I know that it’s very elementary question but I still don’t fully understand difference between maximal element and greatest element. If it’s possible, please explain to me this difference with some examples etc.
I tried to explain this difference to myself using only definition, but maximal element and greatest element still seems almost the same for me.
You are maximal when there is nobody above you.
You are greatest when you are above everyone else.
If nobody has eaten you, it doesn’t follow that you have eaten everyone else.
If nobody is standing on your head, it doesn’t follow that you are standing on everyone else’s head.
If you live on the top floor of your apartment building, it doesn’t follow that you live above everyone else in the city.
So maximal elements need not be greatest.
(Incidentally, although a greatest element is always a unique maximal element, it is a fun exercise to come up with a partial order having a unique maximal element, which is not greatest.)