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

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 ?

• 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. – Anonymous Nov 12 '19 at 16:02
• 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 – Jack Nov 12 '19 at 16:46
• 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. – Anonymous Nov 12 '19 at 16:54
• @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. – Noah Schweber Nov 12 '19 at 17:41
• @NoahSchweber I see. That's unfortunate. – Anonymous Nov 12 '19 at 19:46

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.

• 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 – Jack Nov 12 '19 at 17:56
• @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). – Noah Schweber Nov 12 '19 at 18:04
• (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 $\mathbb{Z}+\mathbb{Z}$.) – Noah Schweber Nov 12 '19 at 18:06
• @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. – Noah Schweber Nov 12 '19 at 18:10
• 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" ? – Jack Nov 12 '19 at 18:42