On the algebra of regular sets

Authors:
Michael Winter;Torsten Hahmann;Michael Grüninger
Affiliations:
Department of Computer Science, Brock University, St. Catharines, Canada L2S 3A1;Department of Computer Science, University of Toronto, Toronto, Canada M5S 3G8;Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Canada M5S 3G8
Venue:
Annals of Mathematics and Artificial Intelligence
Year:
2012

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Abstract

The mereotopology RT驴驴 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驴驴 is incorrect, but is also an important step towards an algebraic representation (up to isomorphism) of the regular sets of topological spaces.