isModularCut -- whether a list of flats of a matroid is a modular cut

Synopsis

• Usage:

  isModularCut(M, K)

• Inputs:
  • M, a matroid
  • K, a list, a list of flats of the matroid
• Outputs:
  • a Boolean value, whether the list of flats is a modular cut

Description

This function is provided by the package Matroids.

A modular cut for a matroid M is a collection K of flats of M with the properties that for any flats F and G of M: (i) if F is in K and F is contained in G, then G is also in K, and (ii) if the rank of union of F and G plus the rank of the intersection of F and G equals the sum of the ranks of F and G, then the intersection of F and G is also in K. Two flats F and G satisfying the rank condition in (ii) above are called a modular pair. If F is contained in G, then F and G trivially form a modular pair.

Each flat in the modular cut may be represented as either a set or a list of elements of the ground set of M. The list of flats K below is a modular cut for the uniform matroid U_{4,5} since the only two incomparable flats are not a modular pair; both flats have rank 3, but their intersection and union have rank 1 and 4 respectively.

i1 : M = uniformMatroid(4, 5);

i2 : K = {set{0,1,2}, set{2,3,4}, set{0,1,2,3,4}};

i3 : isModularCut(M, K)

o3 = true

The list of flats K' below is not a modular cut since the flats {0,1,2} and {0,1,3} are a modular pair but their intersection does not belong to K'.

i4 : K' = {{0,1,2}, {0,1,3}, {0,1,2,3,4}};

i5 : isModularCut(M, K')

o5 = false

See also

• modularCut -- associated to an elementary quotient or linear subclass
• linearSubclass -- associated to an elementary quotient or modular cut