nLab
lower set

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

• whenever $y\le x$ and $x\in U$, then $y\in U$.

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

$$A\downarrow =\{y\mid \exists x,x\in A\wedge y\le x\}.$$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 $x$.