Definition:Algebraic Ordered Set

Definition
Let $\left({S, \preceq}\right)$ be an ordered set.

Then $\left({S, \preceq}\right)$ is algebraic
 * (for all elements $x$ of $S$: $x^{\mathrm{compact} }$ is directed) and
 * $\left({S, \preceq}\right)$ is up-complete and satisfies axiom of K-approximation

where $x^{\mathrm{compact} }$ denotes the compact closure of $x$.