MacNeille completion and profinite completion can coincide on finitely generated modal algebras

Abstract

Following Bezhanishvili & Vosmaer, we confirm a conjecture of Yde Venema by piecing together results from various authors. Specifically, we show that if $\mathbb{A}$ is a residually finite, finitely generated modal algebra such that $\operatorname{HSP}(\mathbb{A})$ has equationally definable principal congruences, then the profinite completion of $\mathbb{A}$ is isomorphic to its MacNeille completion, and $\Diamond$ is smooth. Specific examples of such modal algebras are the free $\mathbf{K4}$-algebra and the free $\mathbf{PDL}$-algebra.