Garrett Birkhoff

Subdirect unions in universal algebra

The number of distinct operations (that is, the range of the variable a) may be infinite, but for our main result (Theorem 2), we shall require every n(a) to be finite—that is, it will concern algebras with finitary operations. The concepts of subalgebra, congruence relation on an algebra, homomorphism of one algebra A onto (or into) another algebra with the same operations, and of the direct union AiX • • • XAr of any finite or infinite class of algebras with the same operations have been developed elsewhere. More or less trivial arguments establish a manyone correspondence between the congruence relations 0i on an algebra A and the homomorphic images Hi — 6i(A) of the algebra (isomorphic images being identified); moreover the congruence relations on A form a lattice (the structure lattice of A). In this lattice, the equality relation will be denoted 0 ; all other congruence relations will be called proper. More or less trivial arguments also show (cf. Lattice theory, Theorem 3.20) that the isomorphic representations of any algebra A as a subdirect union, or subalgebra S^HiX • • • XHr of a direct union of algebras Hi, correspond essentially one-one to the sets of congruence relations 0i on A such that A0t=O. In fact, given such a set of Oi, the correspondence

Von Neumann and lattice theory

1. Introduction. John von Neumanns brilliant mind blazed over lattice theory like a meteor, during a brief period centering around 1935-1937. With the aim of interesting him in lattices, I had called his attention, in 1933-1934, to the fact that the sublattice generated by three subspaces of Hubert space (or any other vector space) contained 28 subspaces in general, to the analogy between dimension and measure, and to the characterization of projective geometries as ir-reducible, finite-dimensional, complemented modular lattices. As soon as the relevance of lattices to linear manifolds in Hubert space was pointed out, he began to consider how he could use lattices to classify the factors of operator-algebras. One can get some impression of the initial impact of lattice concepts on his thinking about this classification problem by reading the introduction of [62J, 1 in which a systematic lattice-theoretic classification of the different possibilities was initiated. In particular, he saw that factors of Type Hi gave rise to continuous-dimensional modular subspace lattices. However, von Neumann was not content with considering lattice theory from the point of view of such applications alone. With his keen sense for axiomatics, he quickly also made a series of fundamental contributions to pure lattice theory. The primary aim of the following paragraphs is to sketch these contributions.

Lattices and their applications

It is my privilege to introduce to this Society a vigorous and promising younger brother of group theory, by name, lattice theory. Among other things, I shall try to bring out the family resemblance. It is generally recognized that some familiarity with the notions of group, subgroup, normal subgroup, inner automorphism, commutator, and their technical properties, in a word, with group theory, is an essential preliminary to the understanding of algebraic equations, of differential equations, of the relation between the different branches of geometry, of automorphic functions, of crystallography, and of many other parts of mathematics and mathematical physics. I shall try to convince you that, in the same way, some familiarity with the notions of lattice, sublattice, the modular identity, dual automorphism, chain, and their technical properties, in a word, with lattice theory, is an essential preliminary to the full understanding of logic, set theory, probability, functional analysis, projective geometry, the decomposition theorems of abstract algebra, and many other branches of mathematics. It is often said that mathematics is a language. If so, group theory provides the proper vocabulary for discussing symmetry. In the same way, lattice theory provides the proper vocabulary for discussing order, and especially systems which are in any sense hierarchies. One might also say that just as group theory deals with permutations, so lattice theory deals with combinations. One difference between the two is that whereas our knowledge of group theory has increased by not more than fifty per cent in the last thirty years, our knowledge of lattice theory has increased by perhaps two hundred per cent in the last ten years. Lattice theory is based on a single undefined relation, the inclusion relation x^y. In this it resembles group theory, which is based on one undefined operation, group multiplication. The relation of inclusion is assumed to satisfy three primary postulates:

The radical of a group with operators

In a recent note (The radical of a non-associative algebra, Bull. Amer. Math. Soc. vol. 48 (1942) pp. 893-897) A. A. Albert has defined the radical of a general linear algebra, and deduced some of its properties from his theory of isotopy. The purpose of the present note is to extend this concept to groups with operators, and to derive similar properties from lattice theory. Let G be a group with a class Q of endomorphisms including all inner automorphisms. By an ideal, We mean a subgroup 5 of G such that sÇ:S and ze/££2 imply swÇ.S. Clearly G and the group identity 0 are ideals; any other ideal is called a proper ideal. We recall that the ideals of G form a modular lattice] we shall assume below that this has finite length. We define G to be prime if and only if it has no proper ideals ; it is well known (and easy to prove) that G/S is prime if and only if 5 is maximal. Now let 3 be any class of groups with operators (specializing to zero algebras) which is invariant under isomorphisms, and let us define a simple group (with operators) to be any prime group not in the exceptional class 3 We define a direct sum of prime groups to be semiprime, and (following Albert and others) a direct sum of simple groups to be semisimple. We further define the

A ternary operation in distributive lattices

-ideal <£ of G as the intersection of all maximal ideals (by analogy with the 0-subgroup of a group), the radical R of G as the intersection of all maximal ideals 5 such that G/S is simple, and denote by Z the intersection of all maximal ideals T such that (G/T)E8-