Quasi-semi-homomorphisms and generalized proximity relations between Boolean algebras

Abstract

In this paper we shall define the notion of quasi-semi-homomorphisms between Boolean algebras, as a generalization of the quasi-modal operators introduced in [3], of the notion of meet-homomorphism studied in [12] and [11], and the notion of precontact or proximity relation defined in [8]. We will prove that the class of Boolean algebras with quasi-semi-homomorphism is a category, denoted by BoQS. We shall prove that this category is equivalent to the category StQB of Stone spaces where the morphisms are binary relations, called quasi-Boolean relations, satisfying additional conditions. This duality extends the duality for meet-homomorphism given by P. R. Halmos in [12] and the duality for quasi-modal operators proved in [3].