Properties of the symmetric difference in lattices with complementation

TL;DR

This paper investigates the properties of symmetric difference in lattices with complementation, characterizing when it coincides with Boolean lattices and exploring related algebraic identities and subvarieties.

Contribution

It introduces conditions under which symmetric difference in lattices with complementation behaves like in Boolean lattices and characterizes related algebraic structures.

Findings

Symmetric difference is associative if and only if the lattice is Boolean.

A lattice with complementation is Boolean iff the symmetric difference satisfies a specific identity.

Characterization of lattices with unary operations satisfying De Morgan's laws.

Abstract

The symmetric difference in Boolean lattices can be defined in two different but equivalent forms. However, it can be introduced also in every bounded lattice with complementation where these two forms need not coincide. We study lattices with complementation and the variety of such lattices where these two expressions coincide and point out explicitly some interesting subvarieties. Using a result of J. Berman we estimate the size of free algebras in these subvarieties. It is well-known that the symmetric difference is associative in every Boolean lattice. We prove that it is just the property of Boolean lattices, namely the symmetric difference in a lattice with complementation is associative if and only if this lattice is Boolean. Similarly, we prove that a lattice with complementation is Boolean if and only if the symmetric difference satisfies a certain simple identity in two variables. We also characterize lattices with a unary operation satisfying De Morgan's laws.