# Algebraic system

A set with operations and relations defined on it. An algebraic system is one of the basic mathematical concepts and its general theory has been developed in depth. This was done in the 1950s, and the work took place on the interface between algebra and mathematical logic.

## Fundamental concepts.

An algebraic system is an object consisting of a non-empty set , a family of algebraic operations (cf. Algebraic operation) () and a family of relations (cf. Relation) () defined on . The indices of the Cartesian powers of under consideration are assumed to be non-negative integers and are said to be the arities of the respective operations and relations. The set is called the carrier or the underlying set of the algebraic system , while its elements are called the elements of this system. The cardinality of is said to be the cardinality or order of the algebraic system . The image of the element under the mapping is called the value of the operation at the point . Similarly, if , then one says that the elements of are in relation , and one writes . The operations () and the relations (), unlike other operations and relations that may be defined on , are called basic or primitive.

The pair of families is called the type of the algebraic system . Two algebraic systems have the same type if , and , for all , . Basic operations , and basic relations , of two algebraic systems , of the same type, with identical indices in , respectively, are called similar.

An algebraic system is called finite if the set is finite and it is called of finite type if the set is finite. An algebraic system of finite type is written as .

An algebraic system is called a universal algebra or an algebra if the set of its basic relations is empty, and it is called a model (in logic) or a relational system if the set of basic operations is empty. Classical algebraic systems are groups, rings, linear spaces, linear algebras, totally ordered sets, totally ordered groups, lattices, etc.

A non-empty subset of the underlying set of an algebraic system is called closed if for any elements of the value of each basic operation also belongs to . By considering the operations of and the relations of on a closed subset , one obtains an algebraic system which is of the same type as the given algebraic system and is called a subsystem of the algebraic system . Subsystems of algebras are called subalgebras, while subsystems of models are called submodels. The concept of a subalgebra strongly depends on the set of basic operations of the algebra under consideration. Thus, a groupoid is an algebra of type , i.e. an algebra with one basic operation . A groupoid with a distinguished unit is an algebra of type , the distinguished element of which has the property for all with respect to the basic operation . For this reason any subgroupoid of a groupoid with a distinguished unit contains , while a subgroupoid of a groupoid does not necessarily contain . Unlike algebras, any non-empty subset of a model may be considered as a submodel.

An algebraic system is isomorphic to an algebraic system of the same type if there exists a one-to-one mapping of the set into the set such that

 (1)
 (2)

holds for all of and for all , . A mapping possessing these properties is called an isomorphism.

In the following, a class of algebraic systems will be understood to mean only an abstract class, i.e. a class of algebraic systems of the same type which, whenever it contains a system , also contains all systems isomorphic to . In considering a class of algebraic systems, all systems in this class are usually written in a definite signature, as follows. Let the type of the class be . To each is assigned some symbol , called a functional, while to each is assigned a symbol , called a predicate. If an algebraic system belongs to the class and if is a basic operation in it, then the element of is written as . Similarly, if is a basic relation in and the element , then one writes (true) or simply . If, on the other hand, , one writes (false) or . Let , and let be the mapping of the union into the set of natural numbers , defined by the formulas (), (). The object is the signature of the class . A finite signature is written as a string or, more briefly, . An algebraic system , written in the signature , is known as an -system and is denoted by .

The conditions (1) and (2) for an isomorphism of the systems and become simplified if the systems are considered in one signature . Thus, if () and () are the signature symbols, equations (1) and (2) assume the form

 (3)
 (4)

A homomorphism of an -system into an -system is any mapping which fulfills both condition (3) and the condition

 (5)

for all and for all of . A homomorphism is called strong if, for any elements of and for any predicate symbol of , the relation implies the existence in of pre-images of the elements such that . The concepts of a homomorphism and of a strong homomorphism of algebras are identical. Models have homomorphisms that are not strong, and have one-to-one homomorphisms that are not isomorphisms. Under a homomorphism , images of subsystems of are subsystems in ; and non-empty complete pre-images of subsystems of are subsystems in .

An equivalence relation is called a congruence of the -system if

for all of and all . For each homomorphism of an algebraic system the binary relation which holds if and only if is a congruence in , called its kernel congruence. For any congruence of an -system and for each element the set is called a coset of the algebraic system by the congruence . On the assumption that, for each , ,

and

if and only if there exist elements in such that and , one obtains the algebraic system , which is of the same type as the given system; it is called the quotient system of the algebraic system by the congruence . For each congruence of an algebraic system the canonical mapping () is a homomorphism of onto the quotient system for which the given congruence is the kernel congruence. If is a homomorphism of an algebraic system into an algebraic system and is the kernel congruence for , then the mapping is a homomorphism of the quotient system into the algebraic system . If the homomorphism is strong, is an isomorphism.

The Cartesian product of -systems , , is the -system in which is the Cartesian product of the underlying sets () and the basic operations and the basic relations on are given by the conditions: (, ) is the element with coordinates (), () if and only if for all .

## A language of the first order.

The basic formal language of the theory of algebraic systems is the first-order language which is constructed as follows. The alphabet of the language in the given signature , , , consists of the object variables the function symbols (), the predicate symbols (), the logical connectives

the quantifiers:

— "for each element xk" ;

— "there exists an element xk" ;

and the auxiliary symbols: brackets and commas. The (first-order) properties of -systems are expressed by finite sequences of alphabet symbols, or words, constructed in accordance with specific rules and known as terms and formulas. It is inductively assumed that each word of the form or is a term if ; if are terms and if , then is also a term.

If is an -system and if is a term of the signature containing object variables , then if one replaces by some elements in , and executes the operations in on the latter elements corresponding to the symbols of forming part of this term, then one obtains an element from , which is called the value of the term for . If is a homomorphism of an -system into an -system , then

The concept of a formula of the signature , and of free and bound object variables in it, is also defined inductively:

1) If is any predicate symbol for or the equality sign , or 2 respectively, and if are arbitrary terms of the signature , then the word is a formula in which all object variables are free.

2) If is a formula, so is . Free (bound) object variables in the formula are those and only those variables which are free (bound) in .

3) If are formulas and if the object variables which form part of both formulas are free in both formulas, then the words

 (6)

are also formulas.

Object variables which are free (bound) in at least one of the formulas , are called free (bound) in the formulas (6) as well.

4) If an object variable occurs free in a formula , then the words , are again formulas in which the variable is bound, while all the other object variables which occur free or bound in the formula remain free or bound in the formulas , as well.

Let an -system and a formula of the signature be given. If one now assigns to all free object variables from some values from and if one interprets the function and the predicate symbols forming part of as the respective basic operations and basic relations in , then one obtains a definite statement which may be true or false. One accordingly assigns to the formula the value or , for denoted by . If is an isomorphic mapping of an -system into an -system , then

for all from .

A formula is called closed if it contains no free object variables. For any closed formula of the signature and any -system one may speak of being true or false in . A set of closed formulas of a given signature is said to be realizable or consistent if there exists an -system in which all formulas from are true.

The compactness theorem or the Gödel–Mal'tsev theorem states: If each finite subset of an infinite set of closed formulas of a given signature is realizable, then the entire set is realizable as well.

## Axiomatizable classes.

Let be some set of closed formulas of a signature . The class of all -systems in which all formulas of are true will be denoted by . The set of all closed formulas of a signature which are true in all -systems of the given class , is called the elementary theory of . In particular, if is the class of -systems isomorphic to the given -system , then is said to be the elementary theory of the -system and is simply denoted by . A class of -systems is said to be axiomatizable if . A class of -systems is axiomatizable if and only if there exists a set of closed formulas of the signature such that .

Along with the general concept of axiomatizability, axiomatizability using first-order formulas of a special kind may also be considered. The special formulas of a given signature which are the most important in algebra are:

Identities — the formulas

where is some predicate symbol from or the equality sign , and are terms of the signature in .

Quasi-identities — the formulas

where are certain predicate symbols from or equality signs, while are terms of the signature in .

Universal formulas — the formulas , where is a formula of the signature not containing quantifiers.

If a set of identities (quasi-identities or universal formulas) of a signature is given, then the class is known as a variety (quasi-variety or universal class) of -systems.

Birkhoff's theorem: A non-empty class of -systems is a variety if and only if it is closed with respect to subsystems, Cartesian products and homomorphic images.

If is some -system, then, by exchanging each function symbol from for a predicate symbol of arity (higher by one) and putting, for elements from

one obtains a model for which . Submodels of the model are called submodels of the -system . For any non-empty finite subsets , , the model is called a finite depletion of the finite submodel of the -system . An -system is said to be locally imbeddable in a class of -systems if for each finite depletion of any finite submodel of the -system there exists in the class an -system (depending on the chosen finite depletion ) such that the model is isomorphic to the model for a suitable subset .

A subclass of a class of -systems is called universal (or universally axiomatizable) in if there exists a set of universal formulas of the signature such that .

The Tarski–Los theorem: A subclass of a class of -systems is universal in if and only if contains all systems from which are locally imbeddable in .

## Filtered products.

Let be the Cartesian product of the -systems (, ) and let be some filter over . The relation

is an equivalence relation on the underlying set of the -system . For each element , let be the coset by this equivalence and let . Putting

one obtains the -system , which is called the product of the -systems () filtered through the filter . The -systems () are said to be the factors of this product. If is an ultrafilter over , the filtered product is known as the ultraproduct of the -systems ().

The ultraproduct theorem: If is the ultraproduct of the -systems () and if is an arbitrary formula of the signature in which are free object variables, then for any elements :

In particular, a closed formula of the signature is true in the ultraproduct of the -systems () if and only if the set of numbers of factors in which the formula is true belongs to the ultrafilter . Therefore, any axiomatizable class of -systems is closed with respect to ultraproducts.

A class of -systems is universally axiomatizable if and only if it is closed with respect to subsystems and ultraproducts.

An -system is said to be a unit if its underlying set consists of a single element, e.g. an element , and if for all .

Mal'tsev's theorem: A class of -systems is a quasi-variety if and only if it contains a unit -system and is closed with respect to subsystems and filtered products (through an arbitrary filter).

## Completeness and categoricity.

A non-empty class of -systems is called categorical if all -systems from are mutually isomorphic. Any categorical axiomatizable class of -systems consists of one (up to an isomorphism) finite -system.

A class of -systems is called categorical in cardinality if it contains an -system of cardinality and if all -systems from of cardinality are mutually isomorphic. For example, the class of algebraically closed fields of fixed characteristic is categorical in any uncountable infinite cardinality. A non-empty class of -systems is called complete if for any -systems from the equality is valid.

Vaught's theorem. If an axiomatizable class of -systems is categorical in some cardinality and if all -systems in are infinite, then is a complete class.

In particular, the class of all algebraically closed fields of fixed characteristic is complete.

#### References

 [1] A.I. Mal'tsev, "Algebraic systems" , Springer (1973) (Translated from Russian) [2] P.M. Cohn, "Universal algebra" , Reidel (1981) [3] G. Grätzer, "Universal algebra" , Springer (1979) [4] J.L. Bell, A.B. Slomson, "Models and ultraproducts: an introduction" , North-Holland (1971) [5] C.C. Chang, H.J. Keisler, "Model theory" , North-Holland (1973)