Let me start with a prototypical situation, with $S \subset \mathbb{R}$. I'd like to divide $S$ into the "smaller" and "bigger" parts, by specifying the boundary $t \in \mathbb{R}$, as (i) $(-\infty, t] \cap S$ and $(t, +\infty) \cap S$, and (ii) in its dual(?) way, into $(-\infty, t) \cap S$ and $[t, +\infty) \cap S$. Forgetting everything about $\mathbb{R}$, cardinality, topology, etc, except it's a totally ordered set, do these two partitions have names?
Now let us be general, by replacing $\mathbb{R}$ with any totally ordered set $T$. Consider partitioning $S$ into two disjoint sets $= A \amalg B$, such that for $\forall a \in A$ and $\forall b \in B$, $a < b$ and $a \le t \le b$, where $t \in T$. There're two ways if $t \in S$. In words, in the above case (i) the smaller part is maximal among the subsets of $S$ for which $t$ is an upper bound.
*My question is*: do such bisections have names in order theory? (They must be final or initial, but I think I can google it once I know the names.)
EDIT: I'd like names that distinguish the two boundary conditions.
Background: This question was inspired when studying the Python language module "bisect". [1] There a sorted array (or a vector, and in Python a sorted "list") a is considered.
One can think of the array a as e.g. $a: [0, 1, ..., n] \to \mathbb{Z}$, an increasing function. (Read [] as {}, sorry. \{ \} didn't work.) Let $A := (-\infty, t],$ then Python's bisect_right returns $a^{-1}(A)$ and its complement $a^{-1}(\mathbb{Z} \setminus A)$.
In programming, the appearance of these two bisections seems natural and ubiquitous, e.g. as a for-loop boundary condition. So they IMHO deserve names.
[1] Python doc: bisect.
In fact, I proposed its revision, but the maintainer rejected it.