Lattice Theory Foundations
The M5 and N5 Forbidden-Sublattice Theorems
Two five-element lattices decide modularity and distributivity by exclusion. These theorems convert conditions stated as identities into a finite, checkable structural test.
Learning objectives
- Describe M5 and N5 precisely
- State both forbidden-sublattice theorems
- Use the theorems to settle modularity and distributivity in examples
The two lattices
<em>N</em><sub>5</sub> — the pentagon
Five elements: 0 < a < b < 1 forming a three-element chain between bottom and top, plus a single element c with 0 < c < 1, incomparable to both a and b.
<em>M</em><sub>5</sub> — the diamond
Five elements: bottom 0, top 1, and three pairwise incomparable elements a, b, c strictly between. Any two of the middle elements meet at 0 and join at 1.
In N5, modularity fails: a ≤ b, yet a ∨ (c ∧ b) = a ∨ 0 = a, while (a ∨ c) ∧ b = 1 ∧ b = b. Since a ≠ b, the modular law is violated.
In M5, modularity holds but distributivity fails: a ∧ (b ∨ c) = a ∧ 1 = a, while (a ∧ b) ∨ (a ∧ c) = 0 ∨ 0 = 0.
The theorems
A lattice is modular if and only if it has no sublattice isomorphic to N5.
A lattice is distributive if and only if it has no sublattice isomorphic to either M5 or N5.
Combining the two gives a clean trichotomy: a lattice is distributive when it excludes both; modular but not distributive when it excludes N5 but contains M5; and non-modular when it contains N5.
Why these theorems are valuable
Distributivity as stated quantifies over all triples of elements — an infinite check in an infinite lattice. The forbidden-sublattice form replaces this with a search for two specific five-element configurations. In a finite lattice the search is finite and mechanical; in an infinite one it is often settled by a structural argument.
Applications
| Lattice | Contains N5? | Contains M5? | Verdict |
|---|---|---|---|
| Subgroup lattice of S3 | Yes | — | Not modular |
| Subgroup lattice of the Klein four-group | No | Yes | Modular, not distributive |
| Subspace lattice of a plane | No | Yes | Modular, not distributive |
| Power set of any set | No | No | Distributive |
| Any chain | No | No | Distributive |
| Divisors of n under lcm, gcd | No | No | Distributive |
The subspace lattice of a vector space of dimension at least two always contains M5: take three distinct lines through the origin in a plane. This is why projective geometry lives in the modular but not distributive world, and it is the origin of the theory of modular geometric lattices.
Frequently asked questions
Does 'sublattice' matter here, or would sub-poset do?
It matters entirely. The theorems require a genuine sublattice — a subset closed under the ambient join and meet. A sub-poset shaped like N5 whose joins are computed differently in the ambient lattice proves nothing.
Are there analogous criteria for other lattice conditions?
Yes, for several. Many lattice properties admit characterisation by excluded sublattices or excluded quotients, though the excluded sets are generally larger and less memorable than these two.
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 13-16.
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.