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 $a \not\prec S-\lbrace b \rbrace$ for some $b \in S$ , then $b \prec (S-\lbrace b \rbrace)\cup\lbrace a \rbrace$ .

Given a dependence relation $\prec$ on $X$ , a subset $S$ of $X$ is said to be independent if $a \not\prec S - \lbrace a \rbrace$ 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 bases 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 spanned by $S$ , is a dependence relatoin. This is equivalent to the definition of linear dependence.
• 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.