dependence relation
Let $X$ be a set. A (binary) relation^{} $\prec $ between an element $a$ of $X$ and a subset $S$ of $X$ is called a dependence relation, written $a\prec S$, when the following conditions are satisfied:

1.
if $a\in S$, then $a\prec S$;

2.
if $a\prec S$, then there is a finite subset ${S}_{0}$ of $S$, such that $a\prec {S}_{0}$;

3.
if $T$ is a subset of $X$ such that $b\in S$ implies $b\prec T$, then $a\prec S$ implies $a\prec T$;

4.
if $a\prec S$ but $a\nprec S\{b\}$ for some $b\in S$, then $b\prec (S\{b\})\cup \{a\}$.
Given a dependence relation $\prec $ on $X$, a subset $S$ of $X$ is said to be independent if $a\nprec S\{a\}$ for all $a\in S$. If $S\subseteq T$, then $S$ is said to span $T$ if $t\prec S$ for every $t\in T$. $S$ is said to be a basis of $X$ if $S$ is independent and $S$ spans $X$.
Remark. If $X$ is a nonempty set with a dependence relation $\prec $, then $X$ always has a basis with respect to $\prec $. Furthermore, any two of $X$ have the same cardinality.
Examples:

•
Let $V$ be a vector space^{} over a field $F$. The relation $\prec $, defined by $\upsilon \prec S$ if $\upsilon $ is in the subspace^{} $S$, is a dependence relatoin. This is equivalent^{} to the definition of linear dependence (http://planetmath.org/LinearIndependence).

•
Let $K$ be a field extension of $F$. Define $\prec $ by $\alpha \prec S$ if $\alpha $ is algebraic over $F(S)$. Then $\prec $ is a dependence relation. This is equivalent to the definition of algebraic dependence.
Title  dependence relation 

Canonical name  DependenceRelation 
Date of creation  20130322 14:19:25 
Last modified on  20130322 14:19:25 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  9 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 03E20 
Classification  msc 05B35 
Related topic  LinearIndependence 
Related topic  AlgebraicallyDependent 
Related topic  Matroid^{} 
Related topic  AxiomatizationOfDependence 