Here is a detailed breakdown of your assignment:

Q1: Sub-lattice Determination

• For the given subsets $L_1, L_2, L_3, L_4$, you are to:

  1. Verify if they are sub-lattices of $L$ (the lattice represented by the diagram).
  2. Justify your answer based on closure under meet ($\wedge$) and join ($\vee$) operations.

• Process:

  1. Check closure under the meet and join operations for each subset.
  2. Provide a formal justification for whether they satisfy sub-lattice properties.

• Additional Task: Identify complements for elements $a, b, c$, if they exist.

Q2: Hasse Diagram of $D_{36}$

• Tasks:

  1. Draw the Hasse diagram for the divisors of 36 ($D_{36}$).
  2. Identify complements of elements $2$ and $9$, if they exist.
  3. Determine whether $D_{36}$ is complemented.

• Hints:

  • $D_{36}$ represents the lattice of divisors of 36 under divisibility.
  • Complemented means every element has a complement such that their meet is the minimum and their join is the maximum.

Q3: Poset $P$

• Poset: $P = \{ \{1,3\}, \{2,5\}, \{4,3\}, \{1,2\}, \{1,4\}, \{2,1,3\}, \{3,4,1\}, \{2,1,4,3\} \}$.

• Tasks:

  1. Identify maximal and minimal elements in $P$.
  2. Find the supremum (least upper bound) of $\{2,3\}$ and $\{5,3\}$, if it exists.
  3. Find the infimum (greatest lower bound) of $\{1,3\}, \{4,3\}, \{2,3\}$, if it exists.

If you'd like, I can solve each of these parts for you step-by-step. Let me know where you'd like to start!

Expansion Questions:

1. How is closure under lattice operations determined for sub-lattices?
2. What are the key properties of a complemented lattice, and how are complements identified?
3. How can the structure of $D_{36}$ be used to find complements or determine if it’s complemented?
4. What steps are involved in identifying maximal and minimal elements in a poset?
5. How do we compute the supremum and infimum for given subsets in a lattice?

Tip:

When working with Hasse diagrams, start by arranging elements from the bottom (minimum) to the top (maximum) based on divisibility or set inclusion.