---
tag: QFE3FKNQQ1
kind: concept
name: Endomorphism monoid
msc: [08A35, 20M20]
sources:
  - url: https://en.wikipedia.org/wiki/Endomorphism
---

## Summary

The endomorphism monoid of a [[structure|983GZBQYT4]] is the set of all [[homomorphisms|KBXSZC6BV1]] from the structure to itself, under composition.
Its identity element is the identity map.
It is a [[monoid|0N6R902FG1]], usually written $\End(\mathbf{A})$, and its group of invertible elements is the [[automorphism group|0K1SCPBNKJ]].
For a vector space it is the monoid underlying the [[algebra of endomorphisms|0E3X2G1ZQB]], whose elements are the [[endomorphisms|0J7H0YBFDJ]] of the space.
Every monoid is isomorphic to the endomorphism monoid of some structure.

## Explanations

### L1 human

An endomorphism is a map from a structure to itself that respects its operations.
Doing one endomorphism after another gives again an endomorphism, and doing nothing is an endomorphism.
So the endomorphisms form a monoid, like the maps of a set to itself, but smaller.
For the integers with addition, the endomorphisms are the maps $x \mapsto nx$, and composing them multiplies the factors $n$.