24.3.1. Definitions

Try these first

1. 1.

   Define a cover in a poset.

   Hint

   Think about how elements relate directly in a Hasse diagram.

2. 2.

   What is a maximal element?

   Hint

   What would happen to an element if nothing is above it?

3. 3.

   What is an element cover in a poset?

   • Element not related
   • Element related with intermediaries
   • Directly related element without intermediaries

   Hint

   Visualize it in a Hasse diagram.

4. 4.

   True or False: A poset can have no minimal elements.

   • True
   • False

   Hint

   Think about the characteristics of infinite sets.

5. 5.

   Create a poset with at least 6 elements, clearly identify the covers, maximal, minimal, greatest, and least elements.

   Hint

   Use a Hasse diagram approach for visualization.

6. 6.

   Questioning a poset with no maximal element: construct and explain how it satisfies the properties of a poset.

   Hint

   Think about the direction and extent of the ordering relation.