Compatibility (mathematics)

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In mathematics , a mapping between two sets, which do not have to be different and which have structures of the same kind, is then compatible with their structures , a homomorphism or a (concrete) morphism of this type of structure, if the elements from the one set are so in the other set depicts that their images behave there in terms of relations and images of the structure in the same way as their archetypes behave in the initial structure.

An important special case for this are the distributive laws as the characterization of two-digit links that are left compatible or right compatible with other connections.

definition

Given are two non-empty sets and as well as any non-empty index sets and for each which can always be infinite in the following. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle I,J,K}$ ${\displaystyle J_{i}}$ ${\displaystyle i\in I,}$

Furthermore, let and be two relations with the same properties as well as and two families of relations and each have the same properties for each index , so that and two structures are of the same kind. ${\displaystyle R\subseteq A^{J}}$ ${\displaystyle S\subseteq B^{J}}$ ${\displaystyle (F_{i})_{i\in I}}$ ${\displaystyle (G_{i})_{i\in I}}$ ${\displaystyle F_{i}\subseteq A^{J_{i}}}$ ${\displaystyle G_{i}\subseteq B^{J_{i}},}$ ${\displaystyle i\in I}$ ${\displaystyle (A,(F_{i})_{i\in I})}$ ${\displaystyle (B,(G_{i})_{i\in I})}$

A relation is then called compatible with the relations and if the following applies to all : ${\displaystyle \varrho \in A\times B}$ ${\displaystyle R}$ ${\displaystyle S,}$ ${\displaystyle (a_{j},b_{j})\in \varrho ,\;j\in J,}$

${\displaystyle (a_{j})_{j\in J}\in R\implies (b_{j})_{j\in J}\in S.}$

Accordingly, a mapping in particular is compatible with the relations and if the following applies: ${\displaystyle \varphi \colon \,A\to B,\,a\mapsto \varphi (a),}$ ${\displaystyle R}$ ${\displaystyle S,}$

${\displaystyle \alpha \in R\implies \varphi \circ \alpha \in S.}$

${\displaystyle \varphi }$is compatible with the structures and if the mapping for each index is compatible with and One then also calls a homomorphism or morphism for short of this type of structure. ${\displaystyle (A,(F_{i})_{i\in I})}$ ${\displaystyle (B,(G_{i})_{i\in I}),}$ ${\displaystyle i\in I}$ ${\displaystyle \varphi }$ ${\displaystyle F_{i}}$ ${\displaystyle G_{i}.}$ ${\displaystyle \varphi }$

Now let an inner connection be on ( may also be infinite) and so that the relation on is given in component terms . then means compatible with if compatible with and ${\displaystyle \chi \colon \,A^{K}\to A}$ ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle R\subseteq A^{J},}$ ${\displaystyle A^{K}}$ ${\displaystyle S:={\bigl \{}\alpha {\bigr |}\,{\hat {\alpha }}(k)\in R\,{\text{ für alle }}\,k\in K\,{\bigl \}}\subseteq {\bigl (}A^{K}{\bigr )}{\bigr .}^{J}}$ ${\displaystyle A}$ ${\displaystyle \chi }$ ${\displaystyle R,}$ ${\displaystyle \chi }$ ${\displaystyle S}$ ${\displaystyle R.}$

Here (and below, for any ) was for the defined by . ${\displaystyle A,K,J,\alpha }$ ${\displaystyle \alpha \in {\bigl (}A^{K}{\bigr )}^{J}}$ ${\displaystyle {\hat {\alpha }}\in {\bigl (}A^{J}{\bigr )}^{K}}$ ${\displaystyle {\hat {\alpha }}(k)(j):=\alpha (j)(k)}$

properties

• Are two relations with the same properties and mappings (i.e. left total and right clear ) and then a map is compatible with the mappings and if and only if ${\displaystyle R_{A}\subseteq A^{I}\!\times A}$ ${\displaystyle R_{B}\subseteq B^{I}\!\times B}$ ${\displaystyle f_{A}\colon \,A^{I}\to A,}$ ${\displaystyle f_{B}\colon \,B^{I}\to B,}$ ${\displaystyle \varphi \colon \,A\to B}$ ${\displaystyle f_{A}}$ ${\displaystyle f_{B},}$

${\displaystyle \varphi \left(f_{A}(\alpha )\right)=f_{B}\left(\varphi \circ \alpha \right)}$   for all ${\displaystyle \alpha \in A^{I}.}$

• Two zero-digit mappings and can always be understood as the one-element single - digit relations and . A picture is therefore if and compatible with the pictures and if the constants and maps to each other: ${\displaystyle f_{A}\colon \,A^{0}\to A,\,()\mapsto f_{A}(),}$ ${\displaystyle f_{B}\colon \,B^{0}\to B,\,()\mapsto f_{B}(),}$ ${\displaystyle R_{A}=\{f_{A}()\}\subseteq A}$ ${\displaystyle R_{B}=\{f_{B}()\}\subseteq B}$ ${\displaystyle \varphi \colon \,A\to B}$ ${\displaystyle f_{A}}$ ${\displaystyle f_{B},}$ ${\displaystyle \varphi }$ ${\displaystyle f_{A}()}$ ${\displaystyle f_{B}()}$

${\displaystyle \varphi (f_{A}())=f_{B}().}$

• ${\displaystyle \chi \colon \,A^{K}\to A}$is if and compatible with a picture if the following applies: ${\displaystyle f_{A}\colon \,A^{I}\to A,}$

${\displaystyle \chi \left(f_{A}\circ {\hat {\alpha }}\right)=f_{A}\left(\chi \circ \alpha \right)}$   for all ${\displaystyle \alpha \in {\bigl (}A^{K}{\bigr )}^{I}.}$

Distributivity

In addition, let a non-empty set be given. A two-digit link is then called left compatible with and if the link transformation for each ${\displaystyle C}$ ${\displaystyle \star \colon \,C\times A\to B,\,(c,a)\mapsto c\star a,}$ ${\displaystyle R_{A}}$ ${\displaystyle R_{B},}$ ${\displaystyle c\in C}$

${\displaystyle \tau _{c\star }\colon \,A\to B,\,a\mapsto \tau _{c\star }(a):=c\star a,}$

is compatible with and according to the above definition . A two-digit link is also called legally compatible with and if for each the legal transformation ${\displaystyle R_{A}}$ ${\displaystyle R_{B}}$ ${\displaystyle *\colon \,A\times C\to B,\,(a,c)\mapsto a*c,}$ ${\displaystyle R_{A}}$ ${\displaystyle R_{B},}$ ${\displaystyle c\in C}$

${\displaystyle \tau _{*c}\colon \,A\to B,\,a\mapsto \tau _{*c}(a):=a*c,}$

is compatible with and . ${\displaystyle R_{A}}$ ${\displaystyle R_{B}}$

If the left is tolerated and quite compatible with pictures and then we also say that linksdistributiv is or rechtsdistributiv is over and ${\displaystyle \star }$ ${\displaystyle *}$ ${\displaystyle f_{A}\colon \,A^{I}\to A}$ ${\displaystyle f_{B}\colon \,B^{I}\to B,}$ ${\displaystyle \star }$ ${\displaystyle *}$ ${\displaystyle f_{A}}$ ${\displaystyle f_{B}\colon }$

${\displaystyle c\star f_{A}(a_{i})_{i\in I}=f_{B}(c\star a_{i})_{i\in I}}$   or       for everyone and for everyone ${\displaystyle f_{A}(a_{i})_{i\in I}*c=f_{B}(a_{i}*c)_{i\in I}}$ ${\displaystyle c\in C}$ ${\displaystyle (a_{i})_{i\in I}\in A^{I}.}$

An inner two-digit link on is called distributive over if left and right distributive is over . ${\displaystyle \cdot \colon \,A\times A\to A}$ ${\displaystyle A}$ ${\displaystyle f_{A},}$ ${\displaystyle \cdot }$ ${\displaystyle f_{A}}$

Examples

• Those with ordered structures and compatible maps are called isotonic or monotonic (increasing) : ${\displaystyle (A,\leq )}$ ${\displaystyle (B,\sqsubseteq )}$ ${\displaystyle \varphi \colon \,A\to B}$

${\displaystyle a_{1}\leq a_{2}\implies \varphi (a_{1})\sqsubseteq \varphi (a_{2})}$   for all ${\displaystyle a_{1},a_{2}\in A.}$

• A congruence relation is an equivalence relation declared on an algebraic structure in such a way that all internal connections are compatible with ${\displaystyle \left(A,(f_{i})\right)}$ ${\displaystyle {\sim }\subseteq A^{2},}$ ${\displaystyle f_{i}}$ ${\displaystyle {\sim }.}$

• The maps that are compatible with algebraic structures are (algebraic) homomorphisms .

• Complete lattice homomorphisms from infinite complete lattices are examples of infinite homomorphisms.

• The topology of a topological space is clearly given by the envelope system of all closed sets of the space and is also clearly determined by the core system , because every open set is the (absolute) complement of a closed set and vice versa. Each closed set can in turn be characterized in that each point then precisely located, if against him a network converges for the topology and the convergence of all the networks are therefore equivalent . ${\displaystyle {\mathcal {O}}}$ ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {O}}}$ ${\displaystyle O\in {\mathcal {O}}}$ ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle a\in X}$ ${\displaystyle A}$ ${\displaystyle (a_{i})_{i\in I}}$ ${\displaystyle a_{i}\in A}$ ${\displaystyle i\in I.}$ ${\displaystyle {\mathcal {O}}}$ ${\displaystyle X}$

With the common topological structure of two topological spaces and therefore a mapping is compatible or continuous if and only if it is compatible for every point with all counter- convergent networks : ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle (Y,{\mathcal {P}})}$ ${\displaystyle \varphi \colon \,X\to Y}$ ${\displaystyle x\in X}$ ${\displaystyle x}$

${\displaystyle (x_{i})_{i\in I}\longrightarrow _{X}x\implies \left(\varphi (x_{i})\right)_{i\in I}\longrightarrow _{Y}\varphi (x)}$   for all networks with for all ${\displaystyle (x_{i})_{i\in I}}$ ${\displaystyle x_{i}\in X}$ ${\displaystyle i\in I.}$

• Representations compatible with categories are called functors .

• The Distributivity plays an important role in many algebraic structures.

literature

• Garrett Birkhoff : Lattice Theory . 3rd edition. AMS, Providence, RI 1973, ISBN 0-8218-1025-1 .
• Marcel Erné: Introduction to Order Theory . Bibliographisches Institut, Mannheim 1982, ISBN 3-411-01638-8 .
• Hans Hermes : Introduction to Association Theory . 2nd Edition. Springer, Berlin - Heidelberg 1967.
• Wilhelm Klingenberg : Linear Algebra and Geometry . Springer, Berlin - Heidelberg 1984, ISBN 3-540-13427-1 .
• Boto von Querenburg : Set theoretical topology . 3., rework. and exp. Edition. Springer, Berlin / Heidelberg / New York 2001, ISBN 3-540-67790-9 .
• F. Reinhardt, H. Soeder: dtv-Atlas Mathematik . 11th edition. tape 1 : Fundamentals, Algebra and Geometry. Deutscher Taschenbuchverlag, Munich 1998, ISBN 3-423-03007-0 .
• Heinrich Werner: Introduction to general algebra . Bibliographisches Institut, Mannheim 1978, ISBN 3-411-00120-8 .

Remarks

1. ↑ The set of all families in with index set , if is finite and contains exactly elements , is also identified with or for with , whereby there is usually no distinction between and . ${\displaystyle A^{J}}$ ${\displaystyle A}$ ${\displaystyle J}$ ${\displaystyle J}$ ${\displaystyle n}$ ${\displaystyle A^{n}=\{(a_{0},\ldots ,a_{n-1})\mid a_{0},\ldots ,a_{n-1}\in A\}}$ ${\displaystyle {\underline {n}}:=\{1,\ldots ,n\}}$ ${\displaystyle A^{\underline {n}}}$ ${\displaystyle A^{n}}$ ${\displaystyle A^{\underline {n}}}$
2. ↑ A structure with a tuple or a family of several carrier sets and with relations in (also different) Cartesian products of these carrier sets can be understood as a structure with the carrier set , since every relation is always a subset of a Cartesian product of . ${\displaystyle ((A_{k})_{k\in K},(R_{i})_{i\in I})}$ ${\displaystyle A_{k}}$ ${\displaystyle R_{i}}$ ${\displaystyle A:=\bigcup (A_{k})_{k\in K}}$ ${\displaystyle R_{i}}$ ${\displaystyle A}$