Abstract
We investigate the class SRaCA n for 4 ≤ n < ω and survey some recent results. We see that RA n — the subalgebras of relation algebras with relational bases — is too weak, and that the class of relation algebras whose canonical extension has an n-dimensional cylindric basis is too strong to define the class. We introduce the notion of an n-dimensional hyperbasis and show that for any relation algebra A the canonical extension A + has such a hyperbasis if and only if A ∈ SRaCA n .We introduce techniques that can be used to show that the hierarchies RA4 ⊃ RA5 ⊃... and SRaCA4 ⊃ SRaCA5 ⊃... are strict and each step is not finitely axiomatisable.We outline a relativized semantics that characterises RA n and another one for the class of subalgebras of relation algebras with n-dimensional cylindric bases.
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