Definition:Arbitrary Join Preserving Mapping

Definition

Let $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ be complete lattices.

Let $\phi: S_1 \to S_2$ be a mapping between the underlying sets of $L_1$ and $L_2$.

Then:

$\phi$ is arbitrary join preserving:

if and only if:

$\forall A \subseteq S_1 : \map \phi {\sup A} = \sup \set{\map \phi a : a \in A}$

Also see

• Definition:Arbitrary Meet Preserving Mapping

• Definition:Finite Join Preserving Mapping

Sources

• 1982: Peter T. Johnstone: Stone Spaces: Chapter II: Introduction to Locales, $\S1.1$ Definition (a)