Basis of a matroid

In mathematics, a basis of a matroid is a maximal independent set of the matroid—that is, an independent set that is not contained in any other independent set.

Examples

As an example, consider the matroid over the ground-set R², with the following independent sets: It has two bases, which are the sets , . These are the only independent sets that are maximal under inclusion.
The basis has a specialized name in several specialized kinds of matroids:

In a graphic matroid, where the independent sets are the forests, the bases are called the spanning forests of the graph.
In a transversal matroid, where the independent sets are endpoints of matchings in a given bipartite graph, the bases are called transversals.
In a linear matroid, where the independent sets are the linearly-independent sets of vectors in a given vector-space, the bases are just called bases of the vector space. Hence, the concept of basis of a matroid generalizes the concept of basis from linear algebra.
In a uniform matroid, where the independent sets are all sets with cardinality at most k, the bases are all sets with cardinality exactly k.
In a partition matroid, where elements are partitioned into categories and the independent sets are all sets containing at most k_c elements from each category c, the bases are all sets which contain exactly k_c elements from category c.
In a free matroid, where all subsets of the ground-set E are independent, the unique basis is ''E.''
Properties

Exchange

All matroids satisfy the following properties, for any two distinct bases and :

Basis-exchange property: if, then there exists an element such that is a basis.
Symmetric basis-exchange property: if, then there exists an element such that both and are bases. Brualdi showed that it is in fact equivalent to the basis-exchange property.
Multiple symmetric basis-exchange property: if, then there exists a subset such that both and are bases. Brylawski, Greene, and Woodall, showed that it is in fact equivalent to the basis-exchange property.
Bijective basis-exchange property: There is a bijection ' from ' to, such that for every, is a basis. Brualdi showed that it is equivalent to the basis-exchange property.
Partition basis-exchange property: For each partition of ' into m parts, there exists a partition of into m parts, such that for every, is a basis.

However, a basis-exchange property that is both symmetric and bijective is not satisfied by all matroids: it is satisfied only by base-orderable matroids.
In general, in the symmetric basis-exchange property, the element need not be unique. Regular matroids have the unique exchange property', meaning that for some, the corresponding b'' is unique.

Cardinality

It follows from the basis exchange property that no member of can be a proper subset of another.
Moreover, all bases of a given matroid have the same cardinality. In a linear matroid, the cardinality of all bases is called the dimension of the vector space.

Neil White's conjecture

It is conjectured that all matroids satisfy the following property: For every integer t ≥ 1, If B and B' are two t-tuples of bases with the same multi-set union, then there is a sequence of symmetric exchanges that transforms B to B'.

Characterization

The bases of a matroid characterize the matroid completely: a set is independent if and only if it is a subset of a basis. Moreover, one may define a matroid to be a pair, where is the ground-set and is a collection of subsets of, called "bases", with the following properties:
implies that, given any two bases A and B, we can transform A into B by a sequence of exchanges of a single element. In particular, this implies that all bases must have the same cardinality.

Duality

If is a finite matroid, we can define the orthogonal or dual matroid by calling a set a basis in if and only if its complement is in. It can be verified that is indeed a matroid. The definition immediately implies that the dual of is .
Using duality, one can prove that the property can be replaced by the following:

If and are distinct bases, and, then there exists an element such that is a basis.

Circuits

A dual notion to a basis is a circuit. A circuit in a matroid is a minimal dependent set—that is, a dependent set whose proper subsets are all independent. The terminology arises because the circuits of graphic matroids are cycles in the corresponding graphs.
One may define a matroid to be a pair, where is the ground-set and is a collection of subsets of, called "circuits", with the following properties:
Another property of circuits is that, if a set is independent, and the set is dependent, then contains a unique circuit, and it contains. This circuit is called the fundamental circuit of w.r.t.. It is analogous to the linear algebra fact, that if adding a vector to an independent vector set makes it dependent, then there is a unique linear combination of elements of that equals.