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The variety of coset relation algebras

Steven Givant, Hajnal Andréka

TL;DR

The present article has shown that the class of coset relation algebras is equationally axiomatizable (that is to say, it is a variety), but that no finite set of sentences suffices to axiom atize the class.

Abstract

A coset relation algebra is one embeddable into some full coset relation algebra, the latter is an algebra constructed from a system of groups, a coordinated system of isomorphisms between quotients of these groups, and a system of cosets that are used to "shift" the operation of relative multiplication. We prove that the class of coset relation algebras is equationally axiomatizable (that is to say, it is a variety), but no finite set of equations suffices to axiomatize the class (that is to say, the class is not finitely axiomatizable).

The variety of coset relation algebras

TL;DR

The present article has shown that the class of coset relation algebras is equationally axiomatizable (that is to say, it is a variety), but that no finite set of sentences suffices to axiom atize the class.

Abstract

A coset relation algebra is one embeddable into some full coset relation algebra, the latter is an algebra constructed from a system of groups, a coordinated system of isomorphisms between quotients of these groups, and a system of cosets that are used to "shift" the operation of relative multiplication. We prove that the class of coset relation algebras is equationally axiomatizable (that is to say, it is a variety), but no finite set of equations suffices to axiomatize the class (that is to say, the class is not finitely axiomatizable).

Paper Structure

This paper contains 4 sections, 20 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Group and coset relation algebras
  3. The variety generated by the class of full coset relation algebras
  4. CRA is not finitely axiomatizable

Key Result

Lemma 2.2

The relations $R_{{{xy}},{\alpha}}$, for $\alpha<\kappa_{{xy}}$, are non-empty and partition the set $G_{x}\times G_{y}$ .

Theorems & Definitions (34)

  • Definition 2.1
  • Lemma 2.2: Partition Lemma
  • Theorem 2.3: Boolean Reduct Theorem
  • Theorem 2.4: Identity Theorem
  • Theorem 2.6: Converse Theorem
  • Lemma 2.8
  • Theorem 2.9: Composition Theorem
  • Theorem 2.10: Image Theorem
  • Definition 2.11
  • Lemma 2.12
  • ...and 24 more