Relation Algebras with n-Dimensional Relational Bases

We study relation algebras with n-dimensional relational bases in the sense of Maddux. Fix n with 3 ≤ n ≤ ω. Write Bn for the class of non-associative algebras with an ndimensional relational basis, and RAn for the variety generated by Bn. We define a notion of relativised representation for algebras in RAn, and use it to give an explicit (hence recursive) equational axiomatisation of RAn, and to reprove Maddux’s result that RAn is canonical. We show that the algebras in Bn are precisely those that have a complete relativised representation of this type. Then we prove that whenever 4 ≤ n < l ≤ ω, RAl is not finitely axiomatisable over RAn. This confirms a conjecture of Maddux. We also prove that Bn is elementary for n = 3, 4 only.