AMERICAN MATHEMATICAL SOCIETY COLLOQUIUM PUBLICATIONS VOLUME XXV LATTICE THEORY BY GARRETT BIRKHOFF PUBLISl'l:BD BY THE AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE lsLAND First Edition, 1940 Second (Revised) Edition, 1948 Third Edition, Second Printing, 1973 COPYRIGHT, 1940, 1948, 1967, BY THE AMERICAN MATHEMATICAL SOCIETY All Rights R68erved No portion of this book may be reproduced without the written permission of the publisher. Library of Congress Catalog Card Number 66--23707 Printed in the United States of America PREFACE TO THE THffiD EDITION The purpose of this edition is threefold: to make the deeper ideas of lattice theory accessible to mathematicians generally, to portray its structure, and to indicate some of its most interesting applications. As in previous editions, an attempt is made to include current developments, including various unpublished ideas of my own; however, unlike previous editions, this edition contains only a very incomplete bibliography. I am summarizing elsewheret my ideas about the role played by lattice theory in mathematics generally. I shall therefore discuss below mainly its logical structure, which I have attempted to reflect in my table of contents. The beauty of lattice theory derives in part from the extreme simplicity of its basic concepts: (partial) ordering, least upper and greatest lower bounds. In this respect, it closely resembles group theory. These ideas are developed in Chapters I-V below, where it is shown that their apparent simplicity conceals many subtle variations including for example, the properties of modularity, semimodularity, pseudo-complements and orthocomplements. At this level, lattice-theoretic concepts pervade the whole of modern algebra, though many textbooks on algebra fail to make this apparent. Thus lattices and groups provide two of the most basic tools of" universal algebra ", and in particular the structure of algebraic systems is usually most clearly revealed through the analysis of appropriate lattices. Chapters VI and VII try to develop these remarks, and to include enough technical applications to the theory of groups and loops with operators to make them convincing. A different aspect of lattice theory concerns the foundations of set theory (including general topology) and real analysis. Here the use of various (partial) orderings to justify transfinite inductions and other limiting processes involves. some of the most sophisticated constructions of all mathematics, some of which are even questionable! Chapters VIII-XII describe these processes from a lattice theoretic standpoint. Finally, many of the deepest and most interesting applications of lattice theory concern (partially) ordered mathematical structures having also a binary addition or multiplication: lattice-ordered groups, monoids, vector spaces, rings, and fields (like the real field). Chapters XIII-XVII describe the properties of such systems, t G. Birkhoff, What can lattices do for you?, an article in Trern.ls in Lattice Theory, James C. Abbot, ed., Van Nostrand, Princeton, N.J., 1967. iii iv PREFACE and also those of positive linear operators on partially ordered vector spaces. The theory of such systems, indeed, constitutes the most rapidly developing part of Ia.ttice theory at the present time. The Ia.bor of writing this book has been enormous, even though I have made no attempt at completeness. I wish to express my deep appreciation to those many colleagues and students who have criticized parts of my manuscript in various stages of preparation. In particular, I owe a very real debt to the following: Kirby Baker, Orrin Frink, George Gratzer, C. Grandjot, Alfred Hales, Paul Halmos, Samuel H. Holland, M. F. Janowitz, Roger Lyndon, Donald MacLaren, Richard S. Pierce, George Raney, Arlan Ramsay, Gian-Carlo Rota, Walter Taylor, and Alan G. Waterman. My thanks are also due to the National Science Foundation for partial support of research in this area and of the preparation of a preliminary edition of notes, and to the Argonne National Laboratory and the Rand Corporation for support of research into aspects of lattice theory of interest to members of their staffs. Finally, I wish to thank Laura Schlesinger a.nd Lorraine Doherty for their skillful typing of the entire manuscript. TABLE OF CONTENTS CHAPTER I TYPES OF LATTICES 1 CHAPTER II POSTULATES FOR LATTICES . 20 CHAPTER III STRUCTURE AND REPRESENTATION THEORY 55 CHAPTER IV GEOMETRIC LATTICES . 80 CHAPTER V COMPLETE LATTICES . 111 CHAPTER VI UNIVERSAL ALGEBRA . . 132 CHAPTER VII APPLICATIONS TO ALGEBRA. . 159 CHAPTER VIII TRANSFINITE INDUCTION . 180 CHAPTER IX APPLICATIONS TO GENERAL TOPOLOGY . 211 CHAPTER X METRIC AND TOPOLOGICAL LATTICES. . 230 CHAPTER XI BOREL ALGEBRAS AND VON NEUMANN LATTICES . 254 CHAPTER XII APPLICATIONS TO LOGIC AND PROBABILITY . 277 v vi CONTENTS CHAPTER XIII LATTICE-ORDERED GROUPS . . 287 CHAPTER XIV LATTICE-ORDERED MONOIDS . 319 CHAPTER XV VECTOR LATTICES . . 347 CHAPTER XVI POSITIVE LL~EAR OPERATORS . 380 CHAPTER XVII LATTICE-ORDERED RINGS . 397 BIBLIOGRAPHY . 411 INDEX . 415 CHAPTER I TYPES OF LAITICES 1. Posets; Chains Pure lattice theory is concerned with the properties of a single undefined binary relation ~ , to be read "is contained in", "is a part of", or "is less than or equal to". This relation is assumed to have certain properties, the most basic of which lead to the following concept of a "partially ordered set ", alias "partly ordered set " or " poset ". DEFINITION. A poset is a set in which a binary relation x ~ y is defined, which satisfies for all x, y, z the following conditions: PL For all x, x ~ x. (Reflexive) P2. If x ~ y and y ;:;; x, then x y. (Antisymmetry) P3. If x ;:;; y and y ;:;; z, then x ~ z. (Transitivity) Ifx ;:;! y and x "!: y, one writes x < y, and says that x "is less than" or" properly contained in" y. The relation x ;:;! y is also written y ~ x, and ready contains x (or includes x). Similarly, x < y is also written y > x. The above notation and terminology are standard. There are countless familiar examples of partly ordered sets-Le., of mathe matical relations satisfying Pl-P3. Three of the simplest are the following. Emmple 1. Let :E(I) consist of all subsets of any class I, including I itself and the void class 0; and let x ;;;:; y mean x is a subset of y. Emmple 2. Let z+ be the set of positive integers; and let x ;:;! y mean that x divides y. . Emmple 3. Let F consist of all real single-valued functions j(x) defined on -1 ~ x ~ l; and let/~ g mean tha.t/(x) ;;;; g(x) for every x with -1 ~ x ~ 1. We now state without proof two familiar laws governing inclusion relations, which follow from Pl-P3. LEMMA 1. In any poset, x < x for no x, while x < y and y < z imply x < z. Conversely, if a binary relation < satisfies the two preceding oondition.s, define x ;:ii y to mean that x < y or x = y; then the relation ~ satisfies Pl-P3. In other words, strict inclusion is characterized by the anti-reflexive and transi tive laws. It is easily shown that a poset P can contain at most one element a which satisfies a ~ x for all x E P. For if a and b are two such elements, then a ~ b and also b ~ a, whence a = b by P2. Such an element, if it exists, is denoted by 1 2 TYPES OF LATTICES I 0, and is called the kast element of X. The dual greate.st element of P, if it exists, is denoted by I. The elements 0and1, when they exist, are called universal bounds of P, since then 0 ;;ii x ;;;;; I for all x E P. LEMMA 2. If X1 ~ X2 ~ ••• ~ x,. ~ Xi, then X1 = X2 = ... = x,.. (.Anti circularity) Example 4. Let R be the set of real numbers, and let x ~ y have its usual meaning for real numbers. The relation of inclusion in this and other important posets satisfies: P4. Given x and y, either x ;'.ii y or y ~ x. DEFINITION. A poset which satisfies P4 is said to be "simply" or "totally" ordered and is called a chain. In other words, of any two distinct elements in a chain, one is less and the other greater. Clearly, the posets of Examples 1-3 are not chains: they contain pairs of elements x, y which are irumnparable, in the sense that neither x ~ y nor y ;;;! x holds. From Examples l-4 above, many other posets can be constructed as subsets. More precisely, let P be any poset, and let 8 be any subset of P; define the relation x ~ y in 8 to mean that x ~yin P. Conditions Pl-P3 being satisfied by ~ in P, they are satisfied a fortiori in 8. A similar observation holds for P4. There follows, trivially, THEOREM l. .Any sithset 8 of a poset Pis itself a poset under the same inclusion rel,ation (restricted to 8 ). .Any subset of a chain is a chain. Thus, the set Z + of positive integers is a chain under the relation ~ of relative magnitude of Example 4, though a poset which is not a chain under the partial ordering of Example 2. Exampk 5. (a) The set {1, 2, · · ·, n} forms a chain n (the ordinal number n) in its natural order. (b) When unordered, so that no two elements are comparable, it forms. another poset (the cardinal number n). The family of all subsets of any class distinguished by any given special property forms a poset under set-inclusion. Thus this is true of the subgroups of any group, the vector subspaces of any vector space, the Borel subsets of any T -space 0 etc. For instance, an ideal is a subset H of a ring R distinguished by the prop erties (i) a, b E R imply a - b E R, and (ii) a EH and b E R imply ab EH and ba EH. Specialized to this case, the principle stated above yields an important example, which will be studied more deeply in Chapters VII and XIV below. Example 6. Let P consist of the ideals H, J, K,· · · of any ring R; let H ~ K mean that H is a subset of K (i.e., that H c K). 2. Isomorphism; Duality A function 8: P ~ Q from a poset P to a poset Q is called order-preserving or isotone if it satisfies (1) x ~ y implies O(x) ~ O(y). §2 ISOMOBPBISM; DUALITY 3 An isotone function which has an isotone two-sided inverse is called an isomorphism. In other words, an isomorphism between two posets P and Q is a bijection which satisfies (1) and a]so (l') B(x) ~ B(y) implies x ~ y. An isomorphism from a poset P to itse1f is called an a'Ui-Orrwrphism. Two posets P and Qare called isomorphic (in symbo]s, P ~ Q), if and on1y if there exists an isomorphism between them. The converse of a re]ation p is, by definition, the re1ation p such that xpy (read, "x is in the re]ation p to y ") if and on]y if ypx. Thus the converse of the relation "includes " is the relation "is inc1uded in"; the converse of "greater than" is "less than ". It is obvious from inspection of conditions Pl-P3 that THEOREM 2 (DUALITY PRINCIPLE). The converse of any partial ordering is itself a partial ordering. DEFINITION. The dual of a poset. X is that poset X defined by the converse par tial ordering relation on the same elements . .f, . Since X ;;;:;: this termino]ogy is legitimate: the relation of duality is symmetric. DEFINITlON. A function 8: P - Q is anlitone if and on1y if (2) x ~ y implies B(x) ~ B(y), (2') B(x) ~ B(y) implies x ~ y. A bijection (one-one correspondence) 8 which satisfies (2)-(2') is called a. dual isomorphism. We shall refer to systems isomorphic with X as" dual" to X. Obviously posets are dual in pairs, whenever they are not self-dual. Similarly, definitions and theorems about posets are dual in pairs whenever they are not se1f-dual; and if any theorem is true for all posets, then so is its dual. As we shall see later, this Duality Principle applies to algebra, to projective geometry, and to logi<). Many important posets are se1f-dual (i.e., anti-isomorphic with themselves). Thus Example l of §2 is self-dual; the correspondence which carries each subset into its complement is one-to-one and inverts inclusion. Similarly the set of all linear subspaces of n-dimensional Euclidean space which contains the origin is se1f-dual: the correspondence carrying each subspace into its orthogonal comple ment is one-to-one and inverts inclusion. In these cases the self-duality is of period two: the image (x')' of the image x' of any xis again x. Such self-dualities (dual automorphisms) are called invol'ldions. Exercises for §§1-2: I. Prove Lemma L 2. Prove Lemme. 2. 3. Show the.t there are exa.ctly three different we.ys of pe.rtly ordering a set of two elements. 4. (a) Show the.t there are just two nonisomorphic posets of two elements, both of which are self-dual. 4 TYPES OF LATTICES I (b) Show that there are five nonisomorphic poaets of three elements, three of which are self-dual. •5. (a) Let G(n) denote the number of nonisomorphic poaets of n elements. Show that G(4) 16, G(5) 63, G(6) = 318. (I. Rose R. T. Sasaki) (b) G*(n) denote the number of different partial orderings of n elements. Show G*(2) = 3, G*(3) 19, G*(4) = 219, G*(5) = 4231, G*(6) = 130,023, G'(7) = 6,129,859. (c) How many of the preceding give self-dual poaets? (d) Is G*(n) odd for all n? Justify. 3. Diagrams; Graded Posets The notion of " immediate superior" in a hierarchy can be defined in any poset, as follows. DEFINITION. By "a covers b" in a poset P, it is meant that a > b, but that a > x > b for no x E P. By the order n(P) of a poset Pis meant the (cardinal) number of its elements. When this number is finite, P is called a "finite" poset. Using the covering relation, one can obtain a graphical representation of any finite poset P as follows. Draw a small circle to represent each element of P, placing a higher than b whenever a > b. Draw a straight segment from a to b whenever a covers b. The resulting figure is called a diagram of P: examples are shown in Figures la-le below. Since a > b if and only if one can move from a to b downward along some broken line, it is clear that any finite poset is defined up to isomorphism by its diagram. It is 8.'lso clear that the diagram of the dual P of a poset P is obtained from that of P by turning the latter upside down. FIGURE 1. Examples of diagrams DEFINITION. By a least element of any subset X of P, we mean an element a EX such that a ~ x for all x EX. By a greatest element of X, we mean an element b EX such that b ~ x for all x EX. The preceding concepts are not to be confused with the concepts of minimal and maximal elements. A minimal element of a subset X of a partly ordered set Pis an element a such that a < x for no x EX; maximal elements are defined dually. Clearly, a least element must be minimi,tl and a greatest element maximal, but the converse is not true. THEOREM 3. Any finite nonempty subset X of a poset has minimal and maximal members.

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