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Bitopological duality for distributive lattices and Heyting algebras

Published online by Cambridge University Press:  18 January 2010

GURAM BEZHANISHVILI ,
NICK BEZHANISHVILI ,
DAVID GABELAIA  and
ALEXANDER KURZ
GURAM BEZHANISHVILI
Affiliation:
Department of Mathematical Sciences, New Mexico State University, Las Cruces NM 88003-8001, U.S.A. Email: gbezhani@nmsu.edu
NICK BEZHANISHVILI
Affiliation:
Department of Computing, Imperial College London, 180 Queen's Gate, London SW7 2AZ, U.K. Email:nbezhani@doc.ic.ac.uk
DAVID GABELAIA
Affiliation:
Department of Mathematical Logic, Razmadze Mathematical Institute, M. Aleksidze Str. 1, Tbilisi 0193, Georgia Email: gabelaia@gmail.com
ALEXANDER KURZ
Affiliation:
Department of Computer Science, University of Leicester, University Road, Leicester LE1 7RH, U.K. E-mail: kurz@mcs.le.ac.uk
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Abstract

We introduce pairwise Stone spaces as a bitopological generalisation of Stone spaces – the duals of Boolean algebras – and show that they are exactly the bitopological duals of bounded distributive lattices. The category PStone of pairwise Stone spaces is isomorphic to the category Spec of spectral spaces and to the category Pries of Priestley spaces. In fact, the isomorphism of Spec and Pries is most naturally seen through PStone by first establishing that Pries is isomorphic to PStone, and then showing that PStone is isomorphic to Spec. We provide the bitopological and spectral descriptions of many algebraic concepts important in the study of distributive lattices. We also give new bitopological and spectral dualities for Heyting algebras, thereby providing two new alternatives to Esakia's duality.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 20 , Issue 3 , June 2010 , pp. 359 - 393
Copyright
Copyright © Cambridge University Press 2010

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