In integral transforms, convolution is defined by
$$ (f*g)(t)=\int_{-\infty}^\infty f(\tau)g(t-\tau)\mathrm d\tau $$
satisfying the commutative, associative property, and
$$ \mathcal F\{f*g\}=\mathcal F\{f\}\cdot\mathcal F\{g\} $$
where $\mathcal F\{\cdot\}$ denotes Fourier transform. Previously, I thought this operation is only dedicated to this property, but when I began learning other branches of mathematics, I found other operations that resembled convolution:
For instance, the product of two power series satisfies
$$ \sum_{n\ge0}a_nx^n\cdot\sum_{k\ge0}b_kx^k=\sum_{m\ge0}\left(\sum_{0\le r\le m}a_rb_{m-r}\right)x^m $$
Likewise, the product between two Dirichlet series also produces a similar pattern:
$$ \sum_{n\ge0}{f(n)\over n^s}\cdot\sum_{k\ge0}{g(k)\over k^s}=\sum_{m\ge0}{1\over m^s}\left[\sum_{d|m}f(d)f\left(\frac md\right)\right] $$
Since this phenomenon occurs in many parts of mathematics, I wonder whether there is a more-abstract, more-generalized algebraic structure that can be extracted from these formulae.