From Contact Relations to Modal Operators, and Back

Abstract

One of the standard axioms for Boolean contact algebras says that if a region x is in contact with the join of y and z, then x is in contact with at least one of the two regions. Our intention is to examine a stronger version of this axiom according to which if x is in contact with the supremum of some family S of regions, then there is a y in S that is in contact with x. We study a modal possibility operator which is definable in complete algebras in the presence of the aforementioned axiom, and we prove that the class of complete algebras satisfying the axiom is closely related to the class of modal KTB-algebras. We also demonstrate that in the class of complete extensional contact algebras the axiom is equivalent to the statement: every region is isolated. Finally, we present an interpretation of the modal operator in the class of the so-called resolution contact algebras.