Notation for half-integers

Question

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.

Answer 1

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: