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Barycentric algebras -- convexity and order

Anna Zamojska-Dzienio

TL;DR

The paper presents barycentric algebras as a unified framework for convexity and order, treating affine spaces and convex sets as algebras and developing a robust structure theory. Central to the approach are semilattice sums and Płonka sums, which decompose and reconstruct barycentric algebras from convex fibers over semilattice replicas, with concrete applications to convex polytopes and complex-system models. It further connects affine geometry to projective geometry by showing the projective lattice arises as a semilattice replica of subspace algebras via a Płonka-sum construction, highlighting deep links between algebraic and geometric viewpoints. Collectively, the results clarify how convexity and order interact in a broad algebraic setting and provide tools for modeling hierarchical or multi-level systems across mathematics and related disciplines.

Abstract

This is the abstract of a series of lectures given during the XIIIth School on Geometry and Physics, Bialystok (Poland), in July 2024. In this minicourse, we first examine the algebraic aspects of barycentric algebras. Then, we focus on various examples and applications, reviewing the pertinence of the barycentric algebra structure.

Barycentric algebras -- convexity and order

TL;DR

The paper presents barycentric algebras as a unified framework for convexity and order, treating affine spaces and convex sets as algebras and developing a robust structure theory. Central to the approach are semilattice sums and Płonka sums, which decompose and reconstruct barycentric algebras from convex fibers over semilattice replicas, with concrete applications to convex polytopes and complex-system models. It further connects affine geometry to projective geometry by showing the projective lattice arises as a semilattice replica of subspace algebras via a Płonka-sum construction, highlighting deep links between algebraic and geometric viewpoints. Collectively, the results clarify how convexity and order interact in a broad algebraic setting and provide tools for modeling hierarchical or multi-level systems across mathematics and related disciplines.

Abstract

This is the abstract of a series of lectures given during the XIIIth School on Geometry and Physics, Bialystok (Poland), in July 2024. In this minicourse, we first examine the algebraic aspects of barycentric algebras. Then, we focus on various examples and applications, reviewing the pertinence of the barycentric algebra structure.
Paper Structure (11 sections, 6 theorems, 18 equations, 3 figures)

This paper contains 11 sections, 6 theorems, 18 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Affine spaces of $\mathbb{R}^n$
  3. Convex subsets of $\mathbb{R}^n$
  4. Barycentric algebras
  5. Examples of barycentric algebras
  6. The structure of barycentric algebras
  7. Semilattice sums
  8. Płonka sums
  9. Semilattice sums for convex polytopes
  10. A toy model for complex systems
  11. The transition from affine to projective geometry

Key Result

Theorem 2.1

OS The variety $\mathbf{A}(\mathbb{R})$ is the class of algebras satisfying the identities:

Figures (3)

  • Figure 1: The barycentric algebra $(T,\underline{I}^{\circ})$
  • Figure 2: The semilattice replica of $(T,\underline{I}^{\circ})$
  • Figure 3: Demographic and ecological levels

Theorems & Definitions (18)

  • Theorem 2.1
  • Remark 3.1
  • Definition 4.1
  • Example 4.2
  • Example 4.3
  • Example 4.4
  • Example 4.5
  • Example 4.6
  • Theorem 5.1
  • Remark 5.2
  • ...and 8 more