If a partial order has just one unique maximal element, must that element be a greatest element of x?

Question

I believe that it does not have to be a greatest element, because to be a greatest element it must be comparable to all the x in X. I just don't know how to go about proving this.

Answer 1

You are correct. For a ready example of a partial order with a unique maximal and minimal element that is neither the least nor the greatest element, take any $x\notin\Bbb Z,$ and let the relation $\precsim\::=\:\leq\cup\:\{\langle x,x\rangle\},$ where $\le$ is the order relation on $\Bbb Z.$ Then $X=\Bbb Z\cup\{x\}$ is partially ordered by $\precsim,$ and $x$ is both the unique maximal and unique minimal element, but there is no greatest or least element of $X$ under $\precsim.$

We can easily adapt this example to a partial order with a unique maximal element and a unique minimal element that are distinct from one another, but which still has no greatest or least element. We can also adapt it to an example with no minimal (maximal) element, which has a unique maximal (minimal) element but no greatest (least) element.