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Question 1
State True or False: Consider the poset (S, ⪯), where S = {1,2,3,12,18,216} and ⪯ denotes the relation {(x,y)∈S×S : x|y}. The poset (S, ⪯) is an example of a lattice.


True



False




**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.

State True or False: Consider the poset (S, ⪯), where S = {1,2,3,12,18,216} and ⪯ denotes the relation {(x,y)∈S×S : x|y}. The poset (S, ⪯) is an example of a lattice.

Analyze whether the poset (S, ⪯), where S = {1,2,3,12,18,216} and ⪯ represents divisibility, forms a lattice. This problem explores concepts from order theory, focusing on least upper bounds and greatest lower bounds. Suitable for advanced high school or university-level students.

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