nLab up set

In a poset or even proset, the up set of an element $x$ is the set

$$x{\uparrow} = \{ y \;|\; x \leq y \} .$$

In a quasiorder, the strict up set of $x$ is the set

$$x\dot{\uparrow} = \{ y \;|\; x \lt y \} .$$

If you think of a poset $P$ as a category, then the up set of $x$ is the coslice category $x / P$.

An up set in the opposite $P^{op}$ of $P$ is a down set in $P$.

Note: The term ‘up set’ is also often used for an upper set, a more general concept. In the terminology above, the up set of $x$ is the upper set generated by $x$.