In order to define cluster algebra one needs to define its ground ring. In most cases, we take a group $P$ (often called a coefficient group) which is taken to be an abelian multiplicative group . Sometimes $P$ is endowed with some additional binary operation of (auxiliary) addition $\oplus$ which is commutative, associative and distributive w.r.t to the multiplication in $P$ and turns $P$ into a semifield. Then the group ring $\mathbb{Z}P$ is used as a ground ring for a cluster algebra $\mathcal{A}$. My question is, why this particular choice of $P$? Is it because we then know that $P$ is torsionfree and so the group ring $\mathbb{Z}P$ is an integral domain and so then we can consider its field of fractions?

1$\begingroup$ Yes, I believe that's the main reason. I'd note that in practice few cluster algebras are left defined integrally in this way: most examples of cluster algebras, especially the geometric ones such as coordinate rings of Grassmannians, are constructed inside the field of fractions of a Laurent polynomial algebra whose indeterminates are the initial cluster variables (both mutable and frozen/coefficient ones) and where the LP coefficients are from a field. Tl;dr: most times people don't worry overly about this... (Maybe they should!) $\endgroup$– Jan GrabowskiOct 3, 2019 at 11:19

$\begingroup$ Thank you @JanGrabowski ! The construction you have outlined, isn't that how we would construct the upper cluster algebras? $\endgroup$– amator2357Oct 3, 2019 at 11:38

$\begingroup$ Are you interested in why there is a semifield P around? Or why given a the semifield we choose ZP as ground ring as opposed to something else involving P? Or both? $\endgroup$– John MachacekOct 3, 2019 at 16:54

$\begingroup$ I would be interesting in both @JohnMachacek $\endgroup$– amator2357Oct 4, 2019 at 7:47

$\begingroup$ @amator2357 I was being a bit brief, as it was a comment. When I said "constructed inside", I meant using mutation to produce the cluster variables and hence the cluster algebra. The upper cluster algebra is given by intersecting the family of LP algebras associated to each cluster, and contains the cluster algebra (by the Laurent phenomenon). $\endgroup$– Jan GrabowskiOct 4, 2019 at 12:42
1 Answer
I will expand a bit in an answer. In the comments the asker expressed interested in all aspects on ground rings for cluster algebras. I have though some about this, but mostly it seems that $\mathbb{Z}P$ is taken as ground ground in practice.
 I think in Cluster algebras I: Foundations one goal was to be as general as possible (Sergey Fomin has expressed this to me in personal communication). At this point the theory was brand new and the authors did not know what generality would be needed. Here is one of few places a ground ring $\mathbb{A} \subseteq \mathbb{Z}P$ is taken and a cluster algebra $\mathcal{A}_{\mathbb{A}}$ is looked at. It most places after $\mathbb{A} = \mathbb{Z}P$ is taken implicitly without any mentioned of $\mathbb{A}$. (Another place "extra" generality occurs in when signskewsymmetric exchagne matrices are used as opposed to just skewsymmetrizable matrices. It is still open to find other conditions which guarantee the signskewsymmetric property after mutating).
 I don't really know of a use of a semifield other than (a) the tropical semifield that shows up in geometric type or (b) the universal semifield of subtraction few rational expressions. But having the semifield generality allows both these choices to be treated at the same type. I would be interested to see an application of some other semifield, I do not know of any such instances.
 Taking ground ring $\mathbb{A} = \mathbb{Z}P$ makes some stuff work nicely, For example, many parts Muller's theory of Locally acyclic cluster algebras depends on frozen variables being invertible. With Bucher and Shapiro we investigate some of what happens if frozen variables are not inverted in Upper cluster algebras and choice of ground ring (I apologize for the self promotion, but I am excited someone is interested in ground ring!). In particular, we give an example where equality with the upper cluster algebra is sensitive to ground ring choice. Our motivation was that GekhtmanShapiroVainshtein have a cluster structure on $3 \times 3$ matrices with frozen variables not inverted (because frozen variables correspond to regular functions) in CremmerGervais cluster structure on $SL_n$. An example I like to give for this idea is in cluster algebra structure on the Grassmannian the frozen vertices correspond to Plücker coordinates $\Delta_{i,i+1, \dots, i+k1}$. Whether or not they are inverted changes between the Grassmannian and an open postroid variety. Goodearl and Yakimov also have a very nice paper titled Cluster algebra structures on Poisson nilpotent algebras Cluster algebra structures on Poisson nilpotent algebras which pays closed attention to if frozen variables are inverted or not.

$\begingroup$ My apologies for a late reply and thank you for such a great answer! I must say that the I feel like I have never paid enough attention to a (choice of) coefficient ring and I also feel like I should really try to understand its role better. Your answer will definitely be a great reference for that! Thanks again! $\endgroup$ Oct 8, 2019 at 8:02

$\begingroup$ No problem! I am glad you found the information useful. There is a lot you can do with just the "default" ground ring (or even just trivial coeffients) in cluster algebras. But in some contexts I think it's interesting to pay attention to the ground ring. $\endgroup$ Oct 13, 2019 at 3:13