# simpleMatroid -- simple matroid associated to a matroid

## Synopsis

• Usage:
S = simpleMatroid M
• Inputs:
• M, ,
• Outputs:
• , the simple matroid associated to M

## 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