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

Sergio A. Celani;


In this paper we shall introduce the notion of quasi-semi-homomorphisms between Boolean algebras, as a generalization of the quasi-modal operators introduced in \cite{Celani}, of the notion of meet-homomorphism studied in \cite{Halmos} and \cite{Graf}, and the notion of precontact or proximity relation defined in \cite{D=0000FCntsch-Vakarelov2007}. We will prove that the class of Boolean algebras with quasi-semi-homomorphism is a category, denoted by $\mathbf{BoQS}$. We shall prove that this category is equivalent to the category $\mathbf{StQB}$ 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 \cite{Halmos} and the duality for quasi-modal operators proved in \cite{Celani}.

Vol. 19 (2018), No. 1, pp. 171-189

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