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Semiframes: the algebra of semitopologies and actionable coalitions

Murdoch J. Gabbay

TL;DR

The paper develops semiframes as compatible complete semilattices and semitopologies as their dual topological counterpart, motivated by decentralized systems where actionable coalitions need not have intersecting openness. It establishes a categorical duality between sober semitopologies and spatial semiframes, and shows how regularity, transitivity, and related well-behavedness notions translate across the duality. Semifilters and abstract points provide an algebraic handle on neighborhoods, leading to a soberification/dualization framework that preserves open-set structure via Op and nbhd maps. The work further organizes the theory into four categories with adjoint functors, clarifying how sobriety and spatiality yield a robust dual equivalence, and situates these ideas within a broader algebraic-topology context for decentralised coherence and coordination.

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.

Semiframes: the algebra of semitopologies and actionable coalitions

TL;DR

The paper develops semiframes as compatible complete semilattices and semitopologies as their dual topological counterpart, motivated by decentralized systems where actionable coalitions need not have intersecting openness. It establishes a categorical duality between sober semitopologies and spatial semiframes, and shows how regularity, transitivity, and related well-behavedness notions translate across the duality. Semifilters and abstract points provide an algebraic handle on neighborhoods, leading to a soberification/dualization framework that preserves open-set structure via Op and nbhd maps. The work further organizes the theory into four categories with adjoint functors, clarifying how sobriety and spatiality yield a robust dual equivalence, and situates these ideas within a broader algebraic-topology context for decentralised coherence and coordination.

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.
Paper Structure (74 sections, 111 theorems, 55 equations, 8 figures)

This paper contains 74 sections, 111 theorems, 55 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The challenge of decentralised systems
  3. Point-set semitopologies
  4. Well-behavedness conditions on semitopologies
  5. Semiframes: the algebraic dual to semitopologies
  6. Map of the paper
  7. Comments on the context of this research
  8. Semitopology
  9. Definition and examples
  10. Semitopologies are not topologies
  11. Examples
  12. Continuity, and its interpretation as agreement
  13. Neighbourhoods of a point
  14. Transitive sets & topens
  15. Some background on sets intersection
  16. ...and 59 more sections

Key Result

Lemma 2.1.2

Figures (8)

  • Figure 1: An example of a point with two minimal open neighbourhoods (Lemma \ref{['lemm.two.min']})
  • Figure 2: Examples of topens (Example \ref{['xmpl.cc']})
  • Figure 3: An unconflicted, irregular space (Proposition \ref{['prop.unconflicted.irregular']}) in which every point is intertwined only with itself (Example \ref{['xmpl.conflicted.points']})
  • Figure 4: Examples of open neighbourhoods (Remarks \ref{['rmrk.no.meet']} and \ref{['rmrk.nbhd.filter']})
  • Figure 5: A semiframe with no abstract points (Lemma \ref{['lemm.no.converge']})
  • ...and 3 more figures

Theorems & Definitions (346)

  • Definition 1.2.2
  • Remark 1.2.3: Justification for semitopologies
  • Remark 1.4.1: Semitopologies are compatible complete join-semilattices in sets
  • Remark 1.6.1: Algebraic topology $\neq$ semitopology and semiframes
  • Remark 1.6.2: Where the interesting properties are
  • Remark 1.6.3
  • Remark 1.6.4
  • Remark 1.6.5: Who should read this paper?
  • Remark 2.1.1
  • Lemma 2.1.2
  • ...and 336 more