Monomorphism from Total Ordering

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Theorem

Let the following conditions hold:

$(1): \quad$ Let $\left({S, \circ, \preceq}\right)$ and $\left({T, *, \preccurlyeq}\right)$ be ordered semigroups.

$(2): \quad$ Let $\phi: S \to T$ be a mapping.

$(3): \quad$ Let $\preceq$ be a total ordering on $S$.

Then $\phi \left({S, \circ, \preceq}\right) \to \left({T, *, \preccurlyeq}\right)$ is a (structure) monomorphism iff:

$(1): \quad \phi$ is strictly increasing from $\left({S, \preceq}\right)$ into $\left({T, \preccurlyeq}\right)$;

$(2): \quad \phi$ is a homomorphism from $\left({S, \circ}\right)$ into $\left({T, *}\right)$.

This follows:

$(1): \quad$ As a direct consequence of Order Monomorphism iff Strictly Increasing;

$(2): \quad$ From the definition of monomorphism as a homomorphism which is an injection.

$\blacksquare$