In a poset or even proset, a lower set is a subset that is ‘downward closed’; that is,
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 .
The characteristic function of a lower set is precisely a (0,1)-presheaf.
An inhabited, open lower set of rational numbers (or equivalently of real numbers) determines precisely a lower real number.
Last revised on July 22, 2022 at 08:26:47. See the history of this page for a list of all contributions to it.