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Stone type representation theorems via games

Tuğba Aslan, Mohamed Khaled

Abstract

The classes of relativized relation algebras (whose units are not necessarily transitive as binary relations) are known to be finitely axiomatizable. In this article, we give a new proof for this fact that is easier and more transparent than the original proofs. We give direct constructions for all cases, whereas the original proofs reduced the problem to only one case. The proof herein is combinatorial and it uses some techniques from game theory.

Stone type representation theorems via games

Abstract

The classes of relativized relation algebras (whose units are not necessarily transitive as binary relations) are known to be finitely axiomatizable. In this article, we give a new proof for this fact that is easier and more transparent than the original proofs. We give direct constructions for all cases, whereas the original proofs reduced the problem to only one case. The proof herein is combinatorial and it uses some techniques from game theory.

Paper Structure

This paper contains 3 sections, 8 theorems, 12 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Atoms in the perfect extensions
  3. Games and networks

Key Result

Lemma 2.2

Let $\mathfrak{A}\in\mathrm{REL}$ be a perfect algebra, and let $a,b\in At(\mathfrak{A})$ be such that $a \breve{\text{ }}\not=0$ and $b \breve{\text{ }}\not=0$. Then the following are true.

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (24)

  • Definition 1.1
  • Definition 1.2: R. Kramer kramer
  • Remark 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 14 more