Two-digit link

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. ${\displaystyle f\colon X\to Y}$ ${\displaystyle g\colon Y\to Z}$ ${\displaystyle g\circ f\colon X\to Z}$ ${\displaystyle \circ \colon \mathrm {Abb} (Y,Z)\times \mathrm {Abb} (X,Y)\to \mathrm {Abb} (X,Z)}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Z}$

Inner two-digit link

A commutative link

An associative link

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 . ${\displaystyle X}$ ${\displaystyle g\circ f}$ ${\displaystyle f,g\colon X\to X}$ ${\displaystyle \mathrm {Abb} (X,X)}$

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. ${\displaystyle A}$ ${\displaystyle V}$ ${\displaystyle A\times A\to V}$ ${\displaystyle (P,Q)\mapsto {\overrightarrow {PQ}}}$

See also

• single digit link

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 .

Web links

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