Axiomatization of Boolean algebras via weak dicomplementations

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.