Name for a function whose image has smaller cardinality than its domain

Question

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.)

Answer 1

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).

Answer 2

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?