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?
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 $p$.