Geometric lattice

In the mathematics of matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the assumption of finiteness. Geometric lattices and matroid lattices, respectively, form the lattices of flats of finite, or finite and infinite, matroids, and every geometric or matroid lattice comes from a matroid in this way. A lattice is a poset in which any two elements and have both a least upper bound, called the join or supremum, denoted by , and a greatest lower bound, called the meet or infimum, denoted by . The following definitions apply to posets in general, not just lattices, except where otherwise stated. For a minimal element , there is no element such that . An element covers another element (written as or ) if and there is no element distinct from both and so that . A cover of a minimal element is called an atom. A lattice is atomistic if every element is the supremum of some set of atoms. A poset is graded when it can be given a rank function mapping its elements to integers, such that whenever , and also whenever . When a graded poset has a bottom element, one may assume, without loss of generality, that its rank is zero. In this case, the atoms are the elements with rank one. A graded lattice is semimodular if, for every and , its rank function obeys the identity A matroid lattice is a lattice that is both atomistic and semimodular. A geometric lattice is a finite matroid lattice. Many authors consider only finite matroid lattices, and use the terms "geometric lattice" and "matroid lattice" interchangeably for both. 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).