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Given a partial order $P$ on a set $S$ does the set of ordered pairs $(x,y)$ in $S\times S\setminus P$ such that $P\cup\{(x,y)\}$ is a partial order have a name? (If so then it would apply to all sorts of orders not just partial orders.)

The answer was "no" in this crosspost: https://cs.stackexchange.com/questions/155928/relation-based-on-a-given-partial-order-does-it-have-a-name

If MathOverflow concurs, then it must not have a name yet. (I named it the envelope of $P.)$

============== edit added 2 June 2023=================

[Some parrot-like ChatGPT answers were added then removed when I learned that ChatGPT content is banned.]

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    $\begingroup$ "Envelope" is a strange name for a set that does not contain the original set. $\endgroup$
    – Andrei Smolensky
    Commented Jun 1, 2023 at 6:17
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    $\begingroup$ The biggest problem with envelope is that it's used for other things in other areas. Was hoping there would already be a name for this. That's looking very unlikely given the comment by @JoelDavidHamkins, which by the way might serve as a better foundation for the name than my definition in the question. $\endgroup$
    – mathematrucker
    Commented Jun 1, 2023 at 17:49
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    $\begingroup$ I think that ChatGPT's answers to this question are irrelevant, at least as part of the question. (I would say that, if you really want to share them, then you can share them as an answer … but ChatGPT-generated answers are banned, so don't do that. In fact the linked post says that ChatGPT-generated content is banned, period.) $\endgroup$
    – LSpice
    Commented Jun 3, 2023 at 10:41
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    $\begingroup$ This is now findstat.org/StatisticsDatabase/St001902 $\endgroup$
    – Martin Rubey
    Commented Jun 4, 2023 at 11:23
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    $\begingroup$ @მამუკაჯიბლაძე assuming $x$ and $y$ are incomparable, the sense in which $x$ is minimal and $y$ maximal is that no point incomparable to $y$ is strictly less than $x$ (thus $x$ is minimal among such points) and no point incomparable to $x$ is strictly greater than $y$. $\endgroup$
    – mathematrucker
    Commented Jun 5, 2023 at 1:03

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It appears that there is no established name for this concept, but if you are looking for a suggestion, "potential covers" might be a reasonable name, since these are precisely the pairs $(x,y)$ which are not in the partial order $P$ but would be a cover in $P\cup \{(x,y)\}$.

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  • $\begingroup$ Thank you for mentioning this good name for the pairs. It arises naturally and shows up in some recent papers. I changed "envelope" to the more available, more descriptive "extender" and will include "potential cover" in the definition. $\endgroup$
    – mathematrucker
    Commented Jun 4, 2023 at 14:39

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