I am wondering if the following kind of objects has some name, or are there any studied examples. I apologize for perhaps too specific definition, this is an adoptation of a situation that arises in physics.

Consider a $n$-polytope $P$ with a distinguished vertex $v_0$. Suppose there exist a map $f:\,V(P)\to A$, where $A$ is a unital commutative algebra over $\mathbb{C}$ and $V(P)$ is the set of vertices of $P$. Moreover, assume that $f(v_0)=1$ and that the image of $f$ forms a basis for $A$ (and in particular is a linearly-independent set).

Edit (in the older version I made a stupid mistake and disregarded a very important grading):

Now we employ the structure of the polytope. We put $v_0$ at the origin of $\mathbb{R}^n$ and introduce the 'grading' on the basis elements that forces the product of two vertices as elements of $A$ to be proportional to the vertex given by their sum as of elements of $\mathbb{R}^n$. And the product is zero if there is no such vertex.

As an example, consider the square with vertices labeled clockwise as 1,a,ab,b and a map to $\mathbb{C}[a,b]/(a^2,b^2)$ given by the names of the vertices.