Dependence relation

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Not to be confused with Dependency relation, which is a binary relation that is symmetric and reflexive.

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

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

  • if , then ;
  • if , then there is a finite subset of , such that ;
  • if is a subset of such that implies , then implies ;
  • if but for some , then .

Given a dependence relation on , a subset of is said to be independent if for all If , then is said to span if for every is said to be a basis of if is independent and spans

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

Examples[edit]

  • Let be a vector space over a field The relation , defined by if is in the subspace spanned by , is a dependence relation. This is equivalent to the definition of linear dependence.
  • Let be a field extension of Define by if is algebraic over Then is a dependence relation. This is equivalent to the definition of algebraic dependence.

See also[edit]

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