Discrete Mathematics | Hasse Diagrams

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Hasse Diagram

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.


To draw a Hasse diagram, the provided set must be a poset. 
A poset or partially ordered set A is a pair, ( B, [Tex]\leq          [/Tex]) of a set B whose elements are called the vertices of A and obeys following rules: 
 

  1. Reflexivity → p [Tex]\leq          [/Tex]p [Tex]\forall          [/Tex]p [Tex]\in          [/Tex]B
  2. Anti-symmetric → p [Tex]\leq          [/Tex]q and q [Tex]\leq          [/Tex]p if p=q
  3. Transitivity → if p [Tex]\leq          [/Tex]q and q [Tex]\leq          [/Tex]r then p [Tex]\leq          [/Tex]r


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

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

example 1


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[Tex]_{12}          [/Tex], /) 

Explanation –

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

example 2


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. 

For regular 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.



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.


Solved Examples:

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.[Tex]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.[/Tex]

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


Example 2:


[Tex]Consider the poset ((P, \leq)) where (P = {a, b, c, d}) and the relation (\leq) is defined as (a \leq b), (a \leq c), (b \leq d), (c \leq d). Elements: a, b, c, d Order Relations: (a \leq b), (a \leq c), (b \leq d), (c \leq d).[/Tex][Tex]Consider the poset ((P, \leq)) where (P = {a, b, c, d}) and the relation (\leq) is defined as (a \leq b), (a \leq c), (b \leq d), (c \leq d). Elements: a, b, c, d Order Relations: (a \leq b), (a \leq c), (b \leq d), (c \leq d).[/Tex]

d / \ b c \ / a

Practice Problems

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

Problem 2: [Tex]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.[/Tex]

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

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

Problem [Tex]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.[/Tex]

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

Problem 7:[Tex]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.[/Tex]

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

Problem [Tex]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.[/Tex]

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

Interlinking of Related Articles:

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.

FAQs on Hasse Diagrams

What is a Hasse diagram?

A Hasse diagram is a type of mathematical diagram used to represent a finite partially ordered set (poset) by displaying its transitive reduction. It shows the order relations between elements without redundant edges.

How is a Hasse diagram constructed?

To construct a Hasse diagram:

1. List the elements of the poset.

2. Determine the ordering relations between the elements.

3. Draw the elements as points in a plane.

4. Connect the points with edges to represent the order relations, ensuring transitive reduction (i.e., if \(a \leq b\) and \(b \leq c\), do not draw a direct line from \(a\) to \(c\)).

What is a poset in the context of Hasse diagrams?

A poset, or partially ordered set, is a set equipped with a partial order relation. In a poset, not every pair of elements needs to be comparable, but the order relation must be reflexive, antisymmetric, and transitive.

What does transitive reduction mean?

Transitive reduction of a relation means removing all redundant edges while maintaining the same reachability between elements. In other words, if \(a \leq b\) and \(b \leq c\), the direct relation \(a \leq c\) is implied and thus not shown in the Hasse diagram.

Can Hasse diagrams be used for infinite sets?

Hasse diagrams are typically used for finite sets because they provide a clear visual representation. For infinite sets, the diagrams can become unwieldy and lose their effectiveness.

How do you identify elements and order relations in a poset?

Elements are the individual items in the set, and order relations define how these elements relate to one another according to the partial order. For example, in a poset of divisibility, if an element \(a\) divides element \(b\), there is an order relation \(a \leq b\).

What are some common applications of Hasse diagrams?

Hasse diagrams are used in various fields of mathematics, including lattice theory, order theory, and graph theory. They help in visualizing the structure of posets, analyzing hierarchical structures, and simplifying complex relationships.

What is the difference between a Hasse diagram and a directed graph?

While both Hasse diagrams and directed graphs (digraphs) represent relations between elements, Hasse diagrams specifically represent posets with transitive reduction. Digraphs can represent any directed relationships, including those that are not transitive reductions.

How does a Hasse diagram help in understanding the structure of a poset?

Hasse diagrams provide a visual representation of the hierarchical structure of a poset, making it easier to see how elements are related. They simplify the analysis of order relations by removing redundant connections.

What are some tips for drawing a Hasse diagram?

Start by listing the elements and determining the order relations. – Arrange elements in levels, with lower elements at the bottom and higher elements at the top. – Draw edges to represent direct order relations, ensuring no transitive edges are included. – Use vertical lines or arrows to maintain clarity and avoid crossing edges.

Are there any tools available for drawing Hasse diagrams?

Yes, there are several software tools and online platforms that can help in drawing Hasse diagrams, such as graph drawing software (e.g., Graphviz), mathematical tools like Mathematica, and specialized diagramming tools.

What is the significance of the Hasse diagram’s name?

The Hasse diagram is named after the German mathematician Helmut Hasse, who made significant contributions to algebra and number theory. His work in posets led to the development of these diagrams for visualizing order relations.



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