Ordered Semigroup Isomorphism is Surjective Monomorphism

Theorem
Let $\left({S, \circ, \preceq}\right)$ and $\left({T, *, \preccurlyeq}\right)$ be ordered semigroups.

Let $\phi: \left({S, \circ, \preceq}\right) \to \left({T, *, \preccurlyeq}\right)$ be a mapping.

Then $\phi$ is an ordered semigroup isomorphism iff:
 * $(1): \quad \phi$ is an ordered semigroup monomorphism
 * $(2): \quad \phi$ is a surjection.

Necessary Condition
Let $\phi: \left({S, \circ, \preceq}\right) \to \left({T, *, \preccurlyeq}\right)$ be an ordered semigroup isomorphism.

Then by definition:


 * $\phi$ is a semigroup isomorphism from the semigroup $\left({S, \circ}\right)$ to the semigroup $\left({T, *}\right)$


 * $\phi$ is an order isomorphism from the ordered set $\left({S, \preceq}\right)$ to the ordered set $\left({T, \preccurlyeq}\right)$.

A semigroup isomorphism is by definition:
 * A semigroup homomorphism

which is:
 * A monomorphism and an epimorphism.

From Order Isomorphism is Surjective Order Embedding, an order isomorphism is an order embedding which is also a surjection.

Putting this all together, we see that an ordered semigroup isomorphism is:
 * A monomorphism
 * An order embedding
 * A surjection.

An ordered semigroup monomorphism is by definition:
 * A monomorphism

which is also
 * An order embedding

Hence $\phi$ is:
 * An ordered semigroup monomorphism
 * A surjection.

Sufficient Condition
Let $\phi$ be:
 * An ordered semigroup monomorphism
 * A surjection.

By definition, that means $\phi$ be:
 * A monomorphism
 * An order embedding
 * A surjection.

From Order Isomorphism is Surjective Order Embedding, an order isomorphism is an order embedding which is also a surjection.

A semigroup isomorphism is by definition:
 * A semigroup homomorphism

which is:
 * A semigroup monomorphism and an semigroup epimorphism.

Thus a semigroup monomorphism which is also a surjection is a semigroup isomorphism.

So $\phi$ is:
 * A semigroup isomorphism from the semigroup $\left({S, \circ}\right)$ to the semigroup $\left({T, *}\right)$


 * An order isomorphism from the ordered set $\left({S, \preceq}\right)$ to the ordered set $\left({T, \preccurlyeq}\right)$.