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\homework{Factorizations}

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\begin{exercise}
Prove that the category of ranked preorder semigroups has a particular $(\alg{E}, \alg{M})$-factorization structure.
A ranked preordered semigroup is a preordered semigroup $\langle M, \leq, * \rangle$ along with a function $r : M \mto \R$ such that $r(m_1 * m_2) = r(m_1) + r(m_2)$, and $m \leq m'$ implies $r(m) \leq r(m')$.
A morphism of ranked preorder semigroups $\mo{f} : \alg{M}_1 \mto \alg{M}_2$ is a morphism of preorded semigroups with the additional property that $r_2(f(m_1)) = r_1(m_1)$.

This category has an underlying functor to $\cat{Prost}$, which we use to define the particular $\alg{E}$ and $\alg{M}$ we want a factorization structure for.
A morphism belongs to $\alg{E}$ iff its underlying morphism is an epimorphism in $\cat{Prost}$.
A source belongs to $\alg{M}$ iff its underlying source is an initial monosource in $\cat{Prost}$.

Show how to construct the category of ranked preorder semigroups from $\cat{Prost}$ using the techniques from class.
I am happy if you give the correct construction, even if you do not do the detailed work of proving the result is isomorphic to the category of ranked preorder semigroups.
\end{exercise}

\begin{remark}
I made up the term \emph{ranked}.
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