# Two digit linkage

a two digit linkage (also binary linkage) is in mathematics a special kind of the linkage, which is characterised by the fact that it possesses exactly two operands. Well-known examples are the basic operations of arithmetic such as addition and division.

## definition

an internal two digit linkage on a quantity of S is an illustration of the cartesian product S × S into the quantity S. The quantity together with an internal two digit linkage is called also magma.

Two digit linkages are an important component of algebraic structures, which are examined in abstract algebra. They step on with half-groups, groups, rings and other structures.

Manybinary linkages, which one regards, are commutative or associative. Many have also a neutral element and inverse elements. Typical examples of binary linkages are the addition and multiplication of numbers and stencils, as well as the composition of functions.

## ways of writing

binary linkagesone often writes b, A in Infix notation , like with A + · b, in place of a function notation as + (A, b). Multiplications are written often completely without symbol: off = A · b.

One can do it in addition, inPrefix or post office fixed notation indicate. A prefix notation e.g. is. the usual function way of writing f (A, b). The most well-known post office fixed notation is the reverse Polish notation, which gets along without clips.

## examples

• through [itex] (x, y) \ mapsto x+y< /math> is an internal linkage on the quantity[itex] \ R< /math> given, since the addition of two real numbers always results in a real number.
• For a given quantity [itex] M< /math> the Durchschnittsbildung is an internal linkage on the power quantity [itex] \ mathcal P (M)< /math>:
[itex] (X, Y) \ mapsto X \ cap Y< /math>

## exterior binary linkage

a two digit function of K × one calls S after S an outside two digit linkage. It differs from a two digit linkage strictly speaking by the fact that K does not have to be a subset of S that thus the first operand from outside comes.

An example of it is the scalar multiplication in linear algebra. Here K is a body and a S a vector space over this body.

One can understand an outside binary linkage often also as operation, K operated then on S.

### exterior (two digit) linkages of first kind

illustrations of the type[itex] A \ times B \ tons of A< /math> one calls outside linkages of first kind. The quantity [itex] B< /math> here operator range one calls.

Example:

• The multiplication of a vector out [itex] \ mathbb {R} ^n< /math> with a scalar out [itex] \ mathbb {R}< /math> an outside linkage on math <\> mathbb {R is} ^n< /math> with operator range [itex] \ mathbb {R}< /math>.

### exterior(two digit) linkages of second kind

exterior linkages of second kind are illustrations of the type [itex] A \ times A \ tons of B< /math>.

Example:

• The dot product in [itex] \ mathbb {R} ^n< /math> out math \ mathbb { <R> arranges ever two vectors} ^n< /math> a real number too and is thus an outside linkage of second kind.
• Is [itex] A< /math> a affiner area, that on a vector space [itex] V< /math> is modelled, then math
<A> is \ times A \ tons of V, \ quad (P, Q) \ mapsto \ overrightarrow {PQ}< /math>
an outside linkage of second kind.

### (general two digit) linkages

general one two digit linkages are illustrations of the type [itex] A \ times B \ tons of C< /math>.

Example:

The composition of illustrations, an illustration[itex] f: X \ tons of Y< /math>, and an illustration [itex] g: Y \ tons of Z< /math> their Hintereinanderausführung [itex] g \ circ f: X \ tons of Z< /math> assigns. The quantities X, Y, and Z can be selected at will.