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Three roads to complete lattices: orders, compatibility, polarity

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Abstract

This note aims to clarify the relations between three ways of constructing complete lattices that appear in three different areas: (1) using ordered structures, as in set-theoretic forcing, or doubly ordered structures, as in a recent semantics for intuitionistic logic; (2) using compatibility relations, as in semantics for quantum logic based on ortholattices; (3) using Birkhoff’s polarities, as in formal concept analysis.

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Acknowledgements

I thank Nick Galatos and Peter Jipsen for conversations at the 4th SYSMICS Workshop that inspired this note and Mai Gehrke for helpful background in an earlier conversation. I also thank an anonymous referee for Algebra Universalis for helpful comments.

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  1. Department of Philosophy and Group in Logic and the Methodology of Science, University of California, Berkeley, Berkeley, CA, 94720, USA

    Wesley H. Holliday

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  1. Wesley H. Holliday
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Correspondence to Wesley H. Holliday.

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Communicated by Presented by P. Jipsen.

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Holliday, W.H. Three roads to complete lattices: orders, compatibility, polarity. Algebra Univers. 82, 26 (2021). https://doi.org/10.1007/s00012-021-00711-y

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