How to create a Hasse Diagram

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Hasse Diagram is created for POSET or Partially Ordered Set. It means that there is a set of elements in which certain element are ordered, sequenced or arranged in some way. It is usually denoted as ≤, this is not “Less than, Equal to”, this symbol shows that elements are ordered.

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Now, there is a relation among all the elements in the partial order set and it is some binary relation R, this relation must satisfy following properties

  1. Reflexive
  2. Anti-Symmetric
  3. Transitive

Example Problem

Suppose (z, |) is a partial order set where | is division operator and a | b means “a divides b”.

Reflexive Property

a | a => a divides itself and this is correct with respect to division. A number can divide itself.

Anti-Symmetric Property

a | b = b | a is correct and works only when a = b. If not, then it is not anti-symmetric.

Transitive Property

a | b => a divides b means b = a * m where m is some number. (1)
b | c => b divides c means c = b * n  where n is some number.  (2)
Using (1) and (2) we get
c = (a * m) * n
c = a * (m * n) which is same as (1). This proves the transitivity property.

Hasse Diagram

Now that we know partial order set means and a Hasse Diagram is graphical representation of posets.

Rules for Hasse Diagram

  1. If x < y, then in the graph x appears lower to y.
  2. We draw line segment between x and y only if x cover y or y cover x, it means some order is maintained between them.
Consider an example
Let A be a set, A = { 1,  3,  5,  12,  15 }, relation between elements are a | b that is., a divides b.

 

Step 1

Due to the reflexive property all elements have direct edge to itself.

 

Hasse Diagram showing all relations
Hasse Diagram showing all relations

In this diagram, it shows the relations removed all the self-directing loops.

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All Self-Directing Edges Removed

Step 2

The green lines shows transitivity, so we remove all the transitive lines. Out diagram look like the following.

Hasse Diagram without Transition and Loops

Step 3

Replace all the vertices with dots and directed edges with ordinary lines. This final diagram is called the Hasse Diagram of poset.

Hasse Diagram for A = { 1, 3, 5, 12, 15 } and relation a | b i.e., a divides b

Exercise

Here is an exercise for you to practice. The prerequisite for Hasse Diagram is to know how to represent relations using graphs.

Let A be a poset, A = { 2,  4,  6,  8 } and the relation a | b is ‘a divides b. Draw a Hasse Diagram for the poset showing all the relations.

 

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