The set of natural numbers $\mathbb{N}$ is not closed under division
The set of natural numbers $\mathbb{Z}$ is not closed under division

$\mathbb{N}$ $\subset$ $\mathbb{Z}$ and $\mathbb{Z}$ is not closed under division by extension $\mathbb{N}$ is also not closed under division. In other words division is only partially defined on $\mathbb{N}$ and $\mathbb{Z}$

If division is partially defined can we say that we have a partial magma or partial groupoid ?

  1. If $\mathbb{Z}$ has negative numbers is correct to say that division is partially defined also for Inverses for Integer Addition while this is not true for $\mathbb{N}$ ?

  2. If this is correct what extra structure is there in the $\mathbb{Z}$ partial groupoid respect to $\mathbb{N}$ partial groupoid if both sets are not closed under division ?

  • $\begingroup$ You might be able to get away with calling $\mathbb{N}$ and $\mathbb{Z}$ partial magmas under division, but they will not be partial groupoids since division is not associative. $\endgroup$ – Anonymous Nov 12 '19 at 16:02
  • $\begingroup$ But is division a N-Z partial magma if division is partially defined on them ? Why you say N partial magma under division when division is a partial magma for N and Z? Axiom of closure maybe make more sense if we are in a field Q where division is closed (except for zero number). I say this because for N and Z division is opened $\endgroup$ – Jack Nov 12 '19 at 16:46
  • $\begingroup$ Yes, I think it's fair to call it a partial magma under division. "under" simply means "with respect to". $\mathbb{Z}$ is a group $\textit{under}$ addition, for example. $\endgroup$ – Anonymous Nov 12 '19 at 16:54
  • $\begingroup$ @Anonymous Note that "groupoid" is sometimes used synonymously with "magma" - see e.g. the OP's link (where "partial groupoid" is exactly what I'd call "partial magma"). I certainly prefer the term "magma" since there's no possible confusion, but the OP is matching existing terminology. $\endgroup$ – Noah Schweber Nov 12 '19 at 17:41
  • $\begingroup$ @NoahSchweber I see. That's unfortunate. $\endgroup$ – Anonymous Nov 12 '19 at 19:46

EDIT: I misread the question, see the comments below.

This is a very incomplete answer, but too long for a comment. As an aside, I'm going to use "magma" exclusively, since "groupoid" has another meaning - namely, associative partial magma.

A partial magma is just a set equipped with a partial binary operation. This is an incredibly broad property, and in particular both $(\mathbb{Z}; \div)$ and $(\mathbb{N}; \div)$ are partial magmas.

But for that exact reason, "partial magmocity" is really silly, and we should (as you do) ask what more we can say.

On either $\mathbb{N}$ or $\mathbb{Z}$, the operation $\div$ is not associative, commutative, or total; the only obvious nice "simple-algebraic" properties it has are that it has an identity element and that it is "almost" left-cancellative (if $a\div b$ and $a\div c$ are each defined and are equal then $b=c$ unless $a=0$) and "really almost" right-cancellative (if $a\div c$ and $b\div c$ are each defined and are equal then $a=b$).

When we go a bit higher up the complexity ladder, we do see a bit more: $(\mathbb{N};\div)$ and $(\mathbb{Z};\div)$ are both "equivalent" in a precise logical sense to $(\mathbb{N};\times)$ and $(\mathbb{Z};\times)$ respectively (that is, division and multiplication in each setting are definable relative to each other). Considerations along these lines belong more to model theory than to algebra, though.

Ultimately though I don't know of a snappy name we can apply here; all these properties are annoyingly still pretty weak.

  • $\begingroup$ Ok, you say that both $(\mathbb{Z}; \div)$ and $(\mathbb{N}; \div)$ are partial magmas but these both partial magmas should be different among them because we have different sets so I think that $\mathbb{N}$ partial magma is not equal to $\mathbb{Z}$ partial magma. Should there be a different structure between these 2 partial magmas $\endgroup$ – Jack Nov 12 '19 at 17:56
  • $\begingroup$ @Jack They are indeed different partial magmas - I don't think I implied they were the same? - and they do indeed have some distinctions (e.g. count the number of things $1$ can be divided by). But again I don't know any snappy names here, nor do I know of a property holding in one and failing in the other which can reasonably considered a "tameness" property (as opposed to merely a curiosity). $\endgroup$ – Noah Schweber Nov 12 '19 at 18:04
  • $\begingroup$ (Incidentally, a useful large-scale term here is "elementary equivalence." Two structures are elementarily equivalent if they satisfy the same first-order sentences. Elementary equivalence is a very strong condition, but at the same time not too strong - there are surprisingly simple examples of structures which are not isomorphic but are elementarily equivalent - for example, the linear orders $\mathbb{Z}$ and $\mathbb{Z}+\mathbb{Z}$.) $\endgroup$ – Noah Schweber Nov 12 '19 at 18:06
  • $\begingroup$ @Jack Oh I see, I misparsed the question - you're not asking what additional structure each has, but what additional structure $\mathbb{Z}$ has compared to $\mathbb{N}$. My apologies! Let me know if you'd like me to delete this. $\endgroup$ – Noah Schweber Nov 12 '19 at 18:10
  • $\begingroup$ Your answer is good, no need to delete it. what I also ask is about partial magmas: "what Z-partial magma additional structure has compared to N-partial magma" ? $\endgroup$ – Jack Nov 12 '19 at 18:42

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