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Istvan Nemeti and Andras Simon

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Relation algebras from cylindric and polyadic algebras

Abstract.
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 CA_3's, and we do the same for the class
SA of semi-associative relation algebras. We also
characterize the class of RA-reducts of PEA_3's. We investigate
the interconnections between the RA-axioms within CA_3 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
[AMN88].

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.

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