Algebraic structure

The concept of algebraic structure (or universal algebra, algebra, or just general algebra ) is a basic concept and a central issue to the mathematical part of the area of universal algebra. An algebraic structure is provided with a lot of links on this amount. A number of cases examined in the abstract algebra structures such as groups, rings and body are special algebraic structures.

Generalizations of algebraic structures are the heterogeneous algebras, the partial algebras and relational structures.

  • 6.1 Example: Groups
  • 6.2 Examples of algebraic structures

Definition of the algebraic structure

An algebraic structure or general algebra is an ordered pair

Consisting of a non-empty set of basic quantity or amount of carrier of the algebra, and a family of inner ( last digit ) links, even basic operations or fundamental operations called on

An inner - digit shortcut is a function of the elements from getting on a ( uniquely determined ) element mapping is then the image of ( notation: ). A zero digit shortcut can be interpreted as a uniquely determined, distinguished element in a constant. Constants are usually referred to with a special symbol (such as a letter or number characters like ). An inner digit link is a function of which is often referred to by an icon which directly (i.e., without additional clamps or separator) before being written back, etc. on the element ( argument).

When the image is a two-digit shortcut the shortcut icon is written to simplify between the two arguments in the rule.

Mostly an algebra has only finitely many fundamental operations you then writes for the algebra just

The ( similarity ) type of an algebra assigns to each index to the respective arity of the fundamental operation, that is, it is a function of the type can also be written as a family:

For example, a group usually seen as structure, the amount of support is a two-digit linking by a constant in a one-digit and linking after a group is thus an algebra of type

Comments

  • Sometimes it proves to be useful to allow the empty set as a support amount of an algebra, is about ensuring that the set of all subalgebras (see below) of an algebra is an algebraic Association.
  • Each non-empty set can make a trivial algebra with the identity mapping also can be used as an algebra with an empty family of links are considered.
  • You could even " unendlichstellige algebras " with unendlichstelligen shortcuts allow (eg σ - algebras ), but this would contradict the usual understanding of " algebraically ".
  • A generalization of general (complete ) algebras are partial algebras, in which not only total functions but also partial functions are allowed as a link. For example, bodies are not strictly complete algebras, because only defined.

Types of algebraic structures

The respective links of algebras of the same type often have more features in common, so you can classify algebras according to their type and after these other properties in different classes. Such properties of the concrete links given an algebra is further specified by axioms in the abstract algebra ( branch of mathematics ) are usually written in the form of equations and determine the type of algebra.

One example is the associative law for an internal binary operation on a set

Now meets the two-digit operation of an algebra of this axiom (replace by and by ), then the algebra belongs to the nature of the semigroup or she is a semigroup.

Substructures ( subalgebras )

Is the basic set of an algebraic structure, so you can define using the links of on a subset of A, a new algebraic structure of the same type, if the amount is chosen so that the links of the original structure does not lead out of the crowd. This means that if one applies the links of the original algebraic structure on the elements of, no elements may arise that are not in - in particular, the constants must already be included. In the actual application, for example are subgroups the substructures of a group. Depending on how you've chosen the equations defining the algebraic structure that substructures can look quite different. So can be defined as, for example, groups that the substructures are normal subgroups.

Homomorphisms

Structure fidelity images, called homomorphisms, algebraic structures, and between each two of the same type (ie they have links from each same arities and given specific properties ) are compatible with the links of the two algebraic structures. Hence each algebraic structure has its own homomorphism - term and therefore defines a category.

Corresponding links in and are usually denoted by the same symbol. Thus, the group operation is uniformly written, eg, about every group under consideration. Must be kept apart in the individual case the two links, the symbols of their basic quantities or the like are usually attached as indices, eg and. A homomorphism is a function that for each link ( with arity ) satisfies the following condition:

The specific notations, the zero-, one - and two -digit shortcuts are taken into account:

  • Are respectively the constants nullstelliger links, then
  • Is each one-digit shortcut, then a one-digit shortcut can also be written as an exponent, index, etc., we obtain with and then, for example,
  • For two-digit shortcuts

A surjective homomorphism is called epimorphism, an injective monomorphism. A homomorphism from within itself (ie if true ) is called endomorphism. A bijective homomorphism whose inverse is also a homomorphism is called an isomorphism. If the isomorphism same time endomorphism, it is called an automorphism.

See also: homomorphism.

Congruence

On algebraic structures to specific types of equivalence relations can be found, which are compatible with the links of an algebraic structure. These are called congruence relations. Using congruence is Faktoralgebren blank form, i.e., it is a structure of the same type generated from the original algebraic structure, the elements of which, however, then the equivalence classes with respect to the congruence. The links are well defined due to the special properties of congruence. In many practical applications, the equivalence classes correspond to the secondary or congruence of certain substructures, for example normal subgroups in groups or ideals in rings, etc.

Products

If one forms the set-theoretic direct product of the basic quantities more general algebras of the same type, so you can in turn get a new algebra of the same type on this product quantity by defining the new links of this algebra componentwise by the links of the original algebras. However, these can have different properties than the original algebra; For example, the product of the bodies does not have to be one body.

For a generalization of the direct product of algebras, see: subdirect product. There also the representation theorem is presented by Birkhoff, after each algebra is a subdirect product subdirectly irreducible algebras.

"Zoo" of algebraic structures

Example: Groups

As an example of the definition of an algebraic structure, we consider a group. Typically, a group is defined as a pair consisting of a set and met a two-digit shortcut so that for all into the following three axioms:

  • ( Associativity ).
  • There is a in such that ( neutral element ).
  • There is an in such that ( inverse element ) for each.

(Sometimes you can still find the requirement of " seclusion ", that again should be in, but from the perspective of Algebraikers the term includes " two -digit shortcut " this property already. )

This definition has the property that the axioms are expressed not only by equations, but also the quantifier " there ... so that" included; in the general algebra one tries therefore to avoid such axioms. The simplification of the axioms to a pure equation form here is not difficult: we add a 0-ary link and a one-digit shortcut to and define a group as a quad with a lot of double-digit combination of a constant and a one-digit shortcut, satisfying the following axioms:

It is now important to consider whether they would actually the definition of a group was reached. It could be that this is not all properties of a group are given or even too many. Indeed, the two definitions of a group match.

Examples of algebraic structures

In the following list (= 1 - digit shortcuts ) all (2- digit ) links, neutral elements ( = 0-ary links), Inverse pictures indicated and operator spaces.

In normal use, for algebraic structures we are, however, only the multi-digit shortcuts and the operator ranges ( occasionally even neutral elements ), for everyone else it 's usually standard notations.

A non-exhaustive list of various algebraic structures:

  • Groupoid or magma, also Binar or Operational (O, *): a non-empty set O with a two-digit shortcut *.
  • Semigroup (S, *): an associative groupoid.
  • Monoid (M, *, 1): a semigroup with a neutral element 1
  • Group (G, *, 1, -1): a monoid with an inverse element a-1 to each element a - or equivalently: an associative quasigroup or an associative loop.
  • Abelian group (G, , 0, -): a commutative group. Abelian groups are usually written additively. The neutral element of an abelian group is denoted by 0 and the " inverse " of an element a as the negative -a.
  • Half-ring (H, , · ): a lot of H are met with two operations (addition) and · (multiplication ) that (H, ) and (H, ·) are semigroups and the distributive laws. Often to (H, ) but also still be commutative and / or a neutral element 0, the zero element of the half ring, possess: The definitions are not uniform here!
  • Dioid (D, , 0, ·, 1): a semi- ring (D, , · ) with an absorbing zero element 0 such that (D, ) is a semilattice and (D, ·, 1 ), as ( D, , 0), a monoid.
  • Boolean association or Boolean algebra (B, 0, 1, c ) a Dioid that an association (B, ) is connected to a complement of the AC to each element a Equivalently: a Boolean ring ( see below).
  • Ring ( R, , 0, -, ·): a semi- ring such that (R, , 0, - ) is an abelian group.
  • The concept of Kleene algebra is a generalization of regular expressions corresponding operations on regular languages ​​, union, concatenation and Kleene star.

Provided with additional structure, internalization

Algebraic structures can be equipped with additional structures, eg with a topology. A topological group is a topological space with a group structure such that the operations of multiplication and inverses are continuous. A topological group has both a topological as well as an algebraic structure. Other common examples are topological vector spaces and Lie groups. Abstractly speaking, the links in such structures are now morphisms in a particular category, such as the topological spaces in the case of topological groups. One speaks of internalization in this category. In the special case of ordinary algebraic structures are the links morphisms in the category of sets, ie functions.

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