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.]

envelopeis 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. $\endgroup$contentis banned, period.) $\endgroup$5more comments