Definition of a Lattice

A lattice is a partially ordered set (poset) where any two elements have a unique least upper bound (supremum) and a greatest lower bound (infimum).

• Supremum (Join): The smallest element that is greater than or equal to both elements.
• Infimum (Meet): The largest element that is smaller than or equal to both elements.

We now give examples of infinite lattices satisfying the four given conditions:

(i) Neither a least nor a greatest element

Example: The set of integers $\mathbb{Z}$ under the usual "less than or equal to" ($\leq$) relation.

• In this lattice:
  • There is no least element because integers extend infinitely in the negative direction.
  • There is no greatest element because integers extend infinitely in the positive direction.
• However, the meet and join of two integers always exist (e.g., the minimum and maximum of any two integers).

(ii) A least but not a greatest element

Example: The set of natural numbers $\mathbb{N} = \{ 0, 1, 2, 3, \dots \}$ under the usual "less than or equal to" ($\leq$) relation.

• In this lattice:
  • $0$ is the least element because $0 \leq n$ for all $n \in \mathbb{N}$.
  • There is no greatest element because natural numbers extend infinitely.

(iii) A greatest but not a least element

Example: The set of negative real numbers $(-\infty, 0)$ under the usual "less than or equal to" ($\leq$) relation.

• In this lattice:
  • $0$ is the greatest element because all elements are less than or equal to $0$.
  • There is no least element because the negative real numbers extend infinitely toward $-\infty$.

(iv) Both a least and a greatest element

Example: A closed interval of real numbers $0, 1$ under the usual "less than or equal to" ($\leq$) relation.

• In this lattice:
  • $0$ is the least element because $0 \leq x$ for all $x \in [0, 1]$.
  • $1$ is the greatest element because $x \leq 1$ for all $x \in [0, 1]$.

Summary Table

Let me know if you need further clarification or examples!

Related Questions

1. What is the difference between a poset and a lattice?
2. Can a finite lattice exist without a least or greatest element?
3. How do meet and join operations work in a lattice?
4. Provide examples of bounded lattices in real life.
5. What are distributive lattices and modular lattices?

Tip: When identifying lattice structures, always check for the existence of the least upper bound (join) and the greatest lower bound (meet) for every pair of elements.