Non-commutative symmetric differences in orthomodular lattices

TL;DR

This paper explores how to define symmetric difference in orthomodular lattices, revealing that non-commutative forms are more similar to Boolean algebra differences and useful for congruence relations.

Contribution

It introduces and analyzes non-commutative symmetric differences in orthomodular lattices, showing their properties and applications in congruence relations.

Findings

Non-commutative forms are closer to Boolean symmetric difference.

Six possible symmetric differences satisfy natural conditions.

Non-commutative differences are useful for congruence relations.

Abstract

We deal with the following question: What is the proper way to introduce symmetric difference in orthomodular lattices? Imposing two natural conditions on this operation, six possibilities remain: the two (commutative) normal forms of the symmetric difference in Boolean algebras and four non-commutative terms. It turns out that in many respects the non-commutative forms, though more complex with espect to the lattice operations, in their properties are much nearer to the symmetric difference in Boolean algebras than the commutative terms. As application we demonstrate the usefullness of non-commutative symmetric differences in the context of congruence relations.