Semilattice Homomorphism is Order-Preserving

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

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

Let $\preceq_1$ be the ordering on $S$ defined by:
 * $a \preceq_1 b \iff \paren {a \circ b} = b$

Let $\preceq_2$ be the ordering on $T$ defined by:
 * $x \preceq_2 y \iff \paren {x * y} = y$

Then:


 * $\phi: \struct {S, \preceq_1} \to \struct {T, \preceq_2}$ is order-preserving.