Germs at infinity of sequence of integers

Question

Consider the $\mathbb Z$-module $\mathcal Z$ obtained as the set of sequences of integers $\mathbb Z ^ \mathbb N$ modulo the relation that two sequences are deemed equivalent when their difference is $0$ almost everywhere. This space is the space of germs at $+\infty$ of sequences of integers. I think this is a quite natural object to study, as for instance it appears naturally when you try to generalize the notion of fundamental group of the circle when dealing with paths with a non-compact source space: imagine the value of $z_n$ as being the "number of turns" done at step $n$, the classical case being recovered from the stationary sequences. In that sense it is a completion of $\mathbb Z$ with a rich structure in the non-finite part. Yet I have not been able to find any literature regarding this object. This may be so because either

• nothing specific/interesting can be said about it, or

• it is an obvious/trivial example of some algebraic concept I'm unaware of (I'm no algebraist myself).

  So my question is: are the basic properties of this module known/interesting? For instance, is it a free module? Does anybody know a classical reference which may help me in studying this object?

  Thanks in advance for your contributions.

Answer 1

This abelian group, which can also be described as the quotient of the direct product $\prod_{\mathbb N}\mathbb Z$ by the direct sum $\sum_{\mathbb N}\mathbb Z$, is isomorphic to the direct sum of the following two pieces. The first piece is a torsion-free, divisible abelian group (so you can view it as a vector space over $\mathbb Q$) of rank (i.e., dimension over $\mathbb Q$) equal to the cardinality of the continuum. The second piece is the direct product, over all primes $p$, of the direct product groups $\prod_{\mathbb N}\mathbb Z_p$, where $\mathbb Z_p$ is the additive group of $p$-adic integers. I believe this result is due to Balcerzyk.

Edit: I believe the Balcerzyk reference is "On factor groups of some subgroups of a complete direct sum of infinite cyclic groups" [Bull. Acad. Polon. Sci., Sér. Sci. Math., Astron., Phys. 7 (1959) 141-142.

Answer 2

Do you mean by "the difference is stationary to 0" that the sequences are eventually equal, i.e., equal from some point on? This structure has been looked at quite a bit in the guise of a quotient of $\mathbb N^{\mathbb N}$.
This structure obviously does not have the algebraic structure you are looking for, but there are still some things that can be said.

The equivalence relation that identifies two sequences if they only disagree on finitely many coordinates is called $E_0$ and play an important role in the study of definable equivalence relations on Polish spaces in descriptive set theory.

Another aspect of this quotient that is often looked at in set theory is the order:

Given $x,y\in\mathbb N^{\mathbb N}$, let $x\leq^{\ast}y$ if for all but finitely many $n\in\mathbb N$, $x(n)\leq y(n)$.
$\leq^{\ast}$ induces a partial order on the quotient $\mathbb N^{\mathbb N}/E_0$. Two important characteristics of this order are the unboundedness number $\mathfrak b$ and the dominating number $\mathfrak d$. $\mathfrak d$ is the least size of a subset $\mathcal D$ of $\mathbb N^{\mathbb N}/E_0$ such that every element of $\mathbb N^{\mathbb N}/E_0$ is below some member of $\mathcal D$. $\mathfrak b$ is the least size of a subset $\mathcal B$ of $\mathbb N^{\mathbb N}/E_0$ such that no element of $\mathbb N^{\mathbb N}/E_0$ is an upper bound for all of $\mathcal B$. Note that for $\mathfrak b$ and $\mathfrak d$ it doesn't matter whether you look at $\mathbb N^{\mathbb N}$ or ${\mathbb Z}^{\mathbb N}$.

These two cardinals $\mathfrak b$ and $\mathfrak d$ show up relatively often in questions about the set theory of the real line.