# Honeycomb product

The **honeycomb product** or **comb product** for short, also known as the **topological product**^{[1]}, 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.

Honeycomb product | |
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The square tiling is the honeycomb product of two apeirogons (outlined in cyan). | |

Symbol | ^{[1]} |

Rank formula | ^{[note 1]}^{[1]} |

Dimension formula | |

Element formula | ^{[note 2]} |

Dual | Self-dual |

Algebraic properties | |

Algebraic structure | Coummutative semigroup^{[note 3]} |

Associative | Yes |

Commutative | Yes |

Identity | Ray^{[note 4]} |

Annihilator | Point |

Uniquely factorizable | Yes^{[note 5]}^{[1]} |

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.

## DefinitionEdit

If is an abstract polytope of rank and is an abstract polytope of rank , then the honeycomb product is defined to be:^{[1]}

with the order:

## NotesEdit

- ↑ For .
- ↑ For .
- ↑ It forms a monoid on partial orders but its identity is not an abstract polytope.
- ↑ Not an abstract polytope.
- ↑ With the exception of the annihilator.

## ReferencesEdit

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}^{1.4}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.

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