# PP-2007-34: MacNeille completion and profinite completion can coincide on finitely generated modal algebras

PP-2007-34: Vosmaer, Jacob (2007) MacNeille completion and profinite completion can coincide on finitely generated modal algebras. [Report]

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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.