a sequence of mutations. Since mutations are involutive (see [@bz-qclust Prop 4.10]), the quantum seeds in $\mathfrak{F}$ can be grouped in equivalence classes, defined by the relation of mutation equivalence. The quantum cluster algebra generated by a seed $(M,B)\subset \mathfrak F$ is the $\CC[t^{\pm 1}]$-subalgebra generated by the cluster variables associated to the seeds in an equivalence class.

There are definitions of quantum lower bounds, upper bounds and quantum upper cluster algebras (see [@bz-qclust Sections 5 and 7]), analogous to the classical case.

Intersections of Ideals with Clusters {#se:key propositions}
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In the present chapter we consider the intersection between Poisson ideals in a cluster algebra and individual clusters. Moreover, we prove quantum analogues of the propositions whenever available.

\[pr:TPP super toric\] Let ${\bf x}$ be a cluster, $rank(\Lambda)=n$ and $\II$ be a non-zero Poisson ideal. Then the ideal $\II$ contains a monomial in $x^m\in \CC[{\bf x}]$.

Notice first that $\II_{\bf x}=\II\cap \CC[x_1,\ldots, x_n]\ne 0$. Indeed, let $0\ne f\in \II$. We can express $f$ as a Laurent polynomial in the variables $x_1,\ldots, x_n$; i.e., $f=x_{1}^{-c_1}\ldots x_n^{-c_n} g$ where $c_1,\ldots, c_n\in \ZZ_{\ge 0}$ and $0\ne g\in\CC[x_1,\ldots, x_n]$. Clearly, $g= x_{1}^{c_1}\ldots x_n^{c_n}f \in \II_{\bf x}$.

We complete the proof by contradiction. Let 