Definition:Mapping Preserves Supremum
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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 $F$.