MacNeille completions of subordination algebras

TL;DR

This paper extends MacNeille completions to S5-subordination algebras, establishing dualities with compact regular frames and providing a choice-free framework for Stone-like dualities involving compact Hausdorff spaces.

Contribution

It generalizes MacNeille completions to S5-subordination algebras and proves their categorical equivalences with de Vries algebras and compact regular frames.

Findings

MacNeille completion functor induces an equivalence with de Vries algebras.

The frame of round ideals yields a dual equivalence with compact regular frames.

Results are choice-free and extend Stone duality to broader morphisms.

Abstract

$\mathsf{S5}$-subordination algebras are a natural generalization of de Vries algebras. Recently it was proved that the category $\mathsf{SubS5^S}$ of $\mathsf{S5}$-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 $\mathsf{S5}$-subordination algebras, and utilize the relational nature of the morphisms in $\mathsf{SubS5^S}$ to prove that the MacNeille completion functor establishes an equivalence between $\mathsf{SubS5^S}$ and its full subcategory consisting of de Vries algebras. We also show that the functor that associates to each $\mathsf{S5}$-subordination algebra the frame of its round ideals establishes a dual equivalence between $\mathsf{SubS5^S}$ 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 $\mathsf{SubS5^S}$ corresponding to continuous relations and continuous functions between compact Hausdorff spaces.