User:Leigh.Samphier/Topology/Definition:Complete Lattice Homomorphism

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 a complete lattice homomorphism from $L_1$ to $L_2$, denoted $\phi: L_1 \to L_2$

$(1)$	$:$		arbitrary join preserving	$\ds \forall A \subseteq S_1:$		$\ds \map \phi {\sup A} $	$\ds = $	$\ds \sup \set{\map \phi x : x \in A} $
$(2)$	$:$		arbitrary meet preserving	$\ds \forall A \subseteq S_1:$		$\ds \map \phi {\inf A} $	$\ds = $	$\ds \inf \set{\map \phi x : x \in A} $

1982: Peter T. Johnstone: Stone Spaces: Chapter $\text {I}$: Preliminaries: $\S4.3$

\((1)\)	$:$		arbitrary join preserving	\(\ds \forall A \subseteq S_1:\)		\(\ds \map \phi {\sup A} \)	\(\ds = \)	\(\ds \sup \set{\map \phi x : x \in A} \)
\((2)\)	$:$		arbitrary meet preserving	\(\ds \forall A \subseteq S_1:\)		\(\ds \map \phi {\inf A} \)	\(\ds = \)	\(\ds \inf \set{\map \phi x : x \in A} \)