LATTICE THEORY.

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2 LATTICE THEORY

3 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

4 PROCEEDINGS OP SYMPOSIA IN PURE MATHEMATICS VOLUME II LATTICE THEORY AMERICAN MATHEMATICAL SOCIETY 190 HOPE STREET, PROVIDENCE, RHODE ISLAND 1961

5 Library of Congress Catalog Number 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.

6 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 By P. M. WHITMAN Generalized partitions and lattice embedding theorems By JURIS HARTMANIS Sublattices of free lattices 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 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 By LEON HENKIN and ALFRED TARSKI Injective and projective Boolean algebras By P. R. HALMOS

7 vi CONTENTS Cardinal and ordinal multiplication of relation types By C. C. CHANG Some questions about complete Boolean algebras By R. S. PIERCE Retracts in Boolean algebras By PHILIP DWINGER PART IV. APPLICATIONS OF LATTICE THEORY Lattices in applied mathematics By GARRETT BIRKHOFF On the lattice of normal subgroups of a group By MARSHALL HALL, JR. Locally compact topological lattices By L. W. ANDERSON Function lattices By F. W. ANDERSON INDEX

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

8 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 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 monograph 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 decompositions, 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.

9 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 progressed, 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-complemented, 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

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

11 204 INDEX Cardinal multiplication, 123 cancellation problem for, 127 Cantor-Bernstein problems for, 126 square root problem for, 127 Cartesian number system, 76 Center, 52 Central cover, 55 decomposition, 55 Chain separating, 200 weakly separating n-, 199 Chain condition, 51 ascending, 9 Close, 24 Compact, 11 Compactly generated, Comparability theorem, Complemented, 51 dimensionally CA a, 98 Complemented modular lattices Baer non-desarguesian, 75 Desarguesian, 71 Moufang non-desarguesian, 72 normal completion of, 78 Complete Boolean algebra, 114 lattice, 51 retract, 141, 145 Completely free lattice, 18, 32 sublattices, 32 Completeness theorem, 107 for predicate calculus, 94 Congruence relations, 3, 90 permuting, 5 Connected, 123 Continuity ascending, 51 descending, 51 Continuous geometry, 51 lattice operations, 51 lattices of functions, 201 part, 55 ring, 59 Convex, locally, 195 Coordinatization procedure, 58 theorem, 57 Countable Boolean algebras, 117 Criticality, 178 Cylindric algebras, 83 a-dimensional, 97 special, 87 Cylindric field, 92, 100 of subsets, 92 Cylindric set algebra of dimension a, 100 relativized, 104 special, 92 Cylindrification, 88 parallel to the kth axis, 92 Decision problem, 19, 96, 107 Decompositions central, 55 irredundant, 6 replaceable, 12 Desargues, Theorem of, 185 Desarguesian complemented lattices, 71 Baer non-, 75 Moufang non-, 72 Descending continuity, 51 Detachment, 84, 86 Diagonal element, 88 set, 92 Dimension function, set, 89, 98 Dimensionally complemented CA Direct product, 118 union, 4 Discrete part, 55 Distributive free lattices, 19 locally, 9 quotients, 192 Domain, 123 Dual ideal, 9 Elementary class, 96 universal, 101 Embedded, 26 neatly, 100, 102 Embedding, 22 neat, 99 theorems, 26 Equational class, 96, 105 Equidimensionality, 54 Equivalence relations, 54 lattices of, 22

12 INDEX 205 Ergodic averaging, 172 quasi hypothesis, 173 theorem, 173 Extension, 115 /-algebra, 167 Factor, 51 Factorization cardinal, 124 ordinal, 125 Field, 123 cylindric, 92, 100 cylindric of subsets, 92 Finite axiomatizability, 96 dimensional lattice, 7 locally dimensional CA a, 98 Finitely generated subalgebra, 101 Finiteness, 54 Frame, normalized, 71 Free algebra, 94, 116 Boolean algebra, 19 completely sublattices, 32 sum, 118 Free lattice, 17 completely, 18, 32 distributive, 19 modular, 18, 190 on n generators, 31 Generalized partitions, 22 Post algebra, 44 regular ring, 67 Geometric lattice, 185 Geometries lattice of, 22, 24 lattice of all, 22 Geometry, continuous, 51 projective, 28-29, 51 Group modular lattice, 191 Hilbert space, 51 Homogeneous, 66 basis, 57 basis of order n, 71 Boolean algebras, 129 Homomorphic images, 90, 104 Homomorphisms, 26, 29 Ideal dual, 9 in an SCA, 91 J in a CA«, 99 principal right, 57 R-, 168 Idempotent, R-, 168 Importance function, 183 Indecomposable directly, 90 subdirectly, 90 Independence, 52 Independent residually, 53 strongly, 53 Injective, 115 retract, 115 Irreducible, 6, 51 completely, 6 r-, 11 subdirectly, 4 Irredundant decompositions, 6 Isomorphism, 123 Isotone operators, 164 i -Boolean algebras, 200 X-function lattice, 199 Z-algebra, 167 (L)-space, abstract, 174, 180 Lattice, 22 complete, 51 continuous operations, 51 finite dimensional, 7 geometric, 185 group modular, 191 identity, 58 X-function, 199 non-modular, 161 normal completion of complemented modular, 78 of all geometries, 22 of continuous functions, 201 of equivalence relations, 22 of geometries, 22, 24 of subspaces, 23-24, 26, orthocomplemented modular, 156 semi-modular, 161 skew, 19 symmetric orthocomplemented, 161 topological, 195 word, 32

13 206 INDEX Lattice, free, 17 completely, 18, 32 distributive, 19 modular, 18, 190 on n generators, 31 Lattices, Desarguesian complemented modular, 71 Baer non-, 75 Moufang non-, 72 Lifting, 115 Linear operator, primitive, 180 Locallyconvex, 195 distributive, 9 finite dimensional CA a, 98 modular, 9 projective, 67 Markoff Hypothesis of, 175 processes, 174 Matrix, primitive, 179 Metric completion, 60 projective, 181 Model, 94, 106 Modular, 51 locally, 9 pair, 56 Modular lattice free, 18, 190 group, 191 non-, 161 orthocomplemented, 156 semi-, 161 Modular lattices, complemented Baer non-desarguesian, 75 Desarguesian, 71 Moufang non-desarguesian, 72 normal completion of, 78 Moufang non-desarguesian complemented modular lattices, 72 Multiplication, cardinal, 123 cancellation problems for, 127 Cantor - Bernstein problems for, 126 square root problem for, 127 Multiplication, ordinal, 123 cancellation problems for, 125 Cantor - Bernstein problems for, 127 square root problem for, 126 Multiplicative processes, 178, 180?i-chain, weakly separating, 199 Non-modular lattice, 161 Normalized frame, 71 Number system, cartesian, 76 Numerical dimension function, 54 Observables, 157 Operators averaging, 163, 165 isotone, 164 primitive linear, 180 regular Reynolds, 169 Reynolds, 165 transition, 178 Order, 66 type, 124 Ordered relation, 123 Ordinal addition, 123 factorization, 125 Ordinal multiplication, 123 cancellation problems for, 125 Cantor-Bernstein problems for, 127 square root problem for, 126 Ore's theorem, 15 Orthocomplemented, 53 modular lattice, 156 symmetric lattice, 161 Partitions, 22 generalized, 22 of type n, 23 superposing, 56 Permuting congruence relations, 5 Perspective, 53 Perspectivity, transitivity of, 53 Positive, uniformly, 180 Post algebra, 43 generalized, 44 Predicate calculus, 85 completeness theorem, 94 with a variables, 105 Predictions, 157 Primitive linear operator, 180 matrix, 179 Principal right ideals, 57 Product direct, 118 subdirect, 98 Projective, 53, 115 algebras, 111 geometry, 28-29, 51

14 INDEX 207 Projective continued locally, 67 metric, 181 retract, 115 spaces, 56 Quantification, universal, 86 Quantifier algebras, 83 a-dimensional, 97 special, 87 Quantum logic, 156 mechanical, 61 Quasi-Ergodic Hypothesis, 173 Quotient algebras, 90 distributive, 192.R-ideal, 168 i?-idempotent, 168 r-irreducible, 14 Range, 123 Rank, 59 Recursive, 96 Reduct, 99 finite, 99 Reflexive relations, 123 Regular Reynolds operators, 169 Regular ring, 57, 65 generalized, 67 Related, 24 Relation algebras, 111 cardinal, 123 ordered, 123 reflexive, 123 square, 123 type of, 123 Relativized cylindric set algebras, 104 Replaceable decompositions, 12 Replacement property, 7 Representable CA«, 100 Representation, subdirect, 56 Representation theorem for BA's, 93 Representation theorem for SCA's, 93 first, 95 second, 95 Retract, 114 a-complete, 141, 144 complete, 141, 145 injective, 115 projective, 115 Reynolds operators, 165 regular, 169 Right ideals, principal, 57 Ring alternative, 58 continuous, 59 regular, 57, 65 regular generalized, 67 Semantical systems, 106 Semiatomic, 13 Semimodular, 8, 10 lattice, 161 Sentential calculus, 83 Separating chain, 200 Separating n-chain, weakly, 199 Set algebra cylindric of dimension a, 100 relativized cylindric, 104 special cylindric, 92 Simple, 90, 100 Simultaneous substitution, 109 for individual variables, Skew lattice, 19 Spaces, projective, 56 Spectral subspaces, 162 Square addition, 123 relation, 123 type, 124 Square root problem for cardinal multiplication, 127 for ordinal multiplication, 126 States, 157 Subalgebra, 90 finitely generated, 101 Subdirect product, 98 representation, 56 union, 4, 198 Subdirectly irreducible, 5 Sublattices, completely free, 32 Subsets, cylindric field of, 92 Subspaces, 23 lattice of, 23-24, 26, Superposing partitions, 56 Symmetric orthocomplemented lattice, 161 Syntactical systems, 106 Transcendental, purely, 59 Transition operators, 178 probabilities, 61 Transitive, 56 Transitivity of perspectivity, 53

15 208 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

Lattice Theory Lecture 4. Non-distributive lattices

Lattice Theory Lecture 4. Non-distributive lattices Lattice Theory Lecture 4 Non-distributive lattices John Harding New Mexico State University www.math.nmsu.edu/ JohnHarding.html jharding@nmsu.edu Toulouse, July 2017 Introduction Here we mostly consider