dependence relation

Let $X$ be a set. A (binary) relation $\prec$ between an element $a$ of $X$ and a subset $S$ of $X$ is called a dependence relation, written $a\prec S$, when the following conditions are satisfied:

1. if $a\in S$, then $a\prec S$;

2. if $a\prec S$, then there is a finite subset $S_0$ of $S$, such that $a\prec S_0$;

3. if $T$ is a subset of $X$ such that $b\in S$ implies $b\prec T$, then $a\prec S$ implies $a\prec T$;

4. if $a\prec S$ but $a\nprec S-\{b\}$ for some $b\in S$, then $b\prec (S-\{b\})\cup \{a\}$.

Given a dependence relation $\prec$ on $X$, a subset $S$ of $X$ is said to be independent if $a\nprec S-\{a\}$ for all $a\in S$. If $S\subseteq T$, then $S$ is said to span $T$ if $t\prec S$ for every $t\in T$. $S$ is said to be a basis of $X$ if $S$ is independent and $S$ spans $X$.

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

Examples:

• Let $V$ be a vector space over a field $F$. The relation $\prec$, defined by $\upsilon \prec S$ if $\upsilon$ is in the subspace $S$, is a dependence relatoin. This is equivalent to the definition of linear dependence (http://planetmath.org/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 algebraic dependence.