# application: matroid

Matroids encode the concept of "(in)dependence" in an abstract way. You can define matroids via vector configurations or graphs, do basic conversions between different descriptions and perform basic operations such as deletions and contractions.

**imports from:**common, graph

**uses:**group, ideal, polytope, topaz

**Objects**

A matroid on the set

*{0,...,n-1}*. Here*n*is the same as N_ELEMENTS.#### Properties of Matroid

**BASES**: common::Array<Set<Int>>Subsets of the ground set which form the bases of the matroid. Note that if you want to define a matroid via its bases, you should also specify N_ELEMENTS, because we allow matroids with loops.

**BINARY_POINTS**: common::Matrix<Int, NonSymmetric>If the matroid is realizable over the field GF(2) with two elements, this property contains coordinates for some realization.

**COCIRCUITS**: common::Array<Set<Int>>Cocircuits (or bonds) of the matroid, i.e., minimal sets intersecting every basis.

**LABELS**: common::Array<String>Unique names assigned to the elements of the matroid.

For a matroid build from scratch, you should create this property by yourself. If you build the matroid with a construction client, (e.g. matroid_from_graph) the labels may be assigend for you in a meaningful way.

**NON_BASES**: common::Array<Set<Int>>All subsets of the ground sets with cardinality RANK that are not bases.

**N_ELEMENTS**: common::IntSize of the ground set. The ground set itself always consists of the first integers starting with zero.

**POINTS**: common::Matrix<Rational, NonSymmetric>If the matroid is realizable over the rationals, this property contains coordinates for some realization. Specifying coordinates is one way to define a matroid.

**POLYTOPE**: polytope::Polytope<Rational>Polytope whose vertices are the characteristic vectors of the bases.

**REGULAR**: common::BoolWhether the matroid is representable over every field, that is the repesentation is unimodular. NOTE: the property is 'undef' when its hard to decide, whether the matroid is ternary.

**TERNARY**: common::BoolWhether the matroid is representable over GF(3) NOTE: the property is 'undef' when its hard to decide.

**TERNARY_POINTS**: common::Matrix<Int, NonSymmetric>If the matroid is realizable over the field GF(3) with three elements, this property contains coordinates for some realization.

## User Functions

- UNDOCUMENTED
**check_matroid**(bases)Tests whether the given

*bases*do actually form the bases of a matroid.##### Parameters

Array<Set<Int>> bases ##### Options

Bool print if set to true the output tells which condition fails; default value is 0**gen_matroid**(ininitial_sets)Returns the set of bases of a matroid, containing

*initial_sets*as basis elements. The ground set is considered to be the union of the entries of*initial sets*##### Parameters

Array<Set<Int>> ininitial_sets ##### Options

Bool print if set to true the output tells because of which condition, sets have been added; default value is 0

- UNDOCUMENTED
**contraction**(m, element) → Matroid

- UNDOCUMENTED
**fano_matroid**() → Matroid**matroid_from_characteristic_vector**(v, r, n) → MatroidCreates the matroid with a given characteristic-plueckervector of rank r and a ground set of n elements.

**matroid_from_graph**(g) → Matroid**matroid_from_matroid_polytope**(p) → MatroidCreates a matroid from the corresponding matroid polytope

*p*.**projective_plane**(p) → Matroid**uniform_matroid**(r, n) → Matroid

- UNDOCUMENTED
**matroid_plueckervector**(m) → ListReturnCreates the characteristic- and the rank-plueckervector of a matroid.