Finite integral relation algebras

The method of forming complex algebras of finite ternary relations will produce all finite nonassociative relation algebras. We use this method to construct many interesting algebras. Every atomic NA determines an involution on atoms, which in turn determines both the cycles of the algebra and also the missing, or forbidden cycles. So we begin by studying the cycles of an arbitrary involution. Suppose is an involution on U , that is, : U → U and u = u for all u ∈ U . For any x, y, z ∈ U let