Definition:Semilattice Homomorphism

Definition
Let $\struct {S, \circ}$ and $\struct {T, *}$ be semilattices.

Let $\phi: S \to T$ be a mapping such that $\circ$ has the morphism property under $\phi$.

That is, $\forall a, b \in S$:
 * $\map \phi {a \circ b} = \map \phi a * \map \phi b$

Then $\phi: \struct {S, \circ} \to \struct {T, *}$ is a semilattice homomorphism.