Abstract
Mereotopology studies relations between regions of space, including the contact relation. It leads to an abstract notion of Boolean contact algebra which has been shown to be representable as an algebra of regular closed subsets of a compact topological space. Here we define mereotopological spaces and their mereomorphisms, and construct a dual equivalence between the category of Boolean contact algebras and a category of mereotopological spaces that have a property we call mereocompactness, strictly stronger than ordinary compactness. This is a further illustration of the kind of duality that has been widely used in the semantic analysis of propositional logics, and which has been a significant theme in the research of J. Michael Dunn.
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Notes
- 1.
For categories A of algebras and S of structures, if the assignments \(A\mapsto A_+\) and \(S\mapsto S^+\) extend to contravariant functors from A to S and vice versa, then this constitutes a dual equivalence when the composition of these functors in either order gives functors that are naturally isomorphic to the identity functors on A and S, respectively. This perspective on models of propositional logic was introduced (for modal algebras and Kripke frames) in the first author’s thesis (Goldblatt 1974). The concept of natural isomorphism is explained in the present article at the end of Sect. 4.
- 2.
Further work on the Region Connection Calculus is surveyed in (Cohn et al. 1997).
- 3.
- 4.
Henkin et al. 1971, 0.2.9.
- 5.
Note that \(a\cdot b\subseteq a\cap b\) for regular closed a and b, so \(a\cap b\ne \emptyset \) is a weaker assertion here than \(a\cdot b\ne 0\).
- 6.
- 7.
See (Mac Lane 1998, I.4) for the theory of natural transformations and isomorphisms.
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Goldblatt, R., Grice, M. (2016). Mereocompactness and Duality for Mereotopological Spaces. In: Bimbó, K. (eds) J. Michael Dunn on Information Based Logics. Outstanding Contributions to Logic, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-29300-4_15
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