Coset relation algebras

TL;DR

This paper extends the class of measurable relation algebras by introducing coset systems to shift relative multiplication, showing many are non-representable and that all atomic measurable algebras are essentially coset relation algebras.

Contribution

It introduces coset relation algebras with shifted multiplication, expanding the class of measurable relation algebras and demonstrating their non-representability and completeness.

Findings

Extended measurable relation algebras with cosets.

Identified non-representable coset relation algebras.

All atomic measurable relation algebras are essentially coset relation algebras.

Abstract

A measurable relation algebra is a relation algebra in which the identity element is a sum of atoms that can be measured in the sense that the "size" of each such atom can be defined in an intuitive and reasonable way (within the framework of the first-order theory of relation algebras). A large class of examples of such algebras, using systems of groups and coordinated systems of isomorphisms between quotients of the groups, has been constructed. This class of group relation algebras is not large enough to exhaust the class of all measurable relation algebras. In the present article, the class of examples of measurable relation algebras is considerably extended by adding one more ingredient to the mix: systems of cosets that are used to "shift" the operation of relative multiplication. It is shown that, under certain additional hypotheses on the system of cosets, each such coset relation algebra with a shifted operation of relative multiplication is an example of a measurable relation algebra. We also show that the class of coset relation algebras does contain examples that are not representable as set relation algebras. In a later article, it is shown that the class of coset relation algebras is adequate to the task of describing all measurable relation algebras in the sense that every atomic measurable relation algebra is essentially isomorphic to a coset relation algebra.