^{}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.

bibitem{BR92} R. A. Brualdi, Introductory Combinatorics, New York, North Holland, Second Edition, 1992.

bibitem{DA81} P. S. Das, {em Fuzzy groups and level subgroups}, J. Math. Anal. Appl., {bf 84} (1981), 264-269.

bibitem{JCLJ2011} S. Jia, Y. Chen, J. Liu and Y. Jiang, {em On the number of fuzzy subgroups of finite abelian p-groups with type $(p^n, p^m)$}, In Proceedings of The 3rd International Conference on Computer Research and Development, China, {bf 4} (2011), 62-64.

bibitem{NG05} S. Ngcibi, {it Studies of equivalent fuzzy subgroups of finite abelian $p$-groups of rank two and their subgroup lattices}, Thesis, (PhD) Rhodes university, 2005.

bibitem{NMM10} S. Ngcibi, V. Murali and B. B. Makamba, {em Fuzzy subgroups of rank two abelian $p$-group}, Iranian Journal of Fuzzy Systems, {bf 7(2)} (2010), 149-153.

bibitem{PS98} E. Pergola and R. A. Sulanke, {em Schr"{o}der triangles, paths, and parallelogram polyominoes}, J. Integer Seq., Article 98.1.7., {bf 1} (1998).

bibitem{SL} N. J. A. Sloane, {it The on-line encyclopedia of integer sequences}, https://oeis.org.

bibitem{TB08} M. Tu{a}rnu{a}uceanu and L. Bentea, {em On the number of fuzzy subgroups of finite abelian groups}, Fuzzy Sets and Systems, {bf 159} (2008), 1084-1096.

bibitem{Tu95} A. Tucker, {it Applied combinatorics}, John Wiley & Sons, Inc., New York, 1995.