Maximal Element vs Greatest Element

Question

Disclaimer: This thread is written in Q&A style. The answer is provided below.

Let $A$ be a poset.

If $a$ is a greatest element then $a$ is a unique maximal element.

Is the converse true as well?

Answer 1

No since still there can be incomparable elements for a unique maximal element.

Consider for example the disjoint union $\mathbb{R}\sqcup [0,1]$ with the partial order $a\leq b$ for $\pi(a)=\pi(b)$. Then $1$ is the only maximal element but it is not a greatest element.