Connections between fuzzy multi-groups and fuzzy multi-graphs

Document Type : Original Article

Authors

1 Department of mathematics, Payame Noor University, P.O. Box 19395-4697, Tehran, Iran

2 Department Mathematics, Payame Noor University, Tehran, Iran.

3 Department of Mathematics, Payame Noor University, Tehran, Iran

Abstract

 In this paper, we introduce the concept of fuzzy multi-graphs and a type of isomorphism on fuzzy multi-graphs and study their properties. We show that how to assign a fuzzy multi-graph to the appropriate fuzzy multi-groups. We also prove that every fuzzy multi-group of group $G$ can be embedded into the fuzzy multi-group of the group $Aut(\Gamma_\wedge^\sigma)$, where $Aut(\Gamma_\wedge^\sigma)$ is a group of automorphisms of the fuzzy multi-graph $\Gamma_\wedge^\sigma$.

Keywords


[1] J.M. Antony, H. Sherwood, Fuzzy groups rede ned, Journal of Analysis and Applications, 69
(1979), 124{130.
[2] J.A. Awolola, A.M. Ibrahim, Some results on multi-groups, Quasigroups and Related Systems,
24(2) (2016), 169{177.
[3] A. Baby, T.K. Shinoj, J.J. Sunil. On abelian fuzzy multi-groups and order of fuzzy multi-
groups, Journal of New Theory, 5(2) (2015), 80{93.
[4] V.K. Balakrishnan, Graph theory, McGraw-Hill, 1997.
[5] P. Bhattacharya, Some remarks on fuzzy graphs, Pattern Recognition Letters, 6 (1987), 297{
302.
[6] K.R. Bhutani, On automorphisms of fuzzy graphs, Pattern Recognition Letters, 9 (1989),
159{162.
[7] W.D. Blizard, multi-set theory, Notre Dame Journal of Formal Logic, 30 (1989), 36{66.
[8] W.D. Blizard, The development of multi-set theory, Modern Logic, 1 (1991), 319{352.
[9] R.A. Borzooei, B. Sheikh Hoseini, M. Mohseni Takallo, Results on t-fuzzy graphs, New Math-
ematics and Natural Computation, 16 (2020), 143{161.
[10] M. Dresher, O. Ore, Theory of multi-groups, American Journal of Mathematics, 60 (1938),
705{733.
[11] P.A. Ejegwa, Homomorphism of fuzzy multi-groups and some of its properties, Applications
and Applied Mathematics, 13(1) (2018), 114{129.
[12] P.A. Ejegwa, On fuzzy multi-groups and fuzzy submulti-groups, Journal of Fuzzy Mathematics,
26(3) (2018), 641{654.
[13] P.A. Ejegwa, On alpha-cuts homomorphism of fuzzy multi-groups, Annals of Fuzzy Mathe-
matics and Informatics, 19(1) (2020), 73{87.
[14] P.A. Ejegwa, Some group's theoretic notions in fuzzy multi-group context, In: Jana, C., Sena-
pati, T. Pal, M. (Eds.); Handbook of Research on Emerging Applications of Fuzzy Algebraic
Structures, pp. 34-62, IGI Global Publisher, Hershey, Pennsylvania 17033-1240, USA, (2020).
[15] D. Knuth, The art of computer programming, semi numerical algorithms, Second Edition 2,
Addison-Wesley, Reading, Massachusetts, 1981.
[16] L. Mao, Topological multi-groups and multi elds, International Journal of Mathematics and
Combinatorics, 1 (2009), 8{17.
[17] S. Mirvakili, H. Naraghi, Connected between reversible regular hypergroups, t-fuzzy subgroups
and t-fuzzy graphs, Algebraic Hyperstructures and Logical Algebras, 1(4) (2020), 71{82.
[18] S. Mirvakili, H. Naraghi, M. Shirvani, P. Ghiasvand, Some fuzzy multi-groups obtained from
fuzzy subgroups, Journal of Mathematical Extension, to appear.
[19] S. Miyamoto, K. Mizutani, multi-set model and methods for nonlinear document clustering
for information retrieval, Springer-Verlag Berlin Heidelberg, (2004), 273{283.
[20] C.V. Negoita, D.A. Ralescu, Applications of fuzzy sets to system analysis, Wiley, New York,
1975.
[21] A. Rosenfeld, Fuzzy groups, Journal of Mathematics and Applications, 35 (1971), 512{517.
[22] A. Rosen eld, Fuzzy sets and their applications, Academic Press, New York, 1975.
[23] S. Samanta, A. Pal, M. Pal, New concepts of fuzzy planar graphs, International Journal of
Advanced Research in Arti cial Intelligence, 3(1) (2014), 52{59.
[24] B.M. Schein, multi-groups, Journal of Algebra, 111 (1987), 114{132.
[25] T.K. Shinoj, A. Baby, J.J. Sunil, On some algebraic structures of fuzzy multi-sets, Annals of
Fuzzy Mathe-matics and Informatics, 9(1) (2015), 77{90.
[26] A. Syropoulos, On generalized fuzzy multi-sets and their use in computation, Iranian Journal
of Fuzzy Systems, 9(2) (2012), 113{125
[27] N.J. Wildberger, A new look at multi-sets, School of Mathematics, UNSW Sydney 2052,
Australia, 2003.
[28] R.R. Yager, On the theory of bags, International Journal of General Systems, 13 (1986), 23{37.
[29] L.A. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338{353.