Modular lattices

Modular lattices are a generalization of distributive lattices, especially useful in the theory of lattices of subalgebras. They can be defined either by the modular law: $a \le b$ implies that $\forall x : a \vee (x \wedge b) = (a \wedge b) \vee b$ or by the forbidden lattice characterization. The following lattice is the simplest non-modular lattice:

It can easily be seen from this that every sublattice of a modular lattice is modular (this is the case for all classes of lattices defined by forbidden sublattices).

Proposition. every sublattice of a modular lattice is modular.

Additionally, complements play a special role in the theory of modular lattices. Suppose that we have an element of a modular lattice, then the set of complements of that element forms an antichain. In other words, complements are unordered. Distributive lattices are a special case in which complements are unique.

While distributive lattices are important in the foundations of mathematics and set theory, modular lattices are especially important in the study of subalgebras of algebraic structures.