Correspondence
relation
A generalization of the notion of a binary relation (usually) between two sets or mathematical structures of the same type. Correspondences are widely used in mathematics and also in various applied disciplines, such as theoretical programming, graph theory, systems theory, and mathematical linguistics.
A correspondence between two sets and
is any subset
of the Cartesian product
. In other words, a correspondence between
and
consists of certain ordered pairs
, where
and
. As a rule, a correspondence is denoted by a triple
and one may write
or
in place of
. Instead of "correspondence" the term "binary relation" , or "relation between setsrelation" is sometimes used (in the general case where
and
need not coincide).
For finite sets, the matrix and graphical representations of a correspondence are commonly used. Suppose that and
have
and
elements, respectively, and let
be some correspondence. One can describe this by using an
matrix the rows and columns of which are labelled with the elements of
and
, respectively, and the intersection of the
-th row with the
-th column contains 1 if
, and
otherwise. Conversely, every
-matrix consisting of zeros and ones describes a unique correspondence between
and
. In the graphical representation, the elements of
and
are represented by points in the plane. These points are usually denoted by the same symbols as the corresponding elements. Then
and
are connected by an arrow (arc) from
to
if
. Thus, the correspondence is represented by an oriented graph.
The set of all correspondences between two sets and
forms a complete Boolean algebra the zero of which is the empty correspondence and the identity of which is the so-called complete correspondence, consisting of all pairs
,
,
. Let
. The set
![]() |
is called the domain of definition of , and the set
![]() |
is called the range, or image, of . The correspondence
is everywhere defined if
, and surjective if
. For every
the set
![]() |
is called the image of with respect to
, and for every
, the set
![]() |
is called the co-image (or pre-image) of with respect to
. Then
![]() |
Any correspondence establishes a Galois correspondence between the subsets of
and those of
. Namely, to any subset
, one assigns the subset
. Together with the dual correspondence
, which assigns to every
the set
, the Galois correspondence defines a closure operator on both
and
.
The inverse or involution , or
, of a correspondence
is defined by the equation
![]() |
This establishes a bijection between the correspondence and
, which is an isomorphism of Boolean algebras. Given two correspondences
and
, their product or composite is given by
![]() |
Multiplication of correspondences is associative, and its identities are the diagonal binary relations. Moreover, , and
implies that
. Therefore the correspondences between a family of sets form an ordered category with involution. Multiplication and involution enable one to express the properties of correspondences in terms of algebraic relations. For example, a correspondence
is everywhere defined if
(
is the diagonal of
), and
is functional, that is, it is the graph of a function from
into
, if
and
.
For any correspondence , there are functional correspondences
and
such that
. Moreover,
. Any difunctional correspondence induces equivalence relations on the domain and on the image whose quotient sets have the same cardinality. This only holds for difunctional correspondences.
Let be a class of mathematical structures of the same type that is closed under finite Cartesian products. By a correspondence between two structures
, one means a substructure
of
. Thus one has group correspondences, module correspondences, ring correspondences, and others. Such correspondences often have useful descriptions of their structure. For example, let
and
be groups and let
be a subgroup of the direct product
. The sets
![]() |
are called the kernel and the indeterminacy of , respectively.
is a normal subgroup of
,
is a normal subgroup of
, and the quotient groups
and
are isomorphic. It follows, in particular that all group correspondences are difunctional.
References
[1] | A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) MR0158000 Zbl 0121.25901 |
[2] | A.I. Mal'tsev, "Algebraic systems" , Springer (1973) (Translated from Russian) Zbl 0266.08001 |
[3] | M.Š. [M.Sh. Tsalenko] Calenko, "Classification of correspondence categories and types of regularities for categories" Trans. Moscow Math. Soc. , 1 (1982) pp. 239–282 Trudy Moskov. Mat. Obshch. , 41 (1980) pp. 241–285 |
Comments
In algebraic geometry correspondences are used widely, [a1], Chapt. 2, 3. They are defined as the following slightly more technical concept. A correspondence between two (projective) varieties
and
is defined by a closed algebraic subset
. It is said to be a rational mapping if
is irreducible and there exists a Zariski-open subset
such that each
is related by
to one and only one point of
(i.e.
). The correspondence
is said to be a birational mapping if both
and
are rational mappings.
References
[a1] | D. Mumford, "Algebraic geometry" , 1. Complex projective varieties , Springer (1976) MR0453732 Zbl 0356.14002 |
Correspondence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Correspondence&oldid=13039