What extra structure is there in the $\mathbb{Z}$ partial magma respect to $\mathbb{N}$ if both sets are not closed under division?

Question

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 ?

Answer 1

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.