Orthomodular lattices, Foulis Semigroups and Dagger Kernel Categories

TL;DR

This paper explores the connections between orthomodular lattices, Foulis semigroups, and dagger kernel categories within quantum logic, providing a categorical framework that broadens understanding of their relationships.

Contribution

It develops a categorical perspective linking orthomodular lattices and Foulis semigroups, extending prior algebraic results to a broader context.

Findings

Reconstructs the relationship between orthomodular lattices and Foulis semigroups categorically

Provides a broader framework for quantum logic structures

Extends previous algebraic results to categorical settings

Abstract

This paper is a sequel to arXiv:0902.2355 and continues the study of quantum logic via dagger kernel categories. It develops the relation between these categories and both orthomodular lattices and Foulis semigroups. The relation between the latter two notions has been uncovered in the 1960s. The current categorical perspective gives a broader context and reconstructs this relationship between orthomodular lattices and Foulis semigroups as special instance.