Suprema Preserving Mapping on Ideals Preserves Directed Suprema
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Theorem
Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be ordered sets.
Let $f: S \to T$ be a mapping.
Let every filter $F$ in $\left({S, \preceq}\right)$, $f$ preserve the infimum on $F$.
Then $f$ preserves directed suprema.
Proof
This follows by mutatis mutandis of the proof of Infima Preserving Mapping on Filters Preserves Filtered Infima.
$\blacksquare$
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_0:73