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latticeOfFlats -- lattice of flats of a matroid

Synopsis

Description

The lattice of flats of a matroid M is the set of flats of M, partially ordered by containment; i.e. $F1 \le F2$ if F1 is contained in F2. The lattice of flats of a matroid is a geometric lattice: i.e. it is atomic (every element is a join of atoms = rank 1 elements) and semimodular ($h(x) + h(y) \ge h(x \vee y) + h(x \wedge y)$ for any x, y, where h is the height function = maximum length of a chain from 0, and all maximal chains have the same length). Conversely, every geometric lattice is the lattice of flats of a matroid.

If M and N are simple matroids (i.e. no loops or parallel classes) with isomorphic lattice of flats, then M and N are isomorphic.

i1 : M = matroid({a,b,c,d},{{a,b},{a,c}})

o1 = a "matroid" of rank 2 on 4 elements

o1 : Matroid
 
i2 : latticeOfFlats M

o2 = Relation Matrix: | 1 1 1 1 |
                      | 0 1 0 1 |
                      | 0 0 1 1 |
                      | 0 0 0 1 |

o2 : Poset

One can also view the lattice of flats, using displayPoset provided by the Posets package (the option SuppressLabels may be useful).

See also

Ways to use latticeOfFlats :

For the programmer

The object latticeOfFlats is a method function.