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).