Bitopological duality for distributive lattices and Heyting algebras. (English) Zbl 1193.06012
A bitopological space is a triple \((X,\tau_1,\tau_2)\), where \(X\) is a (non-empty) set and \(\tau_1,\tau_2\) are two topologies on \(X\).
In this paper, the authors introduce pairwise Stone spaces as a bitopological generalization of Stone spaces and show that they are exactly the bitopological duals of bounded distributive lattices. Thus, it is proved that the category of pairwise Stone spaces is isomorphic to the category of spectral spaces and to the category of Priestley spaces.
Also, the authors provide the bitopological and spectral descriptions of many algebraic concepts important in the study of distributive lattices. So they give new bitopological and spectral dualities for Heyting algebras, thereby providing two new alternatives to Esakia’s duality.
In this paper, the authors introduce pairwise Stone spaces as a bitopological generalization of Stone spaces and show that they are exactly the bitopological duals of bounded distributive lattices. Thus, it is proved that the category of pairwise Stone spaces is isomorphic to the category of spectral spaces and to the category of Priestley spaces.
Also, the authors provide the bitopological and spectral descriptions of many algebraic concepts important in the study of distributive lattices. So they give new bitopological and spectral dualities for Heyting algebras, thereby providing two new alternatives to Esakia’s duality.
Reviewer: Dana Piciu (Craiova)
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