Does anyone know of a term for (equivalently)
- A preorder $P$ where for every $x \in P$ there exists $x_\top \ge x$ such that for any $y \ge x $, $x_\top \ge y $.
- A preorder $P$ where for every $x \in P$, $\uparrow x$ has a greatest element?
for categories this would be a category $C$ where for every object $x$ the coslice $x\backslash C$ has a terminal object or dually every slice has an initial object.
In my applications the preorders/categories involved do not have a global greatest element, just the upper sets/coslices.
My only idea is to call these posets "locally bounded".