# Order-Embedding to Rationals

1. Jun 26, 2012

### heptheorist

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

2. Jun 26, 2012

### Staff: Mentor

First, define a function $f: \mathbb Z \to (0,1) \cap \mathbb Q$, 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 $g: \mathbb Z^\infty \to \mathbb Q$ with
$$g(a_0,a_1,...) = \sum_i \left(f(a_i) \prod_{n=0}^{i-1} d(a_n)\right)$$

I hope this works...
As the sum adds up a finite number of non-zero values, the result is rational.

3. Jun 26, 2012

### heptheorist

This is a great idea, exactly what I wanted. Thank you!!