Relation Algebras with n-dimensional bases

Robin Hirsch and Ian Hodkinson

44 pages. We study relation algebras with n-dimensional relational bases in the sense of Maddux.

Fix n with 3\leq n<\omega. Write Bn for the class of semi-associative algebras with an n-dimensional relational basis, and RAn for the variety generated by Bn. We define a notion of 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 RAn are precisely those that have a complete representation.

Then we prove that whenever 3< n<k\leq\omega, RAk is not finitely axiomatisable over RAn. This confirms a conjecture of Maddux. We also prove that Bn is elementary for n=3,4 only.

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