Strongly representable atom structures of relation algebras

A relation algebra atom structure is said to be strongly rep- resentable if all atomic relation algebras with that atom structure are rep- resentable. This is equivalent to saying that the complex algebra Cm is a representable relation algebra. We show that the class of all strongly repre- sentable 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 undi- rected, loop-free graph , we construct a relation algebra atom structure () and prove, for infinite , that () is strongly representable if and only if the chromatic number of is infinite. A construction of Erdos shows that there are graphs r (r < !) with infinite chromatic number, with a non-principal ul- traproduct Q D r whose chromatic number is just two. It follows that ( r) is strongly representable (each r < !) but Q D ( ( r)) is not.

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