Definition:Mapping Preserves Supremum/Subset
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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.
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 $\map \sup {\map {f^\to} F} = \map f {\sup F}$
Sources
- 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices
- Mizar article WAYBEL_0:def 30