Distributivity and permutability of congruence relations in equational classes of algebras
Alden F. Pixley · Proceedings of the American Mathematical Society · 1963
1. Preliminary concepts. In slightly different terminology and notation, all of the concepts and results discussed in this section (except Theorem 1) are to be found in the work of Birkhoff [2; 3]. Let T be an arbitrary, fixed, index set and let S be a function on T to the set of positive integers. tf is an algebra of species S if 2{ is a mathematical system 5t = (A, K) consisting of a nonempty set A and a family K of primitive operations oa, aCET, where oaGAAS (). We denote by ?a the S(a)-ary operation of any algebra of species S, observing that Oa will have a different interpretation in different algebras. With this in mind we have the following definitions. An S-expression is either an indeterminate or a finite composition of indeterminates with the oa. An S-identity is an equation 4) =4 where 4 and 4' are S-expressions. An algebra 2 of species S satisfies the S-identity 4=41 if for each (single valued) assignment of elements of A to the indeterminates appearing in 4 or 4', 4 yields the same member of A as 4'. Let U be any set of S-identities. By C( U) we denote the equational class of species S satisfying U consisting of those algebras of species S which satisfy all of the )=4, C U. For example, if U and n are taken as the primitive operations, and U consists of the identities: (i) xUy = yUx, xG'y = yG'x, (ii) xuJ(yJz) = (xuy) uz, xfl(y(z) = (xCy)C\z, (iii) xU(xny) =xC(x.JUy) = x, then C(U) consists of the class of all lattices. If k is any cardinal number and U is any set of S-identities, we denote by &k(U) the free algebra of species S, having k generators, determined by the identities U. ?k(U) has as elements classes of Sexpressions in k indeterminates where two expressions 4 and 4' are placed in the same class if 4 =4' is a logical consequence of the identities U. It can be shown [2], and in the sequel we require the fact, that !k( U) C( U).