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In the construction of the integers at Wikipedia, the integers are constructed as differences of natural numbers, with positive, zero and negative dropping out naturally. Additionally, in the article for negative numbers on Wikipedia references the Grothendieck construction, which creates a group from a communative monoid.

I was wondering what is the analougous pre-cursor of the $p$-adic integers?

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    $\begingroup$ Why do you believe there is such a beast? Every commutative monoid gives rise to an abelian group via the Grothendieck construction, but that doesn't imply that every abelian group arises as the 'Grothendieck quotient' or some commutative monoid... $\endgroup$
    – Steven Stadnicki
    May 23, 2017 at 22:11
  • $\begingroup$ Look at the isomorphism $\varphi : \mathbb{Z} \to \mathbb{Z}_p$. Why don't you look at $\varphi(\mathbb{N})$ then take the Grothendieck quotient ? You'll get $\mathbb{Z}_p$ from this after defining the absolute value and the metric and taking the completion. $\endgroup$
    – reuns
    May 23, 2017 at 22:34
  • $\begingroup$ Compare: math.stackexchange.com/questions/49990/… $\endgroup$
    – Chris Culter
    May 23, 2017 at 22:35
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    $\begingroup$ Even when Wikipedia is technically correct, it tends to be so poorly written that it often leaves students of mathematics more confused than before. $\endgroup$
    – Mr. Brooks
    May 23, 2017 at 22:44
  • $\begingroup$ IMO wikipedia is impossible to understand for the $p$-adic numbers $\endgroup$
    – reuns
    May 23, 2017 at 22:45

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If you'd settle for a dense subset of $\Bbb{Q}_p$, you could look at the $p$-adic numbers with expansions that terminate. This should give you positive rational numbers whose denominators are a power of $p$.

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  • $\begingroup$ Or a dense subset of $\mathbb{Z}_p$ for example (the $p$-adic integers corresponding to) $\mathbb{N}$ $\endgroup$
    – reuns
    May 23, 2017 at 22:44

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