Cyclic group representations for relation algebras $57_{65}$ and $63_{65}$

TL;DR

This paper demonstrates finite cyclic group representations for specific relation algebras, expanding the known class of finitely representable symmetric integral relation algebras on four atoms.

Contribution

It provides explicit cyclic group representations for relation algebras 57_{65} and 63_{65}, and clarifies the representability status of related algebras.

Findings

Finite cyclic group representations for $57_{65}$ and $63_{65}$.

All symmetric integral RAs on four atoms with at least one flexible atom are now known to be cyclic group representable, except $33_{65}$.

Abstract

We exhibit finite cyclic group representations for relation algebras $57_{65}$ and $63_{65}$. As a consequence, of the ten symmetric integral RAs on four atoms having at least one flexible atom, all are now known to have a representation over a finite cyclic group except for $33_{65}$, which is not even known to be finitely representable.