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Does anyone know of a term for (equivalently)

  1. 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 $.
  2. 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".

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    $\begingroup$ Or equivalently, $P$ decomposes into a disjoint union of mutually incomparable posets, each of which has a greatest element? $\endgroup$
    – Nik Weaver
    Nov 29, 2017 at 16:59
  • $\begingroup$ So it is a exactly a disjoint sum of bounded-above pre-orders. $\endgroup$ Nov 29, 2017 at 17:06
  • $\begingroup$ Finished the first condition. Can't believe I didn't realize it makes them into a disjoint sum. Makes perfect sense in my application too. $\endgroup$
    – Max New
    Nov 29, 2017 at 17:43

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