nLab down set

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

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

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

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

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

A down set in the opposite $P^{op}$ of $P$ is an up set in $P$.

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