A representation theorem for measurable relation algebras
S. Givant, H. Andréka
TL;DR
The paper addresses representing atomic measurable relation algebras using a group-theoretic synthesis: systems of groups, quotient isomorphisms, and coset shifts. It proves that every atomic, measurable relation algebra ${\mathfrak B}$ is essentially isomorphic to a coset relation algebra ${\mathfrak C}$ in the sense that the completion ${\mathfrak B}^+$ is isomorphic to ${\mathfrak C}$, and that finite ${\mathfrak B}$ are themselves coset relation algebras; complete representability is equivalent to scaffold-coordinated isomorphisms. The results establish a structural bridge from abstract measurable relation algebras to concrete group-theoretic data, illustrating how the induced isomorphisms between quotient groups and coset shifts govern representability. This work extends prior representation theorems (e.g., Maddux, Jónsson–Tarski) to the atomic measurable setting, providing a comprehensive framework for understanding when such algebras admit coset-based representations and how regular elements with normal stabilizers drive the quotient-isomorphism structure.
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.