| Abstract: | The notion of contact algebra is one of the main tools in the region based theory of space. It is an extension of Boolean algebra with an additional relation C called contact. The elements of the Boolean algebra are considered as formal representations of spatial regions as analogues of physical bodies and Boolean operations are considered as operations for constructing new regions from given ones and also to define some mereological relations between regions as part-of, overlap and underlap. The contact relation is one of the basic mereotopological relations between regions expressing some topological nature. It is used also to define some other important mereotopological relations like non-tangential inclusion, dual contact, external contact and others. Most of these definitions are given by means of the operation of Boolean complementation. There are, however, some problems related to the motivation of the operation of Boolean complementation. In order to avoid these problems we propose a generalization of the notion of contact algebra by dropping the operation of complement and replacing the Boolean part of the definition by that of a distributive lattice. First steps in this direction were made in (Düntsch et al. Lect. Notes Comput. Sci. 4136, 135–147, 2006, Düntsch et al. J. Log. Algebraic Program. 76, 18–34, 2008) presenting the notion of distributive contact lattice based on the contact relation as the only mereotopological relation. In this paper we consider as non-definable primitives the relations of contact, nontangential inclusion and dual contact, extending considerably the language of distributive contact lattices. Part I of the paper is devoted to a suitable axiomatization of the new language called extended distributive contact lattice (EDC-lattice) by means of universal first-order axioms true in all contact algebras. EDC-lattices may be considered also as an algebraic tool for a 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 EDC-lattice can be isomorphically 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 EDC-lattices, 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. |
