Orderings & Lattices · step 3 of 13

Minimal, maximal, least & greatest elements

These four words are the most confused pair of pairs in order theory. An element is minimal if nothing is strictly below it; it is least if it is below everything. Minimal elements can be many and often are; a least element is unique when it exists and frequently does not. In a Hasse diagram, minimal elements are those with no downward edges — the bottom row.

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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.

Minimal — look downa is minimal if no b satisfies b ⊏ a. In the diagram, a has no edges going down.
Least — compare with everythinga is least if a ⊑ b for every b in the poset. It must be comparable to all elements.
Maximal and greatest — mirror the aboveMaximal means nothing strictly above; greatest means above everything.
Check uniquenessLeast and greatest are unique when they exist, by antisymmetry. Minimal and maximal elements need not be.

Worked example

For the divisors of 12 excluding 1, ordered by divisibility, find the minimal, maximal, least and greatest elements.

Elements: 2, 3, 4, 6, 12.
Minimal: 2 (nothing in the set divides it properly) and 3 — two minimal elements.
Least: would need to divide everything, including both 2 and 3. No element does.
Maximal: only 12; and since 12 is a multiple of every element, it is also greatest.

Answer. Minimal: 2 and 3. Least: none. Maximal: 12. Greatest: 12.

Where marks get dropped

These are the specific errors that cost credit on minimal, maximal, least & greatest elements questions — QED's rubric penalises each of them separately.

Using "minimal" and "least" interchangeably. The example has two minimal elements and no least element — the distinction is exactly what is being tested.
Assuming a finite poset has a least element. Every finite non-empty poset has at least one minimal element, but a least element may not exist.
Forgetting that a greatest element is automatically the unique maximal one, so finding two maximal elements proves no greatest exists.

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Minimal, maximal, least & greatest elements — frequently asked questions

Can a poset have several least elements?

No. If a and b are both least then a ⊑ b and b ⊑ a, so antisymmetry gives a = b. Uniqueness is automatic.

Does every finite poset have a maximal element?

Yes, provided it is non-empty: walk upward from any element and the process must terminate. Infinite posets like ℤ can fail to have one.

What are top and bottom?

Alternative names for greatest and least, written ⊤ and ⊥. A bounded poset has both, which is required for a complemented lattice.

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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