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arXiv · 2502.06350

Foulis quantales and complete orthomodular lattices

Michal Botur·Jan Paseka·Richard Smolka
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Abstract

Our approach establishes a natural correspondence between complete orthomodular lattices and certain types of quantales. Firstly, given a complete orthomodular lattice X, we associate with it a Foulis quantale Lin(X) consisting of its endomorphisms. This allows us to view X as a left module over Lin(X), thereby introducing a novel fuzzy-theoretic perspective to the study of complete orthomodular lattices. Conversely, for any Foulis quantale Q, we associate a complete orthomodular lattice [Q] that naturally forms a left Q-module. Furthermore, there exists a canonical homomorphism of Foulis quantales from Q to Lin([Q]).

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Michal Botur, Jan Paseka, Richard Smolka. 2025-02-10. Foulis quantales and complete orthomodular lattices. https://arxiv.org/abs/2502.06350

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