Lattice Theory Foundations
Distributive Lattices and their Characterisation
The distributive law for lattices, its self-dual character, and the concrete examples that make distributivity the most important special condition in lattice theory.
Learning objectives
- State the distributive laws and show the two forms are equivalent
- Identify distributive and non-distributive lattices among standard examples
- Explain the role distributivity plays in later chapters
The distributive laws
A lattice is distributive if it satisfies, identically: x ∧ (y ∨ z) ≈ (x ∧ y) ∨ (x ∧ z).
In any lattice, the law above holds identically if and only if its dual — x ∨ (y ∧ z) ≈ (x ∨ y) ∧ (x ∨ z) — holds identically.
So distributivity, though not obviously self-dual from its statement, is self-dual as a condition. Either law may be assumed and the other derived, which is why the class of distributive lattices is closed under passing to duals.
One inequality is free
In any lattice whatsoever, (x ∧ y) ∨ (x ∧ z) ≤ x ∧ (y ∨ z). Distributivity is exactly the assertion that this inequality is always an equality.
The inequality follows from monotonicity: each of x ∧ y and x ∧ z is below both x and y ∨ z, hence below their meet. Knowing which half is automatic makes verification of distributivity a one-sided check.
Examples and non-examples
| Lattice | Distributive? | Reason |
|---|---|---|
| Power set Su(A) under ∪, ∩ | Yes | Set-theoretic distributivity |
| Naturals under lcm, gcd | Yes | Follows from unique factorisation |
| Any chain | Yes | Meets and joins are min and max |
| Every Boolean algebra | Yes | Distributivity is one of the axioms |
| Subgroup lattice of a group | Usually no | Fails already for the Klein four-group |
| M5 | No | The diamond; modular but not distributive |
| N5 | No | The pentagon; not even modular |
| Con(A) for a general algebra | Not in general | Distributivity here defines a major class of varieties |
Why distributivity dominates the later chapters
Congruence-distributive varieties — those in which Con(A) is distributive for every member A — support a structure theory that congruence-modular varieties do not.
If one had to nominate the hypothesis that unlocks the most machinery in the subject, congruence-distributivity would be it. Lattices, Boolean algebras, Heyting algebras and all discriminator varieties satisfy it; groups, rings and modules do not.
Frequently asked questions
Does distributivity imply modularity?
Yes, strictly. Every distributive lattice is modular, but M5 is modular and not distributive, so the implication does not reverse.
Is the subgroup lattice of an abelian group distributive?
Not necessarily. It is always modular, which is a theorem of Dedekind, but the Klein four-group has three subgroups of order two forming a copy of M5 with the trivial subgroup and the whole group — so distributivity fails.
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 12-13.
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.