# latticeOfFlats -- lattice of flats of a matroid

## Synopsis

• Usage:
latticeOfFlats M
• Inputs:
• M, ,
• Outputs:
• an instance of the type Poset,

## 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).