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# latticeOfFlats -- lattice of flats of a matroid

- Usage:
    `latticeOfFlats M`
- Inputs:
  - M, a matroid,
- 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).

## See also

- flats -- flats of a matroid
- rank(Matroid) -- rank of a matroid
- fVector(Matroid) -- f-vector of a matroid

## Ways to use latticeOfFlats:

- `latticeOfFlats(Matroid)`

## For the programmer

The object latticeOfFlats is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Matroids/doc-Matroids.m2:1210:0.