lower set

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

- whenever $y \leq x$ and $x \in L$, then $y \in L$.

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

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

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 $x$ is the lower set generated by $x$.

An inhabited, open lower set of rational numbers (or equivalently of real numbers) determines precisely a lower real number.

Last revised on November 3, 2018 at 22:58:22. See the history of this page for a list of all contributions to it.