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$$\left\{\frac{K}{3^n} : K\in\mathbb N,\ n\in \mathbb N \right\}$$

Any positive integer divided by any power of $3$.

Michael Hardy
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Ben Crossley
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    Those are the positive elements of the ring $\mathbb Z_3$ obtained by localizing $\mathbb Z$ at the multiplicative set generated by $3$. A sensible notation is therefore $\mathbb Z_3^+$, or something similar. – Mariano Suárez-Álvarez Oct 19 '16 at 16:47
  • Isn't that just the set of positive integers? Take any integer, multiply by three, and using the resulting number as an input with n=1 will put the original integer into your set. – Jason Hise Oct 19 '16 at 16:49
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    (On the other hand, the notation $\mathbb Z_3$ is overloaded —it denotes usually the abelian group $\mathbb Z/3\mathbb Z$, the ring of $3$-adic integers, and the localization at $3$ of $\mathbb Z$— so you would in any case have to make explicit what the notation means for you) – Mariano Suárez-Álvarez Oct 19 '16 at 16:49
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    @JasonHise, it contains the positive integers, but also $1/3$. – Mariano Suárez-Álvarez Oct 19 '16 at 16:49
  • Ah, I misinterpreted it as specifying that elements of the set were in N. – Jason Hise Oct 19 '16 at 16:52

2 Answers2

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First, let me say a bit about the same set, but with "$\mathbb{N}$" replaced with "$\mathbb{Z}$" (to allow negatives). I would call these the tryadic rationals in analogy with the dyadic rationals.

In terms of notation, they are a direct limit, and can be written as $$\lim_{\rightarrow}3^{-i}\mathbb{Z}.$$ I think the notation "$3^{-\infty}\mathbb{Z}$" would probably be understandable, in a somewhat strained analogy with the notation for the Prufer groups $\mathbb{Z}(p^\infty)$.

Now, for the nonnegative elements, I would call them the "nonnegative tryadic rationals" - granted, that's not very snappy, but not everything needs to be. In terms of notation, I think replacing $\mathbb{Z}$ with $\mathbb{Z}$ would be good: e.g. "$3^{-\infty}\mathbb{N}$" or "$\lim_\rightarrow 3^{-i}\mathbb{N}$."

Noah Schweber
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The members of $\displaystyle\left\{\frac{K}{3^n} : K\in\mathbb Z,\ n\in \mathbb N \right\}$ are sometimes called "ternary rational numbers" or "ternary rationals", but I don't know of a standard notation for them besides using that phrase or writing the expression that appears here. With $K\in\mathbb N$ rather then $K\in\mathbb Z$, I'd call the members "positive ternary rationals".

Michael Hardy
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