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 $D\mathit{}\mathrm{(}x\mathrm{,}Z\mathrm{)}$.