Definition:Directed Suprema Inheriting

Definition
Let $L = \struct {X, \preceq}$ be an ordered set.

Let $S = \struct {Y, \precsim}$ be an ordered subset of $L$.

Then $S$ inherits directed suprema
 * for all directed subsets $A$ of $Y$: if $A$ admits a supremum in $L$, then $\sup_L A \in Y$

Also see

 * Supremum in Ordered Subset