Lattice Homomorphism is Both Meet and Join Semilattice Homomorphism

This article needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code.

Theorem

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 $\phi: L_1 \to L_2$ be a lattice homomorphsm between $L_1$ and $L_2$.

Then:

• $\phi: \struct{S_1, \vee_1, \preceq_1} \to \struct{S_2, \vee_2, \preceq_2}$
• $\phi: \struct{S_1, \wedge_1, \preceq_1} \to \struct{S_2, \wedge_2, \preceq_2}$

are semilattice homomorphisms

Proof

This follows immediately from:

• Definition:Lattice (Order Theory)
• Definition:Semilattice
• Definition:Lattice Homomorphism
• Definition:Semilattice Homomorphism

$\blacksquare$