Dependence relation

In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence.

Let X be a set. A (binary) relation $\triangleleft$ between an element a of X and a subset S of X is called a dependence relation, written $a\triangleleft S$ , if it satisfies the following properties:

if $a\in S$ , then $a\triangleleft S$ ;
if $a\triangleleft S$ , then there is a finite subset $S_{0}$ of S, such that $a\triangleleft S_{0}$ ;
if T is a subset of X such that $b\in S$ implies $b\triangleleft T$ , then $a\triangleleft S$ implies $a\triangleleft T$ ;
if $a\triangleleft S$ but $a\not \!\triangleleft S-\lbrace b\rbrace$ for some $b\in S$ , then $b\triangleleft (S-\lbrace b\rbrace )\cup \lbrace a\rbrace$ .

Given a dependence relation $\triangleleft$ on X, a subset S of X is said to be independent if $a\not \!\triangleleft S-\lbrace a\rbrace$ for all $a\in S$ . If $S\subseteq T$ , then S is said to span T if $t\triangleleft S$ 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 $\triangleleft$ , then X always has a basis with respect to $\triangleleft$ . Furthermore, any two bases of X have the same cardinality.

Examples

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