Skew Boolean algebras derived from generalized Boolean algebras

摘要

Every skew Boolean algebra S has a maximal generalized Boolean algebra image given by S/D{mathcal{D}} where D{mathcal{D}} is the Green’s relation defined initially on semigroups. In this paper we study skew Boolean algebras w(B)omega({bf B}) constructed from generalized Boolean algebras B by a twisted product construction for which w(B)/D @ Bomega({bf B})/{mathcal{D}} cong {bf B}. In particular we study the congruence lattice of w(B)omega({bf B}) with an eye to viewing w(B)omega({bf B}) as a minimal skew Boolean cover of B. This construction is the object part of a functor w: GB ® LSBomega: {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 W: LSB ® GBOmega: {bf LSB} rightarrow {bf GB}.