nLab
upper set

In a poset or even proset, an upper set UU is a subset that is ‘upwards closed’; that is,

  • whenever xyx \leq y and xUx \in U, then yUy \in U.

Upper sets form a Moore collection and so one can speak of the upper set generated by an arbitrary subset AA:

A={yx,xAxy}. A{\uparrow} = \{ y \;|\; \exists x,\; x \in A \;\wedge\; x \leq y \} .

Sometimes an upper set is called a ‘filter’, but that term can also mean something more specific (and always does in a lattice).

An upper set is also sometimes called an ‘up set’, but that term can also mean something more specific: the up set of xx is the upper set generated by xx.

The upper sets form a topological structure on (the underlying set of) the proset, called the Alexandroff topology.

Revised on September 17, 2011 08:34:53 by Toby Bartels (71.31.209.1)