# 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**for the variety generated by

_{n}**B**. We define a notion of representation for algebras in

_{n}**RA**, and use it to give an explicit (hence recursive) equational axiomatisation of

_{n}**RA**, and to reprove Maddux's result that

_{n}**RA**is canonical. We show that the algebras in

_{n }**RA**are precisely those that have a complete representation.

_{n}Then we prove that whenever 3< n<k\leq\omega, **RA _{k}** is not finitely axiomatisable over

**RA**. This confirms a conjecture of Maddux. We also prove that

_{n}**B**is elementary for n=3,4 only.

_{n}