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

TL;DR

This paper proves that for certain finitely generated modal algebras with specific properties, the MacNeille and profinite completions are isomorphic, and the modal operator is smooth, confirming a conjecture in the field.

Contribution

It establishes conditions under which the MacNeille and profinite completions of finitely generated modal algebras coincide, confirming a conjecture and providing new insights into their structure.

Findings

Profinite and MacNeille completions are isomorphic under specified conditions.

The modal operator $ ext{Diamond}$ is smooth in these cases.

Examples include free $ extbf{K4}$- and $ extbf{PDL}$-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.