Definition:Convolution of Mappings on Divisor-Finite Monoid

Definition
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:
 * $\forall m \in M: \map {\paren {f * g} } m := \displaystyle \sum_{x y \mathop = m} \map f x \times \map g y$

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

Also see

 * Definition:Monoid Ring
 * Definition:Big Monoid Ring

Examples

 * Definition:Dirichlet Convolution