nLab semimodular lattice

Definition

A lattice is semimodular if it satisfies the Jordan-Hölder condition (all maximal chains between two elements $x\lt y$ are of the same length), if it has a finite height and its height function $h$ satisfies the semimodular inequality

$$h(x) + h(y) \geq h(x\wedge y) + h(x\vee y)$$

category: algebra, combinatorics