Relation Algebras from Cylindric and Polyadic Algebras

This paper is a survey of recent results concerning connections between relation algebras (RA), cylindric algebras (CA) and polyadic equality algebras (PEA). We describe exactly which subsets of the standard axioms for RA are needed for axiomatizing RA over the RA-reducts of CA3’s, and we do the same for the class SA of semi-associative relation algebras. We also characterize the class of RA-reducts of PEA3’s. We investigate the interconnections between the RA-axioms within CA3 in more detail, and show that only four implications hold between them (one of which was proved earlier by Monk). In the other direction, we introduce a natural CA-theoretic equation MGR+, a generalization of the well-known Merry-Go-Round equation MGR of CA-theory. We show that MGR+ is equivalent to the RA-reduct being an SA, and that MGR+ implies that the RA-reduct determines the algebra itself, while MGR is not sufficient for either of these to hold. Then we investigate how different CA’s a single RRA can “generate” in the general case. We solve the first part of Problem 11 from the “Problem Session Paper” of [2]. While proving some of the statements, for others we give only outline of proof. The paper contains several open problems. A full version of this paper is under preparation.