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  "body": "\n## Summary\n\nThe ideal completion of a partially ordered set is the set of its order ideals, the nonempty directed down-closed subsets, ordered by inclusion.\nEach element $x$ becomes the principal ideal of all elements below $x$.\nThe ideal completion is a [[directed-complete partial order|7TSXP95QN6]] in which every element is a directed union of principal ideals.\nIt is the free way to add suprema of directed sets.\nIt is a basic construction of [[domain theory|Q2P5D74Q1V]].\n\n## Explanations\n\n### L1 human\n\nAn order ideal is a set of elements that is closed downward and in which any two elements have a common bound inside the set.\nBelow each element $x$ there is the ideal of all elements below $x$, but there may be other ideals that do not have a top.\nAdding these ideals as new elements fills in the missing limits of increasing sequences.\nFor the natural numbers, the only new ideal is $\\N$ itself, so the completion is $\\N$ with one point $\\infty$ added on top.\n",
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