24.1.1. Definitions and Examples

Try these first

1. 1.

   Define what a cover is in a poset.

   Hint

   Consider the relationship without anything in between.

2. 2.

   What distinguishes a maximal element in a poset?

   Hint

   Think about what you cannot find above.

3. 3.

   What defines a cover in a poset?

   • A direct relationship with an intermediate
   • A direct relationship without intermediate
   • An unrelated element

   Hint

   Think about the absence of anything in between.

4. 4.

   True or False: All posets must have both a maximal and a minimal element.

   • True
   • False

   Hint

   Consider cases with one element.

5. 5.

   In a poset where the elements are the set of all divisors of 12, identify the maximal, minimal, greatest, and least elements.

   Hint

   Consider how divisors relate to each other and find the topmost and bottommost elements.

6. 6.

   Construct a Hasse diagram for the set {a, b, c} with the relationships a < b, a < c. Identify any covers and describe maximal/minimal elements.

   Hint

   Think about how to organize the elements visually to find relationships.