Orderings & Lattices · step 12 of 13

Order isomorphism & comparing posets

Two posets are order isomorphic if there is a bijection f between them with a ⊑ b iff f(a) ⊑ f(b) — the structure, not the labels, is what matters. Proving isomorphism means exhibiting the map; refuting it means finding an invariant that differs, such as the number of elements, height, width, or the number of minimal elements.

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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.

Compare the cheap invariants firstCardinality, height, width, number of minimal and maximal elements. A mismatch ends the question immediately.
Match structurally distinguished elementsThe least element must map to the least element, minimal to minimal, and so on. This usually forces most of the map.
Verify both directions of the conditionIt is not enough that f preserves ⊑; f⁻¹ must too. A monotone bijection is not automatically an isomorphism.
Refute with a local invariantIf cardinalities match, count elements covering exactly two others, or compare the multiset of degrees in the Hasse diagram.

Worked example

Are the divisors of 6 and the subsets of {x, y} order isomorphic under divisibility and inclusion?

Divisors of 6: 1, 2, 3, 6. Subsets: ∅, {x}, {y}, {x,y}. Both have 4 elements.
Map 1 ↦ ∅, 2 ↦ {x}, 3 ↦ {y}, 6 ↦ {x,y}.
Check: 2 ∣ 6 and {x} ⊆ {x,y} ✓; 2 and 3 incomparable, {x} and {y} incomparable ✓.
Every divisibility relation matches an inclusion and vice versa.

Answer. Yes — both are the Boolean lattice on two atoms, and the map sends each divisor to its set of prime factors.

Where marks get dropped

These are the specific errors that cost credit on order isomorphism & comparing posets questions — QED's rubric penalises each of them separately.

Accepting a monotone bijection as an isomorphism. The inverse must be monotone too — a bijection from an antichain to a chain is monotone but not an isomorphism.
Comparing only cardinalities. Four-element posets come in many non-isomorphic shapes.
Forgetting that isomorphism must respect incomparability as well as comparability.

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Order isomorphism & comparing posets — frequently asked questions

Which invariants are worth checking?

Size, height, width, counts of minimal and maximal elements, and the number of elements at each height. Together these settle most exam-sized cases.

Is the divisor lattice of a squarefree number always a Boolean lattice?

Yes. If n has k distinct prime factors and is squarefree, its divisors correspond exactly to the subsets of those k primes.

How many posets are there on 4 elements?

16 up to isomorphism. Enumerating small posets is a standard exercise in exactly this invariant-based reasoning.

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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