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$\begingroup$

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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    @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.2011-06-21
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    possible duplicate of [Special arrows for notation of morphisms](http://math.stackexchange.com/questions/20015/special-arrows-for-notation-of-morphisms)2011-06-21
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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.2011-06-21
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    @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.2011-06-21
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    @Asaf: I don't get it. It's exactly the same question in a special context.2011-06-21
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    @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.2011-06-21
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    But an epimorphism in the category of sets is a surjective map, ditto monomoprhism and injective map, no?2011-06-21
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    @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](http://www.indiana.edu/~jfdavis/notes/eckmann.pdf)2011-06-21
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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.2011-06-21
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    @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.2011-06-21
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    @Americo Tavares: The terms were I think popularized by Bourbaki. By the way, in comments, how does one get accents into names?2011-06-21
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    @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.2011-06-21
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    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.2011-06-21

2 Answers 2

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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.

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    Sounds like a good question for our [sister site](http://tex.stackexchange.com/)2011-06-21
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    @Willie, John: $\rightarrowtail$ I assume and it is `\rightarrowtail` (from the commonly used amssymb)2011-06-21
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    @Asaf: I think John wants something like this $\displaystyle\rightarrowtail\!\!\!\!\!\rightarrow$ (I used `\rightarrowtail\!\!\!\!\!\rightarrow` - which is of course an ugly hack)2011-06-21
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    @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!).2011-06-21
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    @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.2011-06-21
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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.2011-06-21
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    @JSchlather Try \mathbin{\rightarrowtail \hspace{-8pt} \twoheadrightarrow} which gives: $\mathbin{\rightarrowtail \hspace{-8pt} \twoheadrightarrow}$2016-11-29
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    I quite like another idea: https://mathoverflow.net/questions/42929/suggestions-for-good-notation/46197#461972018-10-06
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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.

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    seems reasonable, except for dobuble headed bijective arrow which still makes sense.2011-06-21