The question from the image asks for the difference between Minimal elements and Least elements in a poset (partially ordered set), along with examples. Here's a detailed explanation:

Key Definitions

1. Minimal Element:

   • An element $a$ in a poset $(P, \leq)$ is called a minimal element if there is no other element $b \in P$ such that $b < a$ (i.e., $b \leq a$ and $b \neq a$).
   • A poset can have multiple minimal elements.
   • A minimal element is not necessarily related to all other elements.

2. Least Element:

   • An element $l$ in a poset $(P, \leq)$ is called the least element if $l \leq x$ for all $x \in P$.
   • The least element, if it exists, is unique (there cannot be more than one least element).
   • The least element is also a minimal element, but not every minimal element is the least element.

Example to Illustrate the Difference

Example Poset:

Let $P = \{a, b, c, d\}$ with the following order relations:

• $a \leq b$
• $a \leq c$
• $d$ is unrelated to any element (incomparable to $a, b, c$).

Observations:

• Minimal Elements:

  • $a$ is not minimal because $a \leq b$ and $a \leq c$.
  • $b, c, d$ are minimal because there is no element smaller than $b$, $c$, or $d$.

• Least Element:

  • $a$ is the least element because $a \leq b$ and $a \leq c$, and $d$ is unrelated.

Comparison Table

Visual Example

Consider the Hasse diagram:

   a
  / \
 b   c
d

• Minimal Elements: $b, c, d$ (they have no smaller elements below them).
• Least Element: $a$ (it is smaller than $b, c, d$).

Let me know if you'd like a more formal proof, additional examples, or diagrams!

Follow-Up Questions

1. Can a poset exist with only minimal elements but no least element?
2. How do maximal elements compare to greatest elements in posets?
3. In what cases is the least element also the only minimal element?
4. How are Hasse diagrams helpful in identifying minimal and least elements?
5. Can a poset have neither minimal nor least elements?

Tip

When solving poset problems, always examine relationships carefully using Hasse diagrams to visually clarify minimal and least elements!