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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    $\begingroup$ What you call "set of upper bounds" for $A$ looks suspiciosly like the set of lower bounds for the set $A$... $\endgroup$ Feb 23, 2011 at 0:56
  • $\begingroup$ Looks like a kind of order ideal to me, no? $\endgroup$
    – JBL
    Feb 23, 2011 at 1:40
  • $\begingroup$ Google gives a few hits for the obvious "downset generated by $A$". Some people seem to use the notation $A \downarrow$ or $\downarrow A$. $\endgroup$ Feb 23, 2011 at 2:01
  • $\begingroup$ @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 $\endgroup$ Feb 23, 2011 at 11:32

1 Answer 1


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

  • $\begingroup$ I don't know the etiquette on this: is my inclusion of a citation enough to turn Chris Eagle's comment into an answer? $\endgroup$
    – Max
    Feb 23, 2011 at 19:42
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    $\begingroup$ 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 $\endgroup$ Feb 24, 2011 at 0:05

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