A Primer on Chainmails: Structures for Point-free Connectivity

TL;DR

This paper introduces chainmails, a new mathematical structure that generalizes the concept of frames in topology to connected subsets, establishing an equivalence with certain join-semilattices.

Contribution

It proposes the novel concept of chainmails as an abstraction of connected subsets and proves their categorical equivalence with complete join-semilattices.

Findings

Chainmails generalize the notion of frames for connected subsets.

Established an equivalence between chainmails and complete join-semilattices.

Provides a new algebraic framework for point-free topology concepts.

Abstract

In point-free topology, one abstracts the poset of open subsets of a topological space, by replacing it with a frame (a complete lattice, where meet distributes over arbitrary join). In this paper we propose a similar abstraction of the posets of connected subsets in various space-like structures. The analogue of a frame is called a chainmail, which is defined as a poset admitting joins of its mails, i.e., subsets having a lower bound. The main result of the paper is an equivalence between a subcategory of the category of complete join-semilattices and the category of chainmails.