Let's solve this step by step and give an easy explanation.

Definition of a Lattice:

A lattice is a partially ordered set (poset) in which every pair of elements has both:

• a least upper bound (supremum), and
• a greatest lower bound (infimum).

Lattices can be finite or infinite, and they may or may not have least and greatest elements.

Now, we will provide examples for each condition.

Solutions

(i) Neither a least nor a greatest element:

• Example: The set of integers $\mathbb{Z}$ with the usual order $\leq$.
• Explanation:
  • In $\mathbb{Z}$, you can always find a smaller or larger integer, so there is no smallest (least) or largest (greatest) element.
  • For example, if you think $0$ is the smallest, you can always go to $-1$, and similarly, there is no largest integer.

(ii) A least but not a greatest element:

• Example: The set of natural numbers $\mathbb{N} = \{0, 1, 2, 3, \dots\}$ with the usual order $\leq$.
• Explanation:
  • The number $0$ is the least element because no number in $\mathbb{N}$ is smaller than $0$.
  • However, there is no greatest element because you can always find a larger natural number.

(iii) A greatest but not a least element:

• Example: The set $S = \{ x \in \mathbb{Q} \mid x < 0 \}$ (all negative rational numbers) with the usual order $\leq$.
• Explanation:
  • There is no least element because for any negative number, you can always find a smaller one.
  • However, $0$ can act as a greatest element when considering this restricted set since all values in $S$ are less than $0$.

(iv) Both a least and a greatest element:

• Example: The closed interval $[0, 1]$ in the set of real numbers $\mathbb{R}$, with the usual order $\leq$.
• Explanation:
  • The least element is $0$, and the greatest element is $1$.
  • Every other number in the set $[0, 1]$ lies between these two.

Summary Table:

Do you want me to explain any part further? Here are 5 related questions you may want to explore:

1. What are examples of finite lattices with least and greatest elements?
2. How do upper bounds and lower bounds help define lattices?
3. What is the difference between a lattice and a totally ordered set?
4. Can you give an example of a lattice in real-life scenarios?
5. What are sublattices and how are they derived?

Tip: To understand infinite lattices better, always think of sets with no boundaries (like integers) or sets with bounds (like intervals).