Monomorphism_from_Total_Ordering

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 $\left({T, *}\right)$.