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  "body": "\n## Statement\n\nLet $L$ be a [[lattice|WH4SFAACAS]].\nThen $L$ is a [[complete lattice|D7A0ZGZT5S]] if and only if every [[monotone function|0A7KV2WB7A]] $f \\colon L \\to L$ has a fixed point, that is, an element $x \\in L$ with $f(x) = x$.\n\n## Remarks\n\nOne direction is the [[Knaster–Tarski theorem|RHXP3HA64M]]: in a complete lattice, every monotone map has a fixed point, and even a complete lattice of fixed points.\nFor the converse, if $L$ is not complete, Davis constructs a monotone map from $L$ to $L$ without fixed points.\nThe lattice $\\Z$ with its usual order is not complete, and the map $n \\mapsto n + 1$ is monotone without fixed points.\nThe result was published by Anne C. Davis in 1955.\n\n## Explanations\n\n### L1 human\n\nThe Knaster–Tarski theorem says that in a complete lattice every order-preserving map has a fixed point.\nDavis showed that this property characterizes complete lattices among all lattices.\nSo the existence of fixed points is not a lucky extra: it is exactly completeness.\n\n### L2 human\n\nThe result gives a test for completeness that does not mention suprema at all.\nTo show that a lattice is not complete, it is enough to exhibit one monotone map without a fixed point.\nA shift $n \\mapsto n + 1$ on $\\Z$ is such a map.\n",
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