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

Document Type: Research Paper

Author

Mathematics, Gangneung-Wonju National University, Gangneung, Re- public of Korea

Abstract

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, and
gave explicit formulas for the cases when $m$ is any positive
integer and $n=1,2,3$. Even though their method can be used for the
cases when $n=4,5,\ldots$ by using inductive arguments, it does not
provide an explicit formula for that number  for an arbitrarily
given positive integer $n$. In this paper we give a complete answer
to 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.

Keywords


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