\begin{definition}[Definition:Big Monoid Ring]
Let $\struct {M, \cdot}$ be a divisor-finite monoid.
Let $\struct {R, +, \times}$ be an additive semiring.
The '''big monoid ring''' of $R$ over $M$ is the ringoid $\struct {R^M, +, *}$ where:
:$R^M$ be the set of all mappings $M \to R$
:$+$ is the pointwise operation induced by $+$
:$*$ denotes convolution of mappings
\end{definition}