The variety of coset relation algebras

TL;DR

This paper studies coset relation algebras, showing they form a class definable by equations but not finitely axiomatizable, involving systems of groups, isomorphisms, and cosets.

Contribution

It proves that the class of coset relation algebras is a variety but cannot be characterized by a finite set of equations.

Findings

The class of coset relation algebras is equationally axiomatizable.

No finite set of equations can axiomatize this class.

The algebraic structure involves systems of groups, isomorphisms, and cosets.

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).