Modular Lattice - Introduction

Introduction

The modular law can be seen (and memorized) as a restricted associative law that connects the two lattice operations similarly to the way in which the associative law λ(μx) = (λμ)x for vector spaces connects multiplication in the field and scalar multiplication. The restriction x ≤ b is clearly necessary, since it follows from x ∨ (a ∧ b) = (x ∨ a) ∧ b.

It is easy to see that x ≤ b implies x ∨ (a ∧ b) ≤ (x ∨ a) ∧ b in every lattice. Therefore the modular law can also be stated as

Modular law (variant)

x ≤ b implies x ∨ (a ∧ b) ≥ (x ∨ a) ∧ b.

By substituting x with x ∧ b, the modular law can be expressed as an equation that is required to hold unconditionally, as follows:

Modular identity

(x ∧ b) ∨ (a ∧ b) = ∧ b.

This shows that, using terminology from universal algebra, the modular lattices form a subvariety of the variety of lattices. Therefore all homomorphic images, sublattices and direct products of modular lattices are again modular.

The smallest non-modular lattice is the "pentagon" lattice N₅ consisting of five elements 0,1,x,a,b such that 0 < x < b < 1, 0 < a < 1, and a is not comparable to x or to b. For this lattice x ∨ (a ∧ b) = x ∨ 0 = x < b = 1 ∧ b = (x ∨ a) ∧ b holds, contradicting the modular law. Every non-modular lattice contains a copy of N₅ as a sublattice.

Modular lattices are sometimes called Dedekind lattices after Richard Dedekind, who discovered the modular identity.