I'm in a situation where it's useful to think of any subset of $\mathbb{Z}/p\mathbb{Z}$ as being totally ordered (under the order "inherited" from $\mathbb{Z}$). Of course, finite fields can't be totally ordered themselves, which makes we wonder:
Is there a name for sets where every subset (but the set itself) is totally ordered, and the set itself has no notion of order whatsoever? Can such sets even exist?
If $S$ has more than three elements, and the total orders on subsets are compatible, then we can build a total order on the entire set by $a\leq b$ in $S$ iff $a\leq b$ in $\{a,b\}$.
This could fail the transitive property if $|S|=3$, since it would then be entirely possible to have $a\leq b\leq c\leq a$.
Of course, this ordering need have nothing to do with any structure on the set, and definitely won't respect the operations in a finite field the way the canonical order on any subfield of the reals does.