Miklós Ferenczi

On representability of neatly embeddable cylindric algebras

ABSTRACT As is well-known, a classical representation theorem of the theory of cylindric algebras is: A ε IGwsa if and only if A ε SNrαCAα+ε. The part “only if” is trivial. Regarding to the other part “A ε SNrαCAα+ε then A ε IGwsα“ the following question arises: is it possible to replace the class CA in the hypothesis A ε SNrαCAα+ε by a larger class so that the theorem still holds. Such a larger class Kα β is defined. The class Kα β is the best possible, in a sense to be made precise. Representability of set algebras is investigated, too.

Cylindric-like algebras and algebraic logic

Introduction.- H. Andreka and I. Nemeti: Reducing First-order Logic to Df3, Free Algebras.- N.Bezhanishvili: Varieties of Two-Dimensional Cylindric Algebras.- R. Hirsch and I. Hodkinson: Completions and Complete Representations.- J. Madarasz and T. Sayed Ahmed: Amalgamation, Interpolation and Epimorphisms in Algebraic Logic.- T. Sayed Ahmed: Neat Reducts and Neat Embeddings in Cylindric Algebras.- M. Ferenczi: A New Representation Theory: Representing Cylindric-like Algebras by Relativized Set Algebras.- A. Simon: Representing all Cylindric Algebras by Twisting, On a Problem of Henkin.- A. Kurucz: Representable Cylindric Algebras and Many-Dimensional Modal Logics.- T. Sayed Ahmed: Completions, Complete Representations and Omitting Types.- G. Sereny: Elements of Cylindric Algebraic Model Theory.- Y. Venema: Cylindric Modal Logic.- J. van Benthem: Crs and Guarded Logics: A Fruitful Contact.- R. S. Dordevic and M. D. Raskovic: Cylindric Probability Algebras.-I. Duentsch: Cylindric Algebras and Relational Databases. - M. Ferenczi: Probability Measures and Measurable Functions on Cylindric Algebras. - A. Mann: Cylindric Set Algebras and IF Logic. - G. Sagi: Polyadic Algebras. - I. Sain: Definability Issues in Universal Logic. - Bibliography. - Index

The polyadic generalization of the Boolean axiomatization of fields of sets

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).

Representations of polyadic-like equality algebras

Assuming the usual finite axiom schema of polyadic equality algebras, axiom (P10) is changed to a stronger version. It is proved that infinite dimensional, polyadic equality algebras satisfying the resulting system of axioms are representable. The foregoing stronger axiom is not given with a first order schema. The latter is to be expected knowing the negative results for the Halmos schema axiomatizability of the representable, infinite dimensional, polyadic equality algebras. Furthermore, Halmos’ well-known classical theorem that “locally finite polyadic equality algebras of infinite dimension α are representable” is generalized for locally-

Non-standard Stochastics with a First Order Algebraization

{\mathfrak{m}}

On the definition and the representability of quasi-polyadic equality algebras

m polyadic equality algebras, where

Finitary Polyadic Algebras from Cylindric Algebras

{\mathfrak{m}}