We show that the usual axiom system of quasi polyadic equality algebras is strongly redundant. Then, so called non‐commutative quasi‐polyadic equality algebras are introduced (), in which, among others, the commutativity of cylindrifications is dropped. As is known, quasi‐polyadic equality algebras are not representable in the classical sense, but we prove that algebras in are representable by quasi‐polyadic relativized set algebras, or more exactly by algebras in.

It is known that every α-dimensional quasi polyadic equality algebra (QPEA α ) can be considered as an α-dimensional cylindric algebra satisfying the merrygo- round properties . The converse of this proposition fails to be true. It is investigated in the paper how to get algebras in QPEA from algebras in CA. Instead of QPEA the class of the finitary polyadic equality algebras (FPEA) is investigated, this class is definitionally equivalent to QPEA. It is shown, among others, that from every (...) algebra in a β-dimensional algebra can be obtained in QPEA β where , moreover the algebra obtained is representable in a sense. (shrink)