A two-digit link returns the result for the two arguments and .${\ displaystyle \ circ}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle x \ circ y}$

A two-digit link , also called a binary link , is a link in mathematics that has exactly two operands . Two-digit links occur very frequently , especially in algebra, and there is also an abbreviation of linkage without the addition of two-digit . But there are also links with a different arity , which link three or more operands with one another, for example.

## definition

A two-digit link is a mapping of the Cartesian product of two sets and a third set . Such a link assigns an element to each ordered pair of elements and as the two operands with as the result of the link. If the sets , and are equal, the connection is also called an internal connection; otherwise one speaks of an external connection. ${\ displaystyle f \ colon A \ times B \ to C}$ ${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle C}$${\ displaystyle f}$${\ displaystyle (a, b)}$${\ displaystyle a \ in A}$${\ displaystyle b \ in B}$${\ displaystyle f (a, b) = c}$${\ displaystyle c \ in C}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle C}$

### Spellings

Two-digit links are often written in infix notation instead of the usual prefix notation . For example one writes an addition as instead of . A multiplication is often written without any symbol so . The best-known postfix notation is the reverse Polish notation , which does not require brackets. The chosen spelling, whether prefix, infix, or postfix, essentially depends on the usefulness in the given context and the respective traditions. ${\ displaystyle f}$ ${\ displaystyle a \, f \, b}$ ${\ displaystyle f (a, b)}$${\ displaystyle a + b}$${\ displaystyle {+} (a, b)}$ ${\ displaystyle \ cdot}$${\ displaystyle ab = a \ cdot b = \ cdot (a, b)}$

### Examples

• The basic arithmetic operations (addition, multiplication, subtraction and division) on corresponding sets of numbers are two-digit combinations. For example, dividing an integer by a natural number creates a rational number . This corresponds to a link .${\ displaystyle a \ in \ mathbb {Z}}$ ${\ displaystyle b \ in \ mathbb {N} ^ {*} = \ mathbb {N} \ setminus \ {0 \}}$ ${\ displaystyle c = a / b}$${\ displaystyle / \ colon \ mathbb {Z} \ times \ mathbb {N} ^ {*} \ to \ mathbb {Q}}$
• The composition of figures is a two-digit link: it assigns each figure and each figure its execution one after the other . This corresponds to a link . The quantities , and can be chosen as desired. This connection occurs in almost all areas of mathematics and forms the basis of category theory.${\ displaystyle f \ colon X \ to Y}$${\ displaystyle g \ colon Y \ to Z}$${\ displaystyle g \ circ f \ colon X \ to Z}$${\ displaystyle \ circ \ colon \ mathrm {Fig} (Y, Z) \ times \ mathrm {Fig} (X, Y) \ to \ mathrm {Fig} (X, Z)}$${\ displaystyle X}$${\ displaystyle Y}$${\ displaystyle Z}$

An inner two-place link or two-place operation on a set is a two-place link that assigns an element of to every ordered pair of . This corresponds to the above general definition in the special case . The additional attribute inner expresses that all operands are from the set and the link does not lead out. It is said to also be completed with respect . ${\ displaystyle A}$${\ displaystyle f \ colon A \ times A \ to A}$${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle A = B = C}$${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle f}$

Inner two-digit links are an important part of algebraic structures that are studied in abstract algebra . They occur with semigroups , monoids , groups , rings and other mathematical structures .

In general, a set with any internal connection is also called magma . Such links often have other properties, for example they are associative or commutative . Many also have a neutral element and invertible elements . ${\ displaystyle A}$${\ displaystyle * \ colon A \ times A \ to A}$

### Examples

• The addition and the multiplication of whole numbers are internal connections or . The same is true for the natural , rational , real, and complex numbers .${\ displaystyle + \ colon \ mathbb {Z} \ times \ mathbb {Z} \ to \ mathbb {Z}}$${\ displaystyle \ cdot \ colon \ mathbb {Z} \ times \ mathbb {Z} \ to \ mathbb {Z}}$
• The subtraction of whole numbers is an inner link . The same is true for the rational , real, and complex numbers . Note, however, that the subtraction of natural numbers leads out of the set of natural numbers and is therefore not an internal connection. (Here is for example ).${\ displaystyle - \ colon \ mathbb {Z} \ times \ mathbb {Z} \ to \ mathbb {Z}}$${\ displaystyle - \ colon \ mathbb {N} \ times \ mathbb {N} \ to \ mathbb {Z}}$${\ displaystyle 1-2 = -1 \ notin \ mathbb {N}}$
• The division of rational numbers without is an internal connection . The same applies to the real and complex numbers without . Note, however, that the division of whole numbers leads out of the set of whole numbers and is therefore not an internal connection. (Here is for example ).${\ displaystyle 0}$${\ displaystyle / \ colon \ mathbb {Q} ^ {*} \ times \ mathbb {Q} ^ {*} \ to \ mathbb {Q} ^ {*}}$${\ displaystyle 0}$${\ displaystyle / \ colon \ mathbb {Z} \ times \ mathbb {Z} ^ {*} \ to \ mathbb {Q}}$${\ displaystyle 1/2 \ notin \ mathbb {Z}}$
• For a given set the averaging and the union of subsets are internal connections on the power set .${\ displaystyle M}$ ${\ displaystyle X \ cap Y}$ ${\ displaystyle X \ cup Y}$${\ displaystyle X, Y \ subset M}$ ${\ displaystyle {\ mathfrak {P}} (M)}$
• For any crowd , the composition of images is an inner link .${\ displaystyle X}$${\ displaystyle g \ circ f}$${\ displaystyle f, g \ colon X \ to X}$${\ displaystyle \ mathrm {Fig} (X, X)}$

## Outer two-digit links of the first kind

An outer two-place link of the first kind is a two-place link which is called the right operation of on , or which is called the left operation of on . They differ from inner two-digit links in that the set called the operator area , the elements of which are called operators , is not necessarily a subset of , i.e. it can come from outside . We then say operated from the right or from the left on and the elements of hot right or left operators . ${\ displaystyle f \ colon \, A \ times B \ to A,}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle f \ colon \, B \ times A \ to A,}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle B,}$${\ displaystyle A}$${\ displaystyle B}$ ${\ displaystyle A,}$${\ displaystyle B}$

Each operator defines exactly one mapping or which is also called the transformation to . In the case of a multiplication , instead of or , one writes briefly or and there is usually or no longer a distinction between the operator and the associated transformation . One then writes in the so-called operator notation : resp.${\ displaystyle \ beta \ in B}$${\ displaystyle \ vartheta _ {f \ beta} \ colon A \ to A, \, a \ mapsto \ vartheta _ {f \ beta} (a): = a \, f \, \ beta,}$${\ displaystyle \ vartheta _ {\ beta f} \ colon A \ to A, \, a \ mapsto \ vartheta _ {\ beta f} (a): = \ beta \, f \, a,}$${\ displaystyle \ beta}$ ${\ displaystyle f}$${\ displaystyle a \, f \, \ beta}$${\ displaystyle \ beta \, f \, a}$${\ displaystyle a \ beta}$${\ displaystyle \ beta a}$${\ displaystyle \ beta}$${\ displaystyle \ vartheta _ {\ beta} \ colon a \ mapsto a \ beta}$${\ displaystyle \ vartheta _ {\ beta} \ colon a \ mapsto \ beta a}$${\ displaystyle \ beta \ colon A \ to A, \, a \ mapsto a \ beta,}$${\ displaystyle \ beta \ colon A \ to A, \, a \ mapsto \ beta a.}$

### Examples

• For every natural number an inner - digit connection is always also an outer two-digit connection of the first kind, namely both a right and a left operation of on (it is always ). Such internal links are therefore also generally referred to as -digit operations . A zero-digit link can be viewed as an inner link and therefore always count as a zero-digit operation .${\ displaystyle n}$${\ displaystyle n}$ ${\ displaystyle f \ colon A ^ {n} \ to A}$${\ displaystyle A ^ {n-1}}$${\ displaystyle A}$${\ displaystyle A ^ {0} = \ {\ emptyset \}}$${\ displaystyle n}$ ${\ displaystyle f \ colon \ {\ emptyset \} \ to A}$${\ displaystyle f \ colon A ^ {0} \ to A}$
• A group operation is a group and a set. One also demands a certain compatibility of this operation with the group structure namely and for all and the neutral element of${\ displaystyle \ star \ colon \, G \ times X \ to X}$${\ displaystyle (G, *)}$${\ displaystyle X}$${\ displaystyle (G, *),}$${\ displaystyle (g * h) \ star x = g \ star (h \ star x)}$${\ displaystyle e \ star x = x}$${\ displaystyle g, h \ in G, \, x \ in X}$${\ displaystyle e}$${\ displaystyle G.}$
• In linear algebra , the operator domain for scalar multiplication is a field , usually or and an Abelian group, for example or. In addition, a corresponding compatibility of the scalar multiplication with the already given structures and equipped with the operation becomes a vector space${\ displaystyle \ odot \ colon \, K \ times V \ to V}$${\ displaystyle K}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {C},}$${\ displaystyle V}$${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle \ mathbb {C} ^ {n}.}$${\ displaystyle (K, +, \ cdot)}$${\ displaystyle (V, \ oplus).}$${\ displaystyle \ odot}$${\ displaystyle (V, \ oplus, \ odot)}$${\ displaystyle K.}$

### comment

The term operation or operator is used e.g. B. in functional analysis , also for general two-digit links or used. Here are sets with the same (mostly algebraic ) structure , and often the transformation or with the structure should be on and compatible . ${\ displaystyle f \ colon \, A \ times B \ to C}$${\ displaystyle f \ colon \, B \ times A \ to C}$${\ displaystyle A, C}$ ${\ displaystyle \ vartheta _ {f \ beta} \ colon A \ to C}$${\ displaystyle \ vartheta _ {\ beta f} \ colon A \ to C}$${\ displaystyle A}$${\ displaystyle C}$

## Outer two-digit links of the second kind

An outer two-digit link of the second kind is a mapping, that is , a two-digit link on a set but does not have to be closed with respect to it, so it may also apply. ${\ displaystyle f \ colon A \ times A \ to C,}$${\ displaystyle f}$${\ displaystyle A,}$${\ displaystyle A}$${\ displaystyle f}$${\ displaystyle C \ nsubseteq A}$

### Examples

• Every inner two-place link is also an outer two-place link of the second kind.${\ displaystyle f \ colon A \ times A \ to A}$
• The scalar product in the - dimensional - vector space assigns a real number to two vectors from and is thus an outer two-digit combination of the second kind. For the scalar product is also an inner two-digit combination, but not for.${\ displaystyle n}$${\ displaystyle \ mathbb {R}}$ ${\ displaystyle \ mathbb {R} ^ {n}, n \ geq 1,}$${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle n = 1}$${\ displaystyle n> 1}$
• The scalar product in division ring of quaternions is an internal binary operation and thus an external binary operation of the second kind. Summing up against it as a four-dimensional division algebra over on, then the dot product is no internal link more, but it remains an external binary operation of the second kind.${\ displaystyle \ mathbb {H}}$${\ displaystyle \ mathbb {H}}$${\ displaystyle \ mathbb {R}}$
• If an affine space is over a vector space , then with is an outer two-place link of the second kind.${\ displaystyle A}$${\ displaystyle V}$${\ displaystyle A \ times A \ to V}$${\ displaystyle (P, Q) \ mapsto {\ overrightarrow {PQ}}}$

## literature

• Gert Böhme: Algebra (=  application-oriented mathematics . Volume 1 ). 4th, verb. Edition. Springer, Berlin / Heidelberg / New York 1981, ISBN 3-540-10492-5 , pp. 80 .
• F. Reinhardt, H. Soeder: dtv-Atlas Mathematik . 11th edition. tape 1 : Fundamentals, Algebra and Geometry . Deutscher Taschenbuchverlag, Munich 1998, ISBN 3-423-03007-0 , p. 38-41 .
• Günter Scheja, Uwe Storch: Textbook of algebra: including linear algebra . Part 1. Teubner, Stuttgart 1980, ISBN 3-519-02203-6 , pp. 101, 204-207 .
• Bartel Leendert van der Waerden : Algebra I . 9th edition. Springer, Berlin / Heidelberg / New York 1993, ISBN 978-3-662-01514-8 , pp. 146-148 .