Английская Википедия:Dependence relation

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Шаблон:Dablink Шаблон:Unsourced In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence.

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

• if <math>a \in S</math>, then <math>a \triangleleft S</math>;
• if <math>a \triangleleft S</math>, then there is a finite subset <math>S_0</math> of <math>S</math>, such that <math>a \triangleleft S_0</math>;
• if <math>T</math> is a subset of <math>X</math> such that <math>b \in S</math> implies <math>b \triangleleft T</math>, then <math>a \triangleleft S</math> implies <math>a \triangleleft T</math>;
• if <math>a \triangleleft S</math> but <math>a \ntriangleleft S-\lbrace b \rbrace</math> for some <math>b \in S</math>, then <math>b \triangleleft (S-\lbrace b \rbrace)\cup\lbrace a \rbrace</math>.

Given a dependence relation <math>\triangleleft</math> on <math>X</math>, a subset <math>S</math> of <math>X</math> is said to be independent if <math>a \ntriangleleft S - \lbrace a \rbrace</math> for all <math>a \in S.</math> If <math>S \subseteq T</math>, then <math>S</math> is said to span <math>T</math> if <math>t \triangleleft S</math> for every <math>t \in T.</math> <math>S</math> is said to be a basis of <math>X</math> if <math>S</math> is independent and <math>S</math> spans <math>X.</math>

Remark. If <math>X</math> is a non-empty set with a dependence relation <math>\triangleleft</math>, then <math>X</math> always has a basis with respect to <math>\triangleleft.</math> Furthermore, any two bases of <math>X</math> have the same cardinality.

Examples

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

See also

• matroid