Concatenation characters

This article deals with the concatenation symbol as a mathematical operator, for the pipe symbol in IT, see the vertical line .

∘
Mathematical signs
arithmetic
Plus sign	+
Minus sign	- , ./.
Mark	⋅ , ×
Divided sign	: , ÷ , /
Plus minus sign	± , ∓
Comparison sign	< , ≤ , = , ≥ , >
Root sign	√
Percent sign	%
Analysis
Sum symbol	Σ
Product mark	Π
Difference sign , Nabla	∆ , ∇
Prime	′
Partial differential	∂
Integral sign	∫
Concatenation characters	∘
Infinity symbol	∞
geometry
Angle sign	∠ , ∡ , ∢ , ∟
Vertical , parallel	⊥ , ∥
Triangle , square	△ , □
Diameter sign	⌀
Set theory
Union , cut	∪ , ∩
Difference , complement	∖ , ∁
Element character	∈
Subset , superset	⊂ , ⊆ , ⊇ , ⊃
Empty set	∅
logic
Follow arrow	⇒ , ⇔ , ⇐
Universal quantifier	∀
Existential quantifier	∃
Conjunction , disjunction	∧ , ∨
Negation sign	¬

The concatenation symbol ( ), also known as a circle operator , is a mathematical symbol that expresses the concatenation of two functions , relations or words . If there is no risk of confusion with multiplication , it is often left out. ${\displaystyle \circ }$

use

Chaining of functions

The pipe symbol is often used as mathematical operator for the sequential execution of functions used. If sets are and as well as functions, then the concatenation of and through ${\displaystyle A,B,C}$ ${\displaystyle f\colon A\to B}$ ${\displaystyle g\colon B\to C}$ ${\displaystyle f}$ ${\displaystyle g}$

${\displaystyle (g\circ f)(x)=g(f(x))}$

written down. The expression “ ” is read as “ linked with ”, “ composed with ”, “ after ” or “ kringel ”. When using the notation, it is important to ensure that the composition is carried out from right to left, i.e. that it is applied first and then . Occasionally, however, the reverse order is used and written in the literature . ${\displaystyle g\circ f}$ ${\displaystyle g}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle f}$ ${\displaystyle g\circ f}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle (f\circ g)(x)=g(f(x))}$

Chaining of relations

More generally, the concatenation character is used as an operator for concatenating binary relations . If sets and as well as binary relations are again , then the concatenation of and through ${\displaystyle A,B,C}$ ${\displaystyle R\subseteq A\times B}$ ${\displaystyle S\subseteq B\times C}$ ${\displaystyle R}$ ${\displaystyle S}$

${\displaystyle S\circ R=\{(a,c)\in A\times C\mid \exists ~b\in B:(a,b)\in R\land (b,c)\in S\}}$

written down. The result is called a relation product or a relative product. Here, too, the reverse order is occasionally used. ${\displaystyle R\circ S}$

Concatenation of words

In theoretical computer science , the concatenation character is used for the concatenation of words . If and are two words above an alphabet , then the concatenation of and through ${\displaystyle x=(x_{1},x_{2},\dotsc ,x_{n})}$ ${\displaystyle y=(y_{1},y_{2},\dotsc ,y_{k})}$ ${\displaystyle x}$ ${\displaystyle y}$

${\displaystyle x\circ y=(x_{1},x_{2},\dotsc ,x_{n},y_{1},y_{2},\dotsc ,y_{k})}$

written down. Here is the prefix and the suffix of the result of the concatenation. The concatenation of words is a special case of a concatenation of sets . ${\displaystyle x}$ ${\displaystyle y}$

Binary links

In an even more general way, the concatenation symbol is sometimes used alongside the characters , and as a placeholder for binary combinations of general algebraic structures . ${\displaystyle \bullet }$ ${\displaystyle \ast }$ ${\displaystyle \cdot }$

Presentation on the computer

Coding

The concatenation character is defined and coded as follows:

Coding in Unicode, HTML and LaTeX

character	Unicode		designation	HTML			Latex
	position	designation		hexadecimal	decimal	named
∘	U+2218	ring operator	Circle operator	∘	∘		\circ

Similar characters

Similar and related signs - there may be a
risk of confusion with the following signs

character	Unicode		Surname
	position	designation
O	U+006F	Latin small letter or similar	Latin Small Letter O
°	U+00B0	degree sign	Degree sign
ο	U+03BF	greek small letter omicron	Greek Small Letter Omicron
о	U+043E	cyrillic small letter о	Cyrillic lowercase letter о
ₒ	U+2092	Latin subscript small letter or similar	Subscript Latin small letter o
○	U+25CB	white circle	White circle
◦	U+25E6	white bullet	White bullet
⚬	U+26AC	medium small white circle	Middle small white circle
。	U+3002	ideographic full stop	End of sentence in CJK fonts

Individual evidence

1. ↑ Gerd Fischer: Linear Algebra . Springer, 2009, p. 36 .
2. ↑ Ehrhard Behrends: Analysis Volume 1 . Springer, 2014, p. 19 .
3. ↑ George Hoever: Higher Mathematics compact . Springer, 2013, p. 43 .

Web links

• Concatenation characters in Unicode