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 ?
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 $\mathbb{N}$ ?
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 ?