  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 $S$. We define a $\Bbbk$-module homomorphism $\varphi:\Bbbk
S\to\Bbbk\mathcal{C}$ (where $\Bbbk$ is any unital commutative ring). With some
assumptions, we prove that $\varphi$ is an isomorphism of $\Bbbk$-algebras if
and only 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.
