Is there a systematic account of the number systems in the following 3x3 grid?

Question

Consider the following sets of numbers, viewed as number systems with signature $(+,\times,\leq)$.

Let $\mathbb{X} = \{1,2,3,\cdots\}$ denote the nonzero natural numbers. Let the completion of $\mathbb{X}$ with respect to reciprocals and products be denoted $\mathbb{Y}$. So $\mathbb{Y}$ is the set of all strictly positive rational numbers. Let the the completion of $\mathbb{Y}$ with respect to suprema of bounded non-empty subsets be denoted $\mathbb{Z}$. So $\mathbb{Z}$ is the set of alll strictly positive real numbers.

Thus, we so far have three number systems:

• $\mathbb{X} = \{1,2,3,\cdots\}$
• $\mathbb{Y} = $ the set of all strictly positive rational numbers.
• $\mathbb{Z} = $ the set of all strictly positive real numbers.

Lets obtain three more by adjoining a zero element.

• $\mathbb{X}_0 = \{0,1,2,3,\cdots\}$
• $\mathbb{Y}_0 = $ the set of all positive rational numbers, including zero.
• $\mathbb{Z}_0 = $ the set of all positive real numbers, including zero.

Finally, lets adjoin negatives.

• $\mathbb{X}_*= \{\cdots,-3,-2,-1,0,1,2,3,\cdots\}$
• $\mathbb{Y}_* = $ the set of all rational numbers.
• $\mathbb{Z}_* = $ the set of all real numbers.

These 9 systems can be arranged into a 3x3 grid. Adding in the obvious embeddings, the result is a commutative diagram.

So what I'm looking for is a systematic account of these 9 systems. In particular:

1. Nine different axiom systems over the signature $(+,\times,\leq)$, one for each number system.
2. A category whose objects includes all nine number systems.
3. Nine different universal properties, one for each number system.

Answer 1

Six of these nine sets are countable and therefore isomorphic - as sets. So probably you want to put some extra structure on them? I will use the ones which seem to be natural for my experience. If this does not fit to your question, please clarify it. For example, what is a number system? I will try to answer 3 (since I don't know how to say something nontrivial about 1 and 2).

$(X,+)$ is the free semigroup on one generator, $1$.

$(X,*,1)$ is the free monoid on countably many generators, the prime numbers.

$(Y,*,1)$ is the free group on $(X,*,1)$, thus it is the free group on countably many geneators, the prime numbers.

$(Z,<)$ is the completion, i.e. the free complete order on the order $(Y,<)$.

$(X_0,+,0)$ is the free monoid on one generator, $1$.

$(X_0,+,0,*,1)$ is the initial semiring.

$(Y_0,*,1)$ is the localization of $(X_0,*,1)$ at $X_0 \setminus \{0\}$.

$(Z_0,<)$ is the completion of $(Y_0,<)$

$(X_*,+,0,*,1)$ is the initial ring.

From this we get $Y_*$ and $Z_*$ as above. You can always add "commutative".