Dependence Relation

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.