Definition:Algebraic Ordered Set

Definition
Let $\struct {S, \preceq}$ be an ordered set.

Then $\struct {S, \preceq}$ is algebraic
 * (for all elements $x$ of $S$: $x^{\mathrm{compact} }$ is directed)

and:
 * $\struct {S, \preceq}$ is up-complete and satisfies the axiom of $K$-approximation:

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