Suprema Preserving Mapping on Ideals Preserves Directed Suprema

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