True.

A lattice is a partially ordered set (poset) where for any two elements $a$ and $b$, there exists both a least upper bound (join) and a greatest lower bound (meet).

In this case, $S = \{1, 2, 3, 12, 18, 216\}$ with the relation $\preceq$ defined by divisibility (i.e., $x \preceq y$ means $x$ divides $y$).

To check if this poset forms a lattice, we need to verify that for every pair of elements in $S$, there is:

• A least upper bound (LUB): the smallest element that is divisible by both $a$ and $b$.
• A greatest lower bound (GLB): the largest element that divides both $a$ and $b$.

For example:

• The LUB of $2$ and $3$ is $6$ (but since $6$ is not in $S$, we take $12$ as the LUB in this set).
• The GLB of $2$ and $12$ is $2$.

Since such LUB and GLB exist for all pairs of elements in $S$, the poset $(S, \preceq)$ satisfies the conditions of a lattice.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. What are the least upper bound and greatest lower bound for each pair of elements in $S$?
2. How do you verify if a poset is a complete lattice?
3. What are some other examples of posets that are lattices?
4. Can a lattice have elements that are incomparable?
5. What are the differences between distributive and modular lattices?

Tip: When analyzing a poset for lattice properties, always check that every pair of elements has both a least upper bound and a greatest lower bound.