Definition:Rank Function


 * Relation Theory:
 * Rank Function on a relational structure $\struct {S, \RR}$: a mapping $\operatorname {rk}$ from $S$ to a well-ordered set $\struct {T, \prec}$ such that:
 * $\forall x, y \in S: \paren {x \ne y \text { and } \tuple {x, y} \in \RR} \implies \map {\operatorname {rk} } x \prec \map {\operatorname {rk} } y$


 * Matroid theory:
 * Rank Function of a matroid $M = \struct {S, \mathscr I}$: the mapping from the power set of $S$ into the integers defined by:
 * $\forall A \subseteq S : \map \rho A = \max \set {\size X : X \subseteq A \land X \in \mathscr I}$