Definition:Mapping Preserves Supremum/Subset

Definition
Let $F$ be a subset of $S_1$.

$f$ preserves supremum of $F$
 * $F$ admits a supremum in $\left({S_1, \preceq_1}\right)$ implies
 * $f^\to \left({F}\right)$ admits a supremum in $\left({S_2, \preceq_2}\right)$ and $\sup \left({f^\to\left({F}\right)}\right) = f \left({\sup F}\right)$