Revista de la
Unión Matemática Argentina
Factorization of Frieze patterns
Moritz Weber and Mang Zhao
Volume 60, no. 2 (2019), pp. 407–415

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In 2017, Michael Cuntz gave a definition of reducibility of a quiddity cycle of a frieze pattern: It is reducible if it can be written as a sum of two other quiddity cycles. We discuss the commutativity and associativity of this sum operator for quiddity cycles and its equivalence classes, respectively. We show that the sum is neither commutative nor associative, but we may circumvent this issue by passing to equivalence classes. We also address the question whether a decomposition of quiddity cycles into irreducible factors is unique and we answer it in the negative by giving counterexamples. We conclude that even under stronger assumptions, there is no canonical decomposition.