S5-subordination algebras are a natural generalization of de Vries algebras. Recently it was proved that the category SubS5S of S5-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 S5-subordination algebras, and utilize the relational nature of the morphisms in SubS5S to prove that the MacNeille completion functor establishes an equivalence between SubS5S and its full subcategory consisting of de Vries algebras. We also show that the functor that associates to each S5-subordination algebra the frame of its round ideals establishes a dual equivalence between SubS5S and the category of compact regular frames and preframe homomorphisms. Our results are choice-free and provide further insight into Stonelike dualities for compact Hausdorff spaces with various morphisms between them. In particular, we show how they restrict to the wide subcategories of SubS5S corresponding to continuous relations and continuous functions between compact Hausdorff spaces.

MacNeille completions of subordination algebras / M. Abbadini, G. Bezhanishvili, L. Carai. - In: CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES. - ISSN 1245-530X. - 65:2(2024), pp. 151-199.

MacNeille completions of subordination algebras

M. Abbadini
Primo
;
L. Carai
Ultimo
2024

Abstract

S5-subordination algebras are a natural generalization of de Vries algebras. Recently it was proved that the category SubS5S of S5-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 S5-subordination algebras, and utilize the relational nature of the morphisms in SubS5S to prove that the MacNeille completion functor establishes an equivalence between SubS5S and its full subcategory consisting of de Vries algebras. We also show that the functor that associates to each S5-subordination algebra the frame of its round ideals establishes a dual equivalence between SubS5S and the category of compact regular frames and preframe homomorphisms. Our results are choice-free and provide further insight into Stonelike dualities for compact Hausdorff spaces with various morphisms between them. In particular, we show how they restrict to the wide subcategories of SubS5S corresponding to continuous relations and continuous functions between compact Hausdorff spaces.
Compact Hausdorff space; Gleason cover; closed relation; continuous relation; de Vries algebra; subordination relation; proximity; MacNeille completion; ideal completion; compact regular frame
Settore MATH-02/B - Geometria
Settore MATH-01/A - Logica matematica
2024
https://cahierstgdc.com/index.php/volume-lxv-2024/
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/1100848
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