Lattice Homomorphism is Order-Preserving

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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:

$\phi: \struct{S_1, \preceq_1} \to \struct{S_2, \preceq_2}$ is order-preserving

Proof

From Lattice Homomorphism is Both Meet and Join Semilattice Homomorphism:

$\phi: \struct{S_1, \vee_1, \preceq_1} \to \struct{S_2, \vee_2, \preceq_2}$ is a semilattice homomorphism

From Semilattice Homomorphism is Order-Preserving:

$\phi: \struct{S_1, \preceq_1} \to \struct{S_2, \preceq_2}$ is order-preserving

$\blacksquare$