User:Leigh.Samphier/Matroids/Matroid Rank Function Iff Matroid Rank Axioms

This page needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code.

Theorem

Let $S$ be a finite set.

Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers.

Then:

$\rho$ is the rank function of a matroid $M = \struct{S, \mathscr I}$.

if and only if:

$\rho$ satisfies the rank axioms

Proof

Follows immediately from:

• User:Leigh.Samphier/Matroids/Formulation 1 Rank Axioms Implies Rank Function of Matroid

• User:Leigh.Samphier/Matroids/Rank Function of Matroid Satisfies Formulation 1 Rank Axioms

Sources

• 1976: Dominic Welsh: Matroid Theory Chapter $1.$ $\S 6.$ Properties of the rank function

• 2018: Bernhard H. Korte and Jens Vygen: Combinatorial Optimization: Theory and Algorithms (6th ed.) Chapter $13$ Matroids $\S 13.2$ Other Matroid Axion, Theorem $13.10$