Orthomodular lattices can be converted into left residuated l-groupoids

TL;DR

This paper demonstrates a bidirectional correspondence between orthomodular lattices and a specific class of left residuated l-groupoids, enriching the algebraic understanding of quantum logic structures.

Contribution

It establishes a novel equivalence between orthomodular lattices and certain left residuated l-groupoids under specific conditions.

Findings

Orthomodular lattices can be represented as left residuated l-groupoids.

Left residuated l-groupoids satisfying certain conditions can be structured as orthomodular lattices.

The work bridges lattice theory and residuated algebraic structures in quantum logic.

Abstract

We show that every orthomodular lattice can be considered as a left residuated l-groupoid satisfying divisibility, antitony, the double negation law and three more additional conditions expressed in the language of residuated structures. Also conversely, every left residuated l-groupoid satisfying the mentioned conditions can be organized into an orthomodular lattice.