Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping Preserves Directed Suprema

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Theorem

Let $\struct {S, \vee_1, \wedge_1, \preceq_1}$ and $\struct {T, \vee_2, \wedge_2, \preceq_2}$ be complete lattices.

Let $f: S \to T$ be a mapping such that

for all directed sets $\struct {D, \precsim}$ and Moore-Smith sequences $N:D \to S$ in $S$: $\map f {\liminf N} \preceq_2 \map \liminf {f \circ N}$


Then $f$ preserves directed suprema.


Proof

Let $D$ be a directed subset of $S$.

Assume that

$D$ admits a supremum.

Thus by definition of complete lattice:

$f \sqbrk D$ admits a supremum.

Thus by Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Supremum of Image is Mapping at Supremum of Directed Subset:

$\map \sup {f \sqbrk D} = \map f {\sup D}$

$\blacksquare$


Sources