Model companions of distributive p-algebras

Let B n , 0 ≤ n ≤ ω, be the equational classes of distributive p-algebras (precise definitions are given in §1). It has been known for some time that the elementary theories T n of B n possess model companions ; see, e.g., [6] and [14] and the references given there. However, no axiomatizations of were given, with the exception of n = 0 (Boolean case) and n= 1 (Stonian case). While the first case belongs to the folklore of the subject (see [6], also [11]), the second case presented considerable difficulties (see Schmitt [13]). Schmitt's use of methods characteristic for Stone algebras seems to prevent a ready adaptation of his results to the cases n ≥ 2. The natural way to get a hold on is to determine the class E( B n ) of existentially complete members of B n : Since exists, it equals the elementary theory of E( B n ). The present author succeeded [12] in solving the simpler problem of determining the classes A( B n ) of algebraically closed algebras in B n (exact definitions of A( B n ) and E( B n ) are given in §1) for all 0 > n < ω. A( B n ) is easier to handle since it contains sufficiently many "small” algebras-viz. finite direct products of certain subdirectly irreducibles-in terms of which the members of A( B n ) may be analyzed (in contrast, all members of E( B n ) are infinite and ℵ-homogeneous). As it turns out, A( B n ) is finitely axiomatizable for all n, and comparing the theories of A( B 0), A( B 1) with the explicitly known theories of E( B 0), E( B 1)-viz. , , a reasonable conjecture for , 2 ≤ n ≤ ω, is immediate. The main part of this paper is concerned with verifying that the conditions formalized by suffice to describe the algebras in E( B n ) (necessity is easy). This verification rests on the same combinatorial techniques as used in [12] to describe the members of A( B n