Hasse Diagrams | Discrete Mathematics

A Hasse diagram is a graphical representation of the relation of elements of a partially ordered set (poset) with an implied upward orientation. A point is drawn for each element of the partially ordered set (poset) and joined with the line segment according to the following rules: 

• If p<q in the poset, then the point corresponding to p appears lower in the drawing than the point corresponding to q.
• The two points p and q will be joined by line segment if p is related to q.

The Hasse diagram doesn’t include any redundant or obvious connections. For example, if A < B and B < C, the diagram skips the line between A and C because it’s already obvious from the other two. The items higher in the diagram are "greater" than the ones below them. A hasse diagram is a clean and straightforward way to visualize order. To draw a Hasse diagram, the provided set must be a poset. 

Poset

A poset or partially ordered set A is a pair, ( B, \leq ) of a set B whose elements are called the vertices of A and obeys following rules: 

• Reflexivity → p ≤ p \forall          p \in          B
• Anti-symmetric → p ≤ q and q ≤ p if p=q
• Transitivity → if p ≤ q and q ≤ r then p ≤ r

Steps to Draw Hasse Diagram

• List the elements and their relationships : Write down all the elements in the set and figure out how they are related (like "smaller than," "divides," etc.).
• Draw all elements as dots: : Each element is represented by a dot. Start with smaller elements at the bottom and bigger ones at the top.
• Connect related dots: : Draw lines between the dots if one element is directly related to another (like a < b or a divides b).
• Remove unnecessary connections: If a relationship can be figured out through other connections, skip it. For example, if a < b and b < c, you don’t need a line from a to c.

Examples of Hasse Diagram

Example 1: Draw Hasse diagram for ({3, 4, 12, 24, 48, 72}, /) 

Solution -

According to above given question first, we have to find the poset for the divisibility. 
Let the set is A. 
A={(3 \prec 12), (3 \prec  24), (3 \prec 48), (3 \prec 72), (4 \prec 12), (4 \prec 24), (4 \prec 48), (4 \prec 72), (12 \prec 24), (12 \prec 48), (12 \prec          72), (24 \prec 48), (24 \prec 72)} 
So, now the Hasse diagram will be: 

In above diagram, 3 and 4 are at same level because they are not related to each other and they are smaller than other elements in the set. The next succeeding element for 3 and 4 is 12 i.e, 12 is divisible by both 3 and 4. Then 24 is divisible by 3, 4 and 12. Hence, it is placed above 12. 24 divides both 48 and 72 but 48 does not divide 72. Hence 48 and 72 are not joined. 
We can see transitivity in our diagram as the level is increasing. 

Example 2: Draw Hasse diagram for (D_{12} , /) 

Solution:

Here, D_{12} means set of positive integers divisors of 12. 
So, D_{12}   ={1, 2, 3, 4, 6, 12} 
poset A = {(1 \prec 2), (1 \prec 3), (1 \prec 4), (1 \prec 6), (1 
\prec 12), (2 \prec  4), (2 \prec  6), (2 \prec 12), (3 \prec 6), (3 \prec 12), (4 \prec 12), (6 \prec    12)} 
So, now the Hasse diagram will be- 

In above diagram, 1 is the only element that divides all other elements and smallest. Hence, it is placed at the bottom. Then the elements in our set are 2 and 3 which do not divide each other hence they are placed at same level separately but divisible by 1 (both joined by 1). 4 is divisible by 1 and 2 while 6 is divisible by 1, 2 and 3 hence, 4 is joined by 2 and 6 is joined by 2 and 3. 12 is divisible by all the elements hence, joined by 4 and 6 not by all elements because we have already joined 4 and 6 with smaller elements accordingly. 

Note - Greatest and Least element in Hasse diagram are only one. 

In Example-1, 
Maximal elements are 48 and 72 since they are succeeding all the elements. 
Minimal elements are 3 and 4 since they are preceding all the elements. 
Greatest element does not exist since there is no any one element that succeeds all the elements. 
Least element does not exist since there is no any one element that precedes all the elements. 

In Example-2, 
Maximal and Greatest element is 12 and Minimal and Least element is 1.

NOTE: An element can be both maximal and minimal 

Example-

Here a,b,c are maximal as well as minimal.

Example 3: Consider the poset ((P, ≤)) where (P = {1, 2, 3, 4, 6, 12}) and the relation (≤) is divisibility.

Solution:

Elements: 1, 2, 3, 4, 6, 12

Order Relations: 1 divides all, 2 divides 4 and 6, 3 divides 6, 4 divides 12, 6 divides 12.Example 1: Consider the poset ((P, \leq)) where (P = {1, 2, 3, 4, 6, 12}) and the relation (\leq) is divisibility. Elements: 1, 2, 3, 4, 6, 12 Order Relations: 1 divides all, 2 divides 4 and 6, 3 divides 6, 4 divides 12, 6 divides 12.

12
/ \
6 4
| \ |
3 2
\ /
1

Example 4: Consider the poset ((P, ≤)) where (P = {a, b, c, d}) and the relation ≤ is defined as (a ≤ b), (a ≤ c), (b ≤ d), (c ≤ d).

Solution :

Consider the poset ((P, ≤)) where (P = {a, b, c, d}) and the relation (≤) is defined as (a ≤ b), (a ≤ c), (b ≤ d), (c ≤ d). Elements: a, b, c, d Order Relations: (a ≤ b), (a ≤ c), (b ≤ d), (c ≤ d).

Consider the poset ((P, ≤)) where (P = {a, b, c, d}) and the relation (≤) is defined as (a ≤ b), (a ≤ c), (b ≤ d), (c ≤ d). Elements: a, b, c, d Order Relations: (a ≤ b), (a ≤ c), (b ≤ d), (c ≤ d).

d
/ \
b c
\ /
a

Properties of Hasse Diagram 

• Maximal element is an element of a POSET which is not less than any other element of the POSET. Or we can say that it is an element which is not related to any other element. Top elements of the Hasse Diagram.
• Example- In the diagram above, we can say that 4,2,3,6,1 are related to 12 (ordered by division e.g. (4,/) ) but 12 is not related to any other. (As Hasse Diagram is upward directional).
• Minimal element is an element of a POSET which is not greater than any other element of the POSET. Or we can say that no other element is related to this element. Bottom elements of the Hasse Diagram.
• Example- In the diagram above, we can say that 1 is related to 2,3,4,6,12 (ordered by division e.g. (4,/) ) but no element is related to 1. (As Hasse Diagram is upward directional).
• Greatest element (if it exists) is the element succeeding all other elements.
• Least element is the element that precedes all other elements.

Practice Problems on Hasse Diagram

Problem 1:Given the set (P = {1, 2, 4, 8}) with (\leq) being divisibility, draw the Hasse diagram.

Problem 2: Given the set (P = {a, b, c, d, e}) with the relations (a \leq b), (a \leq c), (b \leq d), (c \leq d), (d \leq e), draw the Hasse diagram.

Problem 3: Given the set (P = {1, 3, 9, 27}) with (\leq) being divisibility, draw the Hasse diagram.

Problem 4: Given the set (P = {2, 5, 10, 20}) with (\leq) being divisibility, draw the Hasse diagram

Problem 5:Given the set (P = {a, b, c, d, e}) with the relations (a \leq c), (b \leq c), (c \leq d), (d \leq e), draw the Hasse diagram.

Problem 6:Given the set (P = {x, y, z, w}) with the relations (x \leq y), (y \leq z), (z \leq w), draw the Hasse diagram.

Problem 7:Given the set (P = {m, n, o, p, q}) with the relations (m \leq n), (m \leq o), (n \leq p), (o \leq p), (p \leq q), draw the Hasse diagram.

Problem 8:Given the set (P = {4, 8, 16, 32}) with (\leq) being divisibility, draw the Hasse diagram.

Problem 9:Given the set (P = {r, s, t, u, v}) with the relations (r \leq s), (s \leq t), (t \leq u), (u \leq v), draw the Hasse diagram.

Problem 10:Given the set (P = {1, 2, 5, 10}) with (\leq) being divisibility, draw the Hasse diagram.

Related Articles:

• Graph Theory
• Set Theory
• Partial Orders and Lattices
• Discrete Mathematics

Conclusion

Hasse diagrams are an invaluable tool in discrete mathematics, providing a clear and concise way to visualize the structure of partially ordered sets (posets). By representing the elements and their order relations without redundancy, Hasse diagrams make it easier to analyze and understand complex relationships within a set. Their applications extend to various fields, including lattice theory, graph theory, and beyond. Whether you're a student or a professional, mastering the construction and interpretation of Hasse diagrams can enhance your ability to work with hierarchical data and posets. With practice and the right tools, you can effectively utilize Hasse diagrams to simplify and solve problems in discrete mathematics.