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Math Help - Partially ordered set with no maximal element

  1. #1
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    Partially ordered set with no maximal element

    Could you please help with this?

    Let P be a non-empty partially ordered set and assume that no element of P is maximal. Use the Axiom of Choice to show that there exists a function f: \omega \rightarrow P such that  f(n) < f(n^+) \forall n \in \omega
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  2. #2
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    Here's my idea. We're going to construct a relation that you can apply Axiom of Choice to for obtaining the function you desire.

    Pick some arbitrary  y_{0} \in P

    Construct a set of ordered pairs R s.t. it has two properties.

    (1)  <0,y_{0}> \in R.

    (2) if for some  x \in \mathbb{N} and  y \in P<br />
\ \exists <x,y> \in P  \rightarrow <x^{+}, y'> s.t. y P_{r} y'

    You know that there will always exist greater y's because P has no maximal unit.

    Then from axiom of choice you can create a function from this relation and I believe it should have all the properties you want.
    Last edited by arsenicbear; March 9th 2010 at 12:34 AM. Reason: Did not know how to TeX
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  3. #3
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    Thank you a lot for your help; could you please just edit the equations, as they're given as "Latex Error: Syntax Error"'s now?

    Thank a lot,

    Mimi
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  4. #4
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    I've edited my eqns. Also P_{r} represents the partial order relationship that is assumed in P.
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