\begin{definition}[Definition:Convolution of Mappings on Divisor-Finite Monoid]
Let $\struct {M, \cdot}$ be a divisor-finite monoid.
Let $\struct {R, +, \times}$ be a non-associative ring.
Let $f, g : M \to R$ be mappings.
The '''convolution''' of $f$ and $g$ is the mapping $f * g: M \to R$ defined as:
:$\forall m \in M: \map {\paren {f * g} } m := \ds \sum_{x y \mathop = m} \map f x \times \map g y$
where the summation is over the finite set $\set {\tuple {x, y} \in M^2: x y = m}$.
\end{definition}