Strongly Representable Atom Structures of Relation Algebras

A relation algebra atom structure α 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 Cmα 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 Γ, we construct a relation algebra atom structure α(Γ) and prove, for infinite Γ, that α(Γ) is strongly representable if and only if the chromatic number of Γ is infinite. A construction of Erdös shows that there are graphs Γr (r < ω) with infinite chromatic number, with a non-principal ultraproduct ∏ D Γr whose chromatic number is just two. It follows that α(Γr) is strongly representable (each r < ω) but ∏ D(α(Γr)) is not.