Hi,
I have this problem:
Show that there is exactly one greatest element of a poset, if such element exists.
I dont know if my proof is rigth:
Suppose that there are 2 greatest elements a and b in the poset(P,R).
thus, a R b and b R a -->a=b ( antisymetric). so it is unique.
Can someone someone correct me if I am wrong?
B.
That is a true statement. However, most set theorist distinguish between ‘maximal element’ and ‘greatest element’. The greatest element is preceded by every element in the poset. One the other hand, a maximal element does not precede any other element in the set. If a poset has a greatest element then that element is related to every other element in the set. So indeed by antisymmetry a greatest element is unique.
Now I will grant you that the may very well be a textbook that disagrees with all of the above. That is the curse of mathematical notation: nothing is standard.