nLab lower set

Definition

In a poset or even proset, a lower set $L$ is a subset that is ‘downward 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 $x$.

The characteristic function of a lower set is precisely a (0,1)-presheaf.

Examples

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

See also

• posite
• upper set
• ideal
• filter
• copresheaf
• presheaf