Correspondence (mathematics)

This article deals with correspondence in terms of multi-valued functions. For the term “bijective correspondence” see bijective function .

In mathematics, the concept of correspondence is a more precise specification of the concept of a multi-valued function or multifunction, which is more common in the older mathematical literature . While a function in the usual sense assigns a single element of the target set as a function value to each element of the definition set, several elements of the target set can be assigned to one element of the definition set in a multi-valued function. In the concept of correspondence, these multiple function values ​​are combined to form a set (i.e. a subset of the target set). A correspondence from a set to a set is thus a function that assigns a subset of to each element of . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle B}$

definition

A correspondence from a set to a set is a mapping of into the power set of . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \phi }$ ${\displaystyle A}$ ${\displaystyle B}$

notation

A correspondence from to is written as: ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle \phi \colon A\multimap B}$ or. ${\displaystyle \phi \colon A\to {\mathfrak {P}}(B)}$

Correspondence as a relation

A correspondence from to can be identified with the relation , because the definition gives back the correspondence from the relation . For details see: ${\displaystyle \phi }$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle R=\{(a,b)\in A\times B\mid b\in \phi (a)\}}$ ${\displaystyle R\subseteq A\times B}$ ${\displaystyle \phi (a):=\kappa _{R}(a)=\{b\in B\mid (a,b)\in R\}}$

• Relation §Relations and functions and
• Function §Multi-functions .

Accordingly, relation and correspondence are equivalent terms, but in the case of correspondence, the focus is on the interpretation as mapping one set into the power set of a second.

In the case , the relation represents a transition relation and is the associated transition function. ${\displaystyle B=A}$ ${\displaystyle R}$ ${\displaystyle \Phi =\kappa _{R}}$

Properties of correspondence

If and are topological spaces , then interesting properties of correspondences between and can be defined. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \phi }$ ${\displaystyle A}$ ${\displaystyle B}$

It is called closed (open) when the corresponding relation in the product space is closed ( open ). ${\displaystyle \phi }$

A fixed point of a correspondence from to is a point with . ${\displaystyle \phi }$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle a\in A}$ ${\displaystyle a\in \phi (a)}$

The following, non-constructive Existence Theorem from Shizuo Kakutani ensures the existence of fixed points.

Fixed point theorem of Kakutani

→ Main article : Kakutani Fixed Point Theorem

Formulation of the sentence for ${\displaystyle \mathbb {R} ^{n}}$

Do not be empty, convex and compact , and be a closed correspondence from to such that for each one is convex and not empty. Then has a fixed point. ${\displaystyle A\subset {\mathbb {R} }^{n}}$ ${\displaystyle \phi \colon A\multimap A}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle \phi (a)}$ ${\displaystyle a}$ ${\displaystyle \phi }$

Applications

This fixed point theorem generalizes Brouwer's fixed point theorem , because a mapping can be understood as a correspondence with , and a fixed point of is a fixed point of . ${\displaystyle f\colon A\rightarrow A}$ ${\displaystyle \phi }$ ${\displaystyle \phi (a)=\{f(a)\}}$ ${\displaystyle \phi }$ ${\displaystyle f}$

In mathematical economic theory this theorem leads to interesting existence theorems about equilibrium prices. In mathematical game theory , John Nash used this theorem to show the existence of equilibrium points in certain cooperative two-person games (see Nash equilibrium ).

Correspondences in Algebraic Geometry

In algebraic geometry one describes a correspondence between varieties and a sub- variety of the product . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X\times Y}$

For a body to define the category of correspondences as the category , the objects of which the smooth, projective varieties over and whose morphisms means of Chow groups through are given ${\displaystyle K}$ ${\displaystyle Corr_{K}}$ ${\displaystyle K}$

${\displaystyle Mor(X,Y):=\bigoplus _{i,j}\left\{{\begin{array}{cc}CH^{\mathrm {dim} (X_{i})}(X_{i}\times Y_{j})&\mathrm {dim} (X_{i})=\mathrm {dim} (Y_{j})\\0&sonst\end{array}}\right\}}$

where denotes the decomposition of the varieties into irreducible components . The composition of two morphisms is defined by ${\displaystyle X=\bigcup _{i}X_{i},Y=\bigcup _{j}Y_{j}}$ ${\displaystyle f\in Mor(X,Y),g\in Mor(Y,Z)}$

${\displaystyle g\circ f:=(pr_{13})_{*}(((pr_{12})^{*}(f)\cdot (pr_{23})^{*}(g))\in CH^{\mathrm {dim} (X)}(X\times Z),}$

where the projection of denotes the product of the -th and -th factor. The identity is the diagonal . ${\displaystyle pr_{ij}}$ ${\displaystyle X\times Y\times Z}$ ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle id_{X}}$ ${\displaystyle \Delta _{X}\in CH^{\mathrm {dim} (X)}(X\times X)}$

literature

• Heinz König, Michael Neumann: Mathematical Economic Theory . Publisher Anton Hain Meisenheim GmbH (1986)
• Burkhard Rauhut, Norbert Schmitz, Ernst-Wilhelm Zachow: An introduction to the mathematical theory of strategic games . Teubner Study Books (1979)
• Harro Heuser : Textbook of Analysis - Part 2 . 5th edition, Teubner 1990, ISBN 3-519-42222-0 , p. 609

Web links

• Correspondence , Encyclopedia of Mathematics, Springer