# dual(Matroid) -- dual matroid

## Synopsis

• Function: dual
• Usage:
dual M
• Inputs:
• M, ,
• Outputs:
• , the dual matroid of M

## Description

The dual matroid of a matroid M has the same ground set as M, and bases equal to the complements of bases of M.

Duality is a fundamental operation in matroid theory: for nearly any property/operation of matroids, there is a corresponding dual version, usually denoted with the prefix "co-". For instance, coloops are loops of the dual, and contraction is dual to deletion.

In this package, every dual matroid is created as a matroid-dual matroid pair, and each is cached as the dual of the other. Often the ideal of the dual matroid has a significantly different number of generators, so many algorithms in this package will use an equivalent check for the ideal with fewer generators.

 i1 : M = matroid({a,b,c,d},{{a,b},{a,c}}) o1 = a "matroid" of rank 2 on 4 elements o1 : Matroid i2 : D = dual M o2 = a "matroid" of rank 2 on 4 elements o2 : Matroid i3 : peek D o3 = Matroid{bases => {set {2, 3}, set {1, 3}}} cache => CacheTable{...2...} groundSet => set {0, 1, 2, 3} rank => 2 i4 : M == dual D o4 = true i5 : loops D == coloops M o5 = true i6 : hyperplanes M === apply(circuits D, C -> D.groundSet - C) o6 = true

A matroid that is isomorphic to its dual is called self-dual; and a matroid that is equal to its dual is called identically self-dual.

 i7 : V8plus = specificMatroid "V8+" o7 = a "matroid" of rank 4 on 8 elements o7 : Matroid i8 : V8plus == dual V8plus o8 = true i9 : V = relaxation(V8plus, set{4,5,6,7}) o9 = a "matroid" of rank 4 on 8 elements o9 : Matroid i10 : V == dual V o10 = false i11 : areIsomorphic(V, dual V) o11 = true

If a matroid has a representation stored, then this function will attempt to automatically compute a representation for the dual (whether this works depends on whether reducedRowEchelonForm is implemented for the underlying ring of the matrix).

 i12 : F7 = specificMatroid fano o12 = a "matroid" of rank 3 on 7 elements o12 : Matroid i13 : getRepresentation F7 o13 = | 1 1 1 1 0 0 0 | | 0 0 1 1 1 1 0 | | 0 1 0 1 0 1 1 | ZZ 3 ZZ 7 o13 : Matrix (--) <--- (--) 2 2 i14 : M = dual F7 o14 = a "matroid" of rank 4 on 7 elements o14 : Matroid i15 : getRepresentation M o15 = | 1 1 1 1 0 0 0 | | 1 0 1 0 1 0 0 | | 0 1 1 0 0 1 0 | | 1 1 0 0 0 0 1 | ZZ 4 ZZ 7 o15 : Matrix (--) <--- (--) 2 2