# Canonical Relativized Cylindric Set Algebras and Weak Associativity

## TL;DR

This paper refines the representation theorem for weakly associative relation algebras using canonical relativized cylindric set algebras, ensuring atoms are orbits under permutation groups, building on a 1989 result.

## Contribution

It introduces a sharper version of the relative representation theorem for weakly associative relation algebras using canonical relativized cylindric set algebras.

## Key findings

- Atoms are orbits of sequences under permutation groups.
- The theorem was previously proved for the Resek-Thompson case in 1989.
- Provides a more precise algebraic characterization.

## Abstract

Canonical relativized cylindric set algebras are used to sharpen the relative representation theorem for weakly associative relation algebras, that every complete atomic weakly associative relation algebra is isomorphic with the relativization of a set relation algebra to a symmetric and reflexive binary relation, by insuring that the atoms of the set relation algebra and its relativization are orbits of single sequences under a group of permutations of the underlying set. This sharpening of the relative representation theorem was first proved for the Resek-Thompson Theorem in 1989.