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I am interested in what structures can the space of integers $\mathbb{Z}$ have.

Disclaimer: I am aware that one could of course come with various equivalence relations of the elements or maybe more kind of topologies, but I am more curious about kinds of structures that are different than the ones already mentioned. I just wrote an example for some well known types of math structures, so I am not asking about other topologies, orders etc., because we all know there are more of them.

In particular, I have a question about geometric or measure structure (the 6. and 7. below). I hope this is an interesting question. Maybe I also missed some types of structures from graph theory, set theory, combinatorics, ...?

Mathematical structures of $\mathbb{Z}$:

1. Order structure - We can compare integers with one another. Each number is either less or more than any other number.

2. Lattice structure - Maybe belongs to algebraic structures too, not sure. $\mathbb{Z}$ certainly forms a lattice - every pair of elements has unique supremum and infimum.

3. Topological structure - The usual topology on the integers is the discrete topology — the one where every subset is an open set.

4. Algebraic structure (in particular Abelian group structure) - Zero is the identity element, every number has its inverse (opposite number), it is associative and commutative (with respect to addition).

5. Metric structure - There is a notion of distance between points. We can define the distance between each $x$ and $y$ as $\mid y - x \mid$.

6. Measure structure - ? I am not sure how to precisely determine whether a set has measure structure and whether it is true for integers. My guess is that there is not a measure structure.

7. Geometric structure - ? According to Wikipedia, a set possesses a geometric structure if "it is equipped with a metric and is flat". However, I have found more different definitions of a "geometric structure", for example here. My guess is that integers dont have geometric structure, but I really dont know.

8. Relational structure - The set of integers may be defined as the set of equivalence classes of pairs of natural numbers under the equivalence relation $(n,m)≡(n′,m′)$ if and only if $n+m′=m+n′$.

Tereza Tizkova
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    You can consider the prime spectrum $\text{Spec}(\mathbb{Z})$ equipped with the Zariski topology as a "geometric structure" on $\mathbb{Z}$. – morrowmh Jul 17 '22 at 13:35
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    I think that this question is too broad and undefined to be answerable. For one, what is a "mathematical structure"? For another, each individual entry in your list has infinitely many possibilities: infinitely many different metrics; infinitely many different abelian group structures; and so on. – Lee Mosher Jul 17 '22 at 13:35
  • There are many different types of algebraic structures you can give the integers, for example the form a group under $+$. Adding in multiplication $\times$ they form a ring. Adding in scalar multiplication they form a $\mathbb{Z}$-module.... – David Sheard Jul 17 '22 at 13:36
  • My interpretation of the question is "In what categories does $\mathbb{Z}$ naturally belong?" Naturality is of course ill-defined, but I think the question is worth asking. – user816709 Jul 17 '22 at 13:45
  • @LeeMosher I understand that there are many possibilities (I think I even mention this). The point is not to list infinite possibilities of metrics, groups etc., but rather whether and how could one introduce other kinds of structures, as it has been posted already in the two answers here. Math. structure is set equipped with extra properties (or rules). – Tereza Tizkova Jul 17 '22 at 14:22
  • The question is both too broad and too sloppy: From the context, it appears that you simply treat ${\mathbb Z}$ as a set, but refer to it as the "space of integers", which is an undefined notion. If you are trying to find extra structures on ${\mathbb Z}$ compatible with its ring/group algebraic structure, you have to make it clear. Otherwise, you are effectively asking for a slew of examples from all fields of mathematics of extra structures on countable sets. – Moishe Kohan Jul 19 '22 at 08:25
  • You can take the MathSciNet subject classification from 1 through 60 and, apart from analysis and diff geometry, every field will give you some/many examples. This is totally unfocused. – Moishe Kohan Jul 19 '22 at 08:27

2 Answers2

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Just to answer on the measure, there is the so-called "discrete", or "counting", measure on any set, where any subset is measurable and its measure is its (possibly infinite) cardinality. This is the most natural measure on $\mathbb{Z}$, as it is compatible with its algebraic and topological structure (it is what is called its Haar measure).

Captain Lama
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Your definition of "geometric structure" is indeed a peculiarly restrictive one. From a geometric group theory point of view one could endow the group of integers under $+$ with a geometric structure as follows. Start with a subset $S\subset\mathbb{Z}$ which generates the group - different choices will lead to different structures. Now form the Cayley graph $\textrm{Cay}(\mathbb{Z},S)$ of $(\mathbb{Z},S)$ which has vertex set $\mathbb{Z}$, and a directed edge $n$ to $m$ if $m-n\in S$. Now give each edge length $1$ which defines a metric on the graph (which restricts to a metric on $\mathbb{Z}$) which as a group theorist I would call a geometric structure.

If you are more of a differential geometer and you want your geometric structure to be closer to those definitions you mention in your question, you can specialise this construction. Take $S=\{1\}$, then the Cayley graph is an infinite line (there is a natural identification with $\mathbb{R}$) This is a 1-dimensional manifold, and the standard Riemannian metric is flat. Moreover, the natural action of $\mathbb{Z}$ on the Cayley graph by "left multiplication" (addition in this case) is an action by isometries, giving $X=\textrm{Cay}(\mathbb{Z},\{1\})$ an $(X,\mathbb{Z})$-structure as discussed here.

David Sheard
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