  Let $S$ be a reduced $E$-Fountain semigroup. If $S$ satisfies the congruence
condition, there is a natural construction of a category $\mathcal{C}$
associated with $S$. Under a few conditions, we prove that there is an
isomorphism of algebras $\Bbbk S\simeq\Bbbk\mathcal{C}$ (where $\Bbbk$ is any
unital commutative ring) if some weak form of the right ample identity holds in
$S$. This gives a unified generalization for a result of the author on right
restriction $E$-Ehresmann semigroups and a result of Margolis and Steinberg on
the Catalan monoid.
