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 $$\triangleleft$$ 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:
 * 1) if $$a \in S$$, then $$a \triangleleft S$$;
 * 2) if $$a \triangleleft S$$, then there is a finite subset $$S_0$$ of $$S$$, such that  $$a \triangleleft S_0$$;
 * 3) if $$T$$ is a subset of $$X$$ such that $$b \in S$$ implies $$b \triangleleft T$$, then $$a \triangleleft S$$ implies $$a \triangleleft T$$;
 * 4) if $$a \triangleleft S$$ but  $$a \ntriangleleft S-\lbrace b \rbrace$$ 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 $$a \ntriangleleft S - \lbrace a \rbrace$$ 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 $$X.$$

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.

If $$a \triangleleft S$$ and $$S \subseteq T$$, then $$a \triangleleft T$$, using property 3. and 1.

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.