Bitopological duality for distributive lattices and Heyting algebras
Guram Bezhanishvili, Nick Bezhanishvili, David Gabelaia, Alexander Kurz · Mathematical Structures in Computer Science · 2010
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 categoryPStoneof pairwise Stone spaces is isomorphic to the categorySpecof spectral spaces and to the categoryPriesof Priestley spaces. In fact, the isomorphism ofSpecandPriesis most naturally seen throughPStoneby first establishing thatPriesis isomorphic toPStone, and then showing thatPStoneis isomorphic toSpec. 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.