# Category:Matroid Theory

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This category contains results about Matroid Theory.

Definitions specific to this category can be found in Definitions/Matroid Theory.

**Matroid Theory** is the branch of mathematics which concerns the role which matroids play in disparate branches of combinatorial theory and algebra such as graph theory, lattice theory, combinatorial optimization, and linear algebra.

## Subcategories

This category has the following 7 subcategories, out of 7 total.

### E

### M

## Pages in category "Matroid Theory"

The following 83 pages are in this category, out of 83 total.

### C

### D

### E

- Element Depends on Independent Set iff Union with Singleton is Dependent
- Element is Loop iff Member of Closure of Empty Set
- Element is Loop iff Singleton is Circuit
- Element is Member of Base iff Not Loop
- Element of Matroid Base and Circuit has a Substitute
- Element of Matroid Base and Circuit has a Substitute/Lemma 1
- Element of Matroid Base and Circuit has a Substitute/Lemma 2
- Element of Matroid Base and Circuit has a Substitute/Lemma 3
- Equivalence of Definitions of Matroid
- Equivalent Conditions for Element is Loop

### I

- Independent Set can be Augmented by Larger Independent Set
- Independent Set can be Augmented by Larger Independent Set/Corollary
- Independent Subset is Base if Cardinality Equals Cardinality of Base
- Independent Subset is Base if Cardinality Equals Rank of Matroid
- Independent Subset is Base if Cardinality Equals Rank of Matroid/Corollary
- Independent Subset is Contained in Base
- Independent Subset is Contained in Maximal Independent Subset
- Independent Subset of Matroid is Augmented by Base

### L

- Leigh.Samphier/Sandbox/Bound for Cardinality of Matroid Circuit
- Leigh.Samphier/Sandbox/Cardinality of Maximal Independent Subset Equals Rank of Set
- Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom
- Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Circuit Axioms
- Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Circuit Axioms/Condition 1 Implies Condition 2
- Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Circuit Axioms/Condition 2 Implies Condition 3
- Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Circuit Axioms/Condition 3 Implies Condition 4
- Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Circuit Axioms/Condition 4 Implies Condition 1
- Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Rank Axioms
- Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Rank Axioms/Condition 1 Implies Condition 3
- Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Rank Axioms/Condition 1 Implies Condition 3/Lemma 1
- Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Rank Axioms/Condition 1 Implies Condition 3/Lemma 2
- Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Rank Axioms/Condition 1 Implies Condition 3/Lemma 3
- Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Rank Axioms/Condition 2 Implies Condition 1
- Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Rank Axioms/Condition 3 Implies Condition 2
- Leigh.Samphier/Sandbox/Independent Subset is Contained in Maximal Independent Subset/Corollary
- Leigh.Samphier/Sandbox/Independent Superset of Dependent Set Minus Singleton Doesn't Contain Singleton
- Leigh.Samphier/Sandbox/Matroid Base Axiom Implies Sets Have Same Cardinality
- Leigh.Samphier/Sandbox/Matroid Base Substitution From Fundamental Circuit
- Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom
- Leigh.Samphier/Sandbox/Matroid Satisfies Rank Axioms
- Leigh.Samphier/Sandbox/Matroid Unique Circuit Property
- Leigh.Samphier/Sandbox/Matroid Unique Circuit Property/Proof 1
- Leigh.Samphier/Sandbox/Matroid Unique Circuit Property/Proof 2
- Leigh.Samphier/Sandbox/Matroid with No Circuits Has Single Base
- Leigh.Samphier/Sandbox/Proper Subset of Matroid Circuit is Independent
- Leigh.Samphier/Sandbox/Rank of Matroid Circuit is One Less Than Cardinality
- Leigh.Samphier/Sandbox/Rank of Matroid Circuit is One Less Than Cardinality/Lemma
- Loop Belongs to Every Flat

### M

- Matroid Base Union External Element has Fundamental Circuit
- Matroid Contains No Loops iff Empty Set is Flat
- Matroid Induced by Affine Independence is Matroid
- Matroid Induced by Algebraic Independence is Matroid
- Matroid Induced by Linear Independence in Abelian Group is Matroid
- Matroid Induced by Linear Independence in Vector Space is Matroid
- Matroid Unique Circuit Property
- Matroid Unique Circuit Property/Corollary

### R

### S

- Set with Two Parallel Elements is Dependent
- Singleton is Dependent implies Rank is Zero
- Singleton is Dependent implies Rank is Zero/Corollary
- Singleton is Independent iff Rank is One
- Singleton is Independent implies Rank is One
- Singleton is Independent implies Rank is One/Corollary
- Superset of Dependent Set is Dependent
- Superset of Dependent Set is Dependent/Corollary