Let $X$ be a finite set and let $2^X$ be its power set. Let $Z$ be some ring (e.g. the complex numbers; it doesn't matter).
Suppose $f:2^X\to Z$ and $g:2^X\to Z$ are two functions from $2^X$ to $Z$.
A product $(f\ast g): 2^X\to Z$ can be defined like this: $$\text{For $V\subseteq X$, }~~ (f\ast g)(V) = \sum_{V_1\cup V_2=V} f(V_1)g(V_2).$$ (The sum is over all pairs $V_1,V_2\subseteq X$ such that $V_1\cup V_2=V$.)
My question is: What is this product called?
The binary union operation makes $2^X$ into a semigroup. Now, the product you describe is usually called convolution, and together with pointwise addition in makes the set of functions $2^X \rightarrow Z$ into a new ring, called the semigroup ring $Z[2^X]$. Note that there can be problems in case $X$ is infinite - you'll have to restrict the support of functions involved in order to avoid infinite sums.