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Defining cluster algebras of finite kind $mathrm{A}$ by turbines and relations

Consider a cluster algebra of finite kind $mathrm{A}$. The clique of all (discrete) cluster variables is of finite cardinality, denote it by $okay$, for such algebra. Is it undoubted that, for an capricious altenative of the bottom ring, each such algebra is isomorphic to the polynomial ring (over the bottom ring, I speculate) in (suitably outlined) $okay$ variables moded out by the model generated by the trade relations?

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