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  "body": "\n## Definition\n\nLet $\\mathbf{A}$ be a [[structure|983GZBQYT4]] with domain $A$.\nAn *endomorphism* of $\\mathbf{A}$ is a [[homomorphism|KBXSZC6BV1]] from $\\mathbf{A}$ to $\\mathbf{A}$.\nLet $E$ be the set of endomorphisms of $\\mathbf{A}$, let $\\circ$ be [[composition|0THBK9V825]] and let $\\id_A$ be the identity map of $A$.\nThe *endomorphism monoid* of $\\mathbf{A}$ is the [[monoid|0N6R902FG1]] $\\End(\\mathbf{A}) = (E, \\circ, \\id_A)$.\n\n## Remarks\n\nThe composite of two homomorphisms is a homomorphism and $\\id_A$ is a homomorphism, so $E$ is closed under $\\circ$ and contains $\\id_A$; composition is associative.\n\nExample: let $\\mathbf{A} = (\\Z, +, 0)$.\nEach endomorphism $h$ satisfies $h(x) = x\\,h(1)$ for all $x \\in \\Z$, by induction on $x \\ge 0$ and $h(-x) = -h(x)$.\nSo $h \\mapsto h(1)$ is an isomorphism from $\\End(\\mathbf{A})$ to the monoid $(\\Z, \\cdot, 1)$, since $(g \\circ h)(1) = g(1)\\,h(1)$.\n\nNon-example: the map $x \\mapsto x + 1$ lies in the monoid of all maps $\\Z \\to \\Z$ but not in $\\End(\\Z, +, 0)$, because it sends $0$ to $1$.\n\nThe invertible elements of $\\End(\\mathbf{A})$ are the automorphisms of $\\mathbf{A}$.\n",
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