\begin{definition}[Definition:Directed Suprema Inheriting]
Let $L = \left({X, \preceq}\right)$ be an ordered set.
Let $S = \left({Y, \precsim}\right)$ be an ordered subset of $L$.
Then $S$ '''inherits directed suprema''' {{iff}}
:for all directed subsets $A$ of $Y$: if $A$ admits a supremum in $L$, then $\sup_L A \in Y$
\end{definition}