# Manual Cylindric-like Algebras and Algebraic Logic

Individual models of correspond to simple s and elementary equivalence of models corresponds to isomorphism of s. The elements of an corresponding to a model are best thought of as the relations definable in. The other direction can also be elaborated and then a two-sided duality like Stone duality between s and certain topological spaces can occur; cf. Using the duality theory outlined above, logical properties of can be characterized by algebraic properties of , under some mild assumptions on.

The Beth definability property for is equivalent with surjectiveness of all epimorphisms in. The various definability properties weak Beth, local Beth, etc. A kind of completeness theorem for is equivalent with finite axiomatizability of.

Compactness of is equivalent with being closed under ultraproducts. The above and further equivalence theorems are elaborated in e. Further such results can be found in e. Czelakowski, L. Maksimova, and the references in [a35]. A duality theory for algebraic logic is in [a8]. An overview of duality theories is in [a2] , Chap.

This branch investigates classes of algebras that arise in the algebraization of the most frequently used logics. Below, attention is restricted to algebras of classical quantifier logics, algebras of the finite variable fragments of these logics, relativized versions of these logics, e.

See also Abstract algebraic logic. The objective is to "algebraize" logics which extend classical propositional logic. The algebras of this propositional logic are Boolean algebras cf.

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Boolean algebras are natural algebras of unary relations. One expects the algebras of the extended logics to be extensions of Boolean algebras to algebras of relations of higher ranks. The elements of a Boolean algebra are sets of points; one expects the elements of the new algebras to be sets of sequences since relations are sets of sequences.

They correspond to the -variable fragment of first-order logic. The new operations are cylindrifications. If is a relation defined by a formula , then is the relation defined by the formula. To be precise, one should write for. Assume ,. Then and. This shows that is a natural and simple operation on -ary relations: it simply abstracts from the th argument of the relation. Let ,.

In other words, if is the canonical projection along the th factor, then. The algebra of -ary relations over is. The class of -ary representable cylindric algebras is defined as. Then is a discriminator variety, with an undecidable but recursively enumerable equational theory. Almost all of these theorems remain true if one throws away the constant from and closes up under to make it a universally axiomatizable class. These properties imply theorems about via the duality theory between logics and classes of algebras elaborated in abstract algebraic logic. Further, usual set theory can be built up in and even in the equational theory of.

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For first-order logic with infinitely many variables cf. To generalize to , one needs only a single non-trivial step: one has to brake up the single constant to a set of constants , with. The definition of with an arbitrary ordinal number is practically the same. Most of the theorems about mentioned above carry over to. The greatest element of a "generic" was required to be a Cartesian space.

If one removes this condition and replaces with an arbitrary -ary relation in the definition, one obtains the important generalization of. Many of the negative properties of disappear in. Logic applications of abound, cf. Since is not finite schema axiomatizable, a finitely schematizable approximation was introduced by Tarski. There are theorems to the effect that s approximate s well, cf. The above illustrates the flavour of the theory of algebras of relations; important kinds of algebras not mentioned include relation algebras and quasi-polyadic algebras, cf. II, [a33] , [a9] , [a4] , [a30] , [a29] , [a13].

The theory of the latter two is analogous with that of s. Common generalizations of s, s, relation algebras, polycyclic algebras, and their variants is the important class of Boolean algebras with operators, cf. For category-theoretic approaches, see [a4] and the references therein.