Monomorphism from Total Ordering

Theorem
Let the following conditions hold:


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

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

Proof
This follows:
 * 1) As a direct consequence of Order Monomorphism if Increasing;
 * 2) From the definition of monomorphism as a homomorphism which is an injection.