# Matroid == Matroid -- whether two matroids are equal

## Synopsis

• Operator: ==
• Usage:
M == N
• Inputs:
• M, ,
• N, ,
• Outputs:
• , whether the two matroids are equal

## Description

Two matroids are considered equal if they have the same set of (indexed) bases and same size grounds sets (in particular, the ground sets need not be identical). This happens iff the identity permutation is an isomorphism.

The strong comparison operator === should not be used, as bases (and ground sets) are internally stored as lists rather than sets, so the same matroid with a different ordering on the list of bases (or ground set) will be treated as different under ===. (One might try to sort the list of bases, but this is potentially time-consuming, as the list of bases can grow rapidly with the size of the ground set.)

 i1 : M = matroid completeGraph 3 o1 = a matroid of rank 2 on 3 elements o1 : Matroid i2 : peek M o2 = Matroid{bases => {set {1, 2}, set {0, 2}, set {0, 1}}} cache => CacheTable{...4...} groundSet => set {0, 1, 2} rank => 2 i3 : N = uniformMatroid(2, 3) o3 = a matroid of rank 2 on 3 elements o3 : Matroid i4 : peek N o4 = Matroid{bases => {set {0, 1}, set {0, 2}, set {1, 2}}} cache => CacheTable{...2...} groundSet => set {0, 1, 2} rank => 2 i5 : M == N o5 = true i6 : M === N o6 = false i7 : AG32 = specificMatroid "AG32" -- identically self-dual o7 = a matroid of rank 4 on 8 elements o7 : Matroid i8 : AG32 == dual AG32 o8 = true i9 : AG32 === dual AG32 o9 = false i10 : V = specificMatroid "vamos" -- self-dual, but not identically so o10 = a matroid of rank 4 on 8 elements o10 : Matroid i11 : V == dual V o11 = false i12 : areIsomorphic(V, dual V) o12 = true