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:

1. if , then ;
2. if , then there is a finite subset of , such that ;
3. if is a subset of such that implies , then implies ;
4. 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

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.

If and , then , using property 3. and 1.

Examples

• 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

• matroid

References