# Definition:Directed Suprema Inheriting

## Definition

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 if and only if

for all directed subsets $A$ of $Y$: if $A$ admits a supremum in $L$, then $\sup_L A \in Y$