Star product

The star product (written $$*$$) is an operation that can be applied to any two polytopes. Unlike the prism product, tegum product, pyramid product, and comb product, the star product is not commutative: that is, $$P*Q \neq Q*P$$. The dual of the star product of two polytopes will be the star product of their duals with the order of the polytopes swapped, and the star product of two regular polytopes will also be regular.

The star product generalizes the concept of ditopes and hosotopes, and because of this the product will almost always result in a degenerate polytope. In fact, the ditope of a polytope $$P$$ is the product of $$P$$ with a dyad ($$P*\{\}$$), and the hosotope of $$P$$ is the product of a dyad with $$P$$ ($$\{\}*P$$). The star product of an n-dimensional polytope and an m-dimensional polytope will always be an (n+m)-dimensional polytope.

The number of vertices, edges, ..., facets of the star product of two polytopes $$P*Q$$ will be the same as the number of vertices, edges, ... , facets of $$P$$. In a way, $$P*Q$$ can be thought of as just $$P$$ with some extra higher-rank elements added on top to create a degenerate polytope.

Definition
The star product is almost always defined abstractly. Given two polytopes $$P$$ and $$Q$$, the star product is a poset defined on the set


 * $$P*Q = \{x : x \in P \text{ and } x \text{ is not the maximal element of } P \text{, or } x \in Q \text{ and } x \text{ is not minimal element of } Q \}$$

with the order relation $$x \le y$$ if and only if at least one of these things are true:


 * $$x \in P, y \in P, \text{and } x \le_P y$$
 * $$x \in Q, y \in Q, \text{and } x \le_Q y$$
 * $$x \in P \text{ and } y \in Q$$

To make the product a concrete polytope, the vertices of $$P*Q$$ will just be the vertices of $$P$$.