Abstract. A version of the classical representation theorem for Boolean algebras states that the fields of sets form a variety and that a possible axiomatization is the system of Boolean axioms. An important case for fields of sets occurs when the unit V is a subset of an α-power α U . Beyond the usual set operations union, intersection, and complement, new operations are needed to describe such a field of sets, e.g., the ith cylindrification C i , the constant ijth diagonal D ij , the elementary substitution [i / j] and the transposition [i, j] for all i, j < α restricted to the unit V . Here it is proven that such generalized fields of sets being closed under the above operations form a variety; further, a first order finite scheme axiomatization of this variety is presented. In the proof a crucial role is played by the existence of the operator transposition. The foregoing axiomatization is close to that of finitary polyadic equality algebras (or quasi-polyadic equality algebras).

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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 (CA + α , α ≥ 4). 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 CA + α a β-dimensional algebra can be obtained in QPEA β where β < α (β ≥ 4), moreover the algebra obtained is representable in a sense.

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 (QPENα), 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 QPENα are representable by quasi‐polyadic relativized set algebras, or more exactly by algebras in Gwqα.