Orderings & Lattices · step 5 of 13

Lattices: divisibility & subset orders

A lattice is a poset in which every pair of elements has both a supremum (join, ∨) and an infimum (meet, ∧). The two canonical examples are the divisors of n under divisibility, where join is lcm and meet is gcd, and the power set 𝒫(S) under inclusion, where join is union and meet is intersection. Checking the lattice property means checking every pair, and one bad pair is enough to refute it.

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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 every pair for a joinCompute the upper bounds of {a,b} and check they have a least element. Incomparable pairs are where failures occur.
Test every pair for a meetSame downward. Both must exist for every pair, not just for some.
Recognise the standard latticesDivisors of n: join = lcm, meet = gcd. Subsets: join = ∪, meet = ∩. Both are always lattices.
Refute with a specific pairName a and b, list their upper bounds, and show those bounds have no least element.

Worked example

Is the poset with Hasse diagram bottom 0, two middle elements a and b, and two tops c and d (both above a and b) a lattice?

Meets are fine: a ∧ b = 0, and every pair has a lower bound.
Consider the join of a and b. Upper bounds are c and d.
c and d are incomparable, so the set {c, d} has no least element.
Hence a ∨ b does not exist.

Answer. Not a lattice — a and b have two minimal upper bounds and therefore no supremum. This "diamond with two tops" is the standard counterexample.

Where marks get dropped

These are the specific errors that cost credit on lattices: divisibility & subset orders questions — QED's rubric penalises each of them separately.

Checking only comparable pairs. Comparable elements trivially have meets and joins; the whole question lives with incomparable pairs.
Assuming any finite poset with a top and bottom is a lattice. The example above has both and still fails.
Confusing the lattice operations with set operations in a divisibility lattice — join is lcm, not union of divisors.

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Lattices: divisibility & subset orders — frequently asked questions

Is every finite lattice bounded?

Yes. Taking the join of all elements gives a greatest element and the meet of all gives a least, so finite lattices always have ⊤ and ⊥.

Are divisors of n always a lattice?

Yes, for every positive n, with gcd and lcm as the operations. When n is squarefree with k prime factors the lattice is isomorphic to 𝒫 of a k-element set.

What is a sublattice?

A subset closed under both meet and join. Being a subset that happens to be a lattice is not enough — the operations must agree with the parent’s.

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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