Honeycomb product

The honeycomb product or comb product for short, also known as the topological product, is one of four polytope products along with the prism, tegum and pyramid products. The honeycomb product of two euclidean honeycombs is itself an euclidean honeycomb.

The comb product of two polytopes is known as a duocomb, and a multicomb for more than two polytopes. Polygonal multicombs are regular polytopes, for example the square duocomb.

Definition
If $$A$$ is an abstract polytope of rank $$n$$ and $$B$$ is an abstract polytope of rank $$m$$, then the honeycomb product is defined to be:

$$A\square B=\left\{(a,b)\mid a\in A, b\in B, \text{ either }a\text{ and }b\text{ are proper or the same improper element}\right\}$$

with the order:

$$(a,b)\leq_{A\square B}(a',b') \iff a\leq_A a' \land b\leq_B b'$$