This is a survey article on algebraic logic. It gives a historical background leading up to a modern perspective. Central problems in algebraic logic (like the representation problem) are discussed in connection to other branches of logic, like modal logic, proof theory, model-theoretic forcing, finite combinatorics, and Gödel’s incompleteness results. We focus on cylindric algebras. Relation algebras and polyadic algebras are mostly covered only insofar as they relate to cylindric algebras, and even there we have not told the whole story. (...) We relate the algebraic notion of neat embeddings (a notion special to cylindric algebras) to the metalogical ones of provability, interpolation and omitting types in variants of first logic. Another novelty that occurs here is relating the algebraic notion of atom-canonicity for a class of boolean algebras with operators to the metalogical one of omitting types for the corresponding logic. A hitherto unpublished application of algebraic logic to omitting types of first order logic is given. Proofs are included when they serve to illustrate certain concepts. Several open problems are posed. We have tried as much as possible to avoid exploring territory already explored in the survey articles of Monk [93] and Németi [97] in the subject. (shrink)

We survey various results on the relationship among neat embeddings, complete representations, omitting types, and amalgamation. A hitherto unpublished application of algebraic logic to omitting types of first-order logic is given.

We give a new characterization of the class of completely representable cylindric algebras of dimension 2 #lt; n w via special neat embeddings. We prove an independence result connecting cylindric algebra to Martin''s axiom. Finally we apply our results to finite-variable first order logic showing that Henkin and Orey''s omitting types theorem fails for L n, the first order logic restricted to the first n variables when 2 #lt; n#lt;w. L n has been recently (and quite extensively) studied as a (...) many-dimensional modal logic. (shrink)

SC , CA , QA and QEA stand for the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasipolyadic algebras, and quasipolyadic equality algebras of dimension , respectively. Generalizing a result of Németi on cylindric algebras, we show that for K {SC, CA, QA, QEA} and ordinals , the class Nr K of -dimensional neat reducts of -dimensional K algebras, though closed under taking homomorphic images and products, is not closed under forming subalgebras (i.e. is not a variety) if (...) and only if > 1.From this it easily follows that for 1 , the operation of forming -neat reducts of algebras in K does not commute with forming subalgebras, a notion to be made precise. (shrink)

Let 1 n. We show that the class NrnCAβ of n-dimensional neat reducts of β-dimensional cylindric algebras is not closed under forming elementary subalgebras. This solves a long-standing open problem of Tarski and his co-authors Andréka, Henkin, Monk and Németi. The proof uses genuine model-theoretic arguments.

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)

We give a new characterization of the class of completely representable cylindric algebras of dimension 2 #lt; n ≤ w via special neat embeddings. We prove an independence result connecting cylindric algebra to Martin's axiom. Finally we apply our results to finite-variable first order logic showing that Henkin and Orey's omitting types theorem fails for Ln, the first order logic restricted to the first n variables when 2 #lt; n#lt;w. Ln has been recently (and quite extensively) studied as a many-dimensional (...) modal logic. (shrink)

We give an algebraic version of first order logic without equality in which the class of representable algebras forms a finitely based equational class. Further, the representables are defined in terms of set algebras, and all operations of the latter are permutation invariant. The algebraic form of this result is Theorem 1.1 , while its logical form is Corollary 5.2.For first order logic with equality we give a result weaker than the one for first order logic without equality. Namely, in (...) this case - instead of finitely axiomatizing the corresponding class of all representable algebras - we finitely axiomatize only the equational theory of that class. Subsection 6.1, especially Remark 6.6 there.The proof of Theorem 1.1 is elaborated in Sections 3 and 4. These sections contain theorems which are interesting of their own rights, too, e.g. Theorem 4.2 is a purely semigroup theoretic result. Cf. also 'Further main results' in the Introduction. (shrink)