Abstract
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.
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Acknowledgments
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.
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Presented by I. Hodkinson.
This research was supported by VICI grant 639.073.501 of the Netherlands Organization for Scientific Research (NWO).
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Vosmaer, J. MacNeille completion and profinite completion can coincide on finitely generated modal algebras. Algebra Univers. 61, 449 (2009). https://doi.org/10.1007/s00012-009-0028-9
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DOI: https://doi.org/10.1007/s00012-009-0028-9