Orderings & Lattices · step 1 of 13

Partial vs total orders

A partial order is a relation that is reflexive, antisymmetric and transitive. It is total (or linear) when additionally any two elements are comparable — for all a, b either a ⊑ b or b ⊑ a. Divisibility on ℕ and subset inclusion on 𝒫(S) are the standard partial orders that are not total, and finding a single incomparable pair is all it takes to prove that.

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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.

Check the three axiomsReflexivity: a ⊑ a always. Antisymmetry: a ⊑ b and b ⊑ a force a = b. Transitivity: chains compose.
Test comparability separatelyTotality is a fourth, independent condition. A relation can be a perfectly good partial order without it.
Refute totality with one pairExhibit a and b with neither a ⊑ b nor b ⊑ a. For divisibility, 2 and 3 work.
Watch for strict versus non-strictThe strict order < is irreflexive and transitive. Some courses define orders strictly; check which convention the question uses.

Worked example

Show divisibility on {1,2,3,4,6,12} is a partial order but not total.

Reflexive: a ∣ a for every a, since a = a·1.
Antisymmetric: if a ∣ b and b ∣ a with a, b positive, then a ≤ b and b ≤ a, so a = b.
Transitive: if b = ak and c = bm then c = a(km), so a ∣ c.
Totality: 2 ∤ 3 and 3 ∤ 2, so 2 and 3 are incomparable.

Answer. It is a partial order but not a total order, with 2 and 3 as an incomparable pair.

Where marks get dropped

These are the specific errors that cost credit on partial vs total orders questions — QED's rubric penalises each of them separately.

Assuming antisymmetry needs the elements to be distinct. It says a ⊑ b and b ⊑ a imply equality, so loops never violate it.
Applying divisibility antisymmetry over ℤ. On the negative integers 2 ∣ −2 and −2 ∣ 2 with 2 ≠ −2, so divisibility is only a partial order on the positives.
Claiming a relation is total without checking every pair — it only takes one incomparable pair to break it.

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Partial vs total orders — frequently asked questions

What is a preorder?

A relation that is reflexive and transitive but not necessarily antisymmetric. Quotienting by the induced equivalence "a ⊑ b and b ⊑ a" turns any preorder into a genuine partial order.

Is ≤ on ℝ a total order?

Yes — any two reals are comparable, which is exactly why the real line can be drawn as a line rather than a diagram.

Why do we care about non-total orders?

Because most naturally occurring orderings are partial: task dependencies, subtyping, subsets, divisibility. Totality is the special case, not the norm.

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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