Algebraic systems, class of

A class of algebraic systems of the same type. All systems of a given type are assumed to be written in a given signature $ \Omega $ and are called $ \Omega $- systems. A class $ \mathfrak K $ of $ \Omega $- systems is called abstract if, whenever it contains a system $ \mathbf A $, it also contains all $ \Omega $- systems isomorphic to $ \mathbf A $.

Let $ \mathfrak K $ be an abstract class of $ \Omega $- systems. One says that an $ \Omega $- system $ \mathbf A $ has a local set of $ \mathfrak K $- subsystems if there exists an inclusion-directed set $ \{ {\mathbf A _ \alpha } : {\alpha \in \Lambda } \} $ of subsystems $ \mathbf A _ \alpha $ of the system $ \mathbf A $ which cover $ \mathbf A $( i.e. $ \cup _ \alpha A _ \alpha = A $) and which belong to $ \mathfrak K $. A class $ \mathfrak K $ is called local if each $ \Omega $- system $ \mathbf A $ with a local set of $ \mathfrak K $- subsystems belongs to the class $ \mathfrak K $. Theorems which establish the local nature of given abstract classes are called local (cf. Mal'tsev local theorems).

An $ \Omega $- system $ \mathbf A $ is called $ \mathfrak K $- approximable (or $ \mathfrak K $- residual) if, for any predicate $ P \in \{ \Omega _ {p} , = \} $( i.e. for any basic predicate as well as for the predicate coinciding with the equality relation in $ \mathbf A $) and for any elements $ a _ {1} \dots a _ {n} $ in $ \mathbf A $ for which $ P ( a _ {1} \dots a _ {n} ) = F $, there exists a homomorphism $ \phi : \mathbf A \rightarrow \mathbf B $ of the system $ \mathbf A $ into some system $ \mathbf B $ of the class $ \mathfrak K $ for which, again, $ P ( \phi ( a _ {1} ) \dots \phi ( a _ {n} ) ) = F $. Any subsystem of a $ \mathfrak K $- approximable system is itself $ \mathfrak K $- approximable. If $ \mathfrak K $ is the class of all finite $ \Omega $- systems, a $ \mathfrak K $- approximable system $ \mathfrak A $ is called finitely approximable (or residually finite). If an abstract class $ \mathfrak K $ has a unit system $ E = \langle \{ e \} , \Omega \rangle $, an $ \Omega $- system $ \mathbf A $ is $ \mathfrak K $- approximable if and only if it is isomorphically imbeddable in a Cartesian product of systems from the class $ \mathfrak K $[3]. A class $ \mathfrak K $ is called residual if all $ \mathfrak K $- approximable systems belong to the class $ \mathfrak K $. A class $ \mathfrak K $ is homomorphically closed if it contains, for each of its $ \Omega $- systems $ \mathbf A $, also all $ \Omega $- systems that are homomorphic images of $ \mathbf A $. All residual homomorphically-closed classes are local [5].

A class $ \mathfrak K $ of $ \Omega $- systems is called (finitely) axiomatizable if there exists a (finite) set $ S $ of first-order closed formulas of the signature $ \Omega $ such that $ \mathfrak K $ consists of exactly those $ \Omega $- systems in which all formulas of $ S $ are true. Finitely-axiomatizable classes are also referred to as elementary classes. It has been shown [5] with the aid of the generalized continuum hypothesis that: 1) a class $ \mathfrak K $ of algebraic systems is axiomatizable if and only if it is closed with respect to ultra-products, and its complement (in the class of all $ \Omega $- systems) is closed with respect to ultra-powers; 2) a class $ \mathfrak K $ of algebraic systems is elementary if and only if both it and its complement are closed with respect to ultra-products. The theory of axiomatizable classes of algebraic systems deals with the connection between the structural properties of these classes and the syntactic features of the formal language in which these classes may be specified. Axiomatizable classes which play an especially important role in algebra include varieties (cf. Algebraic systems, variety of) and quasi-varieties (cf. Algebraic systems, quasi-variety of), which are local and residual.

In addition to axiomatizability by first-order closed formulas, axiomatizability by special second-order closed formulas is also considered. To the function and predicate signature symbols $ F _ {i} $( $ i \in I $), $ P _ {j} $( $ j \in I $) of a fixed signature $ \Omega $ are added predicate variables $ R _ {1} , R _ {2} ,\dots $. Let $ \mathfrak F $ be a quantifier-free formula of the first order, consisting of function and predicate signature symbols, predicate variables $ R _ {1} \dots R _ {s} $ and object variables $ x _ {1} \dots x _ {r} $. A second-order formula $ Q \mathfrak F $, where $ Q $ is some sequence of quantifiers of the type $ ( \forall R _ {i} ), ( \exists R _ {i} ) $ or $ ( \forall x _ {k} ) $ is known as crypto-universal. The second-order formulas formed from the crypto-universal formulas without free object variables, with the aid of the logical connectives $ \& , \lor , \rightarrow , \neg $ and with subsequent quantification by $ \forall $ on all free predicate variables encountered in the representations of the crypto-universal formulas, are called Boolean-universal formulas of the signature $ \Omega $. A class $ \mathfrak K $ of $ \Omega $- systems is called quasi-universal if there exists a set $ S $ of Boolean-universal formulas of the signature $ \Omega $ such that $ \mathfrak K $ consists of the $ \Omega $- systems in which all formulas of $ S $ are true, and only of such systems. A quasi-universal class of $ \Omega $- systems is local (Mal'tsev's theorem). A.I. Mal'tsev [4] gave a more detailed definition of a quasi-universal class.

References

[1]	A.I. Mal'tsev, "A general method for obtaining local theorems in group theory" Uchen. Zap. Ivanovsk. Gos. Ped. Inst. , 1 : 1 (1941) pp. 3–9 (In Russian)
[2]	A.I. Mal'tsev, "Model correspondences" Izv. Akad. Nauk SSSR Ser. Mat. , 23 (1959) pp. 313–336 (In Russian)
[3]	A.I. Mal'tsev, "Algebraic systems" , Springer (1973) (Translated from Russian)
[4]	A.I. Mal'tsev, "Some problems in the theory of classes of models" , Proc. 4-th All-Union Math. Congress (1961) , 1 , Leningrad (1963) pp. 169–198 (In Russian) (Transl. in: Amer. Math. Soc. Transl. (2) 83 (1969), 1–48)
[5]	P.M. Cohn, "Universal algebra" , Reidel (1981)
[6]	J.P. Cleave, "Local properties of systems" J. London Math. Soc. , 44 (1969) pp. 121–130

Comments

The characterization (under the generalized continuum hypothesis) of axiomatizable and elementary classes is due to H.J. Keisler [a1].

Translations of the articles [1], [2] and [4] may (also) be found in [a2], Chapts. 2, 11 and 26, respectively.

The term inductive class is sometimes used instead of "local class" .

References

[a1]	H.J. Keisler, "Ultraproducts and elementary classes" Indag. Math. , 23 (1961) pp. 477–495
[a2]	A.I. [A.I. Mal'tsev] Mal'cev, , The metamathematics of algebraic systems. Collected papers: 1936 - 1967 , North-Holland (1971)