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?