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

摘要

Following Bezhanishvili & Vosmaer, we confirm a conjecture of Yde Venema bypiecing 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 principalcongruences, then the profinite completion of $\mathbb{A}$ is isomorphic to itsMacNeille completion, and $\Diamond$ is smooth. Specific examples of such modalalgebras are the free $\mathbf{K4}$-algebra and the free$\mathbf{PDL}$-algebra.