Geometric lattice

Lattices vs. matroids

The geometric lattices are equivalent to (finite) simple matroids, and the matroid lattices are equivalent to simple matroids without the assumption of finiteness (under an appropriate definition of infinite matroids; there are several such definitions). The correspondence is that the elements of the matroid are the atoms of the lattice and an element x of the lattice corresponds to the flat of the matroid that consists of those elements of the matroid that are atoms ${\displaystyle a\le x.}$

Like a geometric lattice, a matroid is endowed with a rank function, but that function maps a set of matroid elements to a number rather than taking a lattice element as its argument. The rank function of a matroid must be monotonic (adding an element to a set can never decrease its rank) and it must be submodular, meaning that it obeys an inequality similar to the one for semimodular ranked lattices:

${\displaystyle r(X)+r(Y)\ge r(X\cap Y)+r(X\cup Y)}$

for sets X and Y of matroid elements. The maximal sets of a given rank are called flats. The intersection of two flats is again a flat, defining a greatest lower bound operation on pairs of flats; one can also define a least upper bound of a pair of flats to be the (unique) maximal superset of their union that has the same rank as their union. In this way, the flats of a matroid form a matroid lattice, or (if the matroid is finite) a geometric lattice.

Conversely, if ${\displaystyle L}$ is a matroid lattice, one may define a rank function on sets of its atoms, by defining the rank of a set of atoms to be the lattice rank of the greatest lower bound of the set. This rank function is necessarily monotonic and submodular, so it defines a matroid. This matroid is necessarily simple, meaning that every two-element set has rank two.

These two constructions, of a simple matroid from a lattice and of a lattice from a matroid, are inverse to each other: starting from a geometric lattice or a simple matroid, and performing both constructions one after the other, gives a lattice or matroid that is isomorphic to the original one.

Duality

There are two different natural notions of duality for a geometric lattice ${\displaystyle L}$: the dual matroid, which has as its basis sets the complements of the bases of the matroid corresponding to ${\displaystyle L}$, and the dual lattice, the lattice that has the same elements as ${\displaystyle L}$ in the reverse order. They are not the same, and indeed the dual lattice is generally not itself a geometric lattice: the property of being atomistic is not preserved by order-reversal. Cheung (1974) defines the adjoint of a geometric lattice ${\displaystyle L}$ (or of the matroid defined from it) to be a minimal geometric lattice into which the dual lattice of ${\displaystyle L}$ is order-embedded. Some matroids do not have adjoints; an example is the Vámos matroid.

Additional properties

Every interval of a geometric lattice (the subset of the lattice between given lower and upper bound elements) is itself geometric; taking an interval of a geometric lattice corresponds to forming a minor of the associated matroid. Geometric lattices are complemented, and because of the interval property they are also relatively complemented.

Every finite lattice is a sublattice of a geometric lattice.