Skew Boolean algebras derived from generalized Boolean algebras

Abstract.

Every skew Boolean algebra S has a maximal generalized Boolean algebra image given by S/${\mathcal{D}}$ where ${\mathcal{D}}$ is the Green’s relation defined initially on semigroups. In this paper we study skew Boolean algebras $\omega({\bf B})$ constructed from generalized Boolean algebras B by a twisted product construction for which $\omega({\bf B})/{\mathcal{D}} \cong {\bf B}$. In particular we study the congruence lattice of $\omega({\bf B})$ with an eye to viewing $\omega({\bf B})$ as a minimal skew Boolean cover of B. This construction is the object part of a functor $\omega: {\bf GB} \rightarrow {\bf LSB}$ from the category GB of generalized Boolean algebras to the category LSB of left-handed skew Boolean algebras. Thus we also look at its left adjoint functor $\Omega: {\bf LSB} \rightarrow {\bf GB}$.