## Common Factor Theorem

(x /\ y) \/ (u /\ v) == (x /\ y /\ v) \/ u - (x /\ y /\ u) \/ v

## Duality

reversion ~

concept | dual | |
---|---|---|

/\ | \/ | |

grade(A) = m | grade(dual(A)) = n - m | |

1 | pseudoscalar |

where:

n = dimension of vector space

## Join and Meet

These are two operations associated with geometric intersection and union of spaces, they are denoted by:

Symbol | Spaces | |
---|---|---|

Meet | /\ | intersection |

Join | \/ | union U |

Confusingly meet and join often use the same symbols '/\' and '\/' as the inner and outer products although the results are slightly different. (some books such as Doran and Lasenby invert this and use \/ for meet and /\ for join so we have to be very careful with terminology and notation).

The mathematical structure of meet and join is an example of a lattice,

- Lattices are explained on this page.
- Use of lattices for vector subspaces are explained on this page.

A line on this diagram means that the element at the bottom of the line is a direct factor of the element at the top. There is not a line from e1 to e123 because we don't need to include e1 when e1 is already a factor of e12 and e31.

To calculate the meet we take a line upwards from both operands until we get to the lowest common denominator.

Meet /\ | 1 | e1 | e2 | e3 | e12 | e31 | e23 | e123 |
---|---|---|---|---|---|---|---|---|

1 | 1 | e1 | e2 | e3 | e12 | e31 | e23 | e123 |

e1 | e1 | e1 | e12 | e31 | e12 | e31 | e123 | e123 |

e2 | e2 | e12 | e2 | e23 | e12 | e123 | e23 | e123 |

e3 | e3 | e31 | e23 | e3 | e123 | e31 | e23 | e123 |

e12 | e12 | e12 | e12 | e123 | e12 | e123 | e123 | e123 |

e31 | e31 | e31 | e123 | e31 | e123 | e31 | e123 | e123 |

e23 | e23 | e123 | e23 | e23 | e123 | e123 | e23 | e123 |

e123 | e123 | e123 | e123 | e123 | e123 | e123 | e123 | e123 |

To calculate the join we take a line downwards from both operands until we get to the highest common factor.

Join \/ | 1 | e1 | e2 | e3 | e12 | e31 | e23 | e123 |
---|---|---|---|---|---|---|---|---|

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

e1 | 1 | e1 | 1 | 1 | e1 | e1 | 1 | e1 |

e2 | 1 | 1 | e2 | 1 | e2 | 1 | e2 | e2 |

e3 | 1 | 1 | 1 | e3 | 1 | e3 | e2 | e3 |

e12 | 1 | e1 | e2 | 1 | e12 | e1 | e2 | e12 |

e31 | 1 | e1 | 1 | e3 | e1 | e31 | e3 | e31 |

e23 | 1 | 1 | e2 | e2 | e2 | e3 | e23 | e23 |

e123 | 1 | e1 | e2 | e3 | e12 | e31 | e23 | e123 |