nLab
down set

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

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

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

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

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

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

Note that the down set of $x$ is the lower set generated by $x$; in fact, it is the (order-theoretic) ideal generated by $x$.