The set $S = \{1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72\}$ with the divisibility relation $\leq_d$ forms a partially ordered set (poset). Here, $a \leq_d b$ means that $a$ divides $b$.

1. Constructing the Hasse Diagram

The Hasse diagram is a graphical representation of a poset where:

• Elements are placed in a hierarchy.
• If $a \leq_d b$ (i.e., $a$ divides $b$), there will be a connection between $a$ and $b$.
• No transitive relations are shown (i.e., if $a \leq_d b \leq_d c$, the diagram only shows $a$ connected to $b$, and $b$ connected to $c$).

Steps for constructing the diagram:

1. List the divisors: For each element in $S$, list all the divisors in the set $S$.
2. Eliminate transitivity: Only include direct connections between divisors and their multiples.
3. Arrange: Elements at lower levels are those that divide the elements at the higher levels.

The Hasse diagram for this set will have 1 at the bottom, as it divides all the other elements. The higher elements (like 72) are divided by the lower ones.

2. Checking if it Forms a Lattice

A lattice is a poset in which:

• Every pair of elements has a least upper bound (LUB) (also called join) and
• Every pair of elements has a greatest lower bound (GLB) (also called meet).

Least Upper Bound (LUB) or Join:

The least upper bound of two elements $a$ and $b$ is the smallest element in the set that is divisible by both $a$ and $b$.

Greatest Lower Bound (GLB) or Meet:

The greatest lower bound of two elements $a$ and $b$ is the largest element that divides both $a$ and $b$.

We need to check for every pair of elements whether both the LUB and GLB exist in the set.

3. LUB and GLB for 18 and 24

• Greatest lower bound (GLB): The divisors of 18 are: $\{1, 2, 3, 6, 9, 18\}$
  The divisors of 24 are: $\{1, 2, 3, 4, 6, 8, 12, 24\}$

  The common divisors of 18 and 24 are $\{1, 2, 3, 6\}$, and the largest common divisor is $6$. Hence, the GLB of 18 and 24 is $6$.

• Least upper bound (LUB): The multiples of 18 in the set are $\{18, 36, 72\}$.
  The multiples of 24 in the set are $\{24, 72\}$.

  The smallest common multiple of 18 and 24 is $72$, so the LUB of 18 and 24 is $72$.

4. Conclusion

Since every pair of elements in $S$ has both a least upper bound (join) and a greatest lower bound (meet), the poset $(S, \leq_d)$ forms a lattice.

Summary:

• GLB of 18 and 24: 6
• LUB of 18 and 24: 72

Would you like me to show the Hasse diagram or provide further details on the lattice structure?

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Here are five follow-up questions:

1. How can we verify the transitive relations in a Hasse diagram?
2. What are some examples of lattices outside number theory?
3. Can the Hasse diagram for this poset be drawn differently but remain valid?
4. What is the significance of a lattice in algebraic structures?
5. How does the concept of divisibility relate to other partial orders in mathematics?

Tip: A lattice where every interval has a unique greatest lower bound and least upper bound is called a "complete lattice."