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 $x$.

Last revised on August 21, 2020 at 00:40:32. See the history of this page for a list of all contributions to it.