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To construct a cluster algebra (of finite type $A$) associated to a convex $m$-gon $P_m$ we first take a triangulation $T$ of $P_m$ and regard it as our initial seed with exchangeable variables being the diagonals in $T$ and with the frozen variables being the edges of $P_m$. I read that if we start with a different triangulation $T'$ of $P_m$ we will end up with an isomorphic (as rings?) cluster algebra. Say I want to construct a map which extends to a ring automorphism between cluster algebras associated to $P_m$ w/ initial seed $T$ and cluster algebra associated to $P_m$ w/ initial seed $T'$. Is it enough to show it maps all exchangeable variables to exchangeable variables and all frozen to frozen, bijectively?

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  • $\begingroup$ Just to be sure: you do not want to assume that your map sends clusters to clusters, but only cluster variables to cluster variables and frozen variables to frozen variables, right? $\endgroup$ Aug 15, 2020 at 22:28

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