5 research outputs found
De Vries powers: a generalization of Boolean powers for compact Hausdorff spaces
We generalize the Boolean power construction to the setting of compact
Hausdorff spaces. This is done by replacing Boolean algebras with de Vries
algebras (complete Boolean algebras enriched with proximity) and Stone duality
with de Vries duality. For a compact Hausdorff space and a totally ordered
algebra , we introduce the concept of a finitely valued normal function
. We show that the operations of lift to the set of all
finitely valued normal functions, and that there is a canonical proximity
relation on . This gives rise to the de Vries power
construction, which when restricted to Stone spaces, yields the Boolean power
construction.
We prove that de Vries powers of a totally ordered integral domain are
axiomatized as proximity Baer Specker -algebras, those pairs ,
where is a torsion-free -algebra generated by its idempotents that is a
Baer ring, and is a proximity relation on . We introduce the
category of proximity Baer Specker -algebras and proximity morphisms between
them, and prove that this category is dually equivalent to the category of
compact Hausdorff spaces and continuous maps. This provides an analogue of de
Vries duality for proximity Baer Specker -algebras.Comment: 34 page
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
Ideal and MacNeille completions of subordination algebras
-subordination algebras were recently introduced as a
generalization of de Vries algebras, and it was proved that the category
of -subordination algebras and compatible
subordination relations between them is equivalent to the category of compact
Hausdorff spaces and closed relations. We generalize MacNeille completions of
boolean algebras to the setting of -subordination algebras, and
utilize the relational nature of the morphisms in to prove
that the MacNeille completion functor establishes an equivalence between
and its full subcategory consisting of de Vries algebras. We
also generalize ideal completions of boolean algebras to the setting of
-subordination algebras and prove that the ideal completion
functor establishes a dual equivalence between and the
category of compact regular frames and preframe homomorphisms. Our results are
choice-free and provide further insight into Stone-like dualities for compact
Hausdorff spaces with various morphisms between them. In particular, we show
how they restrict to the wide subcategories of corresponding
to continuous relations and continuous functions between compact Hausdorff
spaces