Questions: Partially Ordered Sets and Hasse Diagrams

Question 1

In the divisibility poset on {1, 2, 3, 6}, a Hasse diagram does not draw a direct edge from 1 to 6, even though 1 divides 6. Why not?

• A. 1 and 6 are incomparable in the divisibility order
• B. The edge from 1 to 6 is implied by transitivity through the paths 1→2→6 and 1→3→6, so it would be redundant
• C. Hasse diagrams only show edges between adjacent integers
• D. The diagram omits all edges involving the minimum element

Answer: (B)

Hasse diagrams display only *covering relations* — the direct one-step connections where no intermediate element exists between them. Since 2 and 3 both lie strictly between 1 and 6 in the divisibility order, the edge 1→6 is a transitive consequence of existing edges and is deliberately omitted. The full order is recovered by taking the transitive closure of the drawn edges; omitting implied edges is precisely what makes the diagram readable rather than cluttered.

Question 2

A student reads a Hasse diagram that has two elements at the very top with no edges above them, and concludes: 'This poset has two maximum elements.' What is wrong with this reasoning?

• A. A valid poset always has exactly one element at the top of its Hasse diagram
• B. Two elements at the top are both *maximal* (nothing above them), but neither is a *maximum* unless one is above the other — a maximum must be above every element in the poset
• C. The student should call them 'greatest elements,' not 'maximum elements,' which is the correct terminology
• D. Nothing is wrong; two maximal elements and two maximum elements are the same thing

Answer: (B)

Maximal and maximum are distinct concepts. A *maximal* element has nothing strictly above it — but other elements may be incomparable to it. A *maximum* (or greatest) element is above *every* element in the poset. If two elements sit at the top with no edge between them, they are incomparable: neither is above the other, so neither is a maximum. The poset simply has no maximum element. This distinction parallels the difference between 'local maximum' and 'global maximum' in calculus.

Question 3

In a Hasse diagram, if element b appears directly above element a with a line connecting them, then b covers a — meaning there is no element strictly between them in the order.

• T. True
• F. False

Answer: True

Covering relation is exactly what a Hasse diagram encodes. We say b covers a (written a ⋖ b) if a < b and there is no c with a < c < b. The diagram draws exactly these direct connections and nothing else. This is why the diagram is compact and readable: all transitive relationships are implicit, not drawn explicitly.

Question 4

Most finite poset should have at least one maximum element — an element that is greater than or equal to most others.

• T. True
• F. False

Answer: False

A finite poset is guaranteed to have at least one *maximal* element (by finiteness), but not necessarily a *maximum* element. Consider the poset {a, b} where a and b are incomparable — neither divides the other, for instance. Both a and b are maximal (nothing is above either), but neither is a maximum because they cannot be compared. A maximum exists only when there is a single element that sits above every other element in the poset.

Question 5

Why does a Hasse diagram omit transitively implied edges, and what information (if any) is lost by doing so?

Model answer: Transitively implied edges are omitted because they are redundant: the full order can be recovered by taking the transitive closure of the drawn edges. No information is lost — the Hasse diagram is a complete representation of the poset, just compressed. Keeping all implied edges would make the diagram unreadable (a poset with many elements would be a dense tangle of lines), while the covering-relation skeleton gives the same mathematical content in a human-interpretable form.

The key principle is that a partial order is determined by its covering relations: if you know every pair (a, b) where b covers a, you can reconstruct the full order by transitivity. The Hasse diagram is essentially the Hasse graph of covering relations, which is minimal and sufficient. This is analogous to how you can describe a directed acyclic graph by its direct edges rather than listing every path.