# Monoids of endomorphisms

Can one tell whether an (abstract) monoid $M$ is (isomorphic to) the monoid of endomorphisms of a structure $X$?

Or is there a representation theorem saying that for every monoid $M$ there is a structure $X$ such that $M$ is the monoid of endomorphisms of $X$? If not so: what is a simple counter-example?

How can the monoids which are monoids of endomorphisms be characterized?

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Cayley's theorem for monoids asserts that $M$ is isomorphic to the monoid of endomorphisms of $M$ as a right $M$-set. This argument can be internalized to various categories; for example, a ring $R$ is isomorphic to the ring of endomorphisms of $R$ as a right $R$-module.

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Could you please give me a link to Cayley's theorem for monoids? It's not so easily to be found. – Hans Stricker Oct 21 '12 at 23:38
I don't know a reference where it's stated explicitly, but it's a straightforward exercise. The proof is exactly the same as for groups; you don't use inversion anywhere. – Qiaochu Yuan Oct 21 '12 at 23:40
Thanks for the hint. – Hans Stricker Oct 21 '12 at 23:50
@HansStricker I am pretty sure it appears in Jacobson's Basic Algebra I. I think he was pretty meticulous about putting in all versions of Cayley's theorem. Can't find it in the googlebooks preview, but if I remember I'll check again when I find my copy. – rschwieb Oct 22 '12 at 12:32
@rschwieb: Thanks for the hint. I am looking forward to your finding of your copy! – Hans Stricker Oct 23 '12 at 17:25