Existence of partial transposition means representability in cylindric algebras

Abstract

We show that the representability of cylindric algebras by relativized set algebras depends on the scope of the operation transposition which can be defined on the algebra. The existence of "partial transposition" assures this kind of representability of the cylindric algebra (while the existence of transposition assures polyadic representation). Further we characterize those cylindric algebras in which the operator transposition can be introduced.

The polyadic generalization of the Boolean axiomatization of fields of sets

“…Further, the MGR axioms will be postulated. In [10] it is proven that the MGR axioms mean the existence of an operator "weak transposition" (or "partial transposition"). So by Resek-Thompson theorem (see (2.9)), the existence of such an operator yields representability by relativized set algebras.…”

“…where α ≥ 4 (see [9] As regards the representation theory of cylindric-like or polyadic-like algebras by relativized set algebras, see e.g., the references [2], [3], [4], [8], [9], [10], [13]. The references [1], [5], [6], [7], [12], [14], [15], [16] are related indirectly to the topic or are related to the applications.…”

“…By that theorem, the axiomatization of this class of fields of sets is that of the abstract class of cylindic algebras satisfying the so-called merry-go-round properties and a given weakening (axiom (C4)*) of the axiom of commutativity of cylindrifications (class NA + α ). Ferenczi proved in [9] and [10] that (C4)* is equivalent to the commutativity of single substitutions; further, a possible meaning of the merry-go-round properties is a restricted existence of the abstract transposition operator p ij . Considering the Resek-Thompson theorem and our main theorem together we can establish that in the representability of a cylindric-type algebra by relativized set algebra a crucial role is played by the implicit or explicit existence of the transposition operator.…”

Representations of polyadic-like equality algebras

“…The method used in Andréka and Thompson's proof is detailed in [12]. The r-representation of polyadic equality algebras by relativized set algebras was investigated in [6], [7] and [1]. In [6] it is shown that the background of the merry-go-round property is an axiom of transposition algebras (and it is also a property of polyadic algebras) and Resek and Thompson's theorem is closely related to transposition algebras.…”

“…The r-representation of polyadic equality algebras by relativized set algebras was investigated in [6], [7] and [1]. In [6] it is shown that the background of the merry-go-round property is an axiom of transposition algebras (and it is also a property of polyadic algebras) and Resek and Thompson's theorem is closely related to transposition algebras. The direct predecessor of the research here is the investigation of the representability of transposition algebras in [7].…”