Honeycomb product

Honeycomb product
The square tiling is the honeycomb product of two apeirogons (outlined in cyan).
Symbol${\displaystyle \square}$[1]
Rank formula${\displaystyle n+m-1}$[note 1][1]
Dimension formula${\displaystyle n+m}$
Element formula${\displaystyle (n-2)\times(m-2)+2}$[note 2]
DualSelf-dual
Algebraic properties
Algebraic structureCoummutative semigroup[note 3]
AssociativeYes
CommutativeYes
IdentityRay[note 4]
AnnihilatorPoint
Uniquely factorizableYes[note 5][1]

The honeycomb product or comb product for short, also known as the topological product[1], is one of four common 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 polytopes other than honeycombs is skew.

The comb product of two polytopes is known as a duocomb, and a multicomb for more than two polytopes. The comb product of a regular polytope with itself is regular. For the comb product of two squares is the square duocomb, a regular skew polyhedron.

Definition

If ${\displaystyle A}$ is an abstract polytope of rank ${\displaystyle n}$ and ${\displaystyle B}$ is an abstract polytope of rank ${\displaystyle m}$, then the honeycomb product is defined to be:[1]

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

with the order:

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

Notes

1. For ${\displaystyle n,m>1}$.
2. For ${\displaystyle n,m>1}$.
3. It forms a monoid on partial orders but its identity is not an abstract polytope.
4. Not an abstract polytope.
5. With the exception of the annihilator.

References

1. Gleason, Ian; Hubard, Isabel (2018). "Products of abstract polytopes" (PDF). Journal of Combinatorial Theory, Series A. 157: 287–320. doi:10.1016/j.jcta.2018.02.002.