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 coslice category $x / P$.

A down set in the opposite $P^{op}$ 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 June 18, 2016 11:28:57
by ondrejs?
(193.40.13.164)