Computing fuzzy subgroups of some special cyclic groups
Commun. Korean Math. Soc.
Published online August 5, 2019
Babington Makamba and Michael Mbindyo Munywoki
University of Fort Hare, Technical University of Mombasa
Abstract : In this paper, we discuss the number of distinct fuzzy subgroups of the group $\mathbb{Z}_{p^n}\times \mathbb{Z}_{q^m}\times \mathbb{Z}_r$ $m=1,2,3$ where $p,q,r$ are distinct primes for any $n\in \mathbb{Z}^+$ using the criss-cut method that was proposed by Murali and Makamba in their study of distinct fuzzy subgroups. First we briefly revisit the group $\mathbb{Z}_{p^n}\times \mathbb{Z}_{q^m}$ and use the criss-cut method there to arrive at the results that Murali and Makamba obtained using a different method they called the cross-cut method. The criss-cut method first establishes all the maximal chains of the subgroups of a group $G$ and then counts the distinct fuzzy subgroups contributed by each chain. In this paper, all the formulae for calculating the number of these distinct fuzzy subgroups are given in polynomial form.
Keywords : Maximal chain, Equivalence, Fuzzy subgroups
MSC numbers : Primary 20N25; 03E72; Secondary 20K01; 20K27
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