Three classes are introduced which are closely related to the class included in the title. It is proven that the class obtained from by replacing axiom C4 by the commutativity of single substitutions can be considered as the abstract class in the Resek–Thompson theorem, thus it is representable by set algebras. Then the class is defined and it is shown that the necessary and sufficient condition for neat embeddability of an algebra in CAα into is the validity of the merry-go-round (...) properties. Finally, the class is introduced which class is a counterpart of among the polyadic like algebras. (shrink)

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. (shrink)

It is known that every α-dimensional quasi polyadic equality algebra (QPEA α ) can be considered as an α-dimensional cylindric algebra satisfying the merrygo- round properties . The converse of this proposition fails to be true. It is investigated in the paper how to get algebras in QPEA from algebras in CA. Instead of QPEA the class of the finitary polyadic equality algebras (FPEA) is investigated, this class is definitionally equivalent to QPEA. It is shown, among others, that from every (...) algebra in a β-dimensional algebra can be obtained in QPEA β where , moreover the algebra obtained is representable in a sense. (shrink)

We show that the usual axiom system of quasi polyadic equality algebras is strongly redundant. Then, so called non‐commutative quasi‐polyadic equality algebras are introduced (), in which, among others, the commutativity of cylindrifications is dropped. As is known, quasi‐polyadic equality algebras are not representable in the classical sense, but we prove that algebras in are representable by quasi‐polyadic relativized set algebras, or more exactly by algebras in.

Cylindric set algebras are algebraizations of certain logical semantics. The topic surveyed here, i.e. probabilities defined on cylindric set algebras, is closely related, on the one hand, to probability logic (to probabilities defined on logical formulas), on the other hand, to measure theory. The set algebras occuring here are associated, in particular, with the semantics of first order logic and with non-standard analysis. The probabilities introduced are partially continous, they are continous with respect to so-called cylindric sums.

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 . Further we characterize those cylindric algebras in which the operator transposition can be introduced.

Internal sets and the Boolean algebras of the collection of the internal sets are of central importance in non-standard analysis. Boolean algebras are the algebraization of propositional logic while the logic applied in non-standard analysis (in non-standard stochastics) is the first order or the higher order logic (type theory). We present here a first order logic algebraization for the collection of internal sets rather than the Boolean one. Further, we define an unusual probability on this algebraization.

If the language is extended by new individual variables, in classical first order logic, then the deduction system obtained is a conservative extension of the original one. This fails to be true for the logics with infinitary predicates. But it is shown that restricting the commutativity of quantifiers and the equality axioms in the extended system and supposing the merry-go-round property in the original system, the foregoing extension is already conservative. It is shown that these restrictions are crucial for an (...) extension to be conservative. The origin of the results is algebraic logic. (shrink)

There is an extensive literature related to the algebraization of first‐order logic. But the algebraization of full second‐order logic, or Henkin‐type second‐order logic, has hardly been researched. The question arises: what kind of set algebra is the algebraic version of a Henkin‐type model of second‐order logic? The question is investigated within the framework of the theory of cylindric algebras. The answer is: a kind of cylindric‐relativized diagonal restricted set algebra. And the class of the subdirect products of these set algebras (...) is the algebraization of Henkin‐type second‐order logic. It is proved that the algebraization of a complete calculus of the Henkin‐type second‐order logic is a class of a kind of diagonal restricted cylindric algebras. Furthermore, the connection with the non‐standard enlargements of standard complete second‐order structures is investigated. (shrink)