Dependence relation

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 $a \in S$ , 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 $b \in S$ implies $b ◃ T$ , then $a ◃ S$ implies $a ◃ T$ ;
if $a ◃ S$ but $a ⋪ S - {b}$ for some $b \in S$ , then $b ◃ (S - {b}) \cup {a}$ .

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

References

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

Dependence relation

Examples

See also

References