# 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 $\bigcup_{a \in A} \{ x \in P : x \le a \}$?

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What you call "set of upper bounds" for $A$ looks suspiciosly like the set of lower bounds for the set $A$... – Mariano Suárez-Alvarez Feb 23 '11 at 0:56
Looks like a kind of order ideal to me, no? – JBL Feb 23 '11 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 $\downarrow A$. – Chris Eagle Feb 23 '11 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 – Strahinja Popovic Feb 23 '11 at 11:32

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