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We introduce several commutative rings, the  snake rings , that have strong connections to cluster algebras and that are interesting on their own right as combinatorially defined rings. The elements of these rings are residue classes of unions of certain labeled graphs that were instrumental to construct canonical bases in the theory of cluster algebras. We obtain  several  rings by varying the conditions on the structure as well as the labeling of the graphs. A general form of this ring contains  all  cluster algebras of unpunctured surface type.

Snake Graph Calculus and Cluster Algebras from Surfaces III: Band Graphs and Snake Rings