Orthomodular lattices, Foulis Semigroups and Dagger Kernel Categories

TL;DR

This paper explores the connections between orthomodular lattices, Foulis semigroups, and dagger kernel categories to deepen the understanding of quantum logic through a categorical framework.

Contribution

It extends previous work on quantum logic by establishing a categorical relationship between orthomodular lattices and Foulis semigroups, providing a broader context.

Findings

Reconstructed the relationship between orthomodular lattices and Foulis semigroups categorically.

Connected dagger kernel categories with quantum logic structures.

Broadened the understanding of quantum logic through categorical methods.

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.