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I a book the "Upper closure operator" definition is given by:

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I do understand the concept of the Upper closure operator.

But I cannot figure out how to read the last line.

Could someone explain it?

Thank you.

Asaf Karagila
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sky_e
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1 Answers1

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The bottom line shouldn't start with $P$, since that is already being used for the overall poset being considered. Instead it should be $$Q \mapsto \{y \in P \mid \exists x \in Q : x \le_P y\}.$$

Ted
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    Okay, that makes sense. So is it correctly understood that Q is a element of Pow(P). And then the right side is a set builder of the upper set? Though I'm not sure how to read it - could you write the set builder in words? – sky_e Jul 23 '23 at 23:04
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    If $Q$ is a subset of the poset $P$, the right side defines a set called the upper set of $Q$. Since the set builder starts with ${ y \in P \mid \ldots }$, the upper set of $Q$ is a subset of $P$. It is the set of elements $y \in P$ which have the property that for some element $x \in Q$, we have $x \le_P y$. In other words, it is the set of elements of $P$ which are "above" some element of $Q$. Does that help? – Ted Jul 23 '23 at 23:09
  • Yes, Now I understand. Thank you! – sky_e Jul 23 '23 at 23:10