Definition:Mapping Preserves Supremum/Directed
< Definition:Mapping Preserves Supremum(Redirected from Definition:Mapping Preserves Directed 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.
$f$ preserves directed suprema if and only if
- for every directed subset $F$ of $S_1$, $f$ preserves the supremum of $F$.
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 37