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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\nFor $A \\subseteq P$, let $A^u$ be the set of [[upper bounds|0S60DVMVPN]] of $A$ in $P$ and $A^l$ the set of lower bounds of $A$ in $P$.\nThe *Dedekind–MacNeille completion* of $P$ is the set\n$$\\operatorname{DM}(P) = \\set{A \\subseteq P : (A^u)^l = A},$$\nordered by inclusion, together with the map $\\iota \\colon P \\to \\operatorname{DM}(P)$, $\\iota(x) = \\set{y \\in P : y \\le x}$.\n\n## Remarks\n\n$\\operatorname{DM}(P)$ is a [[complete lattice|D7A0ZGZT5S]], whose meets are intersections, and $\\iota$ is an order embedding: $x \\le y$ if and only if $\\iota(x) \\subseteq \\iota(y)$.\nThe image of $\\iota$ is join-dense and meet-dense: each element of $\\operatorname{DM}(P)$ is a supremum and an infimum of elements of $\\iota(P)$.\nThese properties characterize $\\operatorname{DM}(P)$ up to isomorphism, and it embeds in every complete lattice that contains $P$ as an ordered subset.\nExample: for $P = \\Q$, the sets $A = (A^u)^l$ are $\\varnothing$, $\\Q$, and the sets $\\set{q \\in \\Q : q \\le r}$ for $r \\in \\R$.\nSo $\\operatorname{DM}(\\Q)$ is the [[extended real line|0GSZ7B0CGR]], and its proper nonempty elements are closely related to [[Dedekind cuts|0G3RRM41X9]].\nNon-example: in $\\Q$, the set $A = \\set{q \\in \\Q : q \u003c 0}$ is not in $\\operatorname{DM}(\\Q)$, because $A^u$ is the set of nonnegative rationals and $(A^u)^l = \\set{q : q \\le 0} \\ne A$.\n",
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