In most definitions of a cluster algebra of geometric type, it is said to be the subalgebra of the field $\mathbb Q (x_1,\cdots,x_n)$ generated by the cluster variables.
I would like to compute such an algebra but I am not sure of the result. As an example, let's try with the cluster algebra of rank $2$ associated with the seed $((x,y),B)$ where $B$ is the skew-symetric matrix $\begin{pmatrix} 0 & 1 \\-1 & 0\end{pmatrix}$.
There is only a finite number of clusters : $x,y,\dfrac{1+y}{x},\dfrac{1+x}{y},\dfrac{1+x+y}{xy}$ but what is the associated cluster algebra ? What are elements of it ?
