Dependence relation

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

In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence.

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

if $a\in S$ , then $a\triangleleft S$ ;
if $a\triangleleft S$ , then there is a finite subset $S_{0}$ of $S$ , such that $a\triangleleft S_{0}$ ;
if $T$ is a subset of $X$ such that $b\in S$ implies $b\triangleleft T$ , then $a\triangleleft S$ implies $a\triangleleft T$ ;
if $a\triangleleft S$ but $a\not \!\triangleleft S-\lbrace b\rbrace$ for some $b\in S$ , then $b\triangleleft (S-\lbrace b\rbrace )\cup \lbrace a\rbrace$ .

Given a dependence relation $\triangleleft$ on $X$ , a subset $S$ of $X$ is said to be independent if $a\not \!\triangleleft S-\lbrace a\rbrace$ for all $a\in S.$ If $S\subseteq T$ , then $S$ is said to span $T$ if $t\triangleleft S$ 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 non-empty set with a dependence relation $\triangleleft$ , then $X$ always has a basis with respect to $\triangleleft .$ Furthermore, any two bases of $X$ have the same cardinality.

Examples

Let $V$ be a vector space over a field $F.$ The relation $\triangleleft$ , defined by $\upsilon \triangleleft S$ if $\upsilon$ 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 $\triangleleft$ by $\alpha \triangleleft S$ if $\alpha$ is algebraic over $F(S).$ Then $\triangleleft$ is a dependence relation. This is equivalent to the definition of algebraic dependence.

Dependence relation

Examples

See also