linearSubclass -- associated to an elementary quotient or modular cut

Synopsis

• Usage:

  linearSubclass(M, N),

  linearSubclass(M, K)

• Inputs:
  • M, a matroid
  • N, a matroid
  • K, a list, a list of flats of the matroid forming a modular cut
• Optional inputs:
  • CheckWellDefined => ..., default value false, check whether the list is a modular cut
• Outputs:
  • a list, a list of hyperplanes of the matroid forming a linear subclass

Description

This function is provided by the package Matroids.

A modular cut K of the matroid M is determined by the collection of hyperplanes of M that belong to K, which is called the corresponding linear subclass. See isLinearSubclass for more details about linear subclasses. Given a list K of flats of M forming a modular cut, this function produces the corresponding linear subclass.

i1 : A = matrix {{1, 0, 0, 1, 1}, {0, 1, 0, 1, -1}, {0, 0, 1, 0, 0}}

o1 = | 1 0 0 1 1  |
     | 0 1 0 1 -1 |
     | 0 0 1 0 0  |

              3       5
o1 : Matrix ZZ  <-- ZZ

i2 : M = matroid A

o2 = a "matroid" of rank 3 on 5 elements

o2 : Matroid

i3 : K = {{2}, {2, 4}, {2, 3}, {1, 2}, {0, 2}, {0, 1, 2, 3, 4}};

i4 : isModularCut(M, K)

o4 = true

i5 : linearSubclass(M, K)

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

o5 : List

Since every elementary quotient M of a matroid N determines a modular cut of flats of N, we can find the corresponding linear subclass as follows.

i6 : N = matroid completeGraph 5

o6 = a "matroid" of rank 4 on 10 elements

o6 : Matroid

i7 : F = set {0, 1, 4};

i8 : TFN = truncate(F, N);

i9 : isElementaryQuotient(TFN, N)

o9 = true

i10 : linearSubclass(TFN, N)

o10 = {set {0, 9, 1, 4}, set {0, 1, 2, 4, 5, 7}, set {8, 0, 1, 3, 4, 6}}

o10 : List

See also

• elementaryQuotient -- associated to a modular cut or linear subclass
• isElementaryQuotient -- whether a matroid is an elementary quotient of another matroid
• isLinearSubclass -- whether a list of hyperplanes of a matroid is a linear subclass
• isModularCut -- whether a list of flats of a matroid is a modular cut
• modularCut -- associated to an elementary quotient or linear subclass