A representation theorem for measurable relation algebras

TL;DR

This paper characterizes the structure of atomic measurable relation algebras, showing they can be built from systems of groups and isomorphisms, and provides conditions for their complete representability.

Contribution

It introduces a structural construction for atomic measurable relation algebras using groups, isomorphisms, and cosets, and characterizes when they are completely representable.

Findings

Atomic measurable relation algebras are constructed from systems of groups and isomorphisms.

Complete representability requires a stronger coordination via a scaffold.

Measurable relation algebras with finite associated groups are atomic.

Abstract

A relation algebra is called measurable when its identity is the sum of measurable atoms, and an atom is called measurable if its square is the sum of functional elements. In this paper we show that atomic measurable relation algebras have rather strong structural properties: they are constructed from systems of groups, coordinated systems of isomorphisms between quotients of the groups, and systems of cosets that are used to "shift" the operation of relative multiplication. An atomic and complete measurable relation algebra is completely representable if and only if there is a stronger coordination between these isomorphisms induced by a scaffold (the shifting cosets are not needed in this case). We also prove that a measurable relation algebra in which the associated groups are all finite is atomic.