Mapping Preserves Finite and Directed Suprema

Theorem

Let $\left({S_1, \preceq_1}\right)$, $\left({S_2, \preceq_2}\right)$ be join semilattices.

Let $f: S_1 \to S_2$ be a mapping.

Let $f$ preserve finite suprema and preserve directed suprema.

Then $f$ also preserves all suprema

Proof

This follows by mutatis mutandis of the proof of Mapping Preserves Finite and 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_0:74