# What are the generating elements of a cluster algebra?

I'm trying to get into cluster algebras, and so have been looking at the basic stuff to get the foundations settled. A problem I've run into is that I've come across two (seemingly contradictory) definitions of how the cluster algebra is to be generated.

The first source is Lauren Williams' Cluster Algebras: An Introduction, where she writes:

Definition 2.10 (Patterns). Consider the $n$-regular tree $\mathbb{T}_n$ whose edges are labeled by the numbers $1 , \dots , n$, so that the $n$ edges emanating from each vertex receive different labels. A cluster pattern is an assignment of a labeled seed $\sum_{t} =(\mathbf{x}_t , Q)$ to every vertex $t \in \mathbb{T}_n$, such that the seeds assigned to the endpoints of any edge $t \xrightarrow[]{k} t'$ are obtained from each other by the seed mutation in direction $k$. The components of $\mathbf{x}_t$ are written as $\mathbf{x}_t = (x_{1;t} , \dots , x_{n;t} )$.

[...]

Definition 2.11 (Cluster algebra). Given a cluster pattern, we denote \begin{equation*} \mathcal{X} = \cup_{t \in \mathbb{T}_n} \mathbf{x}_t = \{ x_{i;t} : t \in \mathbb{T}_n , 1 \leq i \leq n \} , \end{equation*} the union of clusters of all the seeds in the pattern. The elements $x_{i;t} \in \mathcal{X}$ are called cluster variables. The cluster algebra $\mathcal{A}$ associated with a given pattern is the $\mathbb{Z}[c]$- subalgebra of the ambient field $\mathcal{F}$ generated by all cluster variables: $\mathcal{A} = \mathbb{Z}[c][\mathcal{X}]$.

where the $c$ in $\mathbb{Z}[c]$ are the frozen/coefficient variables $\{ x_{n+1} , \dots , x_m \}$.

Wikipedia gives a somewhat different definition:

A cluster algebra is constructed from a seed as follows. If we repeatedly mutate the seed in all possible ways, we get a finite or infinite graph of seeds, where two seeds are joined if one can be obtained by mutating the other. The underlying algebra of the cluster algebra is the algebra generated by all the clusters of all the seeds in this graph. The cluster algebra also comes with the extra structure of the seeds of this graph.

My supervisor and me believe that these are not in fact conflicting definitions, but that they are in fact equivalent, as we have found them to be for the simplest of cases. However, we have hitherto been unable to figure out how one can prove this for the general case.

If anyone knows how to accomplish this proof, or otherwise know of any source or anywhere where this proof can be found, it would be greatly appreciated.

The fact that the two definitions are equivalent is essentially the observation that a "cluster pattern" is uniquely determined by any seed at any vertex in the tree. This is simply because once you know one seed, all adjacent ones are determined by mutation rules. The wikipedia definition mentions that you get a graph of seeds which are connected if they are related by mutation. This is precisely that $n$-regular tree from the "cluster pattern" definition.