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On the algebra of regular sets

Properties of representable Stonian p-ortholattices

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Abstract

The mereotopology RT  − has in Stonian p-ortholattices its algebraic counterpart. We study representability of these lattices and show that not all Stonian p-ortholattices can be represented by the set of regular sets of a topological space. We identify five conditions that hold in algebras of regular sets and which can be used to eliminate non-representable Stonian p-ortholattices. This shows not only that the original completeness theorem for RT  − is incorrect, but is also an important step towards an algebraic representation (up to isomorphism) of the regular sets of topological spaces.

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Authors and Affiliations

  1. Department of Computer Science, Brock University, St. Catharines, ON, L2S 3A1, Canada

    Michael Winter

  2. Department of Computer Science, University of Toronto, Toronto, ON, M5S 3G8, Canada

    Torsten Hahmann

  3. Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON, M5S 3G8, Canada

    Michael Grüninger

Authors
  1. Michael Winter
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  2. Torsten Hahmann
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  3. Michael Grüninger
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Correspondence to Torsten Hahmann.

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Winter, M., Hahmann, T. & Grüninger, M. On the algebra of regular sets. Ann Math Artif Intell 65, 25–60 (2012). https://doi.org/10.1007/s10472-012-9301-2

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  • DOI: https://doi.org/10.1007/s10472-012-9301-2

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