To determine whether the given posets are lattices, we need to check whether, for every pair of elements in the set, both the least upper bound (supremum) and the greatest lower bound (infimum) exist within the set.

i) Poset: $\{1, 2, 3, 4, 5\}$

Step 1: Define the ordering

Assume the set is ordered by divisibility. That is, $a \leq b$ if and only if $a$ divides $b$.

Step 2: Check pairs for greatest lower bound (infimum) and least upper bound (supremum)

• $1$: Divides every element, so it’s the least element.
• $5$: Divides only itself.
• Pairs:
  • $2$ and $3$: $\text{gcd}(2, 3) = 1$ (infimum = 1), $\text{lcm}(2, 3) = 6$ (not in the set).
  • $2$ and $4$: $\text{gcd}(2, 4) = 2$ (infimum = 2), $\text{lcm}(2, 4) = 4$ (supremum = 4).
  • $3$ and $4$: $\text{gcd}(3, 4) = 1$ (infimum = 1), $\text{lcm}(3, 4) = 12$ (not in the set).
  • $3$ and $5$: $\text{gcd}(3, 5) = 1$ (infimum = 1), $\text{lcm}(3, 5) = 15$ (not in the set).
  • $4$ and $5$: $\text{gcd}(4, 5) = 1$ (infimum = 1), $\text{lcm}(4, 5) = 20$ (not in the set).

Step 3: Conclusion for the poset $\{1, 2, 3, 4, 5\}$

For some pairs like $(2, 3)$, $(3, 4)$, etc., the least upper bound (supremum) does not exist within the set because their least common multiple (lcm) is not in the set. Therefore, the poset $\{1, 2, 3, 4, 5\}$ is not a lattice.

ii) Poset: $\{1, 2, 4, 8, 16\}$

Step 1: Define the ordering

Again, assume the set is ordered by divisibility.

Step 2: Check pairs for greatest lower bound (infimum) and least upper bound (supremum)

• $1$: Divides every element, so it’s the least element.
• $16$: Is divisible by all other elements and is the greatest element.
• Pairs:
  • $2$ and $4$: $\text{gcd}(2, 4) = 2$ (infimum = 2), $\text{lcm}(2, 4) = 4$ (supremum = 4).
  • $2$ and $8$: $\text{gcd}(2, 8) = 2$ (infimum = 2), $\text{lcm}(2, 8) = 8$ (supremum = 8).
  • $4$ and $8$: $\text{gcd}(4, 8) = 4$ (infimum = 4), $\text{lcm}(4, 8) = 8$ (supremum = 8).
  • $8$ and $16$: $\text{gcd}(8, 16) = 8$ (infimum = 8), $\text{lcm}(8, 16) = 16$ (supremum = 16).
  • $4$ and $16$: $\text{gcd}(4, 16) = 4$ (infimum = 4), $\text{lcm}(4, 16) = 16$ (supremum = 16).

Step 3: Conclusion for the poset $\{1, 2, 4, 8, 16\}$

For every pair of elements, both the least upper bound (supremum) and greatest lower bound (infimum) exist within the set. Therefore, the poset $\{1, 2, 4, 8, 16\}$ is a lattice.

Final Summary

• i) The poset $\{1, 2, 3, 4, 5\}$ is not a lattice.
• ii) The poset $\{1, 2, 4, 8, 16\}$ is a lattice.

Would you like to explore these concepts further, or do you have any other questions?

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1. What is the importance of gcd and lcm in determining lattices?
2. Can you explain why a set might not be a lattice?
3. How does ordering by divisibility affect the structure of a poset?
4. What other methods can we use to determine if a poset is a lattice?
5. Can a set with non-divisible elements be a lattice?

Tip: Always ensure that for every pair of elements in a poset, both the infimum and supremum are within the set to determine if it forms a lattice.