# Definition:Mapping Preserves Supremum

## 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.

### Mapping Preserves Supremum on Subset

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

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

$F$ admits a supremum in $\struct {S_1, \preceq_1}$ implies:
$\map {f^\to} F$ admits a supremum in $\struct {S_2, \preceq_2}$ and $\map \sup {\map {f^\to} F} = \map f {\sup F}$

### Mapping Preserves All Suprema

$f$ preserves all suprema if and only if

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

### Mapping Preserves Join

$f$ preserves join if and only if

for every pair of elements $x, y$ of $S_1$, $f$ preserves the supremum of $\left\{ {x, y}\right\}$.

### Mapping Preserves Finite Suprema

$f$ preserves finite suprema if and only if

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

### Mapping Preserves Directed Suprema

$f$ preserves directed suprema if and only if

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