Algebraic structure

In Mathematical, more particularly in Algebra, a algebraic structure is made of a Ensemble combined with one or more laws of composition, possibly supplemented by a order or a topology, the whole satisfying a certain number of Axiome S.

The study of the algebraic structures can be made in a way unified within the framework of the universal Algèbre.

What follows consists primarily of an enumeration and a classification of the majority of the usual algebraic structures, possibly provided of an additional structure.

Pure algebraic structures

These structures comprise only laws of composition.

Basic structures

They comprise only laws of composition internal. Most important are the structures of group, ring and body.

Groupoïdes

The algebraic structures simplest comprising only one internal law of composition.

  • Magma (or groupoïde ): together with only one internal law of composition. (Attention, the term groupoïde has another direction in Théorie of the categories.)

  • Paragroup : a permutatif magma , commutative and regular.

  • Antigroupe : a permutatif magma , regular and involutive on the right.

  • Group : a invertible monoid , i.e. where any element has an opposite ; it is also a buckles associative.

Annélides

These structures comprise two internal laws of composition.

  • Pseudo-ring : a unit provided with a structure of group (the law of composition being named addition) and with a structure of semigroup (the corresponding law of composition being named multiplication), multiplication being distributive on the addition.

  • Ring : a pseudo-ring whose multiplicative semigroup is a monoid. Certain authors call ring what one called pseudo-ring and call unit ring what one called ring .

  • commutative Ring: ring whose multiplication is commutative.

  • Semi-ring : similar to a ring , but for the addition it is a monoid and not necessarily a group.

  • Anneau integrates : a commutative ring not no one and without Diviseur of zero, i.e. any element not no one of the ring is regular for the multiplication.

  • Body : a ring where the neutral element of the addition is not that of the multiplication and where any element not no one has a reverse multiplicative. Because of the English influence (see below), a body is often regarded as implicitly commutative, whereas in the French tradition, it is not it necessarily. To avoid any ambiguity, it is to better indicate:

- “commutative body” for an indeed commutative body,
- and “commutative body or not”, or “unspecified body”, for a body not necessarily commutative.

  • commutative Body , noncommutative body : in the French tradition a “body” is not necessarily commutative; in English, a commutative body is called field , and a noncommutative body division boxing ring . A shift in meaning tends to align the French terminology on the English terminology and to qualify the noncommutative bodies of “rings with (or of) division” and the commutative bodies of very short “body”. This last name is to be avoided because it brings from now on an ambiguity: the “body” considered is it commutative or unspecified?

Structures with external operators

These structures can be considered from an algebraic point of view or geometrical .

Algebraically, a external structure is a Ensemble provided with a external Law of composition on a basic structure, and possibly of one or more laws of composition interns.

Geometrically, it is a unit on which acts a together-operator , or together of operators , known as also scalar . It is thus a unit provided with a action of the together-operator in this unit, i.e. of an application of the together-operator in the whole of the applications of this whole in itself.

The correspondence between the external actions and laws is Bijective; this is why the external laws are often called laws of action .

Homogeneous spaces

These structures comprise only one law, external.

  • Together with operator in a monoid M (M-ensemble): together E provided with an external law of M on E which is exo-associative and exo-unifère

  • Together homogeneous or Together homogeneous on a group G: G-ensemble E for which the operation of G on E is transitive.

Moduloïdes

Structures having at the same time an internal law of composition and an external law of composition.

  • Group with operators (in a unit) : group provided with an external law on a whole of operators, distributive compared to the law of the group

  • Module on the left (on a ring) : abelian group provided with an external law on a ring has , law checking the four following properties:

* Distributivité on the left compared to the law of the group,
* exo-distributivity on the left compared to the first law of the ring,
* exo-associativeness on the left compared to the second law of the ring
* and exo-absorption on the left, the absorbing element being the neutral element of the group, and the scalar ensuring this absorption being the neutral element of the first law of the ring (one can establish an analogy with what occurs in a body, where the neutral element of the first law is absorbent for the second law).

  • Module on the right (on a ring) : modulate on a ring opposed to a ring provided with a module on the left.

  • Module (on a ring) : modulate on the left on a ring has commutative. In fact, in this case, the concepts of module on the right and on the left merge.

  • vector Space on the left (on a body) : module on the left on a body K .

  • vector Space on the right (on a body) : module on the right on a body K . In other words, the vector spaces on the right on K are the vector spaces on the body opposed to K .

  • vector Space (on a body) : module on a body K commutative.

  • Espace refines (on a body) : homogeneous space of a vector space on a body K . If the characteristic of K is different from 2, there exists a definition of spaces closely connected on this body independent of the concept of vector space. A space refines is then a unit provided with two laws:

* one, intern, for whom it is a Paragroupe (in the case of a space refines Euclidean, it acts of the law medium, which at two points, associates their geometrical medium);
* the other, external, which checks properties similar to those of the external law of a module (in the case of a space refines Euclidean, this external law, which depends on the choice of an arbitrary point O , associates with a point P and a scalar X the result of the application to P of the homothety closely connected of report/ratio X and origin O ).

Algebras

Structures having two internal laws and an external law.

  • Algebra (on a commutative ring) : a module (or a vector space) provided in addition to one bilinear internal law of composition.

  • unit Algebra associative : associative algebra having a neutral element for the multiplication.

  • Algebra of Dregs : a particular type of algebra generally nonassociative, important in the study of the groups of Dregs.

  • Algebra of Jordan : a particular type of algebra generally nonassociative.

Bialgèbre S

Structure having two internal laws, an external law, and a law " duale" one of the two internal laws.

Algebraic structures ordinates

Ordered groups and ordered rings

One is interested here in the algebraic structures compatible with a relation of order.

  • a Monoïde ordered is a commutative Monoïde provided with a Relation of order for which the applications partial of the law of intern are increasing. One defines in the same way predetermined Monoïdes by replacing relations of order by relations of order.

  • a Groupe ordered is an ordered monoid which is a commutative group. A Groupe predetermined is a predetermined monoid which is a group.

  • a Anneau ordered is a commutative ring provided with a relation of order for which it is group ordered for the addition and such as the products of two elements equal to or higher than 0 are equal to or higher than 0.

  • a Corps ordered is an ordered ring which is a body and whose relation of order is total.

Lattice

Units provided with two internal laws, which can be also interpreted as the upper limit and the lower limit of the couples within the meaning of the relations of order.

  • Lattice: a unit provided with two commutative, associative internal laws of composition and idempotentes satisfying the principle of absorption.

  • Boolean algebra: a lattice limited, distributive and complémenté.

Topological algebraic structures

Scalar structures and topologies, distances, standards or products

The algebraic structures can also have additional features topological. Thus, while going from the general to the private individual:

  • an algebraic structure can be provided with a Topologie , becoming thus a topological Espace for which each one of its law external and internal are continuous.

* a Semi-goupe topological is a semigroup provided with a topology such as its internal law of composition is continuous.
* a topological Monoïde is a topological semigroup which is a monoid.
* a topological Groupe is a group provided with a topology such as its law of composition interns is continuous as well as the application which with an element associates its reverse is continuous.
* a topological Anneau is a ring provided with topological for which the subjacent additive group is a topological group and the subjacent multiplicative monoid is a topological monoid.
* a topological Corps is a topological ring which is a body such as the multiplicative group of the element not no one is a topological group for induced topology.
* a Corps valué is a body (commutative or not) provided with an absolute value, and is a topological body for the topology defined by this absolute value.
* a topological Module on a topological ring has is a module on has provided with a topology for which it is a topological group and for which the external law is continuous.
* a topological vector Space on a topological body (for example the body of the real numbers or the body of the complex numbers) is a topological module on this topological body.
* a topological Algèbre on a topological Anneau commutative are algebras on this topological ring has for which it is a topological module on has and for which the multiplication is continuous.

  • more particularly, the algebraic structure can be provided with a Distance , becoming a metric Espace :

* an important case is that of the vector spaces having a Norme , which defines the “length” of a vector:
* the spaces pseudo norms (or vector spaces pseudo norms) are real or complex vector spaces (or on a Corps valué nondiscrete) provided with a Semi-norme .
* the normalized spaces (or vector spaces normalized) are real or complex vector spaces (or on a nondiscrete valué body) provided with a Norme . The normalized spaces are metric spaces , because it is always possible to build a distance on the standard basis: one takes as outdistances between two vectors the standard of their difference.
* a Espace of Banach is a vector space normalized Complet.
* a Espace refines normalized is a space closely connected attached to a normalized vector space. It is a metric space: it is possible to define the distance between two points as the standard of the vector which goes from the first point to the second.
* the spaces préhilbertiens are vector spaces real or complexes provided with a scalar Produit . This vector space is a space normalized : the standard of a vector is the square root of its scalar square. Let us note that a normalized space is with more than one manner a space préhilbertien, the standard determining the scalar product. Some important cases received a name:
* a Euclidean vector space is a vector space préhilbertien on \ mathbb {R} of finished size, provided with a scalar Produit whose quadratic Forme corresponding is definite positive. A space refines Euclidean is a Espace closely connected attached to an Euclidean vector space, provided with the distance, known as Euclidean, deduced from the euclidian norm. This space is that of the traditional geometry of Euclide.
* a square vector space is a vector space préhilbertien complex of finished size.
* a Espace of Hilbert (or hilbertien spaces) is a space préhilbertien Complet. It is thus a space of particular Banach . The Euclidean and square vector spaces are examples of spaces of Hilbert.

Structures and differential and algebraic geometry

* a group of Dregs real or complex is a group provided with a structure of analytical Variété real or complex (or of differential Variété in reality, it is sufficient) for which the law of composition is Analytique (or indefinitely Différentiable in the real case), as calls it which with an element associates its reverse. The real and complex groups of Dregs are topological groups. A topological Groupe is the topological group subjacent with more the one real group of Dregs, and thus one can say, without ambiguity, that certain topological groups are real groups of Dregs. One can also define the groups of Dregs on a commutative complete body valué K whose absolute value is noncommonplace (in particular on the body of the numbers '' p '' - adic) by replacing the real or complex analytical varieties by the varieties K - analytical.

* a homogeneous space of Dregs of a real Groupe of Dregs G is a differential Variété X , provided with an external law of G on X which is indefinitely differentiable.

* a algebraic Groupe on a Corps algebraically closed K is a Groupe provided with an algebraic structure of Variété on K for which the law of composition is regular, as well as the application which with an element associates its reverse.

Algebraic structures and categories

Any algebraic structure has its own concept of Homomorphisme, a application compatible with its laws of composition. In this direction, any algebraic structure defines a category.

See too

Simple: Algebraic structure

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