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, dimensionPlanetmathPlanetmath, and the like.

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

Axiom 1

If Y is a subset of S and xY, then D(x,Y).

Axiom 2

If, for some set XS and some y,zS, it happens that D(y,X{z}) but not D(y,X), then D(z,X{y}).

Axiom 3

If, for some sets Y,ZS and some xS, it happens that D(x,Y) and, for every yY, it is the case that D(y,Z), then D(x,Z).

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