Distributive Mereotopology: Extended distributive contact lattices

Abstract

Contact algebra is one of the main tools in the region-based theory of space. It is an extension of Boolean algebra with a relation called contact. The elements of the Boolean algebra are considered as formal representations of physical bodies. The contact relation is used also to define some other important mereotopological relations like non-tangential inclusion, dual contact, external contact. Most of these definitions are given by means of the operation of Boolean complementation. There are, however, some problems related to the motivation of this operation. In order to avoid these problems we propose a generalization of the notion of contact algebra by dropping the operation of complement. In this paper we consider as non-definable primitives the relations of contact, nontangential inclusion and dual contact. Part I of the paper is devoted to a suitable axiomatization called extended distributive contact lattice (EDCL) by means of universal first-order axioms true in all contact algebras. EDCL may be considered also as an algebraic tool for certain subarea of mereotopology, called in this paper distributive mereotopology. The main result of Part I of the paper is a representation theorem, stating that each EDCL can be embedded into a contact algebra, showing in this way that the presented axiomatization preserves the meaning of mereotopological relations without considering Boolean complementation. Part II of the paper is devoted to topological representation theory of EDCL, transferring into the distributive case important results from the topological representation theory of contact algebras. It is shown that under minor additional assumptions on distributive lattices as extensionality of the definable relations of overlap or underlap one can preserve the good topological interpretations of regions as regular closed or regular open sets in topological space.