Scott domain

Remarks

Taking 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. Example: the partial functions from $\N$ to $\N$, ordered by extension. The compact elements are the finite partial functions, and every partial function is the union of its finite restrictions. A family with a common extension consists of compatible functions, and their union is its supremum. Non-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 $d$. It is a finite dcpo, but $\set{a, b}$ has the upper bounds $c$ and $d$ and no least one. So it is not bounded complete. Every algebraic lattice is a Scott domain.