Definition:Depends Relation (Matroid)
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Definition
Let $M = \struct{S, \mathscr I}$ be a matroid.
Let $\rho : \powerset S \to \Z$ be the rank function of $M$.
For $x \in S$ and $A \subseteq S$, $x$ is said to depend on $A$, denoted as $x \sim A$, if and only if:
- $\map \rho {A \cup \set x} = \map \rho A$
Sources
- 1976: Dominic Welsh: Matroid Theory ... (previous) ... (next) Chapter $1.$ $\S 2.$ Axiom Systems for a Matroid