24.3.2. Existence and Uniqueness

Try these first

1. 1.

   Define a cover in a poset.

   Hint

   Consider the direct relationship without gaps.

2. 2.

   What is a maximal element?

   Hint

   Think of elements at the top of the Hasse diagram.

3. 3.

   What does a cover require in a poset?

   • Directly related without gaps
   • Has at least one element above
   • All elements are comparable

   Hint

   Think about what it means to have no barriers.

4. 4.

   True or False: A minimal element can be maximal.

   • True
   • False

   Hint

   Consider a single node in the diagram.

5. 5.

   Construct a poset with at least 5 elements showing both maximal and minimal elements. Prove their existence.

   Hint

   Map relationships that create a hierarchy.

6. 6.

   Given a poset, determine if a greatest or least element exists. If it doesn’t, characterize the elements.

   Hint

   Focus on connectivity and relations.