\begin{definition}[Definition:Mapping Preserves Infimum/Subset]
{{:Definition:Mapping Preserves Infimum}}
Let $F$ be a subset of $S_1$.
$f$ '''preserves infimum of''' $F$ {{iff}}
:$F$ admits a infimum in $\left({S_1, \preceq_1}\right)$ implies
:: $f^\to \left({F}\right)$ admits an infimum in $\left({S_2, \preceq_2}\right)$ and $\inf \left({f^\to \left({F}\right)}\right) = f \left({\inf F}\right)$
\end{definition}