In a poset or even proset, a lower set is a subset that is ‘downwards closed’; that is,
- whenever and , then .
Lower sets form a Moore collection and so one can speak of the lower set generated by an arbitrary subset :
Sometimes a lower set is called an ‘ideal’, but that term can also mean something more specific (and always does in a lattice).
A lower set is also sometimes called a ‘down set’, but that term can also mean something more specific: the down set of is the lower set generated by .
An inhabited, open lower set of rational numbers (or equivalently of real numbers) determines precisely a lower real number.
Revised on November 14, 2016 18:10:07
by Toby Bartels