Definition:Lattice Homomorphism
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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 from $L_1$ to $L_2$, denoted $\phi:L_1 \to L_2$
- both $\vee_1, \wedge_1$ have the morphism property under $\phi$
That is, $\phi$ satisfies the lattice homomorphism axioms:
\((1)\) | $:$ | 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 | \(\ds \forall x, y \in S_1:\) | \(\ds \map \phi {x \wedge_1 y} \) | \(\ds = \) | \(\ds \map \phi x \wedge_2 \map \phi y \) |
Also see
- Results about Lattice Homomorphisms can be found here.
Sources
- 1967: Saunders Mac Lane and Garrett Birkhoff: Algebra: Chapter XIV Lattices : $\S 3$ Sublattices and Products of Lattices
- 1971: George A. Grätzer: Lattice Theory: Chapter $1$: First Concepts, $\S 3$: Some Algebraic Concepts