What's the difference between *not less than*, and *greater than or equal to* particularly in respect of Zorn's Lemma? In the Wikipedia description of Zorn's Lemma, it refers to a maximal element which is not smaller than any other element in $S$ and also to an upper bound, which is greater than or equal to any element in $S$.
Is the difference between these two simply that in the case of the upper bound, this element is not necessarily in $S$ but aside from that it is identical?
Is the upper bound necessarily not in $S$?
 A: In a general poset, it is not the case that for all elements $x,y$ we have $x \leq y$ or $y \leq x$. Neither may hold. Consider for example the situation where $\leq$ is $\subseteq$, the "is a subset of" relation, and $x,y$ are disjoint sets. You can say that $x$ is not less than $y$, but you cannot say that $x$ is greater than $y$.
A: In Zorn's lemma we appeal to two definitions:

Let $(S,<)$ be a partial ordering such that every chain $C$ has an upper bound. Namely, there is an element $s$ in $S$ such that for all $c\in C$, $c\leq s$. Then there is a maximal element in $S$.

The upper bound is for every chain separately(!) whereas the maximal element is a somewhat local definition in the first place.
In general, when we talk about upper bounds and maximal elements we always mean within the partial order of interest. We can always add "artificial"---new---elements and declare them to be the upper bounds or whatever. So it makes little sense for this definition to appeal "outside the domaon of interest".
