The diagram you uploaded illustrates a partial order $R$ with elements $a, b, c, d, e$, and directed arrows indicating the ordering relationships.

Let's break down the information to answer your questions:

1. Maximal Elements:

Maximal elements in a partial order are elements that are not less than any other element (i.e., there are no arrows going out from these elements to other elements, except possibly to themselves).

• In the diagram, the element $d$ has no arrows leading out to other elements (other than possibly to itself). Therefore, $d$ is a maximal element.

2. Minimal Elements:

Minimal elements are elements that are not greater than any other element (i.e., no arrows point to them from other elements, except possibly from themselves).

• In the diagram, element $a$ has no incoming arrows except a loop to itself. So, $a$ is a minimal element.
• Similarly, element $e$ also has no incoming arrows except a loop to itself. Hence, $e$ is also a minimal element.

3. Elements Comparable to $e$:

Elements are said to be comparable to another element if there is a directed path (arrows) between them.

• From the diagram, $e$ has only a loop arrow to itself. This means $e$ is only comparable to itself.

Summary:

• Maximal elements: $d$
• Minimal elements: $a, e$
• Elements comparable to $e$: $e$ (only comparable to itself)

Let me know if you would like more details on any part of this!

Related Questions:

1. What are the maximal and minimal elements of a set in the general case of a poset?
2. What are the comparable elements of $a$ in this partial order?
3. Can there be more than one maximal or minimal element in a partial order?
4. How can one determine if two elements are comparable in a partial order?
5. What are the reflexive, antisymmetric, and transitive properties of a partial order?

Tip:

In a partial order, elements that are incomparable don't have a directed path connecting them in either direction.