User:Leigh.Samphier/Sandbox/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/Sandbox/Independent Superset of Dependent Set Minus Singleton Doesn't Contain Singleton

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

User:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)

User:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 1

User:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 2

User:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 3

User:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 4

User:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 5

User:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 6

User:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 7

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

User:Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom

User:Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Necessary Condition

User:Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition

User:Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma

User:Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma/Lemma 1

User:Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma/Lemma 2

User:Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma/Lemma 3

User:Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom

User:Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Lemma

User:Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 1 Iff Definition 3

References:

User:Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 1 Iff Definition 4

User:Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 4 Iff Definition 5

User:Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 4 Iff Definition 5/Lemma

User:Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 3 Iff Definition 7

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

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

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

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

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

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

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

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

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

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

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

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

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

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

User:Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)

User:Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 1

User:Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 2

User:Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 3

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

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

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

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

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

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

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

User:Leigh.Samphier/Sandbox/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