How to get a Hopf algebra from a monoid?

Question

Let $k$ be a field. Let $S$ be a monoid with netural element $e$.

1. Suppose that all $s \in S$ have only finitely many factoriztions $s=ab$, where $a,b \in S$. Then the free k-module $k[S]$ has a coalgebra structure determined by $$\Delta(s)=\sum \limits_{ab=s}a \otimes b \,\, \text{and} \,\, \varepsilon(s)=\delta_{s, e}$$
2. Define $\Delta(s)= s \otimes s$ and $\varepsilon(s)=1$ for all $s \in S$. Then $k[S]$ is cocommutative bialgebra.

I want to know whether there are other definitions of $\Delta$ on $S$ so that we can get a coalgebra or bialgebra, moreover Hopf algebra structrue? (Maybe for some special semigroups?) Thank you very much.