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dependence relation

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Let $X$ be a set. A (binary) relation $\prec$ between an element $a$ of $X$ and a subset $S$ of $X$ is called a \emph{dependence relation}, written $a \prec S$, when the following conditions are satisfied:
if $a \in S$, then $a \prec S$;
if $a \prec S$, then there is a finite subset $S_0$ of $S$, such that $a \prec S_0$;
if $T$ is a subset of $X$ such that $b \in S$ implies $b \prec T$, then $a \prec S$ implies $a \prec T$;
if $a \prec S$ but $a \not\prec S-\lbrace b \rbrace$ for some $b \in S$, then $b \prec (S-\lbrace b \rbrace)\cup\lbrace a \rbrace$.

Given a \emph{dependence relation} $\prec$ on $X$,  a subset $S$ of $X$ is said to be \emph{independent} if $a \not\prec S - \lbrace a \rbrace$ for all $a \in S$.  If $S \subseteq T$, then $S$ is said to \emph{span} $T$ if $t \prec S$ for every $t \in T$.  $S$ is said to be a \emph{basis} of $X$ if $S$ is \emph{independent} and $S$ \emph{spans} $X$.


\textbf{Remark.}  If $X$ is a non-empty set with a \emph{dependence relation} $\prec$, then $X$ always has a \emph{basis} with respect to $\prec$.  Furthermore, any two {\em \PMlinkescapetext{bases}} of $X$ have the same cardinality.


Let $V$ be a vector space over a field $F$.  The relation $\prec$, defined by $\upsilon \prec S$ if $\upsilon$ is in the subspace \PMlinkescapetext{spanned by} $S$, is a dependence relatoin.  This is equivalent to the definition of {\em \PMlinkname{linear dependence}{LinearIndependence}}.
Let $K$ be a field extension of $F$.  Define $\prec$ by $\alpha \prec S$ if $\alpha$ is algebraic over $F(S)$. Then $\prec$ is a dependence relation.  This is equivalent to the definition of \emph{algebraic dependence}.