I'm confused on how a supremum or infinum is necessarily unique, if they exist. I have a function $f(x)=x^2$, and I want to define a weak partial order on the set $\mathbb{R}$, by $$ \text{$x\succ y$ if $f(x)\geq f(y)$,} $$ and I want to say this is also a totally ordered set, if that's relevant. Consider a subset $S=[0,5]\subset\mathbb{R}$. I think I would be correct to say that the infinum of $S$ is $0$, and that the supremum is $5$.
But if I have a subset $S'=[-5,5]$, then what would the supremum be?
I would say that $-5$ or $5$ are candidates for the supremum, but I heard that the supremum is unique, and I would also say that $-5\neq5$. Does this mean that this set doesn't have a supremum?
Does $S'$ have a supremum? More generally, does the uniqueness of the supremum refer to the uniqueness of the element in the ordered set (or parent set), or the `relative position' that element holds? This proof I've seen implies that it's the former.
Uniqueness of suprema (if they exist) follows from anti-symmetry of the order relation. Typically, we discuss orderings only in an anri-symmetric setting. By working with a weak partial order instead, you lose uniqueness of suprema.