Strongly representable atom structures of relation algebras

Abstract

A relation algebra atom structure α \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 C m α \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 Γ \Gamma , we construct a relation algebra atom structure α ( Γ ) \alpha (\Gamma ) and prove, for infinite Γ \Gamma , that α ( Γ ) \alpha (\Gamma ) is strongly representable if and only if the chromatic number of Γ \Gamma is infinite. A construction of Erdös shows that there are graphs Γ r \Gamma _r ( r > ω r>\omega ) with infinite chromatic number, with a non-principal ultraproduct ∏ D Γ r \prod _D\Gamma _r whose chromatic number is just two. It follows that α ( Γ r ) \alpha (\Gamma _r) is strongly representable (each r > ω r>\omega ) but ∏ D ( α ( Γ r ) ) \prod _D(\alpha (\Gamma _r)) is not.