Foulis quantales and complete orthomodular lattices
Michal Botur, Jan Paseka, Richard Smolka
TL;DR
The paper develops a duality between complete orthomodular lattices and Foulis quantales by associating to each lattice $X$ its endomorphism quantale $Lin(X)$, which makes $X$ a left $Lin(X)$-module, and to each Foulis quantale $Q$ the complete orthomodular lattice $[Q]$, which is a left $Q$-module; there is a canonical homomorphism $h: Q \to Lin([Q])$ linking the two frameworks. This bimodule perspective leverages Sasaki projections as both lattice endomorphisms and quantale-derived projections, situating quantum logic within a fuzzy-logic-inspired, module-theoretic setting. The work integrates dagger category theory, kernels, and involutive quantaloids to provide a robust structural bridge between quantum states (complete orthomodular lattices) and quantum operations (Foulis quantales), with concrete constructions and functorial correspondences. The results offer a conceptual and technical toolkit for interpreting quantum logic through quantale modules, enabling new avenues for semantic modeling and potential applications in substructural logics and categorical quantum mechanics.
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]).