This document contains information about a student named Kiran Kumar Malik enrolled in the first semester of the computer science branch at Bhubaneswar campus. It defines maximal and minimal elements in a partially ordered set (poset) and provides examples to identify these elements from Hasse diagrams. It also contains solutions to questions asking to find the maximal, minimal, greatest and least elements for different posets defined by the divides relation.

1. Name: Kiran Kumar Malik
Registration no: 200301120128
Branch: B-Tech in Computer Science and Engineering's
Section: D
Semester: 1st Sem
Campus: Bhubaneswar

2. Maximal and Minimal elements of a poset
An element of a poset is called maximal if it is not less then any element of the
poset. That is, a is maximal in the poset (S, ≼) if there is no b ∈ S such that a ≺ b.
Similarly, an element of a poset is called minimal if it is not grater than any element
of the poset. That is, a is minimal if there is no element b ∈ S such that b ≺ a.
Maximal and minimal elements are easy to spot using a Hasse diagram. They are the
“top” and “bottom” elements in the diagram
Note:
(i). The minimal and maximal members of a partially ordered set need not unique.
(ii). Maximal and minimal elements are easily spot (calculated) from the Hasse diagram.
They are the ‘top(maximal) and 'bottom' (minimal ) elements in the diagram.

3. Example 1: Which elements of the poset ({2,4,5,10,12,20,25}, |) are
maximal, and which are minimal?
The Hasse diagram in figure 1 for this poset shows that the maximal elements are12, 20,
and 25, and the minimal elements are 2 and 5. As this example shows, a poset can have
more than one maximal elements and more than one minimal element.
12
4
2
20
10
5
25
Figure 1: The Hasse Diagram of a poset.
Solution:

4. Example: Let A= {a, b, c, d, e, f, g, h, i} have the partial ordering
≼ defined by following Hasse diagram. Find all maximal,
minimal, greatest and least elements of A.
Solution: There is just one maximal element g, which is also the greatest
element. The minimal elements are c, d and i, and there is no least element.
a
c
b
g
f
e
h
i
d

5. Question: Answer these questions for the poset ({2, 4, 6, 9, 12, 18, 27, 36, 48, 60, 72}, |).
a) Find the maximal elements.
b) Find the minimal elements.
c) Is there a greatest element?
d) Is there a least element?
Solution:
Given: ({2, 4, 6, 9, 12, 18, 27, 36, 48, 60, 72}, |)
S = {2, 4, 6, 9, 12, 18, 27, 36, 48, 60, 72}
R = {(a,b)| a divides b}
Let us first determine the Hasse diagram
48
60
12
2
4 6
36
72
18
9
27
a) The Maximal elements are all the values in Hasse
diagram that do not have any elements above it.
Maximal elements = 27, 48, 60, 72
b) The Minimal elements are all the values in Hasse
diagram that do not have any elements below it.
Minimal elements = 2, 9
c) The greatest element only exist if there is exactly
one maximal element and is then also equal to the
maximal element.
Greatest element = Does not exist
d) The least element only exist if there is exactly one
minimal element and is then also equal to the
minimal element.
Least element = Does not exist

6. Answer these questions for the poset ({3, 5, 9, 15, 24, 45}, |).
a) Find the maximal elements.
b) Find the minimal elements.
c) Is there a greatest element?
d) Is there a least element?
Solution:
Given: ({3, 5, 9, 15, 24, 45}, |)
S = {3, 5, 9, 15, 24, 45}
R = {(a,b)| a divides b}
Let us first determine the Hasse diagram
45
9
3
15
24
5
a) The Maximal elements are all the values in Hasse
diagram that do not have any elements above it.
Maximal elements = 24, 45
b) The Minimal elements are all the values in Hasse diagram
that do not have any elements below it.
Minimal elements = 3, 5
c) The greatest element only exist if there is exactly one
maximal element and is then also equal to the maximal
element.
Greatest element = Does not exist
d) The least element only exist if there is exactly one
minimal element and is then also equal to the minimal
element.
Least element = Does not exist

7. Submitted by Kiran Kumar Malik
200301120128@cutm.ac.in