Distributive and Modular Lattices: the M5 and N5 Criteria
Two five-element lattices decide everything. M5 and N5 are the complete obstruction set for distributivity and modularity respectively, and no third example is needed.
Engineering · Mathematics10 min readKV-MATH-0205
Learning objectives
State the distributive and modular laws and verify their self-duality.
Show that distributivity implies modularity and that the converse fails.
Recognise M5 and N5 and explain what each one witnesses.
Apply the forbidden-sublattice criteria to classify a given lattice.
Connect these conditions to congruence-distributive and congruence-modular varieties.
01The two laws
Distributivity asserts that meet distributes over join; modularity is a weakened form conditioned on a comparability hypothesis.
Distributive: x ∧ (y ∨ z) ≈ (x ∧ y) ∨ (x ∧ z) Modular: x ≤ z ⟹ x ∨ (y ∧ z) = (x ∨ y) ∧ z
The modular law can be written as a genuine identity by replacing x with x ∧ z, which makes the comparability automatic: (x ∧ z) ∨ (y ∧ z) ≈ ((x ∧ z) ∨ y) ∧ z. Both conditions are therefore equational and define varieties.
Both laws are self-dual, though for different reasons. In the distributive case the dual identity — join distributing over meet — is a consequence of the stated one, not an independent axiom; proving this is a standard exercise using absorption. In the modular case the dual statement is literally the same statement read upside down.
02Distributive implies modular
Assume distributivity and x ≤ z. Then x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) by the dual distributive law, and x ∨ z = z because x ≤ z. So x ∨ (y ∧ z) = (x ∨ y) ∧ z, which is modularity.
Key resultThe implication is strict
M5, the diamond with a bottom, a top and three pairwise incomparable middle elements, is modular but not distributive. It therefore witnesses that modularity is genuinely weaker, and it is the reason the forbidden-sublattice criterion for distributivity needs two excluded configurations rather than one.
The chain of implications runs: distributive ⟹ modular ⟹ arbitrary lattice, with both implications strict. Every chain is distributive; the subgroup lattice of an abelian group is modular; the subgroup lattice of a general group need not be even that.
03The two obstructions
M5 and N5
Lattice
Shape
Modular?
Distributive?
Witnesses
M5 (the diamond)
0 < a, b, c < 1, middle elements pairwise incomparable
Yes
No
Modularity without distributivity
N5 (the pentagon)
0 < a < b < 1 and 0 < c < 1, with c incomparable to a and b
No
No
Failure of modularity
In M5, take x = a, y = b, z = c. Then a ∧ (b ∨ c) = a ∧ 1 = a, whereas (a ∧ b) ∨ (a ∧ c) = 0 ∨ 0 = 0. Distributivity fails at a single triple. Modularity survives because no triple of M5 satisfies the comparability hypothesis in a way that produces a counterexample.
In N5, take x = a, z = b so that x ≤ z, and y = c. Then a ∨ (c ∧ b) = a ∨ 0 = a, while (a ∨ c) ∧ b = 1 ∧ b = b, and a ≠ b. Modularity fails.
04The characterisation theorems
ProcedureClassifying a lattice by forbidden sublattices
in: lattice L → out: distributive / modular / neither
input: lattice L
search for a sublattice of L isomorphic to N5
if found: L is not modular, hence not distributive → STOP
search for a sublattice of L isomorphic to M5
if found: L is modular-or-not, but definitely not distributive
if not found and no N5 either: L is distributive
if N5 absent but M5 present: L is modular and not distributive
Completeness: these are the only obstructions — the theorems assert no third forbidden configuration exists. Caveat: the search is over sublattices, meaning subsets closed under the parent's operations, not merely sub-posets of the right shape.
CautionThe shape must be a sublattice, not a sub-poset
Finding five elements arranged in the pentagon pattern under the inherited order proves nothing. The subset must be closed under the parent lattice's join and meet. A common failure is to spot a pentagon-shaped sub-poset whose join in the parent escapes the five chosen elements — that configuration does not witness non-modularity.
05Why this matters downstream
These conditions reappear immediately as conditions on congruence lattices, and that is where their real weight lies in universal algebra.
Congruence-distributive varieties
Every member has a distributive congruence lattice. Lattices themselves qualify. Jónsson's lemma applies, giving strong control over subdirectly irreducibles.
Congruence-modular varieties
Weaker, but enough to support the commutator theory and the centre. Groups, rings and modules all qualify.
Mal'cev characterisations
Each condition is equivalent to the existence of certain terms — Jónsson terms for distributivity, Day terms for modularity — which converts a lattice-theoretic condition into a checkable syntactic one.
Frequently asked
Is every modular lattice distributive on finite subsets?
No. M5 is finite, modular and not distributive, so finiteness provides no rescue. The relationship between the two conditions is not affected by cardinality at all — both are equational, hence inherited by every subalgebra regardless of size.
Are subgroup lattices always modular?
Only for abelian groups, and more generally for groups in which all subgroups are normal. The lattice of normal subgroups of any group is modular. The full subgroup lattice of a non-abelian group need not be: the symmetric group on three letters has a subgroup lattice containing a pentagon.
How hard is it to search for M5 and N5 in practice?
For a finite lattice it is a finite search over five-element subsets, so it is decidable but grows quickly. In practice one checks the identity directly on triples, which is cubic rather than quintic. For infinite lattices the criterion is a theoretical characterisation rather than an algorithm, and one verifies the identity instead.
Sources and further reading
S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Millennium Edition (a corrected re-typesetting of Springer GTM 78, 1981).
G. Grätzer, Universal Algebra, 2nd edition, Springer.
R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties, Volume I.
Original KEVOS® explanatory article. Written from the topic map of the cited works; no text is reproduced from them.
Handbook application: from concept to controlled practice
Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Distributive and Modular Lattices: the M5 and N5 Criteria. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.
The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.
Treat Distributive and Modular Lattices: the M5 and N5 Criteria as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—distributive, modular, lattices, theorems, algebra—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.
For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.
A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.
Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.
Step-by-step operating method
Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.
Worked-example protocol
Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.
Stage
Record
Quality check
Input
Objects, domain, notation, assumptions
Every symbol is defined
Method
Permitted operation or cited result at each step
All hypotheses hold
Output
Exact result and representation
Correct type, domain and form
Verification
Substitution, invariant or alternative derivation
Independent agreement
Boundary test
Zero, identity, degenerate or failed hypothesis
Scope is understood
Common failure modes and recovery actions
1. Watch for
Using a theorem without checking every hypothesis.
Recovery: Return to the governing definition or requirement and restate the decision in one sentence.
2. Watch for
Treating a suggestive example as a proof of the general case.
Recovery: Separate evidence from assumption, assign an owner and set a date for validation.
3. Watch for
Changing notation or conventions part-way through an argument.
Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.
4. Watch for
Hiding a division-by-zero, convergence, finiteness or commutativity assumption.
Recovery: Record the consequence, decision and rationale, then update the controlled baseline.
5. Watch for
Reporting a computed result without a residual, substitution or structural check.
Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.
Review checklist
Can every symbol be traced to a definition or prior result?
Which hypothesis does each major step use?
Does the method cover zero, identity, degenerate and boundary cases?
Can the conclusion be checked by a second representation or calculation?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?
Questions for deeper application
What is the most important distinction a practitioner must preserve when applying Distributive and Modular Lattices: the M5 and N5 Criteria?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Which assumption about distributive would change the result most if it proved false?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
What evidence would allow an independent reviewer to reproduce or challenge the conclusion?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Which boundary, exception or failure case has not yet been tested?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
What must be handed over, monitored or reviewed after the immediate work is complete?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Authoritative references and use notes
The sources below were selected as institutional or primary guidance for the broader practice. They support the handbook method; they do not imply that every statement or clause in a source applies to every project. Confirm the current edition, jurisdiction, contract and application before treating any requirement as mandatory.
MIT OpenCourseWare — Number Theory I — Massachusetts Institute of Technology. Used for algebraic and analytic number theory. Accessed 2026-08-13.
MIT OpenCourseWare — Algebra I — Massachusetts Institute of Technology. Used for groups, vector spaces, linear transformations and linear groups. Accessed 2026-08-13.
