Q:

Question about posets and maxima/minima

A thought just occurred to me, thinking about posets and maxima/minima... This is a "little" question just to make sure I am really grasping the definitions here: if $E$ is partially ordered by a relation $\preceq$ and $\max_E$ exists, then mustn't $E$ be totally ordered as well?  That is to say, mustn't every pair in $E$ be comparable by $\preceq$ on $E$, since $\max_E$ exists if and only if for every $e \in E$, $e \preceq \max_E$?

A:

No, that is not necessarily true. Consider $E = 2^\mathbb{N}$, that is, all subsets of the natural numbers. $x \preceq y$ if $x \subseteq y$. Clearly there is a maximum, $\mathbb{N}$. But it isn't total, because $\{1\}$ and $\{2\}$ can't be compared (is there a name for that property?).
We can compare everything to $\mathbb{N}$ (and also to $\emptyset$), but not to everything else, so it is not total.

