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=================

I asked ChatGPT the following question:

"Given a partially ordered set, what is a good name for the set of ordered pairs that each produce a new partial order containing the original one?"

Answer 1:

'A common name for the set of ordered pairs that produce a new partial order containing the original one is the "order extension set" or "partial order extension set." This name reflects the idea that each ordered pair extends or expands the original partial order by introducing new order relations while maintaining the existing ones.'

Answer 2:

'The set of ordered pairs that produce a new partial order containing the original one can be referred to as the "composable pairs" or "order-preserving pairs." This name reflects the fact that these pairs can be combined or composed to create a new partial order that preserves the original order.'

Than I asked it this:

"Given a partially ordered set, what is a good name for the set of incomparable pairs that have a minimal point and a maximal point?"

Answer 1:

'The set of incomparable pairs that have a minimal point and a maximal point can be referred to as "antichain pairs" or "unrelated pairs." This name reflects the fact that these pairs represent elements in the partially ordered set that are unrelated or incomparable to each other, and they include a minimal point (no other element is smaller) and a maximal point (no other element is larger).'

Answer 2:

'The set of incomparable pairs that have a minimal point and a maximal point in a partially ordered set can be called the "minimal-maximal incomparable pairs." This name conveys the characteristics of these pairs: they are incomparable, meaning there is no order relation between them, and one element is minimal (it is not greater than any other element) while the other is maximal (it is not smaller than any other element).'

It's at least trying..."A" for effort.

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.]

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 $P.)$

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=================

I asked ChatGPT the following question:

"Given a partially ordered set, what is a good name for the set of ordered pairs that each produce a new partial order containing the original one?"

Answer 1:

'A common name for the set of ordered pairs that produce a new partial order containing the original one is the "order extension set" or "partial order extension set." This name reflects the idea that each ordered pair extends or expands the original partial order by introducing new order relations while maintaining the existing ones.'

Answer 2:

'The set of ordered pairs that produce a new partial order containing the original one can be referred to as the "composable pairs" or "order-preserving pairs." This name reflects the fact that these pairs can be combined or composed to create a new partial order that preserves the original order.'

Than I asked it this:

"Given a partially ordered set, what is a good name for the set of incomparable pairs that have a minimal point and a maximal point?"

Answer 1:

'The set of incomparable pairs that have a minimal point and a maximal point can be referred to as "antichain pairs" or "unrelated pairs." This name reflects the fact that these pairs represent elements in the partially ordered set that are unrelated or incomparable to each other, and they include a minimal point (no other element is smaller) and a maximal point (no other element is larger).'

Answer 2:

'The set of incomparable pairs that have a minimal point and a maximal point in a partially ordered set can be called the "minimal-maximal incomparable pairs." This name conveys the characteristics of these pairs: they are incomparable, meaning there is no order relation between them, and one element is minimal (it is not greater than any other element) while the other is maximal (it is not smaller than any other element).'

It's at least trying..."A" for effort.