Geometric lattice

Remarks

Example: the Boolean lattice of all subsets of a finite set $S$. The atoms are the singletons, and every subset is the union of its singletons. If $A$ covers $A \cap B$, then $A = (A \cap B) \cup \set{s}$ with $s \notin B$, so $A \cup B = B \cup \set{s}$ covers $B$.

Non-example: the chain $0 < a < 1$ with three elements. It is semimodular, but its only atom is $a$, and $1$ is not a join of atoms.

The lattice of flats of a finite matroid is geometric, and every geometric lattice is isomorphic to the lattice of flats of a simple matroid, unique up to isomorphism. Some authors allow infinite geometric lattices, replacing finiteness by finite length or by compactness conditions.