Order Monomorphism into Image is Isomorphism

Theorem

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be posets.

Let $S'$ be the image of a mapping $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$.

Then $\phi$ is an order monomorphism from $\left({S, \preceq_1}\right)$ into $\left({T, \preceq_2}\right)$ iff $\phi$ is an order isomorphism from $\left({S, \preceq_1}\right)$ into $\left({S', \preceq_2 \restriction_{S'}}\right)$.

Proof

• Let $\phi$ be an order monomorphism from $\left({S, \preceq_1}\right)$ into $\left({T, \preceq_2}\right)$.

Then $\phi$ is an injection into $\left({T, \preceq_2}\right)$ by definition.

From Surjection iff Image equals Codomain, any mapping from a set to the image of that mapping is a surjection.

Thus the surjective restriction of $\phi$ onto $S'$ is an order monomorphism which is also a surjection.

Hence the result from Order Isomorphism is Surjective Order Monomorphism.

$\blacksquare$