Final answer:
Transitivity does not work with the 'greater than' relation in a poset.
Step-by-step explanation:
In a partially ordered set (poset), the transitivity property states that if element A is greater than element B, and element B is greater than element C, then element A must be greater than element C as well. When it comes to the 'greater than' relation in a poset, transitivity does not hold. This is because the 'greater than' relation is not always transitive.
A simple counterexample is the set of integers with the 'greater than' relation. If we consider the integers 2, 5, and 8, we have that 2 is greater than 5, and 5 is greater than 8. However, 2 is not greater than 8, violating the transitivity property. Therefore, it is important to note that transitivity does not always work when it comes to the 'greater than' relation in a poset.