The rank of a subset S of elements of the matroid is, similarly, the maximum size of an independent subset of S, and the rank function of the matroid maps sets of elements to their ranks.
The rank functions of matroids defined from certain other types of mathematical object such as undirected graphs, matrices, and field extensions are important within the study of those objects.
In all examples, E is the base set of the matroid, and B is some subset of E. The rank function of a matroid obeys the following properties.
That is, the rank is a submodular set function.
These properties may be used as axioms to characterize the rank function of matroids: every integer-valued submodular set function on the subsets of a finite set that obeys the inequalities
[1][2] The above properties imply additional properties: The rank function may be used to determine the other important properties of a matroid: In graph theory, the circuit rank (or cyclomatic number) of a graph is the corank of the associated graphic matroid; it measures the minimum number of edges that must be removed from the graph to make the remaining edges form a forest.
[6][7] In linear algebra, the rank of a linear matroid defined by linear independence from the columns of a matrix is the rank of the matrix,[8] and it is also the dimension of the vector space spanned by the columns.
In abstract algebra, the rank of a matroid defined from sets of elements in a field extension L/K by algebraic independence is known as the transcendence degree.
[9] Matroid rank functions (MRF) has been used to represent utility functions of agents in problems of fair item allocation.
If the utility function of the agent is an MRF, it means that: The following solutions are known for this setting: The matroid-rank functions are a subclass of the gross substitute valuations.