Matroid

In combinatorics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being in terms of independent sets, bases or circuits, rank functions, closure operators, and closed sets or flats. In the language of partially ordered sets, a finite simple matroid is equivalent to a geometric lattice.

Matroid theory borrows extensively from the terms used in both linear algebra and graph theory, largely because it is the abstraction of various notions of central importance in these fields. Matroids have found applications in geometry, topology, combinatorial optimization, network theory, and coding theory.

Definition

There are many equivalent ways to define a (finite) matroid.

Independent sets

In terms of independence, a finite matroid

M

{\displaystyle M}

is a pair

(

E

,

I

)

{\displaystyle (E,{\mathcal {I}})}

, where

E

{\displaystyle E}

is a finite set (called the ground set) and

I

{\displaystyle {\mathcal {I}}}

is a family of subsets of

E

{\displaystyle E}

(called the independent sets) with the following properties:

(I1) The empty set is independent, i.e.,

∅

∈

I

{\displaystyle \emptyset \in {\mathcal {I}}}

.

(I2) Every subset of an independent set is independent, i.e., for each

A

′

⊆

A

{\displaystyle A'\subseteq A}

, if

A

∈

I

{\displaystyle A\in {\mathcal {I}}}

then

A

′

∈

I

{\displaystyle A'\in {\mathcal {I}}}

. This is sometimes called the hereditary property, or the downward-closed property.

(I3) If

A

{\displaystyle A}

and

B

{\displaystyle B}

are two independent sets (i.e., each set is independent) and

A

{\displaystyle A}

has more elements than

B

{\displaystyle B}

, then there exists

x

∈

A

∖

B

{\displaystyle x\in A\setminus B}

such that

B

∪

{

x

}

{\displaystyle B\cup \{x\}}

is independent.