x_n,y_1\ldots, y_m\}$.

Upper cluster algebras {#se:upper cluster algebras}
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Berenstein, Fomin and Zelevinsky introduced the related concept of upper cluster algebras in [@BFZ].

Let $\mathfrak{A} \subset \mathfrak{F}$ be a cluster algebra with initial cluster $(x_1, \ldots, x_n, B)$ and let, as above, $y_1,
  \ldots, y_m$ be the cluster variables obtained by mutation in the directions $1, \ldots, m$, respectively.

\(a) The upper bound $\UU_{{\bf x},B} ( \mathfrak{A})$ is defined as $$\UU_{{\bf x},B} ( \mathfrak{A}) = \bigcap_{j = 1}^m \CC [x_1^{\pm 1}, \ldots
     x_{j - 1}^{\pm 1}, x_j, y_j, x_{j + 1}^{\pm 1}, \ldots, x_m^{\pm 1},
x_{m+1},\ldots,x_n] \ .$$

\(b) The upper cluster algebra $\UU  ( \mathfrak{A})$ is defined as $$\UU  ( \mathfrak{A})=\bigcap_{({\bf x'},B')}\UU_{\bf x'} ( \mathfrak{A})\ ,$$ where the intersection is over all seeds $({\bf x}',B')$ in the mutation equivalence class of $({\bf x},B)$.

Observe that each cluster algebra is contained in its upper cluster algebra (see [@BFZ]).

Poisson structures {#se:poissonstructure}
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Cluster algebras are closely related to Poisson algebras. In this section we recall some of the related notions and results.

Let $k$ be a field of charactieristic $0$. A Poisson algebra is a pair $(A,\{\cdot,\cdot\})$ consisting of a commutative $k$-algebra $A$ and a bilinear map $\{\cdot,\cdot\}:A\otimes A\to A$, 