# 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.