24.2. Maximal and Minimal Elements in a Poset

Try these first

1. 1.

   Define a maximal element in a poset.

   Hint

   Think about elements without any covers above them.

2. 2.

   Illustrate what a minimal element looks like in a Hasse diagram.

   Hint

   Identify where there are no elements below.

3. 3.

   What defines a maximal element in a poset?

   • It is less than elements
   • It is equal to another element
   • No element is related above it

   Hint

   Think about elements that are at the top.

4. 4.

   True or False: A poset can have multiple maximal elements.

   • True
   • False

   Hint

   Consider an example with multiple elements.

5. 5.

   Given a poset defined by the relations: A < B, A < C, B < D, C < D, determine all maximal and minimal elements.

   Hint

   Look for elements not less than or not greater than others.

6. 6.

   Construct a poset with elements 1, 2, 3 where 1 < 2 and 1 < 3, and illustrate a Hasse diagram for this.

   Hint

   Draw the relationships based on the order given.