# nLab up set

In a poset or even proset, the up 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{\uparrow} = \{ y \;|\; x \leq y \} .

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

$x\stackrel{˙}{↑}=\left\{y\phantom{\rule{thickmathspace}{0ex}}\mid \phantom{\rule{thickmathspace}{0ex}}xx\dot{\uparrow} = \{ y \;|\; x \lt y \} .

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

An up set in the opposite ${P}^{\mathrm{op}}$ 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 $x$.

Revised on July 1, 2010 17:58:23 by Toby Bartels (98.16.139.29)