Scott domain is a non-empty partially ordered set if the following holds:
D is bounded complete, i.e. all subsets of D that have some upper bound have a supremum.
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What would be an example of a set that doesn't have a supremum? I was under impression that the whole point of introducing supremum was to cover cases like 0 < a < 1 when there is no maximum value but there is list upper bond sup(a) = 1.
My reference to partially ordered sets was confusing, because my question was about both ordered and partially ordered sets without supremum.
Best Answer
The set $\{ x \in \mathbb{Q} \mid x^2 < 2 \}$ does not have a supremum in $\mathbb{Q}$, even though it is bounded. However, it does have a supremum if we view it as a subset of $\mathbb{R}$. This is an example of how the least upper bound property (every set bounded from above has a supremum) encodes completeness.