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  "body": "\n## Definition\n\nLet $D$ be a [[directed-complete partial order|7TSXP95QN6]].\nIt is a *Scott domain* if:\n\n1. $D$ is algebraic: for every $x \\in D$, the set of [[compact elements|9F7TYES4KE]] $c \\le x$ is [[directed|D9RY4667K6]] and has [[supremum|0R3H5PANRQ]] $x$;\n2. $D$ is bounded complete: every subset of $D$ that has an [[upper bound|0S60DVMVPN]] in $D$ has a supremum in $D$.\n\n## Remarks\n\nTaking the empty subset in clause 2 gives a least element when $D$ is nonempty; some authors require $D$ nonempty, and some also require countably many compact elements.\nExample: the partial functions from $\\N$ to $\\N$, ordered by extension.\nThe compact elements are the finite partial functions, and every partial function is the union of its finite restrictions.\nA family with a common extension consists of compatible functions, and their union is its supremum.\nNon-example: take the poset with elements $\\bot, a, b, c, d$, where $\\bot$ is below all, and $a, b$ are each below both $c$ and $d$.\nIt is a finite dcpo, but $\\set{a, b}$ has the upper bounds $c$ and $d$ and no least one.\nSo it is not bounded complete.\nEvery algebraic lattice is a Scott domain.\n",
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