Dependence relation

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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.

[edit] Examples

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