# nLab down set

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

$x↓=\left\{y\phantom{\rule{thickmathspace}{0ex}}\mid \phantom{\rule{thickmathspace}{0ex}}x\le y\right\}.$x{\downarrow} = \{ y \;|\; x \leq y \} .

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

$x\stackrel{˙}{↓}=\left\{y\phantom{\rule{thickmathspace}{0ex}}\mid \phantom{\rule{thickmathspace}{0ex}}xx\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 $x$.

Revised on July 24, 2010 17:45:28 by Toby Bartels (75.117.104.156)