Are there names for these subsets of rational numbers?

Question

Rational numbers can be defined as: $$\left\{ \frac{p}{q} | p \in \Bbb{Z}; q \in \Bbb{Z}; q \neq 0 \right\}$$

Are there conventional or existing names for the sets where $q$ is a particular number? For example:

$$\left\{ \frac{p}{4} | p \in \Bbb{Z}\right\}$$

If there aren't commonly used names, are there commonly used notations?

No, these sets of numbers have no particular name, but they are denoted as $\frac{1}{q}\Bbb{Z}$. — Crostul, May 20 '15 at 18:14
$\Bbb Z/4,$ or $,\Bbb Z:4,$ or $,\frac{1}4,\Bbb Z,$ or $,4^{-1}\Bbb Z.,$ What name or notation is used depends on the context, esp.. if it has more structure that just a set, e.g. a group, lattice, etc. — Bill Dubuque, May 20 '15 at 18:24
@Crostul, I edited my question so your comment should now qualify as an answer. — Samuel Edwin Ward, May 22 '15 at 16:53
@BillDubuque, I edited my question so that your comment should now qualify as an answer. — Samuel Edwin Ward, May 22 '15 at 16:53
@BillDubuque : for me $\mathbb Z/4$ are usually the integers mod 4 and $4^{-1} \mathbb Z$ the localization at 4 but I think it's probably matter of convention. — , May 22 '15 at 17:01
@N.H. Yes, such notation is overloaded so is context dependent. — Bill Dubuque, May 22 '15 at 17:52
@N.H. I thought the integers mod 4 were $\mathbb Z/4\mathbb Z$. — Akiva Weinberger, Aug 11 '15 at 04:50

score 2 · Answer 1 · answered May 22 '15 at 16:59

2

I don't think there is a generally-accepted name for such sets. Sometimes they are written as dilations of $\mathbb{Z}$: $\dfrac12\mathbb{Z}$ or the like.

Do not confuse the set $\dfrac12\mathbb{Z}=\{\dfrac n2:\ n\in\mathbb{Z}\}$ with the half-integers $\{n+\dfrac12:\ n\in\mathbb{Z}\}$.

answered May 22 '15 at 16:59

Charles

32,122

The latter would be written as $\mathbb Z+\dfrac12$, right? – Akiva Weinberger Aug 11 '15 at 04:49
@columbus8myhw: I prefer $\mathbb{Z}+{1/2}$ using the Minkowski sum, but each to her/his own. – Charles Aug 11 '15 at 13:39
You just said what I said, unless you meant $\mathbb Z+{1/2}$. (You need to use } for brackets in LaTeX.) – Akiva Weinberger Aug 11 '15 at 13:43
@columbus8myhw: Thanks -- I was typing it out on my phone and missed the extra slashes. – Charles Aug 11 '15 at 14:07
For future reference, on the app, you can see how LaTeX renders your posts by clicking on it, hitting "More," and hitting "Render MathJax."$$e^{d/dx}f(x):=\sum_{k=0}^\infty\frac{d^k}{dx^k}f(x)=f(x+1)$$ – Akiva Weinberger Aug 11 '15 at 14:11

Zelos Malum · Accepted Answer · 2015-05-22T17:43:12.723

1

Notations include $q^{-1}\mathbb{Z}$,$\frac{1}{q}$,$\mathbb{Z}/p$. I would probably call it "Integers over p" or the likes.

edited May 22 '15 at 17:43

answered May 22 '15 at 16:59

Zelos Malum

6,570

Are there names for these subsets of rational numbers?

2 Answers2