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.