Connections between cylindric algebras and relation algebras

We investigate the class SRaCAn for 4 n < ! and survey some recent results. We see that RAn | 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 deene 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 2 SRaCAn. We introduce techniques that can be used to show that the hierarchies RA4 RA5 : : : and SRaCA4 SRaCA5 : : : are strict and each step is not nitely axiomatisable. We outline a relativized semantics that characterises RAn and another one for the class of subalgebras of relation algebras with n-dimensional cylindric bases. This abstract summarises our investigations into the connections between relation algebras and cylindric algebras. The detailed proofs have been omitted, but fuller accounts of the material can be found in HH98, HHM98, HH99a, HH99b]. Our intention here is to provide a concise overview of a number of diierent but related results. Algebraic logic is the study of algebraic counterparts to logical systems and historically an important part of algebraic logic concerns the algebraic treatment of relations. The simplest algebraic logic, boolean algebra, can be thought of as the algebra of unary relations. The correspondence between boolean algebras and elds of unary relations (or just elds of sets) is precise: every eld of sets is a boolean algebra and every boolean algebra is isomorphic to a eld of sets Sto36]. There are a number of algebras that are intended to correspond to relations of higher ranks, though the correspondence is less accurate. For binary relations, the most important kind of algebra is a relation algebra, based on the work of De Morgan, Peirce and Schrr oder and formalised by Tarski. But not every relation algebra is isomorphic to a eld of binary relations (not every relation algebra is representable) and it requires innnitely many axioms to characterise the isomorphism class of the representable relation algebras Lyn50, Mon64]. For higher-order relations there are alternative algebraizations: cylindric algebra, diagonal-free cylindric algebra, polyadic algebra and so on. The one which has received most attention is cylin-dric algebra. As with binary relations, not every n-dimensional cylindric algebra is representable as an n-dimensional generalised cylindric …