or finite and infinite, matroids, and every geometric or matroid lattice comes from a matroid in this way.

Definition

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
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* A matroid lattice is a lattice that is both atomistic and semimodular. A geometric lattice is a finite matroid lattice.

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