5 research outputs found
A generalization of de Vries duality to closed relations between compact Hausdorff spaces
Stone duality generalizes to an equivalence between the categories StoneR of Stone spaces and closed relations and BAS of boolean algebras and subordination relations. Splitting equivalences in StoneR yields a category that is equivalent to the category KHausR of compact Hausdorff spaces and closed relations. Similarly, splitting equivalences in BAS yields a category that is equivalent to the category De VS of de Vries algebras and compatible subordination relations. Applying the machinery of allegories then yields that KHausR is equivalent to De VS, thus resolving a problem recently raised in the literature.The equivalence between KHausR and De VS further restricts to an equivalence between the category KHausR of compact Hausdorff spaces and continuous functions and the wide subcategory De VF of De VS whose morphisms satisfy additional conditions. This yields an alternative to de Vries duality. One advantage of this approach is that composition of morphisms is usual relation composition
Vietoris endofunctor for closed relations and its de Vries dual
We generalize the classic Vietoris endofunctor to the category of compact
Hausdorff spaces and closed relations. The lift of a closed relation is done by
generalizing the construction of the Egli-Milner order. We describe the dual
endofunctor on the category of de Vries algebras and subordinations. This is
done in several steps, by first generalizing the construction of Venema and
Vosmaer to the category of boolean algebras and subordinations, then lifting it
up to -subordination algebras, and finally using MacNeille
completions to further lift it to de Vries algebras. Among other things, this
yields a generalization of Johnstone's pointfree construction of the Vietoris
endofunctor to the category of compact regular frames and preframe
homomorphisms