대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제34권4호
  • /
  • Pages.1049-1067
  • /
  • 2019
  • /
  • 1225-1763(pISSN)
  • /
  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

COMPUTING FUZZY SUBGROUPS OF SOME SPECIAL CYCLIC GROUPS

  • Makamba, Babington (Department of Mathematics University of Fort Hare) ;
  • Munywoki, Michael M. (Department of Mathematics and Physics Technical University of Mombasa)
  • 투고 : 2018.08.10
  • 심사 : 2018.10.12
  • 발행 : 2019.10.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

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. 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.

키워드

참고문헌

  1. B. B. Makamba, Studies in fuzzy groups, PhD Thesis, Rhodes University, Grahamstown, South Africa, 1992.
  2. M. M. Munywoki and B. B. Makamba, Classifying fuzzy subgroups of the abelian group $Zp_1$ x $Zp_2$ x ... x $Zp_-pn}$ for distinct primes $p_1$, $p_2$, ..., $p_n$, Ann. Fuzzy Math. Inform. 13 (2017), no. 2, 289-296. https://doi.org/10.30948/afmi.2017.13.2.289
  3. V. Murali and B. B. Makamba, On an equivalence of fuzzy subgroups. I, Fuzzy Sets and Systems 123 (2001), no. 2, 259-264. https://doi.org/10.1016/S0165-0114(00)00098-1
  4. V. Murali and B. B. Makamba, On an equivalence of fuzzy subgroups. II, Fuzzy Sets and Systems 136 (2003), no. 1, 93-104. https://doi.org/10.1016/S0165-0114(02)00140-9
  5. V. Murali and B. B. Makamba, Counting the number of fuzzy subgroups of an abelian group of order $p^nq^m$, Fuzzy Sets and Systems 144 (2004), no. 3, 459-470. https://doi.org/10.1016/S0165-0114(03)00224-0
  6. V. Murali and B. B. Makamba, On methods of counting preferential fuzzy subgroups, Adv. Fuzzy Sets Syst. 3 (2008), no. 1, 21-42.
  7. O. Ndiweni, The classification of some fuzzy subgroups of finite groups under a natural equivalence and its extension, with particular emphasis on the number of equivalence classes, MS Thesis, University of Fort Hare, Alice, South Africa, 2007.
  8. O. Ndiweni and B. B. Makamba, Distinct fuzzy subgroups of some dihedral groups, Adv. Fuzzy Sets Syst. 9 (2011), no. 1, 65-91.
  9. A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971), 512-517. https://doi.org/10.1016/0022-247X(71)90199-5
  10. H. Sherwood, Products of fuzzy subgroups, Fuzzy Sets and Systems 11 (1983), no. 1, 79-89. https://doi.org/10.1016/S0165-0114(83)80070-0
  11. P. Sivaramakrishna Das, Fuzzy groups and level subgroups, J. Math. Anal. Appl. 84 (1981), no. 1, 264-269. https://doi.org/10.1016/0022-247X(81)90164-5
  12. L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965), 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X