Given a set $A \subset \mathbb{K}$, where $\mathbb{K}$ is a partially ordered set $(\mathbb{K}, \leq)$, the supremum is $\sup{A} = \min\{b\in\mathbb{K}|\forall a \in A: a \leq b\}$. Is there a name for something like $\overset{\sim}{sup} := \min\{b\in\mathbb{K}|\forall a \in A: a \lt b\}$?
For example, with $\mathbb{K} := \mathbb{N}$, $A := \{3, 4, \dots 12\}$, it would be $\overset{\sim}{sup}{A} = 13$.
Another example would be $\mathbb{K} := \mathbb{R}$, $A := \arctan[\mathbb{R}]$ (the image), it’s $\overset{\sim}{sup}{A} = \frac{\pi}{2}$.
It doesn’t always exist though, for example, with $\mathbb{K} := \mathbb{R}$, $A := [3, 13]$. (In $\mathbb{R}$, the supremum is either the maximum or the $\overset{\sim}{sup}$, so only either of the two exists)
For integers, the concept is similar to the upper bound used in for loops which is the number next to the maximum of the relevant set.