Relational Representation Theorems for Extended Contact Algebras
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Relational Representation Theorems for Extended Contact Algebras
Abstract. In topological spaces, the relation of extended contact is a ternary relation that holds between regular closed subsets A, B and D if the intersection of A and B is included in D. The algebraic counterpart of this mereotopological relation is the notion of extended contact algebra which is a Boolean algebra extended with a ternary relation. In this paper, we are interested in the relational representation theory for extended contact algebras. In this respect, we study the correspondences between point-free and point-based models of space in terms of extended contact. More precisely, we prove new representation theorems for extended contact algebras. Keywords: Mereotopology, Point-free theory of space, Contact algebras, Extended contact algebras, Regular closed subsets, Relational representation.
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Introduction
Starting with the belief that the spatial entities like points and lines usually considered in Euclidean geometry are too abstract, de Laguna [19] and Whitehead [30] put forward other primitive entities like solids or regions. Between these entities, they considered relations of “connection” (a ternary relation for de Laguna and a binary relation for Whitehead). They also axiomatically defined sets of properties that these relations should possess in order to provide an adequate analog of the reality we perceive about the connection relation between regions. The ideas of de Laguna and Whitehead about space constitute the basis of multifarious pointless theories of space since the days of Tarski’s geometry of solids. We can cite Grzegorczyk’s theory of the binary relations of “part-of” and “separation” [13] and de Vries’ compingent algebras [29] based on a binary relation that today would be called “non-tangential proper part” [11]. The reason for the success of the axiomatic method in the context of the region-based theories of space certainly lies in the fact that our perception of space inevitably leads us to think about the relative positions of the objects
Presented by Yde Venema; Received May 1, 2018
Studia Logica https://doi.org/10.1007/s11225-020-09923-0
c Springer Nature B.V. 2020
P. Balbiani, T. Ivanova
that occupy space in terms of “part-of” and “separation” or in terms of “part-of” and “connection”. Since the contributions of Clarke [2,3], several region-based theories of space have been developed in artificial intelligence and computer science [4,20,22–24]. In these theories, one generally assumes that regions are regular closed subsets in, for example, the real plane together with its ordinary topology, and one generally studies pointless theories of space based—together with some other relations like “partial overlap”, “tangential proper part”, and so on—on the binary relation of “contact” which holds between two regular closed subsets when they have common points. There are mainly two kinds of results: representability in concrete geometrical structures like the topological spaces associated to abstract algebr
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