Strongly representable atom structures of relation algebras

A relation algebra atom structure $\alpha$ is said to be strongly representable if all atomic relation algebras with that atom structure are representable. This is equivalent to saying that the complex algebra $\mathop{\mathfrak{Cm}} \alpha$ is a representable relation algebra. We show that the class of all strongly representable relation algebra atom structures is not closed under ultraproducts and is therefore not elementary. This answers a question of Maddux (1982).

Our proof is based on the following construction. From an arbitrary undirected, loop-free graph $\Gamma$, we construct a relation algebra atom structure $\alpha(\Gamma)$ and prove, for infinite $\Gamma$, that $\alpha(\Gamma)$ is strongly representable if and only if the chromatic number of $\Gamma$ is infinite. A construction of Erdös shows that there are graphs $\Gamma_r$($r<\omega$) with infinite chromatic number, with a non-principal ultraproduct $\prod_D\Gamma_r$ whose chromatic number is just two. It follows that $\alpha(\Gamma_r)$ is strongly representable (each $r<\omega$) but $\prod_D(\alpha(\Gamma_r))$ is not.