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The investigation and classification of non-unique factorization phenomenahas attracted some interest in recent literature. For finitely generatedmonoids, S.T. Chapman and P.A. Garc\'ia-S\'anchez, together with severalco-authors, derived a method to calculate the catenary and tame degree from themonoid of relations. Then, in [1], the algebraic structure of this approach wasinvestigated and the restriction to finitely generated monoids was removed. Wenow extend these ideas further to the monotone catenary degree and then applyall these results to the explicit computation of arithmetical invariants ofsemigroup rings.  [1] A. Philipp. A characterization of arithmetical invariants by the monoidof relations. Semigroup Forum, 81:424-434, 2010.

A characterization of arithmetical invariants by the monoid of relations II: The monotone catenary degree and applications to semigroup rings