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# linearSubclass -- associated to an elementary quotient or modular cut

## Synopsis

• Usage:
linearSubclass(M, N),
linearSubclass(M, K)
• Inputs:
• M,
• N,
• 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 {2, 3}, set {0, 2}, set {4, 2}, set {1, 2}} 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, 1, 2, 4, 5, 7}, set {8, 0, 1, 3, 4, 6}, set {0, 9, 1, 4}} o10 : List

• 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

## Ways to use linearSubclass :

• linearSubclass(Matroid,List)
• linearSubclass(Matroid,Matroid)

## For the programmer

The object linearSubclass is .