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.)