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 $f:L_1 \to L_2$
- $f$ 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