Algebra universalis

, 61:449 | Cite as

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

  • Jacob Vosmaer
Open Access


Following Bezhanishvili and 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 HSP(\({\mathbb{A}}\)) has equationally definable principal congruences, then the profinite completion of \({\mathbb{A}}\) is isomorphic to its MacNeille completion, and ◊ is smooth. Specific examples of such modal algebras are the free K4-algebra and the free PDL-algebra.

2000 Mathematics Subject Classification

Primary: 06E25 Secondary: 06B23 03B45 22A30 

Keywords and phrases

modal algebra MacNeille completion profinite completion 



The author would like to thank Yde Venema, who suggested that Theorem 4.2 might be true. Additionally, the author is grateful to the Editor and the Referee for their criticisms and suggestions.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.AmsterdamThe Netherlands

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