A Lattice Theoretic Account of Spatial Regions Draft Version { 25th July 1997

There have been several proposals for formal theories of space in which spatial regions are primitives, rather than constructions from sets of more fundamental points. One motivation for some of these region-based theories has been the need to develop formalisms appropriate for the mechanization of`commonsense' spatial reasoning. Such reasoning is applicable to spatial information systems, examples of which include geographic information systems. One widely known region-based spatial formalism is the Region-Connection Calculus (RCC). This paper presents a lattice theoretic account of some models of RCC. Although the account includes some well known point-based models, the signiicant feature of the work is that the proofs make no reference whatsoever to these points. This appears to be the rst point-free account of models of RCC. In providing this point-free account, the paper makes two principal technical contributions. Firstly the concept of a Boolean connection algebra is introduced. This is a pair hA; C i, where A is a Boolean algebra, and C is a relation on A satisfying certain axioms. It is shown that a model of RCC can be constructed from any Boolean connection algebra. Secondly, a construction of Boolean connection algebras from connected and inexhaustible pseudocomplemented distributive lattices is given. This provides a point-free generalization of Gotts' construction of a model of RCC from the regular open sets of a connected and regular topological space. A dual construction, generalizing Gotts' model using regular closed sets is also given. Boolean connection algebras are of independent interest since they provide a clean separation between the mereological and the topological aspects of a set of regions. The paper concludes with some suggestions on how a generalization of Boolean connection algebras might lead to spatial theories admitting vague regions.

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