nLab
lower set

In a poset or even proset, a lower set LL is a subset that is ‘downwards closed’; that is,

  • whenever yxy \leq x and xUx \in U, then yUy \in U.

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

A={y|x,xAyx}. A{\downarrow} = \{ y \;|\; \exists x,\; x \in A \;\wedge\; y \leq x \} .

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 xx is the lower set generated by xx.

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 (64.89.51.24)