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Math Help - poset

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by braddy View Post
    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.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    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.
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  4. #4
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    Quote Originally Posted by Plato View Post
    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.
    That is the curse of mathematical notation: nothing is standard.
    Especially graph theory
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