Dynamic Contact Algebras with a Predicate of Actual Existence: Snapshot Representation and Topological Duality

Abstract

The paper is in the field of region-based theory of space and time (RBTST). This is an extension of the region-based theory of space (RBTS) with time. Its origin goes back to some ideas of Whitehead, De Laguna, and Tarski and is related to the problem of how to build the theory of space without the use of the notion of point. The notion of contact algebra (CA) presents an algebraic formulation of RBTS. CA is an extension of Boolean algebra, considered as an algebra of spatial regions with an additional relation of contact. Dynamic contact algebra (DCA) considered as an algebraic formulation of RBTST is an extension of CA aiming to study regions changing in time. In this paper, we study a version of DCA incorporating an explicit predicate AE of actual existence. We first develop the representation theory of such DCAs by means of the so-called snapshot models. Second, we introduce topological models of DCA and develop the corresponding topological representation and duality theory.