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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]] and let $f \\colon P \\to P$ be a map.\nA *fixed point* of $f$ is an element $x \\in P$ with $f(x) = x$.\nA *least fixed point* of $f$ is a fixed point $x$ of $f$ such that $x \\le y$ for every fixed point $y$ of $f$.\nIn other words, it is the [[least element|0AANKKNHS4]] of the set $\\set{x \\in P : f(x) = x}$.\n\n## Remarks\n\nA least fixed point is unique when it exists, by antisymmetry of $\\le$.\nIt is written $\\operatorname{lfp}(f)$ or $\\mu f$.\nThe *greatest fixed point* is defined dually and written $\\operatorname{gfp}(f)$ or $\\nu f$.\nExample: for $f \\colon \\R \\to \\R$, $f(x) = x^2$, the fixed points are the solutions of $x^2 = x$, namely $0$ and $1$.\nSince $0 \\le 1$, the least fixed point is $0$.\nNon-example: for $g \\colon \\R \\to \\R$, $g(x) = x$, every real number is a fixed point, and $\\R$ has no least element, so $g$ has no least fixed point.\nThe map $x \\mapsto x + 1$ on $\\R$ has no fixed point at all.\n",
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