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Order-Embedding to Rationals

  1. Jun 26, 2012 #1
    For a paper I'm writing: Does anyone know of an explicit order-embedding (i.e. an order-preserving function) from [itex]\mathbb Z^\infty[/itex], the direct sum of infinitely many copies of the integers ordered lexicographically, to [itex]\mathbb Q[/itex], the rationals? It need not be a surjective embedding, but that would be a plus (obviously the two sets are order isomorphic).
  2. jcsd
  3. Jun 26, 2012 #2


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    Staff: Mentor

    What about this?

    First, define a function [itex]f: \mathbb Z \to (0,1) \cap \mathbb Q[/itex], something like
    0 -> 1/2
    n -> 1 - 1/(2n) for i>0
    n -> -1/(2n) for i<0

    This allows to order individual "letters" (I like the analogy, I will keep it).
    Use this function to evaluate the position of all 1-letter-words. In addition, the space to the next word can be used for all words beginning with this letter, in a similar way (0,1) was used for 1-letter-words. Let d(n)=f(n+1)-f(n) be this space.

    Now, let [itex]g: \mathbb Z^\infty \to \mathbb Q[/itex] with
    [tex]g(a_0,a_1,...) = \sum_i \left(f(a_i) \prod_{n=0}^{i-1} d(a_n)\right)[/tex]

    I hope this works...
    As the sum adds up a finite number of non-zero values, the result is rational.
  4. Jun 26, 2012 #3
    This is a great idea, exactly what I wanted. Thank you!!
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