@article {
author = {Oh, Ju-Mok},
title = {An explicit formula for the number of fuzzy subgroups of a finite abelian $p$-group\\ of rank two},
journal = {Iranian Journal of Fuzzy Systems},
volume = {10},
number = {6},
pages = {125-135},
year = {2013},
publisher = {University of Sistan and Baluchestan},
issn = {1735-0654},
eissn = {2676-4334},
doi = {10.22111/ijfs.2013.1335},
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, 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.},
keywords = {Enumeration,Fuzzy subgroup,Subgroup lattice,Abelian $p$-group,Schr\"{o}der's triangle},
url = {https://ijfs.usb.ac.ir/article_1335.html},
eprint = {https://ijfs.usb.ac.ir/article_1335_5968fa05004644552077641dfc99a189.pdf}
}