Let $X,R$ be semigroups. For the sake of this thread a convolution weight $w : X \rightsquigarrow R$ is a function $w : X^2\to R$ such that:
$$w(s\cdot t, r) \cdot w(s,t) = w(s,t\cdot r)\cdot w(t,r)$$
Examples include constant functions $X^2\to R$, indicator functions or e.g. the "binomial weight":
$$w : \mathbb{Z}_{\geq 0}^2 \to R, (s,t) \mapsto \binom{s+t}{t} = \binom{s+t}{s}$$
where $R$ is the multiplicative monoid of a rig (or ring).
What are other (interesting) examples of such functions?
Motivation:
Suppose $R$ is a rig. Given a convolution weight $w$, define its associated convolution $* : (R^X)^2 \to R^X$ by:
$$(f*g)(x) := \sum_{s\cdot t = x} w(s,t)\cdot f(s)\cdot g(t)$$
Then $(R^X,+,*)$ is a rg ("rig without identity"). This includes familiar convolutions like discrete convolution, Dirichlet convolution or binomial convolution.