Finite representations for two small relation algebras

TL;DR

This paper provides two different finite representations for specific small relation algebras, using probabilistic and explicit group methods, and introduces a new finite representation technique for another algebra.

Contribution

It presents novel finite representations for relation algebras 52_{65} and 59_{65} using probabilistic, group, and Comer’s techniques.

Findings

Relation algebra 52_{65} is representable over a finite set.

Relation algebra 59_{65} has a finite representation over 113.

Two distinct proof methods for finite representability are demonstrated.

Abstract

In this note, we give two different proofs that relation algebra $52_{65}$ is representable over a finite set. The first is probabilistic, and uses Johnson schemes. The second is an explicit group representation over $ (\mathbb{Z}/2\mathbb{Z})^{10}$. We also give a finite representation of $59_{65}$ over $\mathbb{Z}/113\mathbb{Z}$ using a technique due to Comer.