Foulis quantales and complete orthomodular lattices

TL;DR

This paper establishes a natural correspondence between complete orthomodular lattices and Foulis quantales, introducing a fuzzy-theoretic perspective and revealing new structural insights into their relationship.

Contribution

It constructs a duality between complete orthomodular lattices and Foulis quantales, linking their structures via module theory and canonical homomorphisms.

Findings

Complete orthomodular lattices can be associated with Foulis quantales of endomorphisms.

Orthomodular lattices can be viewed as modules over their corresponding Foulis quantales.

Existence of a canonical homomorphism from a Foulis quantale to the quantale of endomorphisms of its associated lattice.

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]).