The monoid of all the functions $X^X=\{f:X\to X\}$ has a right inverse : the the surjective function and a left inverse: the injective function.
I remember that it has to do with the set theory, is there a simple and intuition explanation?
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What do you mean by the right-inverse of a set? – Ben Grossmann Dec 11 '19 at 13:35
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3I think what you're trying to say is that elements of the monoid will have a right-inverse iff they are surjective, and a left-inverse iff they are injective. – Ben Grossmann Dec 11 '19 at 13:36
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@Omnomnomnom yes, exactly – newhere Dec 11 '19 at 13:37
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It happens that complete answers are already available, although they are somewhat scattered, especially in "proof verification" problems. This is an attempt to make something a little more canonical. If it happens that there is a more canonical version that I missed in my search, we should link this question as a duplicate to that. Alternatively someone with a gold badge could link the entire collection of questions I'm going to list.
If a function $f:X\to Y$ is surjective, see this question and this question
If a function $f:X\to Y$ is injective, then see this related question.
Proving the converses of both statements is easy to do directly and is done in this post.
rschwieb
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