Dependence relation

In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence.
Let formula_1 be a set. A (binary) relation formula_2 between an element formula_3 of formula_1 and a subset formula_5 of formula_1 is called a "dependence relation", written formula_7, if it satisfies the following properties: 
Given a "dependence relation" formula_2 on formula_1, a subset formula_5 of formula_1 is said to be "independent" if formula_28 for all formula_29 If formula_30, then formula_5 is said to "span" formula_14 if formula_33 for every formula_34 formula_5 is said to be a "basis" of formula_1 if formula_5 is "independent" and formula_5 "spans" formula_39 
Remark. If formula_1 is a non-empty set with a dependence relation formula_2, then formula_1 always has a basis with respect to formula_43 Furthermore, any two bases of formula_1 have the same cardinality. 