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/Equivalence of Definitions of Matroid Base Axioms

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Axiom B1 Iff Axiom B3

References:

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Axiom B3 Iff Axiom B7

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Axiom B1 Iff Axiom B4

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Axiom B4 Iff Axiom B5

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Set of Matroid Bases Iff Axiom B1

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Set of Matroid Bases Implies Axiom B1

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Axiom B1 Implies Set of Matroid Bases

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Lemma 1

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Lemma 2

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Lemma 3

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

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

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Lemma 6

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Lemma 7

Properties of the Rank Function

 * Chapter $1.$ $\S 6.$ Properties of the rank function
 * All Bases of Matroid have same Cardinality

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

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/Proof 1

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

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

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

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

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

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Rank Axioms/Lemma 2

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

User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Rank Axioms/Lemma 4

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