Vietoris endofunctor for closed relations and its de Vries dual (2308.16823v1)

Abstract: 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 $\mathsf{S5}$-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.

• The paper generalizes the Vietoris endofunctor to compact Hausdorff spaces and closed relations using the Egli-Milner order.
• It structurally lifts subordination on boolean algebras to new elements S and S^diamond, mapping them into K-algebras.
• The study leverages MacNeille completions to bridge concepts into dual de Vries algebras, aligning with pointfree frame constructions.

An Examination of the Vietoris Endofunctor for Closed Relations and its De Vries Dual

This paper presents an advanced exploration into the topology and algebraic properties of compact Hausdorff spaces by generalizing the classic Vietoris endofunctor. It extends this notion by incorporating closed relations, extending the range from continuous functions in compact Hausdorff spaces to closed relations. The authors introduce a novel extension of the endofunctor from the category of compact Hausdorff spaces and closed relations (KHaus$^R$) to de Vries algebras with subordinations, utilizing a blend of domain theory, modal logic, and frame theory.

A methodological innovation in this paper is the application of the Egli-Milner order to construct the endofunctor for closed relations. The authors have defined the extension of the dual endofunctor to KRFrm$^P$, predicated upon the foundations established by previous investigations into boolean algebras and subordination relations. Through this work, a new duality, advancing the theory of Vietoris spaces to the field of de Vries algebras, is established.

Key Contributions and Results

1. Generalizing the Vietoris Endofunctor: The authors extend the Vietoris endofunctor to the category of compact Hausdorff spaces and closed relations (KHaus$^R$) through closed relations via the Egli-Milner order. This newly defined endofunctor (V) presents an alternate interpretation of the closed relation framework by preserving the continuous characteristic of functions within a topological structure.
2. Subordination on Boolean Algebras: The subordination on boolean algebras has been structurally lifted to encompass the elements (S) and (S$^\diamond$) through disjunctive and conjunctive normal forms. This comprehensive approach allows the subsequent mapping into K-algebras, thus creating a new horizon in relational and subordinated boolean algebra spaces.
3. MacNeille and Ideal Completions: By leveraging the MacNeille completion of S5-subordination algebras, authors achieved a conceptual leap into the dual field via the analogous completion into de Vries algebras, introducing a new endofunctor, L. Here, the alignment with the continual structure of compact regular frames enables a conceptual expansion of Johnstone’s classic pointfree construction within the dual endofunctorial landscape.
4. Natural Isomorphisms: The paper rigorously establishes a suite of natural isomorphisms that bridge the traditional constructs between functions of Stone spaces and boolean algebras. These constructions form the backbone for the described transformations and illustrate key structural similarities (and differences) between these duals.

Implications and Future Directions

The profound implications of generalizing traditional categorical constructs into new dimensions of closed relations and subordinations hint at a broader applicability across architectonics of domains beyond compact spaces. From a theoretical vantage point, the paper sets a foundation for exploring enriched hyper-topological spaces and algebraic structures.

Practically, this research helps to bridge gaps in fields requiring robust frameworks to handle closed relations and supports further investigation and application in domains such as computational topology and logic. The authors set a clear direction should future investigations explore the interplay of these abstractions with distributive lattice structures, potentially backdrop for enriching categorical algebra avenues.

This exploration notably impacts the theoretical underpinnings of algebraic topology and proposes a new trajectory for research into compactness and continuity via relational mappings, potentially heralding new tools and constructions that can be adapted in AI operations, including constraint satisfaction and hierarchical clustering, where such algebraic structures come into play.