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 coslice category x/Px / P.

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

Note that the down set of xx is the lower set generated by xx; in fact, it is the (order-theoretic) ideal generated by xx.

Revised on June 18, 2016 11:28:57 by ondrejs? (