Hasse Diagram — Definition, Formula & Examples

A Hasse diagram is a drawing of a finite partially ordered set (poset) where elements are represented as points, and an edge connects element

a

below element

b

whenever

a < b

with no element in between. Edges implied by transitivity are omitted, and arrows are unnecessary because the vertical position encodes direction.

Given a finite poset $(S, \leq)$ , the Hasse diagram is the directed graph of the covering relation: $a$ is connected to $b$ (with $b$ drawn above $a$ ) if and only if $a \lessdot b$ , meaning $a < b$ and there is no $c \in S$ with $a < c < b$ . The transitive and reflexive edges are suppressed.

How It Works

To build a Hasse diagram, first list all pairs

(a,b)

where

a < b

. Then remove any pair

(a,b)

for which some

c

satisfies

a < c < b

; the remaining pairs are the covering relations. Draw each element as a node, placing smaller elements lower. Connect each covering pair with a line segment. Because lower always means smaller, no arrowheads are needed.

Worked Example

Problem: Draw the Hasse diagram for the poset

(\{1,2,3,6\}, \mid)

, where

\mid

denotes the divisibility relation.

Step 1: List all divisibility pairs where

a \mid b

and

a \neq b

1\mid 2,\; 1\mid 3,\; 1\mid 6,\; 2\mid 6,\; 3\mid 6

Step 2: Remove pairs implied by transitivity. Since

1\mid 2

and

2\mid 6

, the pair

1\mid 6

is not a covering relation. All other pairs have no element strictly between them.

\text{Covering relations: } 1\lessdot 2,\; 1\lessdot 3,\; 2\lessdot 6,\; 3\lessdot 6

Step 3: Place 1 at the bottom, 2 and 3 at the middle level (side by side), and 6 at the top. Draw edges for each covering pair.

Answer: The Hasse diagram has four nodes arranged in a diamond shape: 1 at the bottom connected to 2 and 3, which are each connected to 6 at the top. No edge connects 1 directly to 6.

Why It Matters

Hasse diagrams appear throughout discrete mathematics, abstract algebra, and lattice theory whenever you need to visualize ordering structures. In computer science, they help illustrate class hierarchies, task dependencies, and the structure of Boolean algebras used in digital logic design.

Common Mistakes

Mistake: Including edges that follow from transitivity, such as drawing a line from 1 to 6 in the divisibility example.

Correction: Only draw covering relations — pairs

a \lessdot b

with no element strictly between

a

and

b

. Transitive edges clutter the diagram and violate the convention.