Axiomatization of Boolean algebras via weak dicomplementations

TL;DR

This paper introduces a new axiomatization of Boolean algebras using weak dicomplementations, providing a single equation that captures distributivity and complementation in lattices.

Contribution

It presents a novel axiomatization of Boolean algebras based on weakly dicomplemented lattices with a single defining equation.

Findings

A Boolean algebra can be characterized by a specific lattice equation.

The approach encodes distributivity and complementation in a unified way.

This axiomatization simplifies the understanding of Boolean algebra structure.

Abstract

In this note we give an axiomatization of Boolean algebras based on weakly dicomplemented lattices: an algebra $(L,\wedge,\vee,\tu)$ of type $(2,2,1)$ is a Boolean algebra iff $(L,\wedge,\vee)$ is a non empty lattice and $(x\wedge y)\vee(x\wedge y\tu)=(x\vee y)\wedge(x\vee y\tu)$ for all $x,y\in L$. This provides a unique equation to encode distributivity and complementation on lattices.