A refinement of Stone duality to skew Boolean algebras

Abstract: We establish two duality theorems which refine the classical Stone duality between generalized Boolean algebras and locally compact Boolean spaces. In the first theorem we prove that the category of left-handed skew Boolean algebras whose morphisms are proper skew Boolean algebra homomorphisms is equivalent to the category of \'{e}tale spaces over locally compact Boolean spaces whose morphisms are \'{e}tale space cohomomorphisms over continuous proper maps. In the second theorem we prove that the category of left-handed skew Boolean $\cap$-algebras whose morphisms are proper skew Boolean $\cap$-algebra homomorphisms is equivalent to the category of \'{e}tale spaces with compact clopen equalizers over locally compact Boolean spaces whose morphisms are injective \'{e}tale space cohomomorphisms over continuous proper maps.