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I asked this question in the comments of this question, whose title would have done just as well for mine. But I suppose it should be a separate question.

Is there a name for functions $f:X\rightarrow Y$ such that $|f(X)|<|X|$? Obviously when $X$ is finite, this is just 'non-injective', but for infinite cardinalities it's a much stronger property.

I am interested to know this because it comes up in studying the full transformation semigroup on an infinite set. For example see this question. (I suppose 'not of maximal rank' is one possible answer to my question.)

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When you talk about infinite cardinalities, does $|f(X)|<|X|$ mean $|f(X)|=\aleph_m$ and $|X|=\aleph_n$ with $m<n$? – AndreasT Mar 2 '13 at 14:59
I meant that there is an injection but no bijection from $f(X)$ to $X$. I suppose that might be equivalent to what you said, provided $m$ and $n$ are ordinals. – Tara B Mar 2 '13 at 15:05

In finite case semigroupists call "singular" the mappings $f:X\to X$ for which the defect $|X\setminus f(X)|$ is positive (P. Higgins, Techniques of semigroup theory). Is this term suitable for you?

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This isn't really what I was after, since I was mainly interested in the infinite case (I'm quite happy with 'not injective' in the finite case, although 'singular' sounds better). But it's still a useful word related to the question, so +1 and thanks. – Tara B Mar 18 '13 at 19:03
Or are you suggesting that 'singular' could be the appropriate word in the infinite case, too? – Tara B Mar 18 '13 at 19:03
Since you cannot use "injective", why not use "singular"? – Boris Novikov Mar 18 '13 at 19:47
Well, I could use it, but I have no idea whether it already has a definition. My question is about whether there is a name already in use for such functions. – Tara B Mar 18 '13 at 22:01

I'd suggest just stating it as you do, and explaining the infinite cardinality case. If you find some reasonable name, use that. Maybe it catches on, meanwhile use the explanation. Mathematics is primarily written to be understood by fellow humans (even if they happen to be mathematicians).

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Yes, it's not that I need a name to refer to it by, because $|f(X)|<|X|$ is very easy to write and understand. I was just curious, because of the first question I linked to, which I thought was going to be asking this question. – Tara B Mar 2 '13 at 16:03
Glad to see somebody who isn't a formalism fetishist ;-) – vonbrand Mar 2 '13 at 16:05

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