In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence.
Let X be a set. A (binary) relation ◃ between an element a of X and a subset S of X is called a dependence relation, written a ◃ S , if it satisfies the following properties:
if a ∈ S , then a ◃ S ;if a ◃ S , then there is a finite subset S 0 of S , such that a ◃ S 0 ;if T is a subset of X such that b ∈ S implies b ◃ T , then a ◃ S implies a ◃ T ;if a ◃ S but a ⧸ ◃ S − { b } for some b ∈ S , then b ◃ ( S − { b } ) ∪ { a } .Given a dependence relation ◃ on X , a subset S of X is said to be independent if a ⧸ ◃ S − { a } for all a ∈ S . If S ⊆ T , then S is said to span T if t ◃ S for every t ∈ T . S is said to be a basis of X if S is independent and S spans X .
Remark. If X is a non-empty set with a dependence relation ◃ , then X always has a basis with respect to ◃ . Furthermore, any two bases of X have the same cardinality.
