The polyadic generalization of the Boolean axiomatization of fields of sets

Abstract

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).

On the representability of neatly embeddable CA’s by cylindric relativized algebras

“…Our results can be adapted to quasi polyadic equality algebras and polyadic equality algebras, with modifications ( [5], [6]). Now we turn to the proofs of Theorem 3.1 and Theorem 3.2.…”

“…An interesting application of our results is the investigation of representability of non-commutative quasi polyadic equality algebras (or transposition algebras TA α 's) by neat-embeddability (see [2] and [5]). It is proven that algebras in TA α are representable by polyadic relativized generalized weak set algebras (class Gwp α ).…”

Existence of partial transposition means representability in cylindric algebras

“…where the elements d ij are constants, c i , s i j , p ij are unary operations, and the axioms (F0-F11) below are valid for every i, j, k < α (see, [6]):…”

“…Axiom (F5) (therefore also the commutativity of cylindrification) fails to be true in Gwp α . It is easy to see that Gwp α = Dp α (see [6]). Now we recall the reader of some representation theorems connected with the above classes.…”

“…So A, s i j , k s(i, j) (k ≤ α) is the suitable extension of A. Conversely, now assume that A is a reduct of B, B ∈ TA α . By the representation theorem for TA α (see [6]), B is representable by a set algebra G in Gwp α .…”