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  "body": "\n## Definition\n\nLet $(L, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\nIt is a *complete lattice* if every subset $S \\subseteq L$ has a [[supremum|0R3H5PANRQ]] $\\bigvee S$ and an [[infimum|0QF1MZ9FRG]] $\\bigwedge S$ in $L$.\n\n## Remarks\n\nIt is enough to ask that every subset has a supremum: then $\\bigwedge S$ is the supremum of the set of lower bounds of $S$.\nTaking $S = \\varnothing$ shows that a complete lattice has a least element $\\bigvee \\varnothing$ and a greatest element $\\bigwedge \\varnothing$; in particular it is nonempty.\nEvery complete lattice is a [[lattice|WH4SFAACAS]], and every finite nonempty lattice is complete.\nExample: for a set $X$, the power set of $X$ ordered by inclusion is complete, with $\\bigvee S = \\bigcup S$ and $\\bigwedge S = \\bigcap S$, where $\\bigcap \\varnothing = X$.\nNon-example: $\\Q$ with its usual order is a lattice but not complete, because $\\set{q \\in \\Q : q^2 \u003c 2}$ has no supremum in $\\Q$.\n",
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