Semimodular lattice

Summary

A semimodular lattice is a lattice in which joining with an element preserves the covering relation: if $a$ covers $a \wedge b$, then $a \vee b$ covers $b$. It is a one-sided weakening of the modular law, and every modular lattice is semimodular. Lattices of flats of matroids and lattices of partitions of a set are semimodular without being modular in general. In lattices of finite length, semimodularity is what makes a well-behaved rank function exist.