Matroid

In mathematics, a matroid or independence space is a structure that generalises the concept of linear and algebraic independence.

An independence structure on a ground set E is a family $\mathcal{E}$ of subsets of E, called independent sets, with the properties

• $\mathcal{E}$ is a downset, that is, $B \subseteq A \in \mathcal{E} \Rightarrow B \in \mathcal{E}$;
• The exchange property: if $A, B \in \mathcal{E}$ with | B | = | A | + 1 then there exists $x \in B \setminus A$ such that $A \cup \{x\} \in \mathcal{E}$.

A basis in an independence structure is a maximal independent set. Any two bases have the same number of elements. A circuit is a minimal dependent set. Independence spaces can be defined in terms of their systems of bases or of their circuits.

Examples

The following sets form independence structures:

Rank

We define the rank ρ(A) of a subset A of E to be the maximum cardinality of an independent subset of A. The rank satisfies the following

$0 \le \rho(A) \le |A| ;\,$
$A \subseteq B \Rightarrow \rho(A) \le \rho(B) ;\,$
$\rho(A) + \rho(B) \ge \rho(A\cap B) + \rho(A \cup B) .\,$

The last of these is the submodular inequality.

A flat is a subset A of E such that the rank of A is strictly less than the rank of any proper superset of A.