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.]



does this relation associated with a poset have a name?
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 $P\cup \{(x,y)\}$.


