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 \(\Omega: {\bf LSB} \rightarrow {\bf GB}\).