A set with a maximal element but not supremum?

Question

I have a problem which is asking me to show by example that a set (with respect some partial ordering) may have a maximal element but still no supremum.

I've been sitting here for hours trying to think of one but I still can't get it. Any help would be greatly appreciated!

Answer 1

For $n \geq 3$ think of the sets with at most $n-1$ elements of $\{1,2,...,n\}$ ordered by inclusion. Then it is clear that this set has no supremum (since two $n-1$ element subsets are not comparable), but every $n-1$ element set is a maximal element.

Answer 2

The sets $n\mathbb Z$ with $n>1$, partially ordered by inclusion, have maximal elements $p\mathbb Z$, where $p$ is a prime, but no supremum.