\begin{definition}[Definition:Mapping Preserves Supremum/Subset]
{{:Definition:Mapping Preserves Supremum}}
Let $F$ be a subset of $S_1$.
$f$ '''preserves supremum of''' $F$ {{iff}}
:$F$ admits a supremum in $\struct {S_1, \preceq_1}$ implies:
::$\map {f^\to} F$ admits a supremum in $\struct {S_2, \preceq_2}$ and $\map \sup {\map {f^\to} F} = \map f {\sup F}$
\end{definition}