Update: In the category of sets, an epimorphism is a surjective map and a monomorphism is an injective map. As is mentioned in the morphisms question, the usual notation is $\rightarrowtail$ or $\hookrightarrow$ for $1:1$ functions and $\twoheadrightarrow$ for onto functions. These arrows should be universally understood, so in some sense, this is a narrow duplicate of the morphisms question.

What are usual symbols for surjective, injective and bijective functions? I think in one of Lang's book I saw an arrow with 1:1 e.g. $A\xrightarrow{\rm 1:1}B$ above it to be understood as a bijective function , what are usual notations for surjective, injective and bijective functions?

Update : maybe following notations make sense and are also easily latexed : $A\xrightarrow{\rm 1:1}B$, $A\xrightarrow{\rm onto}B$, $A\xrightarrow{\rm 1:1,onto}B$

I don't know if these notations make sense with morphisms question, but this question was specific and there was no intent to find an answer for the more general case ( but would definitely be preferred).

  • $\begingroup$ @Arjang: In English, "one to one" meant what we usually nowadays call injective, "onto" meant what we usually now call surjective, so "one to one onto" meant bijective. From the internationalization perspective, the current nomenclature is an improvement. But probably from no other perspective. $\endgroup$ Jun 21 '11 at 11:28
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    $\begingroup$ possible duplicate of Special arrows for notation of morphisms $\endgroup$
    – t.b.
    Jun 21 '11 at 11:38
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    $\begingroup$ @user6312: "From the internationalization perspective, the current nomenclature is an improvement." I agree. The problem for non-native speakers with "onto" and "one to one onto" is that it sounds very idiomatic. $\endgroup$ Jun 21 '11 at 12:26
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    $\begingroup$ @Asaf: I don't get it. It's exactly the same question in a special context. $\endgroup$
    – t.b.
    Jun 21 '11 at 12:31
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    $\begingroup$ @Americo Tavares: But I do prefer short plain words. Mantissa, abscissa, denominator, subtrahend, associative, and so on make it harder for students to know that we are dealing with real things. $\endgroup$ Jun 21 '11 at 12:40

I personnaly use $\hookrightarrow$ to mean injection and $\twoheadrightarrow$ to mean surjection. Although I do not have a particular notation to mean bijection, I use $\leftrightarrow$ to mean bijective correspondance.

  • $\begingroup$ seems reasonable, except for dobuble headed bijective arrow which still makes sense. $\endgroup$
    – jimjim
    Jun 21 '11 at 19:57
  • $\begingroup$ $\hookrightarrow$ is usually used to be elementary embedding. $\endgroup$
    – M. Logic
    Aug 7 at 17:44

My favorites are $\rightarrowtail$ for an injection and $\twoheadrightarrow$ for a surjection. In the days of typesetting, before LaTeX took over, you could combine these in an arrow with two heads and one tail for a bijection. Perhaps someone else knows the LaTeX for this.

  • $\begingroup$ Sounds like a good question for our sister site $\endgroup$ Jun 21 '11 at 12:42
  • $\begingroup$ @Willie, John: $\rightarrowtail$ I assume and it is \rightarrowtail (from the commonly used amssymb) $\endgroup$
    – Asaf Karagila
    Jun 21 '11 at 12:48
  • $\begingroup$ @Asaf: I think John wants something like this $\displaystyle\rightarrowtail\!\!\!\!\!\rightarrow$ (I used \rightarrowtail\!\!\!\!\!\rightarrow - which is of course an ugly hack) $\endgroup$
    – t.b.
    Jun 21 '11 at 12:54
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    $\begingroup$ There's an easy fix to combine the two into one, similar to Theo's but a bit shorter use just \hspace except negative so we can get stuff like $\rightarrowtail \hspace{-8pt} \rightarrow$ and $\hookrightarrow \hspace{-8pt} \rightarrow$, just by doing '\rightarrowtail \hspace{-8pt} \rightarrow' and '\hookrightarrow \hspace{-8pt} \rightarrow'. Although there is an issue with the rightarrowtail being a bit small. $\endgroup$
    – JSchlather
    Jun 21 '11 at 21:22
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    $\begingroup$ @JSchlather Try \mathbin{\rightarrowtail \hspace{-8pt} \twoheadrightarrow} which gives: $\mathbin{\rightarrowtail \hspace{-8pt} \twoheadrightarrow}$ $\endgroup$
    – stranger
    Nov 29 '16 at 9:54

I usually use two types of notations for function, injection, surjection and bijiection as follows.

enter image description here

Note that the \twoheadrightarrowtail is defined as follows, and the others are AMS symbols.

\usepackage{mathtools} \newcommand{\twoheadrightarrowtail}\mathrel{\mathrlap{\rightarrowtail}}\mathrel{\mkern2mu\twoheadrightarrow}}


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