Definition:Mapping Preserves Infimum

Definition

Let $\left({S_1, \preceq_1}\right)$, $\left({S_2, \preceq_2}\right)$ be ordered sets.

Let $f: S_1 \to S_2$ be a mapping.

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

$f$ preserves infimum of $F$ if and only if

$F$ admits a infimum in $\struct {S_1, \preceq_1}$ implies

$\map {f^\to} F$ admits an infimum in $\struct {S_2, \preceq_2}$ and $\map \inf {\map {f^\to} F} = \map f {\inf F}$

$f$ preserves all infima if and only if

for every subset $F$ of $S_1$, $f$ preserves the infimum of $F$.

$f$ preserves meet if and only if

for every pair of elements $x, y$ of $S_1$, $f$ preserves the infimum of $\set {x, y}$.

$f$ preserves finite infima if and only if

$f$ preserves filtered infima if and only if