Is there a Poset which has one maximal element but no greatest element?

Question

If there is exactly one maximal element, then shouldn't that element also be the greatest element of that Poset?

Answer 1

Take $X=\mathbb N\cup \{a\}$ (where $0\in\mathbb N$) and the relation $\leq'$ defined as the standard relation $\leq$ on $\mathbb N$, and additionally, define $0\leq' a$. Then, $a$ is maximal, but not greatest.