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

Abstract:
Following [Bezhanishvili & Vosmaer 2007] 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 the
MacNeille completion of $\mathbb{A}$, and $\Diamond$ is
smooth. Specific examples of such modal algebras are the free
$\mathbf{K4}$-algebra and the free $\mathbf{PDL}$-algebra.