User:Leigh.Samphier/Matroids/Definition:Dual Matroid

Definition

Let $M = \struct {S, \mathscr I}$ be a matroid.

Let $\mathscr B$ be the set of bases of the matroid $M$.

The dual of $M$, denoted $M^* = \struct{S, \mathscr I^*}$, is the matroid whose bases is the set:

$\mathscr B^* = \set{S \setminus B : B \in \mathscr B}$

Also see

• User:Leigh.Samphier/Matroids/Dual Matroid is Matroid

• User:Leigh.Samphier/Matroids/Dual of Dual Matroid Equals Matroid

Sources

• 1976: Dominic Welsh: Matroid Theory Chapter $2.$ $\S 1.$ The Dual Matroid, Theorem $1$

• 2011: James Oxley: Matroid Theory (2nd ed.) Chapter $2.$ Duality, $\S 2.1.$ The definition and basic properties, Theorem $2.1.1$