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 assumptions of finiteness. Geometric lattices and matroid lattices, respectively, form the lattices of flats of finite and infinite matroids, and every geometric or matroid lattice comes from a matroid in this way.

Recall that a lattice is a poset in which any two elements ${\displaystyle x}$ and ${\displaystyle y}$ have a least upper bound (lub), denoted by ${\displaystyle x\vee y}$, and a greatest lower bound (glb), denoted by ${\displaystyle x\wedge y}$.

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