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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|011C29Q1S1]] and let $S \\subseteq P$.\nThe *down-closure* of $S$ is the set\n$$\\downarrow S = \\set{x \\in P : x \\le s \\text{ for some } s \\in S}.$$\nFor $x \\in P$, the set $\\downarrow x = \\downarrow \\set{x} = \\set{y \\in P : y \\le x}$ is the *principal down-set* of $x$.\n\n## Remarks\n\nIn the subsets of $\\set{1, 2, 3}$ ordered by inclusion, $\\downarrow \\set{\\set{1, 2}, \\set{3}} = \\set{\\emptyset, \\set{1}, \\set{2}, \\set{1, 2}, \\set{3}}$.\nEach listed set is contained in $\\set{1, 2}$ or in $\\set{3}$, and no other subset is.\nThe set $\\downarrow S$ is a [[lower set|BWYFKCWRW5]] that contains $S$, and it lies inside each lower set that contains $S$.\nSo $S$ is a lower set if and only if $\\downarrow S = S$.\nThe map $S \\mapsto \\downarrow S$ is a [[closure operator|17FVAN6HXX]] on the subsets of $P$.\nDually, the up-closure is $\\uparrow S = \\set{x \\in P : s \\le x \\text{ for some } s \\in S}$.\n\n## Explanations\n\n### L1 human\n\nThe down-closure fills in everything below the given elements.\nIt turns any subset into a lower set in the most economical way.\n",
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  "name": "Down-closure",
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