This paper is a survey of recent results concerning connections between relation algebras , cylindric algebras and polyadic equality algebras . 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 . In the other direction, we introduce a natural CA-theoretic equation MGR+, generalization of the well-known Merrry-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.Keywords: relation algebras, cylindric algebras, polyadic algebras, algebraic logic, arrow logic, proof theory, finite variable fragments, provability with 3 variables, non-finitizability, twisting, non-standard models, neat reducts, representability. (shrink)

We study algebras whose elements are relations, and the operations are natural “manipulations” of relations. This area goes back to 140 years ago to works of De Morgan, Peirce, Schröder . Well known examples of algebras of relations are the varieties RCAn of cylindric algebras of n-ary relations, RPEAn of polyadic equality algebras of n-ary relations, and RRA of binary relations with composition. We prove that any axiomatization, say E, of RCAn has to be very complex in the following sense: (...) for every natural number k there is an equation in E containing more than k distinct variables and all the operation symbols, if 2 < n < ω. Completely analogous statement holds for the case n ω. This improves Monk's famous non-finitizability theorem for which we give here a simple proof. We prove analogous nonfinitizability properties of the larger varieties SNrnCAn + k. We prove that the complementation-free subreducts of RCAn do not form a variety. We also investigate the reason for the above “non-finite axiomatizability” behaviour of RCAn. We look at all the possible reducts of RCAn and investigate which are finitely axiomatizable. We obtain several positive results in this direction. Finally, we summarize the results and remaining questions in a figure. We carry through the same programme for RPEAn and for RRA. By looking into the reducts we also investigate what other kinds of natural algebras of relations are possible with more positive behaviour than that of the well known ones. Our investigations have direct consequences for the logical properties of the n-variable fragment Ln of first order logic. The reason for this is that RCAn and RPEAn are the natural algebraic counterparts of Ln while the varieties SNrnCAn + k are in connection with the proof theory of Ln. This paper appears in two parts. The first part contains the non-finite axiomatizability results. The present second part contains finite axiomatizations of some fragments together with a figure summarizing the finite and non-finite axiomatizability results in this area and the problems left open. (shrink)