Mathematical structure

A mathematical structure is a set with certain properties. These properties result from one or more relations between the elements ( structure first stage) or the subsets of the set (second level structure). These relations and thus also the structure that define them, can be of very different kinds. Such a type can be defined by certain axioms which have to satisfy the defining relations. The main big guy, inspiration in the classified structures are algebraic structures, organizational structures and topological structures. Many important quantities even have multiple structures, ie mixed structures of these basic structures. For example, numerical ranges have both an algebraic, a trim and a topological structure, which are interconnected. In addition, there are also geometric structures.

Algebraic Structures

An algebraic structure or just a ( general ) algebra is a ( first-order ) structure, which is defined only by one or more links ( as functions are shortcuts special relations).

Structures with an inner join: groups and similar

The fundamental algebraic structures have one or two double-digit internal links. The taxonomy, thus classifying these structures depends on which apply the following group axioms in the set M with respect to the link:

The following structures with a two-digit inner join generalize or specialize the fundamental concept of the group:

Structures with two inner joins: rings, fields, and similar

Other axioms that concern both links are:

The valid axioms are the following in the order (additive axioms | multiplicative axioms | mixed axioms ) marked.

• Semiring: axioms (EA | EA | D ), two semigroups

• Dioid: axioms ( EAN | EAN | D) two monoids

• Fast Ring: axioms ( Eani | EA | Dr): An additive group, a multiplicative semigroup and the right - distributive.

• ( Left ) quasi- Body: axioms ( EANIK | ENI | DLU): An additive abelian group, a multiplicative loop.

• Ring: axioms ( EANIK | EA | D): An additive abelian group, a multiplicative semigroup.

• Commutative ring: axioms ( EANIK | EWC | D): ring with commutative multiplication.

• Ring with one or unitary ring: axioms ( EANIK | EAN | D): Ring with neutral element of multiplication.

• Zero divisor Free Ring: axioms ( EANIK | EA | DT): ring in which from a · b = 0, it follows that a = 0 or b = 0

• Integrity range: axioms ( EANIK | EANK | DTU): commutative, unitary, zero-divisor -free ring with 1 ≠ 0

• Half body: axioms (EA | Eani * | D) half-ring with multiplicative group on the set ( without the leading 0 if it exists ).

• Alternative Body: axioms ( EANIK | ENI * | DTU): unitary, zero divisors, 1 ≠ 0 and multiplicative inverses, except for the element 0 instead of the associative law of multiplication takes the alternativeness.

• (Legal) Fast Body: axioms ( Eani (k ) | Eani * | DrTU ) near-ring with multiplicative group on the set without the 0 The addition of each body is almost commutative.

• Skew field: axioms ( EANIK | Eani * | DTU): Unitary, zero-divisor -free ring with 1 ≠ 0 and multiplicative inverses, except for the element 0

• Body: axioms ( EANIK | Eani * K | DTU): commutative division ring, integral domain with multiplicative inverses, except for the element 0 - Every body is also a vector space ( with himself as the underlying Skalarkörper ). If a standard or a scalar product is defined in the body, a body will then receive the topological properties of a normed space or an inner product space. See below. - Examples: the number of areas, and.

Important subsets are:

• Ideals, see Ideal (mathematics).

Structures with two inner joins: associations, and similar Mengenalgebren

An association is an algebraic structure having two internal links in the general case, can not be considered as addition and multiplication:

With this axiom we obtain the structures:

• Association: axioms (EWC (re) | EWC (re) | V).
• Distributive lattice: axioms (EWC (re) | EWC (re) | V, D).

A Boolean algebra is an association in which the two links per a neutral element, a 0 = a and a 1 = a, and in which each element a with respect to both links has matching complement,

Note that the complement is no inverse element, since it provides the neutral element of the other link.

• Boolean algebra: axioms ( EAKN (re) | EAKN (re) | V, D, C).
• Algebra: a Boolean algebra whose elements are sets, namely subsets of a ground set X, the set operators and as links with the zero element and the unit element ø X.
• σ - algebra with respect to a countably infinite number of connections closed set algebra.
• Measurement space and measure space are special σ - algebras.
• Borel algebra makes a topological space to measure space: it is the smallest σ -algebra containing a given topology.

• Divalent Boolean algebra has only the elements 0 and 1

Structures with internal and external link: vector spaces and similar

These structures consist of an additively written Magma (usually an abelian group) V and a range of numbers ( a structure with two inner joins, usually a body ) K, whose group action on V as a left multiplication *: K × V → V or right multiplication *: V × K → V written and (as seen from V out) is interpreted as outer join. The elements of K are called scalars, the outer join accordingly scalar multiplication. It satisfies the following compatibility axioms ( in notation for left multiplication ):

Thus we obtain the following structures in the notation (V | K | Compatibility axioms ):

• Links module: ( Abelian group | Ring | AL, DL).
• Law Module: ( Abelian group | Ring | AR, DR) with scalar multiplication from the right instead of left.
• Module: ( Abelian group | commutative ring | ALR, DLR) with interchangeable left or right multiplication.
• Vector space: ( Abelian group | body | ALR, DLR) with interchangeable left or right multiplication.

Additional algebraic structure on vector spaces

• Lie algebra: vector space with the Lie bracket as an additional anti-symmetric bilinear link.

• Associative algebra: vector space with a bilinear associative link.

Launched in the following scalar product and norm inner links help a vector space (which may be especially aufzufassender as a vector space body ) to a topological structure.

• A Bilinearraum is almost an inner product space (see below) - except that the inner product need not be positive definite. An important example is the Minkowski space of special relativity.

• Inner product space: vector space with a scalar product ( a positive definite bilinear form or by sesquilinear after ). The Euclidean space is a special inner product space.

• Unitary space: inner product space over whose scalar product is a Hermitian form.

• Normed space: vector space with a norm.

• Locally convex space: vector space with a system of semi-norms. Every normed space is a locally convex space with.

Go down and to the right the specialization of vector spaces increases. The below in the table vector spaces have the properties of the above it as a scalar product induces a norm and a norm a distance.

Order structures

An order structure is a ( first-order ) structure, which is equipped with an order relation, that is, it is a relational structure or just a relative.

• Quasi-ordering: reflexive and transitive. Example: For a, b ​​in, a R b if | a | ≤ | b | (see absolute value ).

• Partial order ( partial order, partial order. Warning: sometimes simply called order ): reflexive, antisymmetric and transitive. Examples: The subset relation in a power set; the relation " component-wise less than or equal " on the vector space n

• Strict partial order: irreflexive and transitive. Examples: The relation "Genuine subset " in a power set; the relation " component-wise less than or equal, but not equal to" on the vector space n

• Total order ( linear order ): complete partial order. Example: " Less than or equal " to.

• Strict total ordering: total, irreflexive and transitive. Example: " Little " on.

• Founded ordering: a partial order in which every non-empty subset has a minimal element. For example, the relation " equal or element of" in a set of sets.

• Well-ordering: total order in which each non-empty subset has a minimal element. Example: " Little " on.

Topological structures

The geometric notion of distance ( metric ) makes it possible to handle in metric spaces, the basic concept of modern analysis, convergence. Topological areas are the result of an effort to deal with the convergence in a general sense ( each metric space is a topological space with the topology induced by the metric ). The different topological spaces, they can be classified by their possible local structures to get their structure through the labeling of certain subsets as open or, equivalently, when completed ( second-level structures).

Geometric Structures

A geometric structure comes through properties such as the congruence of figures expressed. Their classification according to the valid axioms (see the article geometry, Euclidean geometry, Euclid's Elements ):

• Parent Geometry: Each geometry in which the first two of the five Euclidean postulates: apply ( more precisely, the Hilbert logic and axioms of order ).
• Projective geometry
• Affine geometry
• Absolute Geometry: Each geometry in which the first four of the five Euclidean postulates are ( more precisely, the Hilbert 's axioms except the parallel postulate ).
• Euclidean geometry: Absolute geometry in which the parallel postulate is true. Or: geometry, in which all Hilbert's axioms.
• Non-Euclidean geometry: Absolute geometry in which the parallel postulate does not apply. Or: geometry in which the Hilbert 's axioms of groups I, II, III and V, and the negation of the axiom of parallels apply.

Their classification according to the groups of transformations under which certain geometric properties remain invariant ( Felix Klein, in Erlangen program ):

• Projective geometry, invariants: point, line.
• Affine geometry, additional invariants: parallelism, split ratio, surface area ratio.
• Similarity geometry, additional invariants: distance ratio, angle.
• Kongruenzgeometrie, additional invariant: route length.

Number ranges

Numerical ranges are the quantities which are expected usually. The basis is the set of natural numbers. As an algebraic shortcut used addition and multiplication. By requiring that the inverse operations subtraction and division should always be possible to extend the set of natural numbers to the set of integers and the set of all fractures. The real numbers are introduced as limits of sequences of numbers; they provide ( among other things) the square root of any positive numbers. The roots of negative numbers lead to the complex numbers.

• The set of natural numbers is used for counting and stands at the beginning of the axiomatic structure of mathematics. We understand the following as included in; but the opposite convention is also common. and are monoids with the neutral elements or. Addition and multiplication, as well as for all other speed ranges, distributive.

• The set of integers is formed from, by constructing as negative numbers relative to the inverse of addition. is an abelian group is a monoid, is a ring.

• The amount of non-negative results from openings by constructing fractions as inverse with respect to the multiplication. is therefore a group; is a monoid.

• The amount of breaks or rational numbers is created by adding the inverse with respect to addition or by adding the inverse with respect to multiplication. and are abelian groups. Addition and multiplication are distributive; is a field.

• The set of real numbers generated by topological completion: a real number is an equivalence class of rational Cauchy sequences. is a field.

• The set of complex numbers consisting of pairs of real numbers which satisfy the notation with the usual processing laws. In every algebraic equation is solvable. is a field.

• Quaternions, Cayley numbers and beyond advanced numerical ranges are not commutative with respect to multiplication.

Important are also some limited number of areas:

• The residue class ring can be considered as a limitation on the amount of the natural numbers. All arithmetic operations are performed modulo. is a ring; if a prime number, even a body. In low-level programming languages ​​unsigned integers are represented as residue class rings for example or.