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Brock TR # CS-02-21 Abstract

A Proximity Approach to Some Region-Based Theories of Space    [PDF]
D. Vakarelov, G. Dimov, I. Düntsch, B. Bennett, November 2002.

This paper is a continuation of earlier work (Dimiter Vakarelov, Ivo Düntsch, and Brandon Bennett. A note on proximity spaces and connection based mereology. In C. Welty and B. Smith, editors, Proceedings of the 2nd International Conference on Formal Ontology in Information Systems (FOIS'01), pages 139-150. ACM, 2001). The notion of local connection algebra, based on the primitive notions of connection and boundedness, is introduced. It is slightly different but equivalent to Roeper's notion of region-based topology. The similarity between the local proximity spaces of Leader and local connection algebras is emphasized. Machinery, analogous to that introduced by Efremovic, Smirnov and Leader for proximity and local proximity spaces, is developed. This permits us to give new proximity-type models of local connection algebras, to obtain a representation theorem for such algebras and to give a new shorter proof of the main theorem of Roeper's paper. Finally, the notion of MVD-algebra is introduced. It is similar to Mormann's notion of enriched Boolean algebra, based on a single mereological relation of interior parthood. It is shown that MVD-algebras are equivalent to local connection algebras. This means that the connection relation and boundedness can be incorporated into one, mereological in nature relation. In this way a formalization of the Whiteheadian theory of space based on a single mereological relation is obtained.