Lattice Theory Foundations
Modular Lattices and the Modular Law
The modular law as a conditional weakening of distributivity, Dedekind's theorem that subgroup lattices of abelian groups are modular, and the reason modularity is the right hypothesis for a large part of algebra.
Learning objectives
- State the modular law in both conditional and identity form
- Verify that every distributive lattice is modular
- Identify the algebraic settings where modularity holds but distributivity fails
The modular law
A lattice is modular if it satisfies the conditional identity: whenever x ≤ z, then x ∨ (y ∧ z) = (x ∨ y) ∧ z.
The condition can be written as a genuine identity by substituting x ∧ z for x, since x ∧ z ≤ z always holds:
(x ∧ z) ∨ (y ∧ z) ≈ ((x ∧ z) ∨ y) ∧ z
Because it can be stated as an identity, the class of modular lattices is a variety. Birkhoff's HSP theorem therefore applies: modular lattices are closed under homomorphic images, sublattices and products, and free modular lattices exist.
Distributive implies modular
Assume distributivity and x ≤ z. Then x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) by the dual distributive law, and x ∨ z = z since x ≤ z. The right side is therefore (x ∨ y) ∧ z, as required.
The converse fails: M5, the diamond, is modular but not distributive. So modularity sits strictly between arbitrary lattices and distributive lattices.
One inequality is again free
In any lattice, if x ≤ z then x ∨ (y ∧ z) ≤ (x ∨ y) ∧ z. Modularity asserts equality, so as with distributivity the verification is one-sided.
Where modularity comes from
Modularity is not an arbitrary weakening. It is the lattice-theoretic shadow of a phenomenon pervasive in algebra.
| Setting | Lattice | Status |
|---|---|---|
| Abelian group | Subgroup lattice | Modular (Dedekind) |
| Module over a ring | Submodule lattice | Modular |
| Vector space | Subspace lattice | Modular; distributive only in dimension ≤ 1 |
| Group (general) | Normal subgroup lattice | Modular |
| Ring | Ideal lattice | Modular |
| Group (general) | Full subgroup lattice | Not modular in general |
Groups, rings and modules are all congruence-modular. The commutator theory developed for congruence-modular varieties — which generalises the group commutator — is one of the major achievements of the subject since 1981, and it is precisely modularity that makes it possible.
The Jordan–Hölder connection
Modularity is what makes composition-series arguments work. In a modular lattice of finite length, any two maximal chains between the same pair of elements have the same length, and their factors correspond in pairs.
This is the abstract content of the Jordan–Hölder theorem for groups and the invariance of dimension for vector spaces. Both are instances of a single lattice-theoretic fact, which is a good illustration of what the universal-algebraic viewpoint buys.
Frequently asked questions
Is the modular law self-dual?
Yes. The dual of the modular law, after renaming variables, is the modular law again — so the dual of a modular lattice is modular.
Why is the full subgroup lattice of a group not modular?
The symmetric group on three letters provides a counterexample: its subgroup lattice contains a copy of N5 formed by the trivial subgroup, a subgroup of order two, the alternating subgroup, and the whole group.
Source. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, The Millennium Edition — a corrected re-typesetting of Springer-Verlag Graduate Texts in Mathematics 78 (1981). Section I.3, book pages 12-14.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates and reorganises mathematical results; it is not a reproduction of the source text.