down set

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

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

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

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

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 July 24, 2010 17:45:28 by Toby Bartels (