15A03-AxiomatizationOfDependence.tex

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\pmtitle{axiomatization of dependence}
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\newtheorem{axiom}{Axiom}
\begin{document}
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:

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

\begin{axiom}
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 $D(z, X \cup \{y\})$.
\end{axiom}

\begin{axiom}
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 $D(x,Z)$.
\end{axiom}
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