up set

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

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

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

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

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

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

Note: The term ‘up set’ is also often used for an upper set, a more general concept. In the terminology above, the up set of xx is the upper set generated by xx.

Last revised on July 1, 2010 at 17:58:23. See the history of this page for a list of all contributions to it.