nLab down set

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

x={y|yx}. x{\downarrow} = \{ y \;|\; y \leq x \} .

In a quasiorder, the strict down set of xx is the set

x˙={y|y<x}. x\dot{\downarrow} = \{ y \;|\; y \lt x \} .

If you think of a poset PP as a category, then the down set of xx is the slice category P/xP / x.

A down set in the opposite P opP^{op} of PP is an up set in PP.

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 xx is the lower set generated by xx.

Last revised on August 21, 2020 at 04:40:32. See the history of this page for a list of all contributions to it.