Definition:Convolution of Mappings on Divisor-Finite Monoid

Definition
Let $\left({M, \cdot}\right)$ be a divisor-finite monoid.

Let $\left({R, +, \times}\right)$ 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:
 * $ \left({f * g}\right) \left({m}\right) = \displaystyle \sum_{x y \mathop = m} f \left({x}\right) \times g \left({y}\right)$

where the summation is over the finite set $\left\{ {\left({x, y}\right) \in M^2: x y = m}\right\}$.

Also see

 * Definition:Monoid Ring
 * Definition:Big Monoid Ring

Examples

 * Definition:Dirichlet Convolution