A representation theorem for measurable relation algebras with cyclic groups

TL;DR

The paper proves that a measurable relation algebra whose associated groups $G_x$ are finite and cyclic is essentially isomorphic to a cyclic group relation algebra built from a group frame, and thus is completely representable. The authors develop a scaffold construction using regular elements and index gcd properties, then derive a full group relation algebra $\mathfrak{G}[\mathcal F]$ via a group frame $\mathcal F$; this yields a structural, representational description of the algebra. The result provides a definitive classification for the finite cyclic case and connects to prior work on atomic, complete, and pair-dense relation algebras, including extensions and special cases such as one element or two element groups. The approach gives a concrete Cayley representation framework and clarifies when measurable algebras coincide with group relation algebras.

Abstract

A relation algebra is measurable if the identity element is a sum of atoms, and the square x;1;x of each subidentity atom x is a sum of non-zero functional elements. These functional elements form a group Gx. We prove that a measurable relation algebra in which the groups Gx are all finite and cyclic is completely representable. A structural description of these algebras is also given.

Figures (2)

• Figure 1: A graphical example of verifying the index conditions.
• Figure 2: Diagram of the levels of construction of the scaffold.

Theorems & Definitions (42)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Definition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• Lemma 3.4
• proof
• Lemma 3.5