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  "body": "\n## Statement\n\nLet $(L, \\le)$ be a [[complete lattice|D7A0ZGZT5S]] and let $f \\colon L \\to L$ be a [[monotone|0A7KV2WB7A]] map, so that $x \\le y$ implies $f(x) \\le f(y)$.\nLet $\\operatorname{Fix}(f) = \\set{x \\in L : f(x) = x}$.\nThen $\\operatorname{Fix}(f)$, with the order of $L$, is a complete lattice.\nIn particular $f$ has a [[least fixed point|DBD64TH52D]] and a greatest fixed point.\nThey are given by\n$$\\operatorname{lfp}(f) = \\inf \\set{x \\in L : f(x) \\le x}, \\qquad \\operatorname{gfp}(f) = \\sup \\set{x \\in L : x \\le f(x)},$$\nwhere $\\inf$ and $\\sup$ are the [[infimum|0QF1MZ9FRG]] and the [[supremum|0R3H5PANRQ]] in $L$.\n\n## Remarks\n\nProof idea for the least fixed point: let $D = \\set{x : f(x) \\le x}$ and $m = \\inf D$.\nFor $x \\in D$, $m \\le x$ gives $f(m) \\le f(x) \\le x$, so $f(m)$ is a lower bound of $D$ and $f(m) \\le m$.\nThen $f(f(m)) \\le f(m)$, so $f(m) \\in D$ and $m \\le f(m)$; hence $f(m) = m$, and every fixed point lies in $D$, so $m$ is the least one.\nLet $W$ be a set of fixed points with supremum $a$ in $L$.\nThe map $f$ sends the complete lattice $\\set{x \\in L : a \\le x}$ into itself.\nIts least fixed point there is the join of $W$ in $\\operatorname{Fix}(f)$, and meets are handled dually.\nThe join of a set of fixed points in $\\operatorname{Fix}(f)$ need not be its join in $L$.\nExample: on the power set of $\\N$, the map $f(S) = \\set{0} \\cup \\set{n + 2 : n \\in S}$ is monotone, and its least fixed point is the set of even numbers.\nMonotonicity is needed: on the two-element lattice $\\set{0, 1}$, the map that swaps $0$ and $1$ has no fixed point.\nCompleteness is needed too: on the lattice $\\Z$ the monotone map $x \\mapsto x + 1$ has no fixed point.\n\n## Explanations\n\n### L1 human\n\nIf a rule moves the points of a complete lattice and never reverses the order, then some points stay put.\nThere is a smallest such point and a largest one, and the fixed points even form a complete lattice themselves.\nThe smallest fixed point is the meet of all points that the rule pushes downward.\nThis is why inductive definitions, such as \"the smallest set containing $0$ and closed under adding $2$\", always make sense.\n\n### L2 human\n\nThe theorem is the basic existence result for fixed points in order theory.\nIn logic and computer science it gives the semantics of inductive and coinductive definitions as least and greatest fixed points.\nIt also underlies the proof of the Cantor–Schröder–Bernstein theorem through a monotone map on a power set.\nWhen $f$ is moreover Scott-continuous, the [[Kleene fixed-point theorem|DDAHG7QRY8]] computes the least fixed point as the supremum of the iterates $f^n(\\bot)$.\nA converse holds: a lattice in which every monotone self-map has a fixed point is complete.\n",
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