In a poset or even proset, the down set of an element is the set
In a quasiorder, the strict down set of is the set
If you think of a poset as a category, then the down set of is the coslice category .
A down set in the opposite of is an up set in .
Note that the down set of is the lower set generated by ; in fact, it is the (order-theoretic) ideal generated by .
Revised on June 18, 2016 11:28:57