Dependence relation

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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:

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.


See also

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