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  "body": "\n## Summary\n\nThe endomorphism monoid of a [[structure|983GZBQYT4]] is the set of all [[homomorphisms|KBXSZC6BV1]] from the structure to itself, under composition.\nIts identity element is the identity map.\nIt is a [[monoid|0N6R902FG1]], usually written $\\End(\\mathbf{A})$, and its group of invertible elements is the [[automorphism group|0K1SCPBNKJ]].\nFor a vector space it is the monoid underlying the [[algebra of endomorphisms|0E3X2G1ZQB]], whose elements are the [[endomorphisms|0J7H0YBFDJ]] of the space.\nEvery monoid is isomorphic to the endomorphism monoid of some structure.\n\n## Explanations\n\n### L1 human\n\nAn endomorphism is a map from a structure to itself that respects its operations.\nDoing one endomorphism after another gives again an endomorphism, and doing nothing is an endomorphism.\nSo the endomorphisms form a monoid, like the maps of a set to itself, but smaller.\nFor the integers with addition, the endomorphisms are the maps $x \\mapsto nx$, and composing them multiplies the factors $n$.\n",
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