# Name for union of upsets/downsets

Set of lower bounds in poset is defined like $A^l = \{ x \in P : \forall a \in A . x \le a \} = \bigcap_{a \in A} \{ x \in P : x \le a \}$.

Is there in literature a name for union $\bigcup_{a \in A} \{ x \in P : x \le a \}$?

• What you call "set of upper bounds" for $A$ looks suspiciosly like the set of lower bounds for the set $A$... Feb 23, 2011 at 0:56
• Looks like a kind of order ideal to me, no?
– JBL
Feb 23, 2011 at 1:40
• Google gives a few hits for the obvious "downset generated by $A$". Some people seem to use the notation $A \downarrow$ or $\downarrow A$. Feb 23, 2011 at 2:01
• @Mariano Suárez-Alvarez - Fixed, thank you for correction @JBL - Yes, that is it, thank you very much. @Chris Eagle - Yes, I'm googling now with your keywords, thank you Feb 23, 2011 at 11:32

Introduction to Lattices and Order by B. A. Davey and H. A. Priestly calls this $$\mathord{\downarrow}A$$ or the downset of $$A$$ and also uses $$\mathord{\downarrow} a$$ for $$\{x \in P : x \le a\}$$