# Parallel product

The **parallel product** may refer to one of several closely related concepts. The parallel product is closely related to the mixing of abstract polytopes and the two are generalized by the parallel product on maniplexes.^{[1]}

## Groups[edit | edit source]

Parallel product | |
---|---|

Symbol | ^{[2]} |

Algebraic properties | |

Algebraic structure | Monoid |

Associative | Yes |

Commutative | Yes |

Identity | Trivial group |

Idempotent | Cyclic groups |

The parallel product of two groups G and H generated by the sets and respectively is a group generated by the set under the action:

This action is the same action as the direct product, and thus .

The resulting group depends on the choice of generating sets. Since every group generates itself, the direct product is a special case of the parallel product, although usually a different generating set is chosen.

### Examples[edit | edit source]

Let be a cyclic group of order n. Such a group always has an element g which generates the entire group. is then a group generated by in the direct product group. This group is isomorphic to . Thus cyclic groups are idempotent.

## Actions[edit | edit source]

Parallel product | |
---|---|

Algebraic properties | |

Algebraic structure | Monoid |

Associative | Yes |

Commutative | Yes |

Identity | Yes |

If we have some group G and two group actions , and , then we can define the parallel product to be an action of G on any orbit of the direct product . The selection of orbit is akin to the selection of generators in the parallel product on groups. In fact the parallel product on actions is a generalization of the product on groups.^{[3]}

## Maps[edit | edit source]

## Maniplexes[edit | edit source]

Parallel product | |
---|---|

Symbol | ^{[1]} |

Algebraic properties | |

Algebraic structure | Semigroup |

Associative | Yes |

Commutative | Yes |

Idempotent | Regulars |

There are multiple equivalent ways to define maniplexes, thus there are multiple ways to define the parallel product on maniplexes.

### Actions[edit | edit source]

A n-maniplex can be defined as a group with a sequence of n generators acting on a set of flags. The parallel product then corresponds to an orbit of the parallel product of the groups with respect to their generators acting on the cartesian product of their flags.

### Graphs[edit | edit source]

For two n-colored graphs with vertex sets and , their direct product is an n-colored graph with vertex set such that there is an i-edge between two vertices and iff there is an i-edge between and in the first graph and an i-edge between and in the second graph.

The parallel product of two maniplexes is then a connected component of the direct product of the maniplexes.

## References[edit | edit source]

- ↑
^{1.0}^{1.1}Garza-Vargas & Hubard (2018:19) - ↑ Wilson (1991)
- ↑ Wilson (1991:540)

## Biliography[edit | edit source]

- Garza-Vargas, Jorge; Hubard, Isabel (7 July 2018). "Polytopality of Maniplexes" (PDF). arXiv:1604.01164.
- Wilson, Stephen (1991). "Parallel Products in Groups and Maps".
*Journal of Algebra*. doi:10.1006/jabr.1994.1200.

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