Given two vectors $\vec{u}$, $\vec{v}$ indexed by $2^X$ for some finite set $X$, define $\vec{u} \star \vec{v}$ as the vector of similar type whose dimension indexed by $S \subseteq X$ is: $$\sum_{\begin{array}{c}S = A \cup B\\[-2em]A\cap B = \emptyset\end{array}} u_A v_B\enspace.$$ Question: Does this operation have a name? Am I seeing that from the wrong point of view?
As a concrete example, if $X = {1, 2}$, $\vec{u} = (x, x_1, x_2, x_{12})$ and $\vec{v} = (y, y_1, y_2, y_{12})$, where the components are, in order, indexed by $\emptyset, \{1\}, \{2\}, \{1, 2\}$, then: $$\vec{u} \star \vec{v} = (xy,\quad x_1y + xy_1, \quad x_2y + xy_2, \quad x_{12}y + x_1y_2 + x_2y_1 + xy_{12})\enspace.$$
This is a version of the exterior product, but missing some signs. It can be described as the multiplication in a certain ring, namely the ring
$$\mathbb{R}[x_1, x_2, \dots x_n]/(x_1^2 = x_2^2 = \dots = 0)$$
where the set $X$ has $n$ elements, and for convenience we'll identify it with the set $\{ 1, 2, \dots n \}$. A "vector indexed by $2^X$," which I'll interpret as a function $f : 2^X \to \mathbb{R}$, is sent to the element
$$\sum_S f(S) \prod_{i \in S} x_i$$
in this ring. The point of the relations $x_i^2 = 0$ is that the product of two monomials $x_S = \prod_{i \in S} x_i$ and $x_T = \prod_{i \in T} x_i$ vanishes as soon as $S$ and $T$ have nontrivial intersection, and if their intersection is trivial then the result is $x_{S \cup T}$.