Bitopological duality for distributive lattices and heyting algebras

  • Authors:

  • Guram Bezhanishvili;Nick Bezhanishvili;David Gabelaia;Alexander Kurz

  • Affiliations:

  • Department of mathematical sciences, new mexico state university, las cruces nm 88003-8001, u.s.a. email: gbezhani@nmsu.edu;Department of computing, imperial college london, 180 queen's gate, london sw7 2az, u.k. email:nbezhani@doc.ic.ac.uk;Department of mathematical logic, razmadze mathematical institute, m. aleksidze str. 1, tbilisi 0193, georgia email: gabelaia@gmail.com;Department of computer science, university of leicester, university road, leicester le1 7rh, u.k. e-mail: kurz@mcs.le.ac.uk

  • Venue:
  • Mathematical Structures in Computer Science
  • Year:
  • 2010

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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.