The lattice of flats of a matroid M is the set of flats of M, partially ordered by containment; i.e. F1 <= 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) >= h(x ∨ y) + h(x ∧ y) for any x, y, where h is the height function = maximum length of a chain from 0). 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 (together with the option SuppressLabels => false).
The object latticeOfFlats is a method function.