Orderings & Lattices · step 9 of 13

Distributive & complemented lattices

A lattice is distributive when a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) for all elements — and remarkably, that identity holds if and only if its dual does. It is complemented when it is bounded and every element a has some b with a ∧ b = ⊥ and a ∨ b = ⊤. A lattice that is both is a Boolean algebra, which is why 𝒫(S) with ∩, ∪ and complement is the model case.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Test distributivity on a tripleCompute both sides for a well-chosen a, b, c. Incomparable triples in the middle of the diagram are where it fails.
Use the forbidden sublatticesA lattice is distributive iff it contains neither the diamond M₃ nor the pentagon N₅ as a sublattice. Spotting one is a complete disproof.
Find complements pair by pairFor each a, look for b with meet ⊥ and join ⊤. Elements strictly between ⊥ and ⊤ in a chain never have complements.
Check uniquenessIn a distributive lattice complements are unique when they exist; in M₃ each element has two, which is how distributivity fails.

Worked example

Show the diamond M₃ — bottom 0, three incomparable middles a, b, c, top 1 — is complemented but not distributive.

Complements: a ∧ b = 0 and a ∨ b = 1, so b complements a; likewise c complements a.
Every element has a complement, so the lattice is complemented.
Distributivity test: a ∧ (b ∨ c) = a ∧ 1 = a.
But (a ∧ b) ∨ (a ∧ c) = 0 ∨ 0 = 0, and a ≠ 0.

Answer. M₃ is complemented but not distributive — and the failure shows up as a having two different complements.

Where marks get dropped

These are the specific errors that cost credit on distributive & complemented lattices questions — QED's rubric penalises each of them separately.

Testing distributivity only on comparable elements, where it holds automatically.
Assuming complements are unique. Uniqueness needs distributivity, and M₃ is the standard counterexample.
Confusing complemented with bounded. Bounded just means ⊤ and ⊥ exist; complemented is a much stronger condition on every element.

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Distributive & complemented lattices — frequently asked questions

What are M₃ and N₅?

The diamond (three incomparable elements between ⊥ and ⊤) and the pentagon (a two-element chain alongside a single element). A lattice is distributive iff it contains neither as a sublattice — Birkhoff’s criterion.

Is every distributive lattice a Boolean algebra?

No — a chain of three elements is distributive but the middle element has no complement. Boolean algebras need both properties.

Is the divisor lattice of n distributive?

Yes, always, since gcd and lcm distribute over each other. It is complemented exactly when n is squarefree.

The rest of Orderings & Lattices

Partial orders, Hasse diagrams, bounds, lattices. Each subtopic below has its own method, worked example and mark-losing traps.

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