Is there a convention for notating a subset of the rationals with restrictions on the denominators? I'd prefer there to be a relatively intuitive and concise notation for the set $\{\frac{n}{10^m}:n,m\in\mathbb{Z}\}$ in the ballpark of something along the lines of the following made-up notation: $\mathbb{Q}_{10}$
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1$\begingroup$ You could call it $\frac{\mathbb Z}{10^{\mathbb N_0}}$. There isn't a specific convention for that (that I know of), but there's a general convention that applying operations to sets that are usually applied to its elements yields the set of all elements you could get with any elements from the sets, e.g. $A+B=\{a+b\mid a\in A\land b\in B\}$. (Note that you shouldn't use $ \mathbb Z/10^{\mathbb N_0}$, since the quotient is defined for rings over ideals and thus might be misunderstood (though in this case that wouldn't make sense since $10^{\mathbb N_0}$ is not an ideal of $\mathbb Z$).) $\endgroup$– jorikiApr 30, 2020 at 8:45
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I think ${\Bbb Z}[\frac{1}{10}]$ is quite standard. It is used for instance in the definition of the Prüfer group:
... where ${\Bbb Z}[1/p]$ denotes the group of all rational numbers whose denominator is a power of p, using addition of rational numbers as group operation
See also this question, where the same notation is used.