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  "body": "\n## Summary\n\nAn omega-complete partial order, or $\\omega$-cpo, is a partially ordered set in which every increasing sequence has a supremum.\nIt is a weaker form of a [[directed-complete partial order|7TSXP95QN6]] that is enough for the fixed-point semantics of recursive programs.\nLet the $\\omega$-cpo have a least element $\\bot$.\nThen a monotone map that preserves suprema of increasing sequences has a least fixed point, the supremum of its iterates on $\\bot$.\n\n## Explanations\n\n### L1 human\n\nA computation that runs step by step produces a sequence of better and better approximations $x_0 \\le x_1 \\le x_2 \\le \\cdots$.\nIn an $\\omega$-cpo such a sequence always has a limit, its least upper bound.\nFor example, in $\\N \\cup \\set{\\infty}$ every increasing sequence has a supremum, either its largest value or $\\infty$.\nOnly countable increasing sequences are required, which makes the notion easy to use in programming language semantics.\n",
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