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Let $P$ be a partial order on $X.$ Does the relation $E(P)=$ { $(x,y)\in (X\times X)\setminus P:P$ $\cup$ { $(x,y)$ } is a partial order on $X$ } have a name? If not, what's a good thing to call it?

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I searched online. Also checked several articles/papers on poset topological sorting and linear extension of poset. No name has been found for that relation.

$E(P)$ can be described as the set of ordered pairs compatible to $P$ or the set of ordered pairs extending $P$ minus $P$.

As the asker noted, if an ordered pair belong to an extension of $P$, it does not belong to $E(P)$ necessarily. For example, let $P$ on $X=\{a,b,c\}$ be defined by $a\prec b$. Then $P$ can be extended by letting $b\prec c$ and $a\prec c$. However, only letting $b\prec c$ does not extend $P$ to a bigger poset since $a\prec c$ would be missing. Hence we cannot describe $E(P)$ as the union of all extensions of $P$ minus $P$.

Some names that might be good for $E(P)$ are "the $P$-extending relation", "the extending relation of $P$", or "the $P$-compatible relation".

Once defined at the start of your post/article, the notation $E(P)$ could be the best name for that set, where $E$ is, as you intended probably, shorthand for extension or extending. For the sake of clarity in references, $\text{Ext}(P)$ might be better ($\text{Extension}(P)$ looks a bit long).

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  • $\begingroup$ Your answer is helpful - thank you. "E" was indeed for "extension". Perhaps I should have mentioned this question: mathoverflow.net/questions/418092 The answer shows that for $X$ finite, every extension of $P$ contains at least one pair in $E(P)$ (maybe this holds for any $X$). Naming is discussed briefly in the comments. I am now thinking of possibly calling it the "envelope" of $P$ and using "Env($P$)". $\endgroup$ Dec 5, 2022 at 17:07
  • $\begingroup$ Your description "union of all extensions of $P$ minus $P$" isn't accurate, by the way. $\endgroup$ Dec 5, 2022 at 17:20
  • $\begingroup$ @mathematrucker Not sure what you mean by not accurate. What I mean is $E(P)=\{r\in X\times X\mid \exists Q\ (Q \text{ is a poset that extends }Q) \land (r\in Q)\}\setminus P$. $\endgroup$
    – John L.
    Dec 5, 2022 at 21:56
  • $\begingroup$ The pair $(4,7)$ satisfies your condition for the poset at math.stackexchange.com/questions/3480391 but it's not in $E(P)$ for that poset because, for example, the pair $(2,7)$ is not in the poset. $\endgroup$ Dec 5, 2022 at 22:27
  • $\begingroup$ ...hence $P\cup\{(4,7)\}$ fails to satisfy the transitive property. $\endgroup$ Dec 5, 2022 at 22:40

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