The question of characterizing the (finite) representable relation algebras in a ``nice" way is open. The class $\mathbf{RRA}$ is known to be not finitely axiomatizable in first-order logic. Nevertheless, it is conjectured that ``almost all'' finite relation algebras are representable. All finite relation algebras with three or fewer atoms are representable. So one may ask, Over what cardinalities of sets are they representable? This question was answered completely by Andr\'eka and Maddux (``Representations for small relation algebras,'' \emph{Notre Dame J. Form. Log.}, \textbf{35} (1994)); they determine the spectrum of every finite relation algebra with three or fewer atoms. In the present paper, we restrict attention to cyclic group representations, and completely determine the cyclic group spectrum for all seven symmetric integral relation algebras on three atoms. We find that in some instances, the spectrum and cyclic spectrum agree; in other instances, the spectra disagree for finitely many $n$; finally, for other instances, the spectra disagree for infinitely many $n$. The proofs employ constructions, SAT solvers, and the probabilistic method.

翻译：刻画（有限）可表示关系代数在“优美”意义上的特征这一问题尚未解决。已知类$\mathbf{RRA}$在一阶逻辑中不是有限可公理化的。尽管如此，人们推测“几乎所有”有限关系代数都是可表示的。所有具有三个或更少原子的有限关系代数都是可表示的。因此，一个问题自然浮现：它们在何种集合基数上可表示？Andréka和Maddux（“Representations for small relation algebras,” \emph{Notre Dame J. Form. Log.}, \textbf{35} (1994)）完整回答了该问题；他们确定了每个具有三个或更少原子的有限关系代数的谱。本文中，我们聚焦于循环群表示，并完全确定了三个原子的全部七个对称整关系代数的循环群谱。我们发现：在某些情形下，谱与循环谱一致；在另一些情形下，两者对有限多个$n$不一致；最后，还有情形下，两者对无穷多个$n$不一致。证明采用了构造方法、SAT求解器以及概率方法。