Set of lower bounds not containing a greatest member

Question

The Wikipedia page for Supremum and Infimum contains this quote.

Infima and suprema do not necessarily exist. Existence of an infimum of a subset S of P can fail if S has no lower bound at all, or if the set of lower bounds does not contain a greatest element. However, if an infimum or supremum does exist, it is unique.

I understand how a suprema cannot exist in the example set S1 containing all natural numbers or how infima cannot exist in the example set S2 containing all rational numbers. But the next sentence,

if the set of lower bounds does not contain a greatest element

doesn't make sense to me. How can a set not contain a greatest element? This doesn't mean some kind of 2D vector space like the complex numbers (where there is no natural ordering), does it? What would be an example of a set that has no infima or suprema for this reason rather than the much more easy to understand "no lower bound" reason?

I've not formally studied Analysis so I would appreciate a more "simple" answer.

Answer 1

You just have to read a little further in the article:

As an example, let $S$ be the set of all finite subsets of natural numbers and consider the partially ordered set obtained by taking all sets from $S$ together with the set of integers $\mathbb{Z}$ and the set of positive real numbers $\mathbb{R} ^+$, ordered by subset inclusion [as above]. Then clearly both $\mathbb{Z}$ and $\mathbb{R}^+$ are greater than all finite sets of natural numbers. Yet, neither is $\mathbb{R}^+$ smaller than $\mathbb{Z}$ nor is the converse true: both sets are minimal upper bounds but none is a supremum.

The problem is that for partially ordered sets different upper bounds may not be comparable. If your set is totally ordered, such as $\mathbb{R}$ or $\mathbb{Z}$, then there is no issue. Or in other words, such examples can only happen in sets that are not totally ordered.