Trends in LATTICE THEORY J. C. ABBOTT, GENERAL EDITOR United States Naval Academy Contributors GARRETT BIRKHOFF HENRY CRAPO and GIAN-CARLO ROTA SAMUEL S. HOLLAND, JR. GEORGE GRATZER VAN NOSTRAND REINHOLD COMPANY NEW YORK CINCINNATI TORONTO LONDON MELBOURNE UNIVERSAL ALGEBRA by George Grlitzer 1. Introduction. During my first course in abstract algebra and even for a long while after that I had difficulty remembering what computational rules can be applied in the different branches of algebra: rings, integral domains, fields, division rings, and so on. I wished there were a branch of algebra where I did not have to remember any axiom systems. I did not know that there was one, and that it was called universal algebra, or abstract algebra (as if algebra were not abstract enough), or general algebraic systems. Universal algebra can be defined as the study of properties that such diverse algebraic systems as groups, rings, lattices, and algebras (over fields)have in common. Have we not just defined the empty set? Not quite. Each of these examples can be considered as a set with a family of finitary operations, and this is the defini­ tion of a universal algebra. A universal algebra ~{ is an ordered pair <A; F> where A is a nonvoid set and F is a family of fini­ tary operations on A. In most cases F is finite. However if, for example, we want to consider a vector space as a universal algebra, then F will have to consist of the binary addition and of the set of unary operations fa(x) = ax, one for each element a of the un­ derlying field. Thus if the field is infinite, so is F. If ~{ = <A; F> is a universal algebra and f f F, then for some non-negative inte- 173 174 TRENDS IN LATTICE THEORY ger n, 1 is an n-ary operation, that is, a mapping of An into A. In case n = 0, AD is the set whose only element is 0, the void set. Thus 1 is determined by 1(0). In other words, a nullary operation picks out an element of A. Examples of nullary operations are the o and 1 in any usual axiom system of Boolean algebras. For vari­ ous purposes it is very convenient to have nullary operations. We would not gain much by excluding them. Thus having convinced ourselves that the above definition of a universal algebra is the right one, there are two basic questions we have to answer. If we assume so little, as we do in the defini­ tion of universal algebras, is it still possible to develop a non­ trivial theory? And if the answer to the first question is in the af­ firmative, what has all this to do with a symposium on lattice theory and related subjects? It is the thesis of this lecture that the answer to the first question is indeed in the affirmative. I try to prove my case by re­ viewing what I consider the major accomplishments of this field, which is abundant in very deep results. There is a very close connection between universal algebra and lattice theory, partly for personal and partly for mathematical reasons. Professor G. Birkhoff, who invented lattice theory, was the first to publish non-trivial results in universal algebra. He is also responsible for popularizing the subject by lectures as well as by ample references to it in his widely read book on lattice theory. Thus it is not very surprising that quite a few mathematicians who started in lattice theory took interest in universal algebra (the pres­ ent author as one example) and vice-versa (B. J6nsson). There are lattice-theoretical results galore in universal algebra, and there are many results in lattice theory inspired by universal algebra (to which the major part of the last section is devoted). UNIVERSAL ALGEBRA 175 2. Basic concepts. If <Rj+, •> is a ring, then + and, are both binary operations, however different in nature. To make sure that we do not confuse them we use different symbols to indicate these operations. Similarly, in a universal algebra 9f = <A,' F> it is useful to have a fixed well-ordering of Fj F = <fo"'" fy'"'' >y<f3' Then the indices will serve as "names" for the operations. We as- sociate with 9f the sequence r =:: <no"'" ny'''' >y<[3' called the type of 2(, where ny is the "arity" of fy' [3 will be denoted by o (r). Given two algebras 9f, 1a with the same type r, the opera­ tions will be denoted by the same symbols fy' though (fy}9( and (fy}m would be more appropriate. If we talk about two or more alge­ bras, we will always assume them to be of the same type unless otherwise specified. Now the basic concepts can be defined. A homomorphism ¢> of the algebra ~l into the algebra 1a is a mapping of A into B which preserves all the operations, i.e., f/ao"'" any. I)¢> =:: f/ao ¢>"'" any_l ¢>}, for any ao'..., any _I f A and y < 0 (r). 1a is a subalgebra of ~ ~ if B A, and the operation fy of 1a is the restriction of the opera­ tion f of 9f to Bny, and B is closed under all the f ' A congru­ v y ence relation 6 of ~ is an equivalence relation on A satisfying the substitution property for all f ; that is, a. == b.(8}, i =:: 0, ..., y II nY - 1 implies fy(ao"'" any· I} fy(bo'"'' bny- 1}(8}. Polynomials (over ~O and polynomial symbols (of type r) also play an important role. Using the operation symbols f y' Y < Xl' ..., ' ... o (r), and the symbols xo' xn we build up symbols from the Xi by substituting them in operation symbols. Thus if no = 2, n l =:: 1, the following are examples of polynomial symbols: 176 TRENDS IN LATTICE THEORY XO' Xi' fO(xO' Xi)' f 1(fO(fO(XO' Xl)' f1(xO))) , and so on. If we use only xO"'" Xn -1 we get the tt-ary polynomi­ al symbols. Let ~ be an algebra, e an tt-ary polynomial symbol and a '"'' a _ € A. We can define p(a "'" a _ ) in a natural manner. o n 1 o n 1 Indeed, if p ". xi' let p(ao"'" an_ 1) == ai • If, e.g., P ". [l(fO(£O(Xo' Xl)' [1(x2))), then p(a a1' a ) ". [l(£O(£O(a a ), [1(a Thus p induces an n­ o' 2 o' 1 2»). ary function p over A, called a polynomial over~. The collection of all (of all tt-ary) polynomials over ~l is denoted by p(~n (p(n)(~». The set of all (of all n-ary) polynomial symbols is denoted by P(r) (p(n)(T». If p and q are polynomial symbols, then the expression p". q is an identity. The algebra ~{ satisfies p ". q if P and q induce the same function over A. These are the basic tools of an algebraist. And as we shall see in the subsequent sections, quite a bit can be accom­ plished using only these. These basic concepts can be defined with little or no changes for various generalizations of the concept of universal al­ gebra. Such generalizations consider partial operations, infinitary operations, relations, and so on. Of these, partial algebras are most useful in contributing to the theory of universal algebras. In this report, I will refrain from reviewing these other theories, but occasionally I will mention extensions of known results to the in­ finitary case. 3. From the Homomorphism Theorem to the Jordan-Holder Theorem. Once Kurosh remarked that there is no point in giving UNIVERSAL ALGEBRA 177 various generalizations of the concept of groups if the theory does not do more than extend the homomorphism theorem of groups. In­ deed, the homomorphism theorem holds for any algebra: every homo­ morphic image of ~{ is isomorphic to some quotient algebra ~/a (a homomorphic image 18 is any algebra such that there is a homomor­ phism from i"!l onto 18, and the quotient algebra ~(fa is the algebra defined in the usual way on the set of equivalence classes under e, where a is a congruence on ~{). Similarly, the two isomorphism theorems of groups hold for any algebra. The first states that there is a I-I correspondence between the congruence relations of ~ua and the congruence relations ell of ~ of which a is a refinement (i.e., x 5; y{a) implies x E y(eIl)), while the second states: where 18 is a subalgebra of ~r intersecting every congruence class modulo a, and eB is the restriction of a to B. The latter isomorphism can be strengthened to a rather use­ ful one. Let m be a subalgebra of ~(, a a congruence relation of ~r, and ell a congruence relation of m such that aB is a refine­ ment of ell. Let [B] a denote the union of the equivalence classes under a which intersect B. Then [B]a is closed under the op­ erations and it defines, therefore, a subalgebra [18]a of ~. On £B]a we can define the relation a(eIl) by the rule: x E y(a(eIl)) if there exist bo' bi l B with x E bo(e), bo bI (eIl), bi E y(a). Then a (ell) is a congruence relation on [.\8] a and The celebrated Zassenhaus lemma is a simple corollary of this isomorphism. All we have to do is to apply it twice. This proof of the Zass~haus lemma is as simple as, if not simpler than, 178 TRENDS IN LATTICETHEORY any known proof for groups or rings.