modular lattices ...... 51 By ISRAEL HALPERIN

Extensions of von Neumann's coordinatization theorem . . 65 By B. JONSSON

Coordinates in non-Desarguesian complemented modular lattices . 71 By K. D. FRYER

The normal completion of a complemented modular point lattice . 78 By J. E. MCLAUGHLIN

PART III. BOOLEAN ALGEBRAS Cylindric algebras ...... 83 By LEON HENKIN and ALFRED TARSKI

Injective and projective Boolean algebras . . . . .114 By P. R. HALMOS vi CONTENTS

Cardinal and ordinal multiplication of relation types . . .123 By C. C. CHANG

Some questions about complete Boolean algebras . . .129 By R. S. PIERCE

Retracts in Boolean algebras ...... 141 By PHILIP DWINGER

PART IV. APPLICATIONS OF LATTICE THEORY Lattices in applied mathematics ...... 155 By GARRETT BIRKHOFF

On the lattice of normal subgroups of a group . . . .185 By MARSHALL HALL, JR.

Locally compact topological lattices ...... 195 By L. W. ANDERSON

Function lattices ...... 198 By F. W. ANDERSON

INDEX ...... 205 PREFACE

This volume contains the papers presented at the Symposium on Partially Ordered Sets and Lattice Theory held in conjunction with the Monterey meeting of the American Mathematical Society in April 1959. The Symposium was sponsored by the American Mathematical Society and supported by a grant from the National Science Foundation. The interest and support of these organizations is gratefully acknowledged. Some twenty-one years earlier, on April 15, 1938, the first general symposium on lattice theory was held in Charlottesville in conjunction with a regular meeting of the American Mathematical Society. The three principal addresses on that occasion were entitled: Lattices and their Applications, On the Application of Structure Theory to Groups, and The Representation of Boolean Algebras. It is interesting to observe that the first and last of these titles appear again as section titles for the present Symposium. Furthermore the second title is still of current interest as evidenced by the paper of Marshall Hall. Nevertheless there have been major changes in emphasis and interest during the intervening years and thus some general comments concerning the present state of the subject and its relationship to other areas of mathematics appear to be appropriate. The theory of groups provided much of the motivation and many of the technical ideas in the early development of lattice theory. Indeed it was the hope of many of the early researchers that lattice-theoretic methods would lead to the solution of some of the important problems in group theory. Two decades later, it seems to be a fair judgment that, while this hope has not been realized, lattice theory has provided a useful framework for the formulation of certain topics in the theory of groups (for example, generalizations of the Jordan-Holder theorem) and has produced some interesting and difficult group-theoretic problems (cf. the excellent mono­ graph of M. Suzuki). On the other hand, the fundamental problems of lattice theory have, for the most part, not come from this source but have arisen from attempts to answer intrinsically natural questions concerning lattices and partially ordered sets; namely, questions concerning the de­ compositions, representations, imbedding, and free structure, of such systems. It should be pointed out that group theory and other areas of mathematics have furnished concepts and methods which have proved to be useful in the study of these questions. Thus the techniques associated with the study of composition series and chief series in group have been successfully applied to the structure of modular and semi-modular lattices. Set topology and ring theory have been the source of many fruitful ideas in the study of Boolean algebras. Also the theory of linear vector spaces and projective vii viii PREFACE geometries have contributed some of the basic methods for the development of the theory of complemented modular lattices and, in particular, continuous geometries. Nevertheless, as the study of these basic questions has pro­ gressed, there has come into being a sizable body of technical ideas and methods which are peculiarly lattice-theoretic in nature. These conceptual tools are intimately related to the underlying order relation and are particularly appropriate for the study of general lattice structure. At the 1938 Symposium, lattice theory was described as a " vigorous and promising younger brother of group theory". In the intervening years it has developed into a full-fledged member of the algebraic family with an extensive body of knowledge and a collection of exciting problems all of its own. Such outstanding problems as the construction of a set of structure invariants for certain classes of Boolean algebras, the characterization of the lattice of congruence relations of a lattice, the imbedding of finite lattices in finite partitions lattices, the word problem for free modular lattices, and the construction of a dimension theory for continuous, non-comple­ mented, modular lattices, have an intrinsic interest independent of the problems associated with other algebraic systems. Furthermore, these and other current problems are sufficiently difficult that imaginative and ingenious methods will be required in their solutions. A vigorous group of mathematicians are attacking these problems and the results of some recent progress may be found in the papers included in this volume.

R. P. DILWORTH

CALIFORNIA INSTITUTE OF TECHNOLOGY INDEX

Abstract (L)-space, 174, 180 Averaging Addition ergodic, 172 cardinal, 123 operators, 163, 165 ordinal, 123 Axiomatizability, finite, 96, 107 square, 123 Algebra Baer a-dimensional cylindric, 97 algebra, 76 a-dimensional quantifier, 97 non-Desarguesian complemented mod­ Baer, 76 ular lattices, 75 cylindric, 83 Base, 92, 100 cylindric set of dimension a. Basis, 66 100 homogeneous, 57 /-, 167 homogeneous — of order n, 71 free, 94, 116 Bernstein (See Cantor) generalized Post, 44 Boolean algebra, 10, 23, 25, 83, 85, 114, Z-, 167 174 of formulas, 84, 86 cardinal property of, 130 Post, 43 complete, 114 projective, 111 countable, 117 quantifier, 83 free, 19 quotient, 90 homogeneous, 129 relation, 111 K-y 200 relativized cylindric set, 104 special cylindric, 87 CAa special cylindric set, 92 atomistic, 100 special quantifier, 87 dimensionally complemented, 98 Algebra, Boolean, 10, 23, 25, 83, 85, 114, ideal J in, 99 174 locally finite dimensional, 98 cardinal property of, 130 representable, 100 complete, 114 Calculus countable, 117 predicate, 85 free, 19 predicate with a variables, homogeneous, 129 105 K-, 200 sentential, 83 Algebraic, 59 Cancellation problem a-complete retract, 141, 144 for cardinal multiplication, 127 a-dimensional cylindric algebra, 97 for ordinal multiplication, 125 a-dimensional quantifier algebra, 97 Canonical forms, 32 Alternative ring, 58 Cantor-Bernstein problems Arguesian, 65 for cardinal multiplication, 126 Arithmetical class, 96 for ordinal multiplication, 127 Ascending Cardinal chain condition, 9 addition, 123 continuity, 51 factorization, 124 Atomistic CAa, 100 property of Boolean algebras, 130 Atoms, 55 relation, 123 Automorphism, 60 type, 124 204 INDEX

Cardinal multiplication, 123 Cylindric algebras, 83 cancellation problem for, 127 a-dimensional, 97 Cantor-Bernstein problems for, 126 special, 87 square root problem for, 127 Cylindric field, 92, 100 Cartesian number system, 76 of subsets, 92 Center, 52 Cylindric set algebra Central of dimension a, 100 cover, 55 relativized, 104 decomposition, 55 special, 92 Chain Cylindrification, 88 separating, 200 parallel to the kth axis, 92 weakly separating n-, 199 Chain condition, 51 Decision problem, 19, 96, 107 ascending, 9 Decompositions Close, 24 central, 55 Compact, 11 irredundant, 6 Compactly generated, 11-12 replaceable, 12 Desargues, Theorem of, 185 Comparability theorem, 54-55 Desarguesian complemented Complemented, 51 lattices, 71 dimensionally CA , 98 a Baer non-, 75 Complemented modular lattices Moufang non-, 72 Baer non-Desarguesian, 75 Descending continuity, 51 Desarguesian, 71 Detachment, 84, 86 Moufang non-Desarguesian, 72 Diagonal normal completion of, 78 element, 88 Complete set, 92 Boolean algebra, 114 Dimension lattice, 51 function, 55-56 retract, 141, 145 set, 89, 98 Completely free Dimensionally complemented CA lattice, 18, 32 Direct sublattices, 32 product, 118 Completeness theorem, 107 union, 4 for predicate calculus, 94 Discrete part, 55 Congruence relations, 3, 90 Distributive permuting, 5 free lattices, 19 Connected, 123 locally, 9 Continuity quotients, 192 ascending, 51 Domain, 123 descending, 51 Dual ideal, 9 Continuous geometry, 51 Elementary class, 96 lattice operations, 51 universal, 101 lattices of functions, 201 Embedded, 26 part, 55 neatly, 100, 102 ring, 59 Embedding, 22 Convex, locally, 195 neat, 99 Coordinatization theorems, 26 procedure, 58 Equational class, 96, 105 theorem, 57 Equidimensionality, 54 Countable Boolean algebras, 117 Equivalence relations, 54 Criticality, 178 lattices of, 22 INDEX 205

Ergodic Ideal averaging, 172 dual, 9 quasi hypothesis, 173 in an SCA, 91 theorem, 173 J in a CA«, 99 Extension, 115 principal right, 57 R-, 168 /-algebra, 167 Idempotent, R-, 168 Factor, 51 Importance function, 183 Factorization Indecomposable cardinal, 124 directly, 90 ordinal, 125 subdirectly, 90 Field, 123 Independence, 52 cylindric, 92, 100 Independent cylindric of subsets, 92 residually, 53 Finite strongly, 53 axiomatizability, 96 Injective, 115 dimensional lattice, 7 retract, 115 Irreducible, 6, 51 locally dimensional CAa, 98 Finitely generated subalgebra, 101 completely, 6 Finiteness, 54 r-, 11 Frame, normalized, 71 subdirectly, 4 Free Irredundant decompositions, 6 algebra, 94, 116 Isomorphism, 123 Boolean algebra, 19 Isotone operators, 164 completely sublattices, 32 sum, 118 i£-Boolean algebras, 200 Free lattice, 17 X-function lattice, 199 completely, 18, 32 distributive, 19 modular, 18, 190 Z-algebra, 167 on n generators, 31 (L)-space, abstract, 174, 180 Lattice, 22 complete, 51 Generalized continuous — operations, 51 partitions, 22 finite dimensional, 7 Post algebra, 44 geometric, 185 regular ring, 67 group modular, 191 Geometric lattice, 185 identity, 58 Geometries X-function, 199 lattice of, 22, 24 non-modular, 161 lattice of all, 22 normal completion of complemented Geometry, 22-23 modular, 78 continuous, 51 of all geometries, 22 projective, 28-29, 51 of continuous functions, 201 Group modular lattice, 191 of equivalence relations, 22 of geometries, 22, 24 Hilbert space, 51 of subspaces, 23-24, 26, 28-29 Homogeneous, 66 orthocomplemented modular, 156 basis, 57 semi-modular, 161 basis of order n, 71 skew, 19 Boolean algebras, 129 symmetric orthocomplemented, 161 Homomorphic images, 90, 104 topological, 195 Homomorphisms, 26, 29 word, 32 206 INDEX

Lattice, free, 17 ?i-chain, weakly separating, 199 completely, 18, 32 Non-modular lattice, 161 distributive, 19 Normalized frame, 71 modular, 18, 190 Number system, cartesian, 76 on n generators, 31 Numerical dimension function, 54 Lattices, Desarguesian complemented modular, 71 Observables, 157 Baer non-, 75 Operators Moufang non-, 72 averaging, 163, 165 Lifting, 115 isotone, 164 Linear operator, primitive, 180 primitive linear, 180 Locally- regular Reynolds, 169 convex, 195 Reynolds, 165 distributive, 9 transition, 178 finite dimensional CAa, 98 Order, 66 modular, 9 type, 124 projective, 67 Ordered relation, 123 Ordinal Markoff addition, 123 Hypothesis of, 175 factorization, 125 processes, 174 Ordinal multiplication, 123 Matrix, primitive, 179 cancellation problems for, 125 Metric Cantor-Bernstein problems for, 127 completion, 60 square root problem for, 126 projective, 181 Ore's theorem, 15 Model, 94, 106 Orthocomplemented, 53 Modular, 51 modular lattice, 156 locally, 9 symmetric lattice, 161 pair, 56 Modular lattice Partitions, 22 free, 18, 190 generalized, 22 group, 191 of type n, 23 non-, 161 superposing, 56 orthocomplemented, 156 Permuting congruence relations, 5 semi-, 161 Perspective, 53 Modular lattices, complemented Perspectivity, transitivity of, 53 Baer non-Desarguesian, 75 Positive, uniformly, 180 Desarguesian, 71 Post algebra, 43 Moufang non-Desarguesian, 72 generalized, 44 normal completion of, 78 Predicate calculus, 85 Moufang non-Desarguesian complemented completeness theorem, 94 modular lattices, 72 with a variables, 105 Multiplication, cardinal, 123 Predictions, 157 cancellation problems for, 127 Primitive Cantor - Bernstein problems for, linear operator, 180 126 matrix, 179 square root problem for, 127 Principal right ideals, 57 Multiplication, ordinal, 123 Product cancellation problems for, 125 direct, 118 Cantor - Bernstein problems for, subdirect, 98 127 Projective, 53, 115 square root problem for, 126 algebras, 111 Multiplicative processes, 178, 180 geometry, 28-29, 51 INDEX 207

Projective continued Right ideals, principal, 57 locally, 67 Ring metric, 181 alternative, 58 retract, 115 continuous, 59 spaces, 56 regular, 57, 65 regular generalized, 67 Quantification, universal, 86 Quantifier algebras, 83 Semantical systems, 106 a-dimensional, 97 Semiatomic, 13 special, 87 Semimodular, 8, 10 Quantum lattice, 161 logic, 156 Sentential calculus, 83 mechanical, 61 Separating chain, 200 Quasi-Ergodic Hypothesis, 173 Separating n-chain, weakly, 199 Quotient Set algebra algebras, 90 cylindric of dimension a, 100 distributive, 192 relativized cylindric, 104 special cylindric, 92 .R-ideal, 168 Simple, 90, 100 i?-idempotent, 168 Simultaneous substitution, 109 r-irreducible, 14 for individual variables, 86-87 Range, 123 Skew lattice, 19 Rank, 59 Spaces, projective, 56 Recursive, 96 Spectral subspaces, 162 Reduct, 99 Square finite, 99 addition, 123 Reflexive relations, 123 relation, 123 Regular Reynolds operators, 169 type, 124 Regular ring, 57, 65 Square root problem generalized, 67 for cardinal multiplication, 127 Related, 24 for ordinal multiplication, 126 Relation States, 157 algebras, 111 Subalgebra, 90 cardinal, 123 finitely generated, 101 ordered, 123 Subdirect reflexive, 123 product, 98 square, 123 representation, 56 type of, 123 union, 4, 198 Relativized cylindric set algebras, 104 Subdirectly irreducible, 5 Replaceable decompositions, 12 Sublattices, completely free, 32 Replacement property, 7 Subsets, cylindric field of, 92 Representable CA«, 100 Subspaces, 23 Representation, subdirect, 56 Representation theorem for BA's, 93 lattice of, 23-24, 26, 28-29 Representation theorem for SCA's, 93 Superposing partitions, 56 first, 95 Symmetric orthocomplemented lattice, second, 95 161 Retract, 114 Syntactical systems, 106 a-complete, 141, 144 Transcendental, purely, 59 complete, 141, 145 Transition injective, 115 operators, 178 projective, 115 probabilities, 61 Reynolds operators, 165 Transitive, 56 regular, 169 Transitivity of perspectivity, 53 208 INDEX

Type Unit, 59 cardinal, 124 set, 92 of a relation, 123 Universal order, 124 elementary class, 101 square, 124 quantification, 86

Uniformly positive, 180 Weakly separating n-chain, 199 Union Word problems, 17 direct, 4 subdirect, 4, 198