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 B_n for the class of semi-associative algebras with an n-dimensional relational basis, and RA_n for the variety generated by B_n. We define a notion of 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 RA_n are precisely those that have a complete representation.

Then we prove that whenever 3< n<k\leq\omega, RA_k 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.