Endomorphism monoid

Summary

The endomorphism monoid of a structure is the set of all homomorphisms from the structure to itself, under composition. Its identity element is the identity map. It is a monoid, usually written $\End(\mathbf{A})$, and its group of invertible elements is the automorphism group. For a vector space it is the monoid underlying the algebra of endomorphisms, whose elements are the endomorphisms of the space. Every monoid is isomorphic to the endomorphism monoid of some structure.