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Semitopological Barycentric Algebras

Jean Goubault-Larrecq

TL;DR

This work extends Keimel’s functional-analytic framework to the realm of barycentric algebras, developing a comprehensive theory of semitopological and topological barycentric algebras. It constructs free semitopological cones over barycentric algebras, characterizes embeddability, and analyzes barycenters via valuation spaces, Schröder–Simpson decompositions, and Choquet-type representations. The paper establishes local convexity, duality with valuations, and robust separation results, culminating in general barycenter existence theorems for convenient (pointed) barycentric algebras. These results unify domain-theoretic and convex-analytic perspectives, enabling systematic treatment of valuations, open/closed convex hulls, and power-constructs (Smyth) in a non-Hausdorff setting. The framework has potential applications in abstract convexity, non-Hausdorff topology, and domain-theoretic approaches to probability-like measures.

Abstract

Barycentric algebras are an abstraction of the notion of convex sets, defined by a set of equations. We study semitopological and topological barycentric algebras, in the spirit of a previous study by Klaus Keimel on semitopological and topological cones (2008), which are special cases of semitopological and topological barycentric algebras.

Semitopological Barycentric Algebras

TL;DR

This work extends Keimel’s functional-analytic framework to the realm of barycentric algebras, developing a comprehensive theory of semitopological and topological barycentric algebras. It constructs free semitopological cones over barycentric algebras, characterizes embeddability, and analyzes barycenters via valuation spaces, Schröder–Simpson decompositions, and Choquet-type representations. The paper establishes local convexity, duality with valuations, and robust separation results, culminating in general barycenter existence theorems for convenient (pointed) barycentric algebras. These results unify domain-theoretic and convex-analytic perspectives, enabling systematic treatment of valuations, open/closed convex hulls, and power-constructs (Smyth) in a non-Hausdorff setting. The framework has potential applications in abstract convexity, non-Hausdorff topology, and domain-theoretic approaches to probability-like measures.

Abstract

Barycentric algebras are an abstraction of the notion of convex sets, defined by a set of equations. We study semitopological and topological barycentric algebras, in the spirit of a previous study by Klaus Keimel on semitopological and topological cones (2008), which are special cases of semitopological and topological barycentric algebras.

Paper Structure

This paper contains 45 sections, 110 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Barycentric algebras
  4. Plain barycentric algebras
  5. Preordered and ordered barycentric algebras
  6. Semitopological and topological barycentric algebras
  7. The free semitopological cone over ${\mathbf V}_1 X$
  8. Barycenters, part 2
  9. Pointed barycentric algebras
  10. Pointed barycentric algebras and preordered pointed barycentric algebras
  11. Telescopes
  12. Semitopological telescopes
  13. The free semitopological cone over ${\mathbf V}_{\leq 1} X$
  14. The free pointed barycentric algebra over a barycentric algebra
  15. The free pointed barycentric algebra over ${\mathbf V}_1 X$
  16. ...and 30 more sections

Key Result

Lemma 3.2

For every map barycentric algebra $\mathfrak{B}$ and every preordered cone $\mathfrak{C}$, for every function $f \colon \mathfrak{B} \to \mathfrak{C}$, $f^\leftslice$ is positively homogeneous; if $f$ is concave then $f^\leftslice$ is superlinear, if $f$ is convex then $f^\leftslice$ is sublinear, a

Theorems & Definitions (311)

  • Remark 2.1
  • Example 2.2
  • Example 2.3: Example 3.5 in GLJ:Valg
  • Example 2.4: Section 6.1 in heckmann96
  • Example 2.5
  • proof
  • Lemma 3.2
  • proof
  • Definition 3.3: $\leq^{\mathrm{cone}}$
  • Proposition 3.4
  • ...and 301 more