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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\nIt is a *directed-complete partial order*, or *dcpo*, if every [[directed subset|D9RY4667K6]] $D \\subseteq P$ has a [[supremum|0R3H5PANRQ]] $\\bigvee D$ in $P$.\nHere $D$ is directed if it is nonempty and every two elements of $D$ have an upper bound in $D$.\n\n## Remarks\n\nA dcpo with a least element $\\bot$ is called pointed, and is often simply called a cpo.\nExample: the set of partial functions from $\\N$ to $\\N$, ordered by inclusion of graphs.\nIf $D$ is a directed family, any two members agree where both are defined, so $\\bigcup D$ is again a partial function and is the supremum.\nNon-example: $\\N$ with its usual order is not a dcpo, because $\\N$ itself is directed and has no upper bound.\nEvery finite poset is a dcpo, since a finite directed set has a greatest element.\nEquivalently, by a theorem that uses the axiom of choice, a poset is a dcpo if and only if every nonempty chain has a supremum.\n",
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