Definition:Convolution of Mappings on Divisor-Finite Monoid

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

Let $(R, +, \times)$ be an nonassociative 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:
 * $(f*g)(m) = \displaystyle \sum_{xy = m}f(x)\times g(y)$

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

Also see

 * Definition:Monoid Ring
 * Definition:Big Monoid Ring

Examples

 * Definition:Dirichlet Convolution