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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
up vote 1 down vote accepted

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 $\{x \in P : x \le a\}$

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I don't know the etiquette on this: is my inclusion of a citation enough to turn Chris Eagle's comment into an answer? – Max Feb 23 '11 at 19:42
You gave an (accepted, so presumably good) answer, and Chris Eagle gave a good comment. It's Chris's responsibility to post an answer if he wants, but a good rule is to mention in your answer something like "Inspired by Chris Eagle's comment above, I went and found ...", or "Following Chris Eagle's observation in an earlier comment, ..." . That way Chris can (vicariously) enjoy the acceptance/added reputation. Gerhard "Ask Me About System Design" Paseman, 2011.02.23 – Gerhard Paseman Feb 24 '11 at 0:05

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