We have of course:

$2\Bbb Z-1$, or $2\Bbb N+1$

These I think of as the integers reduced by the congruence $x\cong x+1$ where in one case $x$ is even and in the other $x$ is odd (it makes no difference really).

But I want to think specifically of reducing integers by the congruence $x\cong 2x$ perhaps more clearly the transitive closure of the equivalence relation $x\sim 2x$.

My random stab in the dark is to write it $\Bbb Z/\langle 2\rangle$

Expressed like this, I guess the set includes $0$.

Since this is a multiplicative group modulo a prime, perhaps it's better to exclude $0$ something like: $\Bbb Z^\times/\langle2\rangle$?

What's acceptable / usual here?

  • $\begingroup$ Exactly what's your aim? $\Bbb Z/(2)$ has only two elements: (the representatives of) $0$ and $1$ $\endgroup$
    – Berci
    Jun 21, 2018 at 8:14
  • $\begingroup$ @Berci I intended $\langle2\rangle=\{1,2,4,8,16,\ldots\}$ rather than $(2)$ by which I think you mean something different. My understanding is a quotient is based on equivalence classes. I want to set $\forall x:x\sim2^nx$, so $3\sim6\sim12\sim24\ldots$ and I want a notation for the set of such classes. $\endgroup$ Jun 21, 2018 at 8:33
  • $\begingroup$ If I understand correctly, you want your equivalence classes to be $\{1, 2, 4, 8, \dots \}, \{ 3, 6, 12, 24, \dots \}, \{ 5, 10, 20, 40, \dots \}, \dots, \{ 15, 30, 60, 120, \dots \}$, and so on. This is not a usual quotient ring (since, for example $2 + 4 = 6$ and $2 + 2 = 4$ and the left hand sides are sums of equivalent elements but the right hand sides are not equivalent). Your group idea doesn’t work neither because $\Bbb Z^\times = \{-1, 1\}$. $\endgroup$ Jun 21, 2018 at 8:37
  • $\begingroup$ You might want to look into localization, in particular localization of $\Bbb Z$ by $2$ (written $\Bbb Z_2$). In that ring, to elements are equivalent in the way you want when they are associated, i.e. if one is the other multiplied by a unit. $\endgroup$ Jun 21, 2018 at 8:48
  • $\begingroup$ @EikeSchulte yes. I have rudimentary 2-adic theory in my armoury but I want to restrict closer to $\Bbb {Z}$ and not necessarily go to a field. Maybe the answer is to go to $\Bbb Z[\frac{1}{2}]^\times/\langle2\rangle$ which I believe is then a quotient of the multiplicative group. $\endgroup$ Jun 21, 2018 at 8:57

1 Answer 1


If I understand the question right, we are looking at $\mathbb N$ and the relation $x \sim y \iff $ there exists $i \in \mathbb Z$ such that $x = y \cdot 2^i$. This is an equivalence relation, and - apart from the equivalence class {0} - each equivalence class contains a unique odd integer.

$\mathbb N$ with multiplication is a monoid (like a group, but without the requirement for inverses to exist), and it's easy to show that for any monoid $A$, if you have an equivalence relation $\sim$ such that $a \sim b$ and $x \sim y$ implies $ax \sim by$ - which holds in the case we are considering - then there is a quotient monoid structure on $A\ / \sim$, and it is this quotient monoid structure you are thinking of.

I don't know of any particular standard notation for the equivalence relation $\sim$ or for $\Bbb N \ / \sim$.

  • $\begingroup$ Thank-you. Not pushing for extensive further information right now, but I guess the quotient monoid has some form of addition - what does that look like and is it a reasonable question to ask whether multiplication and exponentiation can be defined? $\endgroup$ Jun 21, 2018 at 9:31
  • $\begingroup$ A monoid has a single operation - in this case it is multiplication. Exponentiation - in a monoid $A$ you can define $a^n$ for $a \in A$, $n \in \Bbb N$ but not more general exponentiation. $\endgroup$ Jun 21, 2018 at 9:33
  • 1
    $\begingroup$ Btw, as monoid, $\Bbb N/\sim\cong\Bbb N$, maybe that's a reason why it didn't receive any particular name. (Both are free commutative monoids generated by countably infinite elements, and attached a $0$.) $\endgroup$
    – Berci
    Jun 21, 2018 at 11:55
  • 3
    $\begingroup$ Yes, exactly, the primes generate it freely. $\endgroup$
    – Berci
    Jun 21, 2018 at 12:09
  • 1
    $\begingroup$ @RobertFrost yes, though the morphism defined by $p_n \mapsto p_{n+1}$ goes $\Bbb N \to \Bbb N \ / \sim$ not the other way around $\endgroup$ Jun 21, 2018 at 12:11

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