Definition:Mapping Preserves Supremum/Subset

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