. We refer to the elements of the clusters as the [*cluster variables*]{}.

Notice that the coefficients, resp. frozen variables $x_{m+1},\ldots, x_n$ will never be mutated. Of course, that explains their name.

We have the following fact, motivating the definition of cluster algebras in the study of total positivity phenomena and canonical bases.

[@FZI Section 3](Laurent phenomenon) Let $\mathfrak{A}$ be a cluster algebra with initial extended cluster $(x_1,\ldots, x_n)$. Any cluster variable $x$ can be expressed uniquely as a Laurent polynomial in the variables $x_1,\ldots, x_n$ with integer coefficients.

Moreover, it has been conjectured for all cluster algebras, and proven in many cases (see e.g. [@MSW] and [@FST],[@FT]) that the coefficients of these polynomials are positive.

Finally, we recall the definition of the lower bound of a cluster algebra $\AA$ corresponding to a seed $({\bf x}, B)$. Denote by $y_i$ for $1\le i\le m$ the cluster variables obtained from ${\bf x}$ through mutation at $i$; i.e., they satisfy the relation $x_iy_i=P_i$.

[@BFZ Definition 1.10] \[def:lower bounds\] Let $\AA$ be a cluster algebra and let $({\bf x}, B)$ be a seed. The lower bound $ \mathfrak{L}_B \subset \AA$ associated with $({\bf x}, B)$ is the algebra generated by the set $\{x_1,\ldots 