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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\nAn *order ideal* of $P$ is a subset $I \\subseteq P$ such that:\n\n1. $I$ is nonempty;\n2. $I$ is a down-set: if $x \\in I$ and $y \\le x$, then $y \\in I$;\n3. $I$ is [[directed|D9RY4667K6]]: for all $x, y \\in I$ there is $z \\in I$ with $x \\le z$ and $y \\le z$.\n\nThe *ideal completion* of $P$ is the set $\\operatorname{Idl}(P)$ of all order ideals of $P$, ordered by inclusion.\nIt comes with the map $x \\mapsto {\\downarrow} x = \\set{y \\in P : y \\le x}$ from $P$ to $\\operatorname{Idl}(P)$.\n\n## Remarks\n\n$\\operatorname{Idl}(P)$ is a [[directed-complete partial order|7TSXP95QN6]]: the supremum of a directed family of ideals is its union.\nThe map $x \\mapsto {\\downarrow} x$ is an order embedding, and the principal ideals ${\\downarrow} x$ are exactly the [[compact elements|9F7TYES4KE]] of $\\operatorname{Idl}(P)$, so $\\operatorname{Idl}(P)$ is an algebraic dcpo.\nWhen $P$ is a lattice, the order ideals are its [[lattice ideals|5GW2T2AM1A]] and $\\operatorname{Idl}(P)$ is an algebraic lattice if $P$ has a least element.\nExample: for $P = \\N$, the ideals are the sets $\\set{0, \\ldots, n}$ and $\\N$ itself, so $\\operatorname{Idl}(\\N) \\cong \\N \\cup \\set{\\infty}$.\nNon-example: in $\\N$, the set $\\set{0, 2}$ is not an ideal, because $1 \\le 2$ but $1 \\notin \\set{0, 2}$.\n",
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