In this paper, we introduce a notion of unistructural cluster algebras, for which the set of cluster variables uniquely determines the clusters. We prove that cluster algebras of Dynkin type and cluster algebras of rank 2 are unistructural, then prove that if A is unistructural or of Euclidean type, then f : A → A is a cluster automorphism if and only if f is an automorphism of the ambient field which restricts to a permutation of the cluster variables. In order to prove this result, we also

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... estigate the Fomin-Zelevinsky conjecture that two cluster variables are compatible if and only if one does not appear in the denominator of the Laurent expansions of the other. Conjecture 1.2. Any cluster algebra is unistructural. c c c c c · · ? ? where we identify along the vertical dashed lines and T r−1 lies on the mouth of T . Lemma 4.2. With the notation above, we have (a) dim Hom C (T 1 , M ) = 2 for every M ∈ ∆. (b) dim Hom C (U 1 , M ) = 2 if M ∈ ∆ does not lie on the sectional path from the mouth to τ U 1 .