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Arithmetical invariants---such as sets of lengths, catenary and tamedegrees---describe the non-uniqueness of factorizations in atomic monoids. Westudy these arithmetical invariants by the monoid of relations and bypresentations of the involved monoids. The abstract results will be applied tonumerical monoids and to Krull monoids.

Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids