If b is the maximum element, must it be the supremum?

[The Student]

The question in the headline is all that I hope to know in this OP. If the answer is "yes", then that makes sense to me. But if it is "no", please explain why, thanks!

 

[Bob Brown MSEE]

The question in the headline is all that I hope to know in this OP. If the answer is "yes", then that makes sense to me. But if it is "no", please explain why, thanks!

Yes, if b exists and you have identified a suitable (ordered) super-set. If b does not exist then look to the ordered super-set.

In Wiki:, the supremum (sup) of a subset S of a totally or partially ordered set T is the least element of T that is greater than or equal to all elements of S

Example:
Let S be the set of all rational numbers < Pi
S has no maximum element.
However S has a super-set (reals)
S has a supremum in the reals which is Pi

Conundrum: "I am my own grandpa", oldtime'n song
1) If I define Pi as S
2) S is the supremum of S

 

[HallsofIvy]

That follow pretty readily from the definitions of "maximum" and "supremum" of a set.