Axiom:Lattice Homomorphism Axioms

Definition

Let $L_1 = \struct{S_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct{S_2, \vee_2, \wedge_2, \preceq_2}$ be 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 lattice homomorphism

if and only if:

$\phi$ satisfies:

\((1)\)	$:$		join morphism property	\(\ds \forall x, y \in S_1:\)		\(\ds \map \phi {x \vee_1 y} \)	\(\ds = \)	\(\ds \map \phi x \vee_2 \map \phi y \)
\((2)\)	$:$		meet morphism property	\(\ds \forall x, y \in S_1:\)		\(\ds \map \phi {x \wedge_1 y} \)	\(\ds = \)	\(\ds \map \phi x \wedge_2 \map \phi y \)

These criteria are called the lattice homomorphism axioms

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