24.1. Cover of an Element in a Poset

Try these first

1. 1.

   Define what it means for one element to cover another in a poset.

   Hint

   Think of the direct relationship without anything in between.

2. 2.

   What is a maximal element?

   Hint

   It’s at the top of the hierarchy in the poset.

3. 3.

   What defines a cover in a poset?

   • x is less than y with intermediates
   • x is related to y with no intermediates
   • y is less than x with intermediates

   Hint

   Think of direct connections in Hasse diagrams.

4. 4.

   In a poset, can an element be both a maximal and minimal element?

   • True
   • False

   Hint

   Consider simple cases with one item in a poset.

5. 5.

   Given the elements {1, 2, 3, 4, 5}, with the following relationships: 1 < 2, 1 < 3, 2 < 4, 3 < 4, and 3 < 5, define the covers of each element.

   Hint

   Visualize with Hasse diagrams to track relations directly.

6. 6.

   Construct a Hasse diagram for the elements A, B, C, D with the following relation: A < B, A < C, B < D, C < D. Identify maximal and minimal elements.

   Hint

   Remember; stack based on relationships.