Dependence relation

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 ${\displaystyle X}$ be a set. A (binary) relation ${\displaystyle ◃}$ between an element ${\displaystyle a}$ of ${\displaystyle X}$ and a subset ${\displaystyle S}$ of ${\displaystyle X}$ is called a dependence relation, written ${\displaystyle a◃S}$, if it satisfies the following properties:

• if ${\displaystyle a\in S}$, then ${\displaystyle a◃S}$;
• if ${\displaystyle a◃S}$, then there is a finite subset ${\displaystyle S_0}$ of ${\displaystyle S}$, such that ${\displaystyle a◃S_0}$;
• if ${\displaystyle T}$ is a subset of ${\displaystyle X}$ such that ${\displaystyle b\in S}$ implies ${\displaystyle b◃T}$, then ${\displaystyle a◃S}$ implies ${\displaystyle a◃T}$;
• if ${\displaystyle a◃S}$ but ${\displaystyle a⋪S-\{b\}}$ for some ${\displaystyle b\in S}$, then ${\displaystyle b◃(S-\{b\})\cup \{a\}}$.

Given a dependence relation ${\displaystyle ◃}$ on ${\displaystyle X}$, a subset ${\displaystyle S}$ of ${\displaystyle X}$ is said to be independent if ${\displaystyle a⋪S-\{a\}}$ for all ${\displaystyle a\in S.}$ If ${\displaystyle S\subseteq T}$, then ${\displaystyle S}$ is said to span ${\displaystyle T}$ if ${\displaystyle t◃S}$ for every ${\displaystyle t\in T.}$ ${\displaystyle S}$ is said to be a basis of ${\displaystyle X}$ if ${\displaystyle S}$ is independent and ${\displaystyle S}$ spans ${\displaystyle X.}$

Remark. If ${\displaystyle X}$ is a non-empty set with a dependence relation ${\displaystyle ◃}$, then ${\displaystyle X}$ always has a basis with respect to ${\displaystyle ◃.}$ Furthermore, any two bases of ${\displaystyle X}$ have the same cardinality.

Examples

• Let ${\displaystyle V}$ be a vector space over a field ${\displaystyle F.}$ The relation ${\displaystyle ◃}$, defined by ${\displaystyle \upsilon ◃S}$ if ${\displaystyle \upsilon }$ is in the subspace spanned by ${\displaystyle S}$, is a dependence relation. This is equivalent to the definition of linear dependence.
• Let ${\displaystyle K}$ be a field extension of ${\displaystyle F.}$ Define ${\displaystyle ◃}$ by ${\displaystyle \alpha ◃S}$ if ${\displaystyle \alpha }$ is algebraic over ${\displaystyle F(S).}$ Then ${\displaystyle ◃}$ is a dependence relation. This is equivalent to the definition of algebraic dependence.

See also

• matroid

0.00 (0 votes) Original source: https://en.wikipedia.org/wiki/Dependence relation. Read more