Definition:Bounded Lattice Homomorphism

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 $f: \struct{S_1, \vee_1, \wedge_1, \preceq_1} \to \struct{S_2, \vee_2, \wedge_2, \preceq_2}$ be a lattice homomorphism.

Then:

$f$ is a bounded lattice homomorphism from $L_1$ to $L_2$, denoted $f:L_1 \to L_2$

if and only if:

$f$ preserves the identities:

\((3)\)	$:$		Preserve smallest element			\(\ds \map f {\bot_1} \)	\(\ds = \)	\(\ds \bot_2 \)
\((4)\)	$:$		Preserve greatest element			\(\ds \map f {\top_1} \)	\(\ds = \)	\(\ds \top_2 \)

That is, $f$ is a bounded lattice homomorphism from $L_1$ to $L_2$ if and only if

the bounded lattice homomorphism axioms are satisfied:

\((1)\)	$:$		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			\(\ds \map f {\top_1} \)	\(\ds = \)	\(\ds \top_2 \)

Also known as

Some authors insist that a lattice have identity elements, and so refer to a bounded lattice homomorphism simply as a lattice homomorphism.

Other authors denote the smallest element and greatest element of a bounded lattice as $0$ and $1$ respectively, and so refer to a bounded lattice homomorphism as a $\set{0, 1}$-lattice homomorphism

Sources

• 1971: George A. Grätzer: Lattice Theory: Chapter $1$: First Concepts, $\S 6$: Special Elements