Modular Lattice

In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition:

Modular law

x ≤ b implies x ∨ (a ∧ b) = (x ∨ a) ∧ b,

where ≤ is the partial order, and ∨ and ∧ (called join and meet respectively) are the operations of the lattice. Modular lattices arise naturally in algebra and in many other areas of mathematics. For example, the subspaces of a vector space (and more generally the submodules of a module over a ring) form a modular lattice.

Every distributive lattice is modular.

In a not necessarily modular lattice, there may still be elements b for which the modular law holds in connection with arbitrary elements a and x (≤ b). Such an element is called a modular element. Even more generally, the modular law may hold for a fixed pair (a, b). Such a pair is called a modular pair, and there are various generalizations of modularity related to this notion and to semimodularity.