# axiomatization of dependence

As noted by van der Waerden, it is possible to define the notion of dependence axiomatically in such a way that one can deal with linear dependence, algebraic dependence, and other sorts of dependence via a general theory. In this general theoretical framework, one can prove results about bases, dimension^{}, and the like.

Let $S$ be a set. The basic object of this theory is a relation^{} $D$ between $S$ and the power set^{} of $S$. This relation satisfies the following three axioms:

###### Axiom 1

If $Y$ is a subset of $S$ and $x\mathrm{\in}Y$, then $D\mathit{}\mathrm{(}x\mathrm{,}Y\mathrm{)}$.

###### Axiom 2

If, for some set $X\mathrm{\subseteq}S$ and some $y\mathrm{,}z\mathrm{\in}S$, it happens that $D\mathit{}\mathrm{(}y\mathrm{,}X\mathrm{\cup}\mathrm{\{}z\mathrm{\}}\mathrm{)}$ but not $D\mathit{}\mathrm{(}y\mathrm{,}X\mathrm{)}$, then $D\mathit{}\mathrm{(}z\mathrm{,}X\mathrm{\cup}\mathrm{\{}y\mathrm{\}}\mathrm{)}$.

###### Axiom 3

If, for some sets $Y\mathrm{,}Z\mathrm{\subseteq}S$ and some $x\mathrm{\in}S$, it happens that $D\mathit{}\mathrm{(}x\mathrm{,}Y\mathrm{)}$ and, for every $y\mathrm{\in}Y$, it is the case that $D\mathit{}\mathrm{(}y\mathrm{,}Z\mathrm{)}$, then $D\mathit{}\mathrm{(}x\mathrm{,}Z\mathrm{)}$.

Title | axiomatization of dependence |
---|---|

Canonical name | AxiomatizationOfDependence |

Date of creation | 2013-03-22 16:27:46 |

Last modified on | 2013-03-22 16:27:46 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 8 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 15A03 |

Related topic | DependenceRelation |