Ordered Semigroup Isomorphism is Surjective Monomorphism

Theorem
Let $\struct {S, \circ, \preceq}$ and $\struct {T, *, \preccurlyeq}$ be ordered semigroups.

Let $\phi: \struct {S, \circ, \preceq} \to \struct {T, *, \preccurlyeq}$ be a mapping.

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

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

Then by definition:


 * $\phi$ is a semigroup isomorphism from the semigroup $\struct {S, \circ}$ to the semigroup $\struct {T, *}$


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

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 $\struct {S, \circ}$ to the semigroup $\struct {T, *}$


 * An order isomorphism from the ordered set $\struct {S, \preceq}$ to the ordered set $\struct {T, \preccurlyeq}$.