Modular lattice

Summary

A modular lattice is a lattice that satisfies the modular law: if $a \le b$, then $a \vee (x \wedge b) = (a \vee x) \wedge b$ for every $x$. The lattices of subspaces of a vector space, of submodules of a module and of normal subgroups of a group are modular. Every distributive lattice is modular, but not conversely. Modularity is what makes dimension and length behave well, as in the Jordan–Hölder theorem for modular lattices.