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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.

See also

matroid

This article incorporates material from Dependence relation on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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