An explicit formula for the number of fuzzy subgroups of a finite abelian p-group\\ of rank two

Ngcibi, Murali and Makamba [Fuzzy subgroups of rank two abelianp-group, Iranian J. of Fuzzy Systems {\bf ۷} (۲۰۱۰), ۱۴۹-۱۵۳]considered the number of fuzzy subgroups of a finite abelianp-group \mathbb{Z}_{p^m}\times \mathbb{Z}_{p^n} of rank two, andgave explicit formulas for the cases when m is any positiveinteger and n=۱,۲,۳. Even though their method can be used for thecases when n=۴,۵,\ldots by using inductive arguments, it does notprovide an explicit formula for that number  for an arbitrarilygiven positive integer n. In this paper we give a complete answerto this problem. Thus for arbitrarily given positive integers mand n, an explicit formula for the number of fuzzy subgroups of\mathbb{Z}_{p^m}\times \mathbb{Z}_{p^n}  is given.

Keywords:

Enumeration , Fuzzy subgroup , Subgroup lattice , Abelian p-group , Schr {o}der's triangle