Axiom:Bounded Lattice Homomorphism Axioms

Definition

Let $L_1 = \struct{S_1, \vee_1, \wedge_1, \preceq_1}$ be a bounded lattice with greatest element $\top_1$ and smallest element $\bot_1$.

Let $L_2 = \struct{S_2, \vee_2, \wedge_2, \preceq_2}$ be a bounded lattice with greatest element $\top_2$ and smallest element $\bot_2$.

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

Then:

$f$ is a bounded lattice homomorphism

if and only if:

$f$ satisfies:

\((1)\)	$:$		Join morphism property	\(\ds \forall x, y \in S_1:\)		\(\ds \map f {x \vee_1 y} \)	\(\ds = \)	\(\ds \map f x \vee_2 \map f y \)
\((2)\)	$:$		Meet morphism property	\(\ds \forall x, y \in S_1:\)		\(\ds \map f {x \wedge_1 y} \)	\(\ds = \)	\(\ds \map f x \wedge_2 \map f y \)
\((3)\)	$:$		Preserve smallest element			\(\ds \map f {\bot_1} \)	\(\ds = \)	\(\ds \bot_2 \)
\((4)\)	$:$		Preserve greatest element			\(\ds \map f {\top_1} \)	\(\ds = \)	\(\ds \top_2 \)

These criteria are called the bounded lattice homomorphism axioms

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