Foulis quantales and complete orthomodular lattices

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]).