Foulis m-semilattices and their modules

TL;DR

This paper introduces a new fuzzy-theoretic approach to orthomodular lattices by constructing Foulis m-semilattices and modules, revealing their categorical and algebraic structures.

Contribution

It constructs Foulis m-semilattices for orthomodular lattices and demonstrates their role as quantales and modules, extending the theoretical framework.

Findings

The category OMLatLin forms a dagger category.

Foulis m-semilattices act as quantales.

Orthomodular lattices can be regarded as modules over these structures.

Abstract

Building upon the results of Jacobs, we show that the category OMLatLin of orthomodular lattices and linear maps forms a dagger category. For each orthomodular lattice X, we construct a Foulis m-semilattice Lin(X) composed of endomorphisms of X. This m-semilattice acts as a quantale, enabling us to regard X as a left Lin(X)-module. Our novel approach introduces a fuzzy-theoretic dimension to the theory of orthomodular lattices.