Dependence relation

Short description: Relation showing how elements or variables rely on or determine each other.

Not to be confused with Dependency relation, which is a binary relation that is symmetric and reflexive.

This article does not cite any external source. HandWiki requires at least one external source. See citing external sources. Please help improve this article. Unsourced material may be removed in future.
Find sources: "Dependence relation" – news · newspapers · books · scholar · JSTOR (March 2023) (Learn how and when to remove this template message)

|

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:

1. if a∈S, then a◃S;
2. if a◃S, then there is a finite subset S0 of S, such that a◃S0;
3. if T is a subset of X such that b∈S implies b◃T, then a◃S implies a◃T;
4. 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.

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.

If a◃S and S⊆T, then a◃T, using property 3. and 1.

## Examples

* Let V be a vector space over a field F. The relation ◃, defined by υ◃S if υ is in the subspace spanned by S, is a dependence relation. This is equivalent to the definition of linear dependence.
* Let K be a field extension of F. Define ◃ by α◃S if α is algebraic over F(S). Then ◃ is a dependence relation. This is equivalent to the definition of algebraic dependence.

## See also

* matroid

## References

This article incorporates material from Dependence relation on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Dependence relation. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Dependence_relation&oldid=4296302"