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Math Help - Partial Order

  1. #1
    MHF Contributor chiph588@'s Avatar
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    Partial Order

    Let the set S = {a, b, c, d, e}. Draw the diagram of a partial order with two
    maximal elements on the set S.

    My question is what does it mean to be a maximal element here?
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  2. #2
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    Quote Originally Posted by chiph588@ View Post
    Let the set S = {a, b, c, d, e}. Draw the diagram of a partial order with two maximal elements on the set S.
    My question is what does it mean to be a maximal element here?
    A maximal element has no successor. A maximal element proceeds no other element.
    In the diagram a & b are maximal.
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  3. #3
    MHF Contributor chiph588@'s Avatar
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    but isn't c the successor of a & b?

    and is that the answer? I'm sorry, I'm still confused on this subject.
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  4. #4
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    if you have a partially ordered set (S, \leq), then an element s \in S is called a maximum element of S, if s \leq t implies that s=t. for example, in your set, we can put this partial order:

    a \leq b, \ b \leq c, \ a \leq c, \ d \leq e. (of course a \leq a, \ b \leq b, \ c \leq c, \ d \leq d, \ e \leq e is also a part of definition.) see that \{c, \ e \} is the set of maximal elements of S.
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