# Concatenation characters

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
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 . ${\ displaystyle x}$ ${\ displaystyle y}$ 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
U+2218 ring operator Circle operator &#x2218; &#8728; \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
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 .