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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).

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jimjim
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  • @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. – André Nicolas Jun 21 '11 at 11:28
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    possible duplicate of Special arrows for notation of morphisms – t.b. Jun 21 '11 at 11:38
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    @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. – Américo Tavares Jun 21 '11 at 12:26
  • @Theo: I do not think that this is a duplicate question. The one you say is about morphisms, and while functions are morphisms this is not the point of this question. – Asaf Karagila Jun 21 '11 at 12:27
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    @Asaf: I don't get it. It's exactly the same question in a special context. – t.b. Jun 21 '11 at 12:31
  • @Theo: While general cases are useful, in a site like this we need not close all specific cases. All this is moot now as there is an accepted answer. – Asaf Karagila Jun 21 '11 at 12:34
  • But an epimorphism in the category of sets is a surjective map, ditto monomoprhism and injective map, no? – Willie Wong Jun 21 '11 at 12:34
  • @Asaf: I don't insist that it should be closed, I don't care much about that. I just don't understand your point about the point of this question. By the way: Arrows are a very young invention and only the needs of algebra and topology led to their introduction and success, see the first article here – t.b. Jun 21 '11 at 12:39
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    @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. – André Nicolas Jun 21 '11 at 12:40
  • @user6312: Back in 1968 I learned the portuguese equivalent to "injective" (injectiva), "surjective" (sobrejectiva) and "bijective" (bijectiva). That's why this terminology is easier for me. – Américo Tavares Jun 21 '11 at 13:07
  • @Americo Tavares: The terms were I think popularized by Bourbaki. By the way, in comments, how does one get accents into names? – André Nicolas Jun 21 '11 at 13:20
  • @user6312: most likely. In Portugal there was a huge influence of the french terminology at that time. // Since I have a keyboard with the portuguese accents there's no problem for me. You might copy the accented name, I guess. Or use TeX for that purpose. – Américo Tavares Jun 21 '11 at 13:27
  • after reading the comments I updated the question, if the more general notations for morphisms can be used in this case, then yes this question would be a special case and a narrowed duplicate. – jimjim Jun 21 '11 at 20:32
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    $A\xrightarrow{\rm bij}B$ is nice and concise – john Dec 06 '21 at 04:28
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    I'm glad multiple other people associate double headed arrows with bijection. I was trying to figure out if I could get away with it, since it seems like by far the most intuitive notation, except that it might get confused with similar concepts/notation. – Mr. Nichan Mar 08 '23 at 16:55

4 Answers4

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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.

Thomas Connor
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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.

John Bentin
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    Sounds like a good question for our sister site – Willie Wong Jun 21 '11 at 12:42
  • @Willie, John: $\rightarrowtail$ I assume and it is \rightarrowtail (from the commonly used amssymb) – Asaf Karagila Jun 21 '11 at 12:48
  • @Asaf: I think John wants something like this $\displaystyle\rightarrowtail!!!!!\rightarrow$ (I used \rightarrowtail\!\!\!\!\!\rightarrow - which is of course an ugly hack) – t.b. Jun 21 '11 at 12:54
  • @Theo: The the two non-tail arrowheads on the bijection arrow should be close together, exactly as on the surjection, with the arrow body the same length as the others; that is, the bijection is exactly the superimposition of injection and surjection (which is why I like it!). – John Bentin Jun 21 '11 at 17:56
  • @John: I agree, it was just a demonstration for Asaf. I have never seen such an arrow in an official LaTeX font, but you can produce these things in diagrams using the xypic-package. – t.b. Jun 21 '11 at 17:58
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    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. – JSchlather Jun 21 '11 at 21:22
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    @JSchlather Try \mathbin{\rightarrowtail \hspace{-8pt} \twoheadrightarrow} which gives: $\mathbin{\rightarrowtail \hspace{-8pt} \twoheadrightarrow}$ – stranger Nov 29 '16 at 09:54
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    I quite like another idea: https://mathoverflow.net/questions/42929/suggestions-for-good-notation/46197#46197 – Mr Pie Oct 06 '18 at 09:05
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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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2

Since the authors of preceding answers seem to have gotten away with presenting notation as they (individually) like it, allow me to present notation I like instead: I'm used to denoting the relation between domain and codomain as

$ \large \unicode{x1f814} \hspace{-0.3em} \unicode{x1f816} $ for bijections, i.e. for functions which are both injective and surjective; and

$ \large \! \style{display: inline-block; transform: translateY(-1px)}{\unicode{xFF0D}} \hspace{-0.8em} \style{display: inline-block; transform: translateY(-1px)}{\unicode{xFF0D}} \hspace{-0.5em} \unicode{x1f816} $ for injections which are not bijections, i.e. which are not surjective as well.
(Since other answers seem to attach different meaning to arrows pointing only in the one direction from domain to codomain, I've tried to draw my arrows consistently in a separate style.)

For functions which are in general "many-to-one" relations (and thus not injective) I'd symbolize the relation between domain and codomain correspondingly as

$ \large \unicode{5171} \hspace{-0.2em} \unicode{x1f816} {\hspace{-2.em} \style{display: inline-block; transform: rotate(153deg) translateY(-6px)}{\unicode{x1f816}}} {\hspace{-2.em} \style{display: inline-block; transform: rotate(-153deg) translateY(4px)}{\unicode{x1f816}}} $ for surjective (and not injective) functions; and

$ \large \unicode{5171} \hspace{-0.3em} \unicode{x1f816} $ for functions which are neither surjective, nor injective.

Readily added can be symbols for relating domain and codomain of maps which are in general "one-to-many", and which are therefore not functions at all:

$ \large \unicode{x1f814} \hspace{-0.2em} \unicode{5176} {\hspace{-0.5em} \style{display: inline-block; transform: rotate(-27deg) translateY(-6px)}{\unicode{x1f816}}} {\hspace{-1.em} \style{display: inline-block; transform: rotate(27deg) translateY(5px)}{\unicode{x1f816}}}$ if the mapping is to each element of the codomain, or

$ \large \! \style{display: inline-block; transform: translateY(-1px)}{\unicode{xFF0D}} \hspace{-0.75em} \style{display: inline-block; transform: translateY(-1px)}{\unicode{xFF0D}} \hspace{-0.4em} \unicode{5176} {\hspace{-0.5em} \style{display: inline-block; transform: rotate(-27deg) translateY(-6px)}{\unicode{x1f816}}} {\hspace{-1.em} \style{display: inline-block; transform: rotate(27deg) translateY(5px)}{\unicode{x1f816}}}$ otherwise.

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