Orderings & Lattices · step 10 of 13

Product & lexicographic orders

Two natural orders live on A × B. The product order compares componentwise — (a₁,b₁) ⊑ (a₂,b₂) iff a₁ ⊑ a₂ AND b₁ ⊑ b₂ — and is generally only partial. The lexicographic order compares first coordinates and only consults the second on a tie; it is total whenever both factors are. Dictionary order and version numbers are lexicographic; Pareto dominance is the product order.

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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.

Product order — both must agree(1,5) and (2,3) are incomparable in the product order, since 1 < 2 but 5 > 3.
Lex order — first coordinate winsCompare a₁ with a₂ first. Only if they are equal does b decide. So (1,5) ≺ (2,3) lexicographically.
Check totalityLex order on totally ordered factors is total. The product order almost never is.
Check well-ordering separatelyLex order on ℕ × ℕ is a well-order; lex order on ℕ × ℤ is not, since the second coordinates descend forever.

Worked example

On ℕ × ℕ, compare (2,5) and (3,1) in the product order and in the lexicographic order.

Product: need both coordinates to compare the same way. 2 < 3 ✓ but 5 > 1 ✗.
So (2,5) and (3,1) are incomparable in the product order.
Lex: compare first coordinates, 2 < 3.
That decides it — the second coordinates are never consulted.

Answer. Incomparable in the product order; (2,5) ≺ (3,1) in the lexicographic order.

Where marks get dropped

These are the specific errors that cost credit on product & lexicographic orders questions — QED's rubric penalises each of them separately.

Treating the product order as total. It is partial even when both factors are chains — that is exactly why Pareto optimality admits many incomparable solutions.
Comparing the second coordinate in lex order when the first coordinates differ. It is only a tie-breaker.
Assuming lex order preserves well-ordering. It does for ℕ × ℕ but fails as soon as a factor has infinite descending chains.

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Product & lexicographic orders — frequently asked questions

Which order does a dictionary use?

Lexicographic, on strings rather than pairs: compare position by position and the first difference decides, with a shorter prefix coming first.

Is the product order a lattice order?

Yes, if both factors are lattices: meets and joins are computed componentwise. This makes ℕⁿ under componentwise ≤ a lattice.

Where does the product order appear in practice?

Multi-objective optimisation. A solution dominates another only if it is at least as good in every objective, which is precisely the product order, and the maximal elements form the Pareto frontier.

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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