# Linkage (math)

In mathematics , **linkage is used** as a generic term for various **operations** : In addition to the basic arithmetic arithmetic **operations** ( addition , subtraction , etc.), it also covers geometric operations (such as mirroring , rotation, etc.) and other arithmetic operations and occasionally logical operators . A link defines how mathematical objects of the same or similar type determine another object with one another. In the case of a relatively small number of elements and a link with only a few, such as two places, where elements can be used as operands, this definition is clearly possible using a link table in which, for example, B. for a 2-digit link all possible pairings are listed and their result is given, the result of the calculation.

The word *linkage* is also used to denote the sequential execution (concatenation) of functions .

## general definition

For a natural number, let sets and another set be given. Then every mapping of the Cartesian product is referred to as a *-digit link* . Such a link uniquely assigns an element of the set to each tuple with . Of course, the quantities and can partly or wholly coincide.
* *

In the special case that only occurs, the link becomes

*inner -ary link* or *-ary operation* on called. Occurs at least once among them, for example

- and

for a with so the link is called the *outer **-digit link* on with the operator area . The elements of are then called operators.

An *inner* -digit link on can also be viewed as an *outer* two-digit link on with the operator area.

Every -digit link can be understood as a -digit relation .

### Examples

- By

- defined mapping from to is a three-digit link or inner three-digit link to .

- If a picture is from to , then it is through

- (each of the image and an element of the formed pair of the image of this element is below the figure assigned)

- given an outer two-digit link on with operator scope and the only operator .

## Zero-digit links

As a **zero-digit combination** of a lot for a lot one can figure out of to be considered. It applies

therefore, each of these figures can be specified as follows:

- for a

Each zero-digit link is therefore constant and can in turn be understood as the constant .

Since it always applies, every zero-digit link can be viewed as an inner link on :

## One-digit links

Single-digit links are mappings of a set after a set .

### Examples

- A lot is given . For each element of the power set , i.e. for each subset of , let us define:

- ( Complement of ).

- The sine function

- is a one-digit shortcut.

## Two-digit (binary) links

The term “link” is particularly often used in the sense of a two-digit link. Important special cases are internal and external connections. Two-digit links are often noted in infix notation, i.e. with a symbol such as a plus sign between the two operands.

## Three- and multi-digit links

Rarely one speaks of **three- and multi-digit links** . Examples of a three-digit link are:

- the mapping, which each assigns three vectors from the their late product (from ) and
- the
*ternary link*in a ternary body .

## Partial links

If in the above definition for (total) links the term of the (totally understood) *mapping is* replaced by *partial mapping* , then one speaks of a **partial link** : It is then permitted that a link value ( ie image value, function value) is assigned.

## Links in algebra

In algebra , links are used to define algebraic structures . The links must meet certain conditions ( axioms ). In the case of *partial algebras* , partial links are also permitted.

For example, a semigroup is a set with an interior two-digit link that satisfies the associative law. The requirement that the result of the connection should again be an element of the given set *(isolation)* is already contained in the definition of the *inner* connection.

## Web links

**Wikibooks: Math for Non-Freaks: Linking**- Learning and Teaching Materials