فهرست نشریات علمی دسترسی آزاد ایران

Ngcibi, Murali and Makamba [Fuzzy subgroups of rank two abelian$p$-group, Iranian J. of Fuzzy Systems {\bf 7} (2010), 149-153]considered the number of fuzzy subgroups of a finite abelian$p$-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 $n=1,2,3$. Even though their method can be used for thecases when $n=4,5,\ldots$ 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 $m$and $n$, an explicit formula for the number of fuzzy subgroups of$\mathbb{Z}_{p^m}\times \mathbb{Z}_{p^n}$  is given.