Semiframes: the algebra of semitopologies and actionable coalitions

TL;DR

This paper introduces semiframes, an algebraic structure dual to semitopologies, to model decentralized systems and actionable coalitions, extending topological ideas to non-intersecting open sets.

Contribution

It defines semiframes and establishes a categorical duality with semitopologies, providing a new algebraic-topological framework for decentralized computing systems.

Findings

Established a duality between semiframes and semitopologies.

Defined categorical notions of morphisms and properties.

Analyzed how properties transfer across the duality.

Abstract

We introduce semiframes (an algebraic structure) and investigate their duality with semitopologies (a topological one). Both semitopologies and semiframes are relatively recent developments, arising from a novel application of topological ideas to study decentralised computing systems. Semitopologies generalise topology by removing the condition that intersections of open sets are necessarily open. The motivation comes from identifying the notion of an actionable coalition in a distributed system -- a set of participants with sufficient resources for its members to collaborate to take some action -- with open set; since just because two sets are actionable (have the resources to act) does not necessarily mean that their intersection is. We define notions of category and morphism and prove a categorical duality between (sober) semiframes and (spatial) semitopologies, and we investigate how key well-behavedness properties that are relevant to understanding decentralised systems, transfer (or do not transfer) across the duality.