# poset

• Nov 9th 2006, 05:45 AM
poset
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.
• Nov 9th 2006, 07:46 AM
ThePerfectHacker
Quote:

Hi,
I have this problem:
Show that there is exactly one greatest element of a poset, if such element exists.

That is not true. Consider the Partially ordered set of all ideals of a ring. Ordered by inclusion. It has a maximal element, "maximal ideal", but not necessarily unique.
• Nov 9th 2006, 08:25 AM
Plato
Quote:

Originally Posted by ThePerfectHacker
That is not true. …a maximal element, not necessarily unique.

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.
• Nov 9th 2006, 08:27 AM
ThePerfectHacker
Quote:

Originally Posted by Plato
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.

So you mean an upper bound for the entire set.
Then his antisymettrical proof was correct.
Quote:

That is the curse of mathematical notation: nothing is standard.
Especially graph theory