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 $x\in Y$, then $D(x,Y)$.

Axiom 2

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

Axiom 3

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

Title axiomatization of dependence AxiomatizationOfDependence 2013-03-22 16:27:46 2013-03-22 16:27:46 rspuzio (6075) rspuzio (6075) 8 rspuzio (6075) Definition msc 15A03 DependenceRelation