One-variable equational compactness in partially distributive semilattices with pseudocomplementation

Abstract

A universal algebraAis called one-variable equationally compact if every system of equations with constants inAinvolving a single variablex, every finite subsystem of which has a solution inA, has itself a solution inA. The one-variable equationally compact semilattices with pseudocomplementation⟨S;∧,∗,0⟩\langle S; \wedge {,^ \ast },0\ranglewhich satisfy the partial distributive lawx∧(y∧z)∗=(x∧y∗)∨(x∧z∗)x \wedge {(y \wedge z)^ \ast } = (x \wedge {y^ \ast }) \vee (x \wedge {z^ \ast })are characterized, and as a consequence we are able to describe the one-variable compact Stone semilattices. Similar considerations yield a characterization of the one-variable equationally compact Stone algebras, extending a well known result for distributive lattices.