Short Answer 
In the poset, the maximal elements identified are 27, 48, 60, and 72, while the minimal elements are 2 and 9. Upper bounds of the set {2, 9} include 18, 36, and 72, with 18 being the least bound, and the lower bounds of {60, 72} are 2, 4, 6, and 12. Notably, there is no greatest or least element in this poset, which is significant for analyzing weak orderings.
Step 1: Identify Maximal and Minimal Elements
In a poset, maximal elements are those that are not less than any other element. For this specific case, the maximal elements are:
- 27
- 48
- 60
- 72
On the other hand, minimal elements do not exceed any other elements. The minimal elements identified here are:
- 2
- 9
Step 2: Understand Boundaries in the Set
Bounds provide important relationships among elements in a poset. Notably, there are several types of bounds based on the sets provided:
- Upper bounds of {2, 9} include: 18, 36, and 72.
- Least bound of {2, 9} is identified as: 18.
- Lower bounds of {60, 72} consist of: 2, 4, 6, and 12.
Step 3: Recognition of Existence of Greatest and Least Elements
In certain posets, there may not be definitive greatest or least elements. In this particular case:
- There is no greatest element, meaning no single element is greater than all others.
- Additionally, there is no least element, indicating that no element is less than all others.
This lack of defining elements is an important characteristic when analyzing weak orderings in posets.