Abstract
For beta an ordinal, let PEA(beta) (SetPEA(beta)) denote the class of polyadic equality (set) algebras of dimension beta. We show that for any infinite ordinal alpha, if A is an element of PEA(alpha) is atomic, then for any n < omega, the n-neat reduct of A, in symbols NrnA, is a completely representable PEA(n) (regardless of the representability of A). That is to say, for all non-zero a is an element of N(rn)A, there is a B-a is an element of SetPEA(n) and a homomorphism f(a) : N(rn)A -> B such that f(a)(a) not equal 0 and f(a)(Sigma X) = boolean OR(x is an element of X) f(a)(x) for any X subset of A for which Sigma X exists. We give new proofs that various classes consisting solely of completely representable algebras of relations are not elementary; we further show that the class of completely representable relation algebras is not closed under equivalent to(infinity,omega). Various notions of representability (such as 'satisfying the Lyndon conditions', weak and strong) are lifted from the level of atom structures to that of atomic algebras and are further characterized via special neat embeddings. As a sample, we show that the class of atomic CA(n)s satisfying the Lyndon conditions coincides with the class of atomic algebras in ElS(c)Nr(n)CA(omega), where El denotes 'elementary closure' and S-c is the operation of forming complete subalgebras.