This is one of those things where people think that because they learned something in set theory 101 or intro to computers that it maps perfectly to real life situations.

Ok, let me translate it into non-math for you - there can be a goal that is a "highest priority" for the organisation in the sense that no other goal is a higher priority, and yet there are other goals for the organization that are equal to it in priority.

Isn't "a Maximal Element need not be Unique" a more elegant and much more succinct way of saying the same thing?

Look, that's a great proof and all but it has nothing to do with the situation at hand. Just because mathematics say there can be two maximal elements doesn't mean morality agrees - and this is the domain of morality, not mathematics.

I disagree. When you say “I just had the most exhilarating experience”, that doesn’t mean that there can only be a single most exhilarating experience. And, independent of that, morality also doesn’t imply that there can be only one most fundamental duty.

Ridiculous. Stop playing pointless language games.

But it says "its" not "one of its", that suggests only one maximum.

That doesn’t follow. “John failed in his duty to regularly check the logs” doesn’t imply that John can’t simultaneously have had any other duties. You have to parse it as “its (most fundamental duty to notify Jessie’s family of her death)”, not as “(its most fundamental duty) to (notify Jessie’s family of her death)”, so to speak.