In a poset or even proset, an upper set is a subset that is ‘upwards closed’; that is,
- whenever and , then .
Upper sets form a Moore collection and so one can speak of the upper set generated by an arbitrary subset :
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 is the upper set generated by .
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