  Since Ref. [1] shows the emergence of non-Abelian fusion rules in some
examples of a class of Abelian models, but does not prove whether these rules
also exist in other cases, the purpose of this paper is to present such proof
emphasizing the importance of the existence of these rules. By the way, as the
ground state of these models can be degenerate as a function of their algebra
and, hence, they can support some symmetry-protected topological (SPT) phases,
we prove that these non-Abelian fusion rules are always necessary for these SPT
phase transitions to occur via a condensation mechanism or/and some global
symmetry breaking.
