partial order, maximal but not greatest

Question

How do I create a R over {a,b,c,d} such that a is maximal but not the greatest element?

I know that something like R{(a,a)(b,b)(c,c)(d,d)}, all the element are maximal.

Answer 1

To clarify, the relation you propose

$$R \equiv \{(a,a), (b,b), (c,c), (d,d)\}$$

has four maximal elements $a,b,c,d$ and no greatest element.

---

I think you're looking for a relation $R$ where $a$ is a maximal element and no other element is maximal. But on a finite set, if you have a maximal element $a$, and no other element is maximal, then $a$ is the greatest element. You can't have a single maximal element on a finite set without having a greatest element.

My understanding is that your problem is only asking you to find a relation $R$ in which $a$ is a maximal— but not greatest— element, and that it's alright (indeed necessary) if there are other maximal elements as well.