The semilattices with distinguished endomorphisms which are equationally compact

Abstract:

We consider universal algebras $(S;\{ \wedge \} \cup E)$ in which E is a set of endomorphisms of the semilattice $(S; \wedge )$. It is proved in this paper that such an algebra is equationally compact iff (i) every nonempty subset of S has an infimum, (ii) every up-directed subset of S has a supremum, (iii) for every $s \in S$ and every up-directed family $({d_i})$ in S the equality $s \wedge \vee {d_i} = \vee s \wedge {d_i}$ holds, (iv) for each $f \in E,f( \wedge {s_i}) = \wedge f({s_i})$ holds for every family $({s_i})$ in S, and (v) for each $f \in E,f( \vee {d_i}) = \vee f({d_i})$ holds for every up-directed family $({d_i})$ in S. In addition, it is shown that every equationally compact algebra of this type is a retract (algebraic) of a compact, Hausdorff, 0-dimensional topological one. These results reduce to known ones for semilattices without additional structure.