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Relation Algebras Compatible with $\mathbb{Z}_2$-sets

Jeremy F. Alm, John W. Snow

TL;DR

This work classifies relation algebras that are isomorphic to the algebra of binary relations compatible with a ${oldsymbol Z}_2$-set, introducing the group-action representable RA (GARRA) framework. It proves a complete characterization: a relation algebra is a ${oldsymbol Z}_2$-GARRA precisely when it is simple, pair-dense, and every atom or its converse is a function, and it is shown that such algebras are representable as ${ m Rel}({oldsymbol U})$ for a suitable ${oldsymbol Z}_2$-set ${oldsymbol U}$. The paper also proves that this ${oldsymbol Z}_2$-GARRA class is finitely axiomatizable in first-order logic over relation algebras, via three explicit axioms plus the standard RA axioms. Overall, the results connect Maddux’s structure theory for pair-dense algebras with a concrete group-action representation, providing both a sharp classification and a path for extending to other groups and representations.

Abstract

We provide a characterization of those relation algebras which are isomorphic to the algebras of compatible relations of some $\Z_2$-set. We further prove that this class is finitely axiomatizable in first-order logic in the language of relation algebras.

Relation Algebras Compatible with $\mathbb{Z}_2$-sets

TL;DR

This work classifies relation algebras that are isomorphic to the algebra of binary relations compatible with a -set, introducing the group-action representable RA (GARRA) framework. It proves a complete characterization: a relation algebra is a -GARRA precisely when it is simple, pair-dense, and every atom or its converse is a function, and it is shown that such algebras are representable as for a suitable -set . The paper also proves that this -GARRA class is finitely axiomatizable in first-order logic over relation algebras, via three explicit axioms plus the standard RA axioms. Overall, the results connect Maddux’s structure theory for pair-dense algebras with a concrete group-action representation, providing both a sharp classification and a path for extending to other groups and representations.

Abstract

We provide a characterization of those relation algebras which are isomorphic to the algebras of compatible relations of some -set. We further prove that this class is finitely axiomatizable in first-order logic in the language of relation algebras.

Paper Structure

This paper contains 6 sections, 5 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The Structure of Pair-Dense Algebras
  4. Examples
  5. Pair-Dense Relation Algebras and Group Actions
  6. Open Problems

Key Result

Theorem 5.1

Suppose that ${\mathbf{R}}$ is the algebra of binary relations compatible with a $G$-set ${\mathbf{A}}$ where $G$ is a cyclic group of prime order $p$. If $r$ is an atom of ${\mathbf{R}}$, then either $r$ or ${\breve{r}}$ is a function.

Theorems & Definitions (13)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 5.1
  • proof
  • Theorem 5.2
  • Theorem 5.3
  • proof
  • Theorem 5.4
  • ...and 3 more