modular lattice (original) (raw)

A lattice L is said to be _modular_if x∨(y∧z)=(x∨y)∧zfor all x,y,z∈L such that x≤z. In fact it is sufficient to show thatx∨(y∧z)≥(x∨y)∧zfor all x,y,z∈L such that x≤z, as the reverse inequality holds in all lattices (see modular inequality).

There are a number of other equivalent conditions for a lattice L to be modular:

• •
  (x∧y)∨(x∧z)=x∧(y∨(x∧z))for all x,y,z∈L.
• •
  (x∨y)∧(x∨z)=x∨(y∧(x∨z))for all x,y,z∈L.
• •
  For all x,y,z∈L, if x<z then either x∧y<z∧y or x∨y<z∨y.

The following are examples of modular lattices.

• •
  All distributive lattices (http://planetmath.org/DistributiveLattice).
• •
• •
  The lattice of submodules of any module (http://planetmath.org/Module). (See modular law.)

A finite lattice L is modular if and only if it is graded and its rank function ρ satisfies ρ⁢(x)+ρ⁢(y)=ρ⁢(x∧y)+ρ⁢(x∨y) for all x,y∈L.