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