User:Leigh.Samphier/Matroids

Matroids

 * Chapter $13$ Matroids $\S 13.1$ Independence Systems and Matroids, Definition $13.1$

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

Let $\rho: \powerset S \to \Z$ denote the rank function of $M$.

Let $\sigma: \powerset S \to \powerset S$ denote the closure operator of $M$.

FIX
Definition:Closure Axioms (Matroid)

Properties of Independent Sets and Bases

 * Chapter $1.$ $\S 5.$ Properties of independent sets and bases
 * All Bases of Matroid have same Cardinality

User:Leigh.Samphier/Matroids/Matroid Base Substitution From Fundamental Circuit

User:Leigh.Samphier/Matroids/Axiom:Base Axioms (Matroid)

User:Leigh.Samphier/Matroids/Axiom:Base Axioms (Matroid)/Axiom 1

User:Leigh.Samphier/Matroids/Axiom:Base Axioms (Matroid)/Axiom 2

User:Leigh.Samphier/Matroids/Axiom:Base Axioms (Matroid)/Axiom 3

User:Leigh.Samphier/Matroids/Axiom:Base Axioms (Matroid)/Axiom 4

User:Leigh.Samphier/Matroids/Axiom:Base Axioms (Matroid)/Axiom 5

User:Leigh.Samphier/Matroids/Axiom:Base Axioms (Matroid)/Axiom 6

User:Leigh.Samphier/Matroids/Axiom:Base Axioms (Matroid)/Axiom 7

User:Leigh.Samphier/Matroids/Matroid Base Axiom Implies Sets Have Same Cardinality

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Lemma

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Condition 1 Iff Condition 3

References:

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Condition 1 Iff Condition 4

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Condition 4 Iff Condition 5

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Condition 4 Iff Condition 5/Lemma

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Condition 3 Iff Condition 7

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Condition 1 Iff Condition 8

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Condition 1 Iff Condition 8/Necessary Condition

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Condition 1 Iff Condition 8/Sufficient Condition

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Condition 1 Iff Condition 8/Sufficient Condition/Lemma

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Condition 1 Iff Condition 8/Sufficient Condition/Lemma/Lemma 1

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Condition 1 Iff Condition 8/Sufficient Condition/Lemma/Lemma 2

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Condition 1 Iff Condition 8/Sufficient Condition/Lemma/Lemma 3

Properties of the Rank Function
User:Leigh.Samphier/Matroids/Independent Subset is Contained in Maximal Independent Subset/Corollary

User:Leigh.Samphier/Matroids/Cardinality of Maximal Independent Subset Equals Rank of Set

User:Leigh.Samphier/Matroids/Matroid Satisfies Rank Axioms

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Rank Axioms

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Rank Axioms/Condition 1 Implies Condition 3

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Rank Axioms/Condition 1 Implies Condition 3/Lemma 1

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Rank Axioms/Condition 1 Implies Condition 3/Lemma 2

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Rank Axioms/Condition 1 Implies Condition 3/Lemma 3

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Rank Axioms/Condition 3 Implies Condition 2

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Rank Axioms/Condition 2 Implies Condition 1

Circuits
User:Leigh.Samphier/Matroids/Proper Subset of Matroid Circuit is Independent

User:Leigh.Samphier/Matroids/Rank of Matroid Circuit is One Less Than Cardinality

User:Leigh.Samphier/Matroids/Bound for Cardinality of Matroid Circuit

User:Leigh.Samphier/Matroids/Matroid with No Circuits Has Single Base

User:Leigh.Samphier/Matroids/Axiom:Circuit Axioms (Matroid)

User:Leigh.Samphier/Matroids/Axiom:Circuit Axioms (Matroid)/Axioms 1

User:Leigh.Samphier/Matroids/Axiom:Circuit Axioms (Matroid)/Axioms 2

User:Leigh.Samphier/Matroids/Axiom:Circuit Axioms (Matroid)/Axioms 3

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Circuit Axioms

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Circuit Axioms/Condition 1 Implies Condition 2

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Circuit Axioms/Condition 2 Implies Condition 3

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Circuit Axioms/Condition 3 Implies Condition 4

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Circuit Axioms/Condition 4 Implies Condition 1

User:Leigh.Samphier/Matroids/Matroid Unique Circuit Property

User:Leigh.Samphier/Matroids/Matroid Unique Circuit Property/Proof 1

User:Leigh.Samphier/Matroids/Matroid Unique Circuit Property/Proof 2

The Greedy Algorithm
Maximization Problem (Greedy Algorithm)
 * Chapter $19.$ $\S 1.$ The greedy algorithm

Complete Greedy Algorithm yields Maximal Set

Complete Greedy Algorithm may not yield Maximum Weight

Complete Greedy Algorithm guarantees Maximum Weight iff Matroid