To determine whether the given posets (partially ordered sets) are lattices, we need to check if, for any two elements $a$ and $b$ in the poset, both the least upper bound (supremum) and the greatest lower bound (infimum) exist.

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

Assume the elements are ordered by divisibility (i.e., $a \leq b$ if $a$ divides $b$).

• Comparing pairs of elements:
  • $1$ divides all other elements, so it is the least element.
  • $5$ is prime and doesn't divide any other element except itself.
  • For any two elements, check their greatest common divisor (gcd) as the greatest lower bound (infimum) and least common multiple (lcm) as the least upper bound (supremum).

For pairs like $(2, 3)$, $(2, 5)$, $(3, 5)$, etc., there is no common divisor greater than 1 (their gcd is 1), but the lcm is not an element of the set.

• Conclusion:
  • This poset is not a lattice because not all pairs have both a supremum and infimum within the set.

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

Assume the elements are ordered by divisibility.

• Comparing pairs of elements:

  • Each pair of elements has a greatest lower bound (gcd) within the set and a least upper bound (lcm) also within the set.
  • For example, for $a = 4$ and $b = 8$:
    • $\text{gcd}(4, 8) = 4$, which is in the set.
    • $\text{lcm}(4, 8) = 8$, which is also in the set.

• Conclusion:

  • This poset is a lattice because for every pair of elements, both the supremum and infimum exist within the set.

Summary:

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

Would you like to explore the concepts of lattice properties further or need any clarifications?

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1. What are the properties of a lattice?
2. How do gcd and lcm relate to lattices in posets?
3. What is the significance of ordering by divisibility in posets?
4. Can a poset be a lattice if it has elements with no common divisors?
5. How do we find the supremum and infimum in different types of posets?

Tip: When dealing with lattices in posets, always verify that both the least upper bound and greatest lower bound exist for all pairs of elements in the set.