Dependence relation

In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence.
Let be a set. A 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
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.