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I'm wondering if there is any sort of colloquial notation for the set of half integers (i.e. $\{ \frac{n}{2} \mid n\in \mathbb{Z}\}$), or any sort of set of fractions of integers, (i.e. $\{\frac{n}{a}\mid $ $n\in \mathbb{Z}$ and $a$ is some rational number$\}$). I'd also consider adopting a-not-so-colloquial notation. Thanks.

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"Half-integer" usually refers to $\frac n2$ only when $n$ is odd. –  MJD Jan 4 '13 at 3:10
    
Ah you are right! Thanks for pointing this out. –  PatEugene Jan 4 '13 at 3:22
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I think most people would understand the notation $\frac{a}{b}\mathbb{Z}$ for the set $\bigl\{\frac{na}{b} \mid n\in\mathbb{Z}\bigr\}$. Intuitively, this notation indicates to me "take the set $\mathbb{Z}$, and scale it by $\frac{a}{b}$", which produces the correct result.

Alternatively, I think the notation $\mathbb{Z}\frac{a}{b}$ is just as good. Intuitively, it indicates to me "take the $\mathbb{Z}$-span of $\frac{a}{b}$", which again is the correct set.

More generally, given an integral domain $R$ and its fraction field $K$, a common notation for the principal fractional ideal generated by some $x\in K^\times$ is $xR$ (I've also seen $Rx$, and $(x)$ when the context is clear). See for example this blurb by Keith Conrad (about midway down the page).

Here is an example of the other two alternatives, in Bourbaki's Algebra:

enter image description here

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I don't think I've ever seen this, and I'm not sure I would understand it without an explanation. (I agree that it is quite clear, if the explanation you gave is included.) –  MJD Jan 4 '13 at 3:14
    
Wikipedia says for half-integers to use $\mathbb{Z} +\frac{1}{2}$, but I like your notation better, since it extends to the set of rational numbers. I'll go ahead and use it, at least for my notes. Edit: It's worth noting that wikipedia's notation lets us translate the entire set. –  PatEugene Jan 4 '13 at 3:16
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@PatEugene: $\mathbb Z+\frac{1}{2}$ and $\mathbb Z\frac12$ are different sets. Your question appears to be about the latter. –  Jonas Meyer Jan 4 '13 at 3:53
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