Monomorphism from Total Ordering

Theorem
Let the following conditions hold:


 * $(1): \quad$ Let $\struct {S, \circ, \preceq}$ and $\struct {T, *, \preccurlyeq}$ be ordered semigroups.
 * $(2): \quad$ Let $\phi: S \to T$ be a mapping.
 * $(3): \quad$ Let $\preceq$ be a total ordering on $S$.

Then $\phi \struct {S, \circ, \preceq} \to \struct {T, *, \preccurlyeq}$ is a (structure) monomorphism :
 * $(1): \quad \phi$ is strictly increasing from $\struct {S, \preceq}$ into $\struct {T, \preccurlyeq}$
 * $(2): \quad \phi$ is a homomorphism from $\struct {S, \circ}$ into $\struct {T, *}$.

Proof
This follows:
 * $(1): \quad$ As a direct consequence of Mapping from Totally Ordered Set is Order Embedding iff Strictly Increasing
 * $(2): \quad$ From the definition of monomorphism as a homomorphism which is an injection.