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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.

Samuel Han
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    You may want to use a different type of inequality such as $\prec$, since it seems you are defining a nonstandard inequality $<$ for the reals in terms of the standard inequality $<$ for the reals (using the same symbol $<$). So I find your definitions to be confusing. – Michael May 20 '23 at 18:39
  • Thank you; I've just edited my question to make the distinction clearer! – Samuel Han May 20 '23 at 18:43
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    But in any case, a "supremum" is generally a "least upper bound" for a set or sequence, and so it does not matter if that value occurs in the set/sequence of interest or not (or how many times that value occurs). I do not think the concept of supremum and infimum will be useful in your problem because a general ordering does not need to assign a "value" to the elements that are ordered. – Michael May 20 '23 at 18:43
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    If it helps, I observe that both $1\prec -1$ and $-1 \prec 1$ fail. – Michael May 20 '23 at 18:52
  • That's true, so I've changed it to a weak partial order to make it a total order. – Samuel Han May 20 '23 at 18:56
  • So now $1\prec -1$ and $-1 \prec 1$ are both true. – Michael May 20 '23 at 19:01
  • That's my intention, so does that mean that this is not an example of a poset? – Samuel Han May 20 '23 at 22:42
  • Let us continue this discussion in chat. – Samuel Han May 20 '23 at 22:50

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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.

Arno
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