Mathematical structure

A mathematical structure is a set with certain properties. These properties result from one or more relations between the elements ( structure of the first level ) or the subsets of the set (structure of the second level). These relations and thus also the structure that they define can be of very different types . Such a kind can be determined by certain axioms which the defining relations have to fulfill. The most important large types into which structures can be classified are algebraic structures , relational structures such as, in particular, order structures , and topological structures . Many important sets even have multiple structures , that is, mixed structures from these basic structures. For example, number ranges have an algebraic, an order, and a topological structure that are interconnected. There are also geometric structures .

Algebraic structures

An algebraic structure, or a (general) algebra for short, is a structure (first level) that is only defined by one or more links (as functions , links are special relations).

Structures with an internal link: groups and the like

A hierarchical compilation of the basic algebraic structures

The fundamental algebraic structures have one or two two- digit internal connections . The taxonomy , i.e. the classification of these structures, depends on which of the following group axioms apply in the set with regard to the link : ${\displaystyle M}$ ${\displaystyle \circ }$

(E) Existence and uniqueness (also isolation ): ${\displaystyle \forall a,b\in M:a\circ b\in M.}$

(A) Associative law : ${\displaystyle \forall a,b,c\in M:(a\circ b)\circ c=a\circ (b\circ c).}$

(N) existence of a neutral element : ${\displaystyle \exists e\in M:\forall a\in M:a\circ e=e\circ a=a.}$

(I) existence of an inverse element : ${\displaystyle \forall a\in M:\exists a^{-1}\in M:a\circ a^{-1}=a^{-1}\circ a=e.}$

(K) Commutative law : ${\displaystyle \forall a,b\in M:a\circ b=b\circ a.}$

(Ip) Idempotency Law : ${\displaystyle \forall a\in M:a\circ a=a.}$

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

Surname	Axioms	description
Groupoid (also magma)	E.	A set with a two-digit internal link.
Half group	EA	A groupoid with associative law . Example: . ${\displaystyle (\mathbb {N} ,+)}$
Semi-association	EAKIp	A semigroup with commutative law and idempotency law . Example: ${\displaystyle (\mathbb {N} ,{\text{max}}).}$
Monoid	EAN	A semi-group with a neutral element . Example: with . ${\displaystyle e}$ ${\displaystyle (\mathbb {N} _{0},+)}$ ${\displaystyle e=0}$
Loop with inverse property	ENI	A groupoid with a neutral element, in which there is a (unique) inverse for each element .
group	EANI	Simultaneously a monoid and a quasi-group. Groups were introduced in the early 19th century to describe symmetries and have proven to be fundamental to the entire structure of algebra. Examples of number ranges that form a group: , . Examples of transformation groups that describe symmetries: the point groups for describing molecular symmetries , the symmetrical groups for describing permutations , the Lie groups for describing continuous symmetries. ${\displaystyle (\mathbb {Z} ,+)}$ ${\displaystyle (\mathbb {Q} \setminus \lbrace 0\rbrace ,\cdot )}$
Abelian group	EANIK	A group with a commutative link.

Structures with two internal links: rings, bodies and the like

The following structures have two interconnections, usually written as addition and multiplication; these structures are of the speed ranges (such as , , ) abstracted, which is usually considered. The compatibility of the multiplicative with the additive combination is ensured by the following axioms: ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$

(Dl) Left distributive law : . ${\displaystyle \forall a,b,c\in M:a\cdot (b+c)=a\cdot b+a\cdot c}$

(Dr) right distributive : . ${\displaystyle \forall a,b,c\in M:(a+b)\cdot c=a\cdot c+b\cdot c}$

(D) distributive : apply Dl and Dr .

Further axioms that apply to both connections are:

(U) The neutral elements relating to addition and multiplication, and , are not the same. ${\displaystyle 0}$ ${\displaystyle 1}$

(T) a zero divisor freedom : If the identity element of the additive referred to link, then it follows from all of that or applies. ${\displaystyle 0}$ ${\displaystyle a\cdot b=0}$ ${\displaystyle a,b}$ ${\displaystyle M}$ ${\displaystyle a=0}$ ${\displaystyle b=0}$

(I ^* ) For every element, with the exception of the neutral element of the additive link, there is the inverse element with regard to the multiplicative link. Formal: . ${\displaystyle \forall a\in M\setminus \lbrace 0\rbrace :\exists a^{-1}\in M:a\cdot a^{-1}=a^{-1}\cdot a=e}$

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

• Half- ring : Axioms ( EA | EA | D ) two semigroups

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

• Fastring : Axioms ( EANI | EA | Dr ): An additive group, a multiplicative semigroup and the right distributive law.

• (Left) quasi-bodies : 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 | EAK | D ): ring with commutative multiplication.

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

• Zero-divisor ring : Axioms ( EANIK | EA | DT ): Ring in which it follows that or . ${\displaystyle a\cdot b=0}$ ${\displaystyle a=0}$ ${\displaystyle b=0}$

• Integrity domain : Axioms ( EANIK | EANK | DTU ): Commutative, unitary, zero divisor ring with . ${\displaystyle 1\neq 0}$

• Half-body : Axioms ( EA | EANI ^* | D ) Half- ring with a multiplicative group on the set (without which, if it exists). ${\displaystyle 0}$

• Alternative field : Axioms ( EANIK | ENI ^* | DTU ): Unitary, zero divisors, and with a multiplicative inverse, except for the element . Instead of the associative law , there is the alternative of multiplication. ${\displaystyle 1\neq 0}$ ${\displaystyle 0}$

• ( Right- ) fast body : axioms ( EANI (k) | EANI ^* | DrTU ) fast ring with multiplicative group on the set without the . The addition of each fast body is commutative. ${\displaystyle 0}$

• Inclined body : Axioms ( EANIK | EANI ^* | DTU ): Unitary, zero-divisor-free ring with and with a multiplicative inverse, except for the element . ${\displaystyle 1\neq 0}$ ${\displaystyle 0}$

• Field : Axioms ( EANIK | EANI ^* K | DTU ): Commutative skew field, integrity domain with multiplicative inverse, except for the element . Each body is also a vector space (with itself as the underlying scalar body). If a norm or a scalar product is defined in the body, a body thereby receives the topological properties of a normalized space or an interior product space. See below. Examples: the number ranges , and . ${\displaystyle 0}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$

Important subsets are:

• Ideals, s. Ideal (math) .

Structures with two internal links: lattices, set algebras and the like

A lattice is an algebraic structure, the two internal connections of which in the general case cannot be understood as addition and multiplication:

(V) Laws of Merger (also called Laws of Absorption): and . ${\displaystyle a\cap (a\cup b)=a}$ ${\displaystyle a\cup (a\cap b)=a}$

With this axiom we get as structures:

• Association : Axioms ( EAK (regarding ) | EAK (regarding ) | V ). ${\displaystyle \cup }$ ${\displaystyle \cap }$
• Distributive Association : Axioms ( EAK (regarding ) | EAK (regarding ) | V, D ). ${\displaystyle \cup }$ ${\displaystyle \cap }$

In a distributive union one only has to demand one of the two merging laws; the other follows from the distributive law.

A Boolean algebra is an association in which the two links each have a neutral element, and , and in which each element a with respect to both links matching complement has ${\displaystyle a\cup 0=a}$ ${\displaystyle a\cap 1=a}$

(C) Existence of a complement: for each there is one for which applies and . ${\displaystyle a}$ ${\displaystyle \lnot a}$ ${\displaystyle a\cup \lnot a=1}$ ${\displaystyle a\cap \lnot a=0}$

Note that the complement is not an inverse element, as it provides the neutral element of the other link.

• Boolean algebra : axioms ( EAKN (re ) | EAKN (re ) | V, D, C ). ${\displaystyle \cup }$ ${\displaystyle \cap }$
• Set algebra : a Boolean algebra whose elements are sets, namely subsets of a basic set , with the set operators and as links, with the zero element and the one element . ${\displaystyle X}$ ${\displaystyle \cup }$ ${\displaystyle \cap }$ ${\displaystyle \emptyset }$ ${\displaystyle X}$
• σ-algebra : a closed set algebra with respect to countable-infinite connections.
• Measurement space and measurement space are special σ-algebras.
• Borel algebra turns a topological space into a measure space : it is the smallest σ-algebra that contains a given topology.

• Divalent Boolean algebra : only the elements and . ${\displaystyle 0}$ ${\displaystyle 1}$

Structures with internal and external connections: vector spaces and the like

These structures consist of an additively written magma (usually an abelian group) and a speed range (a structure with two internal threads, usually a body) , whose group action on the left multiplication or right multiplication written (from viewed from) the outer construed link is . The elements of are called scalars , the outer connection accordingly also scalar multiplication . It satisfies the following compatibility axioms (in notation for left multiplication): ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle \ast \colon K\times V\to V}$ ${\displaystyle \ast \colon V\times K\to V}$ ${\displaystyle V}$ ${\displaystyle K}$

(AL) associative law: for from and out of true . ${\displaystyle a,b}$ ${\displaystyle K}$ ${\displaystyle v}$ ${\displaystyle V}$ ${\displaystyle (a\cdot b)\ast v=a\ast (b\ast v)}$

(DL) distributive laws: for out and out applies and . ${\displaystyle a,b}$ ${\displaystyle K}$ ${\displaystyle v,w}$ ${\displaystyle V}$ ${\displaystyle a\ast (v+w)=a\ast v+a\ast w}$ ${\displaystyle (a+b)\ast v=a\ast v+b\ast v}$

This gives us the following structures in the notation ( | | Compatibility axioms): ${\displaystyle V}$ ${\displaystyle K}$

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

Additional algebraic structure on vector spaces

Relationships between mathematical spaces

• K-Algebra : algebra over a field (also outdated: Linear Algebra (structure)): vector space with additional bilinear link . ${\displaystyle [\cdot ,\cdot ]\colon V\times V\rightarrow V}$

• Lie algebra : vector space with the Lie bracket , as an additional anti-symmetric bilinear link . ${\displaystyle [\cdot ,\cdot ]\colon V\times V\rightarrow V}$

• associative algebra : vector space with a bilinear associative link . ${\displaystyle V\times V\rightarrow V}$

The inner connections of scalar product and norm introduced in the following help a vector space (this can in particular also be a body to be understood as a vector space) to a topological structure.

• A bilinear space is almost an interior product space (see below) - except that the interior product does not have to be positive definite. An important example is the Minkowski space of special relativity.

• Inner product space: vector space with a scalar product (a positively defined bilinear form according to or sesquilinear form according to ) . The Euclidean space is a special interior product space . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \langle \cdot ,\cdot \rangle \colon V\times V\rightarrow K}$ ${\displaystyle \mathbb {R} ^{n}}$

• unitary space : interior product space above , the scalar product of which is a Hermitian form. ${\displaystyle \mathbb {C} }$

• normalized space : vector space with a norm . ${\displaystyle \|\cdot \|\colon V\rightarrow K}$

• locally convex space : vector space with a system of semi-norms . Every normalized space is a locally convex space with . ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}=\{\|\cdot \|\}}$

Vector space with	general	+ Completeness
Metric	metric space	complete space
standard	normalized space	Banach space
Scalar product	Prähilbertraum (interior product room)	Hilbert dream

The specialization of the vector spaces increases downwards and to the right. The vector spaces in the table below have the properties of those above, since a scalar product induces a norm and a norm induces a distance . ${\displaystyle \left\|v\right\|={\sqrt {\left\langle v,v\right\rangle }}}$ ${\displaystyle d(v,w)=\left\|v-w\right\|}$

Organizational structures

An order structure is a structure (first level) that is equipped with an order relation , i. that is, it is a relational structure or, for short, a relative .

• Quasi-order : reflexive and transitive . Example: For off , if (see absolute amount ) applies . ${\displaystyle a,b}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle a~R~b}$ ${\displaystyle |a|\leq |b|}$

• 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 smaller or equal” on the vector space . ${\displaystyle \mathbb {R} ^{n}}$

• strict semi-order : irreflexive and transitive. Examples: The relation “real subset” in a power set ; the relation “component-wise less than or equal, but not equal” on the vector space . ${\displaystyle \mathbb {R} ^{n}}$

• total order (linear order): total partial order. Example: "Smaller or equal" on . ${\displaystyle \mathbb {Z} }$

• strict total order : total, irreflexive and transitive. Example: "Smaller" on . ${\displaystyle \mathbb {Z} }$

• well-founded order : a partial order in which every non-empty subset has a minimal element. Example: The relation “equal to or member of” in a set of sets.

• Well-order : total order in which every non-empty subset has a minimal element. Example: "Smaller" on . ${\displaystyle \mathbb {N} }$

Topological structures

The geometric concept of distance (the metric ) makes it possible to use the basic concept of modern analysis , convergence , in metric spaces . Topological spaces have emerged from an effort to treat 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 with their possible local structures classify , retain their structure by the award of certain subsets as open or, equivalent to a complete (structures the second stage).

See also : topology (mathematics) , topological space and axiom of separation

Geometric structures

A geometric structure is expressed through properties such as the congruence of figures. Their classification according to the valid axioms (compare the articles Geometry , Euclidean Geometry , Euclid's Elements ):

• Projective geometry
• Affine geometry
• Absolute geometry : Any geometry in which the first four of the five Euclidean postulates apply (more precisely: Hilbert's axioms with the exception of the axiom of parallels).
• Euclidean geometry : Absolute geometry, in which the parallel postulate applies. Or also: Geometry in which all Hilbert's axioms apply.
• Non-Euclidean geometry : Absolute geometry in which the parallel postulate does not apply. Or also: Geometry in which Hilbert's axioms of groups I, II, III and V as well as the negation of the axiom of parallels apply.

Their classification according to the transformation groups under which certain geometric properties remain invariant ( Felix Klein , Erlanger program ):

• Projective geometry , invariants: point, straight line.
• Affine geometry , additional invariants: parallelism, partial ratio, area ratio.
• Similarity geometry , additional invariants: distance ratio, angle.
• Congruence geometry , additional invariant: segment length.

Number ranges

→ Main article : Number

Number ranges are the amounts that are usually expected. The basis is the set of natural numbers. Addition and multiplication serve as algebraic connections. By requiring that the inverse operations subtraction and division should always be possible, one extends the set of natural numbers to the set of whole numbers and to the set of all fractions. The real numbers are introduced as limit values ​​for number sequences; they allow (among other things) the extraction of 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 is at the very beginning of the axiomatic structure of mathematics. Below the zero should not in be included, but the opposite convention is also common. and are commutative semigroups . As with all other number ranges, addition and multiplication are distributive . ${\displaystyle \mathbb {N} }$ ${\displaystyle 0}$ ${\displaystyle \mathbb {N} }$ ${\displaystyle (\mathbb {N} ,+)}$ ${\displaystyle (\mathbb {N} ,\cdot )}$

• The set of integers is created by constructing zero as a neutral element and negative numbers as the inverse of the addition. is an Abelian group with the neutral element and is a commutative monoid with the neutral element . is a commutative ring with one. ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {N} }$ ${\displaystyle (\mathbb {Z} ,+)}$ ${\displaystyle 0}$ ${\displaystyle (\mathbb {Z} ,\cdot )}$ ${\displaystyle 1}$ ${\displaystyle (\mathbb {Z} ,+,\cdot )}$

• The set of positive fractions is obtained by constructing fractions as the inverse of the multiplication. is therefore a group and is a semigroup (both commutative). ${\displaystyle \mathbb {Q} ^{+}}$ ${\displaystyle \mathbb {N} }$ ${\displaystyle (\mathbb {Q} ^{+},\cdot )}$ ${\displaystyle (\mathbb {Q} ^{+},+)}$

• The set of fractions or rational numbers is created by adding the neutral element and 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 body . ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} ^{+}}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle (\mathbb {Q} ,+)}$ ${\displaystyle (\mathbb {Q} \setminus \{0\},\cdot )}$ ${\displaystyle \mathbb {Q} }$

• The set of real numbers results from topological completion : a real number is an equivalence class of rational Cauchy sequences . is a body . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$

• The set of complex numbers consists of pairs of real numbers that comply with the usual arithmetic rules when written . In any algebraic equation is solvable. is a body . ${\displaystyle \mathbb {C} }$ ${\displaystyle (a,b)}$ ${\displaystyle a+bi}$ ${\displaystyle i^{2}=-1}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} }$

• Quaternions , Cayley numbers and also extended number ranges are no longer commutative with regard to multiplication.

Some restricted number ranges are also important:

• The remainder class ring is the restriction of the whole numbers to the set . All arithmetic operations are carried out modulo . is a ring ; if is a prime number , even a field . In machine-level programming languages, unsigned whole numbers are represented as residual class rings, for example with or . ${\displaystyle \mathbb {Z} _{m}}$ ${\displaystyle \{0,1,\ldots ,m-1\}}$ ${\displaystyle m}$ ${\displaystyle \mathbb {Z} _{m}}$ ${\displaystyle m}$ ${\displaystyle m=2^{16}}$ ${\displaystyle m=2^{32}}$

literature

• Nicolas Bourbaki : The Architecture of Mathematics I . In: Physical sheets . tape 17 , no. 4 , 1961, pp. 161-166 , doi : 10.1002 / phbl.19610170403 . The architecture of mathematics II . In: Physical sheets . tape  17 , no. 5 , 1961, pp. 212–218 , doi : 10.1002 / phbl.19610170503 (French: Les grands courants de la pensée mathématique . Marseille 1948. Translated by Karl Strubecker, Helga Wünsch).
• Fritz Reinhardt, Heinrich Soeder: dtv atlas mathematics . 11th edition. tape 1 : Fundamentals, Algebra and Geometry. Deutscher Taschenbuchverlag, Munich 1998, ISBN 3-423-03007-0 , p. 36-37 .

Individual evidence

1. ↑ Nicolas Bourbaki : The architecture of mathematics I. S. 165 f.
2. ^ Nicolas Bourbaki : The Architecture of Mathematics II. P. 212-214.
3. ↑ Nicolas Bourbaki : The architecture of mathematics II. P. 215.
4. ↑ Closely related to the concept of relational structure is that of the graph in the graph-theoretical sense. The set of supports is called the set of nodes, the place of the relation is taken by the set of edges. If not stated otherwise, graphs are finite.