Abstract
The theory of Boolean contact algebras has been used to represent a region based theory of space. Some of the primitives of Boolean algebras are not well motivated in that context. One possible generalization is to drop the notion of complement, thereby weakening the algebraic structure from Boolean algebra to distributive lattice. The main goal of this paper is to investigate the representation theory of that weaker notion, i.e., whether it is still possible to represent each abstract algebra by a substructure of the regular closed sets of a suitable topological space with the standard Whiteheadean contact relation.
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References
Allwein, G., MacCaull, W.: A Kripke semantics for the logic of Gelfand quantales. Studia Logica 61, 1–56 (2001)
Balbes, R., Dwinger, P.: Distributive Lattices. University of Missouri Press, Columbia (1974)
Cornish, W.H.: Crawley’s completion of a conditionally upper continuous lattice. Pac. J. Math. 51(2), 397–405 (1974)
Dimov, G., Vakarelov, D.: Contact algebras and region–based theory of space: A proximity approach. Fundamenta Informaticae (2006) (to appear)
Dimov, G., Vakarelov, D.: Topological Representation of Precontact algebras. In: MacCaull, W., Winter, M., Düntsch, I. (eds.) RelMiCS 2005. LNCS, vol. 3929, pp. 1–16. Springer, Heidelberg (2006)
Düntsch, I., Vakarelov, D.: Region–based theory of discrete spaces: A proximity approach. Discrete Applied Mathematics (2006) (to appear)
Düntsch, I., Winter, M.: Lattices of contact relations (2005) (Preprint)
Düntsch, I., Winter, M.: A representation theorem for Boolean contact algebras. Theoretical Computer Science (B) 347, 498–512 (2005)
Düntsch, I., Winter, M.: Weak contact structures. In: MacCaull, W., Winter, M., Düntsch, I. (eds.) RelMiCS 2005. LNCS, vol. 3929, pp. 73–82. Springer, Heidelberg (2006)
Engelking, R.: General topology, PWN (1977)
MacCaull, W., Vakarelov, D.: Lattice-based paraconsistent logic. In: MacCaull, W., Winter, M., Düntsch, I. (eds.) RelMiCS 2005. LNCS, vol. 3929, pp. 173–187. Springer, Heidelberg (2006)
Stone, M.: Topological representations of distributive lattices and Brouwerian logics. Casopis Pest. Mat. 67, 1–25 (1937)
Vakarelov, D., Düntsch, I., Bennett, B.: A note on proximity spaces and connection based mereology. In: Welty, C., Smith, B. (eds.) Proceedings of the 2nd International Conference on Formal Ontology in Information Systems (FOIS 2001), pp. 139–150. ACM, New York (2001)
Vakarelov, D., Dimov, G., Düntsch, I., Bennett, B.: A proximity approach to some region-based theory of space. Journal of applied non-classical logics 12(3-4), 527–559 (2002)
Wallman, H.: Lattices and topological spaces. Math. Ann. 39, 112–136 (1938)
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Düntsch, I., MacCaull, W., Vakarelov, D., Winter, M. (2006). Topological Representation of Contact Lattices. In: Schmidt, R.A. (eds) Relations and Kleene Algebra in Computer Science. RelMiCS 2006. Lecture Notes in Computer Science, vol 4136. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11828563_9
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DOI: https://doi.org/10.1007/11828563_9
Publisher Name: Springer, Berlin, Heidelberg
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