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

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

2012-02-15

arxiv.org/abs/1202.3271v1

Algebra Universalis 61 pp. 449-453, 2009

Mathematics

Logic

5 pages

Scientific paper

10.1007/s00012-009-0028-9

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.

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