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# 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 $D(x,Z)$.

## Mathematics Subject Classification

15A03*no label found*

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