simpleMatroid -- simple matroid associated to a matroid

Description

The simple matroid associated to a matroid M is obtained from M by deleting all loops, and all but one element from each parallel class.

In a simple matroid, the lattice of flats has the empty set as minimal element, and all atoms are singletons.

i1 : M = uniformMatroid(0, 2) ++ uniformMatroid(1, 2) ++ uniformMatroid(2, 4)

o1 = a "matroid" of rank 3 on 8 elements

o1 : Matroid

i2 : isSimple M

o2 = false

i3 : S = simpleMatroid M

o3 = a "matroid" of rank 3 on 5 elements

o3 : Matroid

i4 : latticeOfFlats M == latticeOfFlats S

o4 = true

i5 : select(flats S, f -> rank(S, f) <= 1)

o5 = {set {}, set {4}, set {3}, set {2}, set {1}, set {0}}

o5 : List

i6 : AG32 = affineGeometry(3, 2)

o6 = a "matroid" of rank 4 on 8 elements

o6 : Matroid

The source of this document is in Matroids/doc-Matroids.m2:2764:0.