  Let $\A$ be a finitely generated semigroup with 0. An $\A$-module over $\fun$
(also called an $\A$--set), is a pointed set $(M,*)$ together with an action of
$\A$. We define and study the Hall algebra $\H_{\A}$ of the category $\C_{\A}$
of finite $\A$--modules. $\H_{\A}$ is shown to be the universal enveloping
algebra of a Lie algebra $\n_{\A}$, called the \emph{Hall Lie algebra} of
$\C_{\A}$. In the case of the $\fm$ - the free monoid on one generator $\fm$,
the Hall algebra (or more precisely the Hall algebra of the subcategory of
nilpotent $\fm$-modules) is isomorphic to Kreimer's Hopf algebra of rooted
forests. This perspective allows us to define two new commutative operations on
rooted forests. We also consider the examples when $\A$ is a quotient of $\fm$
by a congruence, and the monoid $G \cup \{0\}$ for a finite group $G$.
