# relation based on a given partial order - does it have a name?

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?

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).

• 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($P$)". Dec 5, 2022 at 17:07
• Your description "union of all extensions of $P$ minus $P$" isn't accurate, by the way. Dec 5, 2022 at 17:20
• @mathematrucker Not sure what you mean by not accurate. What I mean is $E(P)=\{r\in X\times X\mid \exists Q\ (Q \text{ is a poset that extends }Q) \land (r\in Q)\}\setminus P$. Dec 5, 2022 at 21:56
• 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. Dec 5, 2022 at 22:27
• ...hence $P\cup\{(4,7)\}$ fails to satisfy the transitive property. Dec 5, 2022 at 22:40