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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 αU . 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 Ci, the constant ijth diagonal Dij , 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).

The polyadic generalization of the Boolean axiomatization of fields of sets