The polyadic generalization of the Boolean axiomatization of fields of sets

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 $\alpha$-power $^{\alpha }U$. Beyond the usual set operations union, intersection, and complement, new operations are needed to describe such a field of sets, e.g., the $i$th cylindrification $C_{i},$ the constant $ij$th diagonal $D_{ij},$ the elementary substitution $\left [ i\;/\;j\right ]$ and the transposition $\left [ i,\;j\right ]$ for all $i,j<\alpha$ 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).