  The investigation and classification of non-unique factorization phenomena
has attracted some interest in recent literature. For finitely generated
monoids, S.T. Chapman and P.A. Garc\'ia-S\'anchez, together with several
co-authors, derived a method to calculate the catenary and tame degree from the
monoid of relations. Then, in [1], the algebraic structure of this approach was
investigated and the restriction to finitely generated monoids was removed. We
now extend these ideas further to the monotone catenary degree and then apply
all these results to the explicit computation of arithmetical invariants of
semigroup rings.
  [1] A. Philipp. A characterization of arithmetical invariants by the monoid
of relations. Semigroup Forum, 81:424-434, 2010.
