Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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A NOTE ON IMPRECISE GROUP AND ITS PROPERTIES

  • JABA RANI NARZARY (Department of Mathematics, Central Institute of Technology Kokrajhar) ;
  • SAHALAD BORGOYARY (Department of Mathematics, Central Institute of Technology Kokrajhar)
  • Received : 2023.08.23
  • Accepted : 2024.03.13
  • Published : 2024.05.30
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Abstract

In this paper, using the notion of the imprecise set, the idea of an imprecise group is introduced including some examples. The two key rules of classical set theory are obeyed by this extended version of fuzzy sets, which the existing complement definition of a fuzzy set failed to do. With the support from general group theory, the paper also provides some fundamental properties of an imprecise group here. Additionally, it includes a few characteristics of imprecise subgroups, and abelian imprecise group.

Keywords

Acknowledgement

The first author acknowledges the financial support received from the University Grant Commission, New Delhi under the Scheme of the National Fellowship for Higher Education (NFHE) vide award letter-number 202021-NFST-ASS-01210, Dated 20th September 2021 to carry out this research work.

References

  1. A. Piegat, A new definition of the fuzzy set, Int. J. Appl. Math. Comput. Sci. 15 (2005), 125-140. 
  2. A. Prasanna, M. Premkumar, S.I. Mohideen & D.K. Shukla, K-Q-fuzzy orders relative to K-Q-fuzzy subgroups and cyclic group on various fundamental aspects, Materials Today: Proceedings (2020), 1-4. https://doi.org/10.1016/j.matpr.2020.12.1063 
  3. A. Rosenfeld, Fuzzy groups, Journal of mathematical analysis and applications 35 (1971), 512-517.  https://doi.org/10.1016/0022-247X(71)90199-5
  4. B. Basumatary, A note on fuzzy closure of a fuzzy set, Journal of Process Management and New Technologies 3 (2015), 35-39. 
  5. B. Basumatary & D.D. Mwchahary, A Note on Intuitionistic Fuzzy Set on the Basis of Reference Function, International Journal of Applied Engineering Research 13 (2018), 11240-11241. 
  6. B. Basumatary, S. Borgoyary, K.P. Singh & H.K. Baruah, Towards Forming the Field of Fuzzy Boundary on the Basis reference Function, Global Journal of Pure and Applied Mathematics 13 (2017), 2703-2716. 
  7. B. Basumatary, Towards forming the field of fuzzy closure with reference to fuzzy boundary, Journal of Process Management and New Technologies 4 (2016), 30-40.  https://doi.org/10.5937/JPMNT1601030B
  8. C. Bejines, M.J. Chasco & J. Elorza, Aggregation of fuzzy subgroups, Fuzzy Sets and Systems 418 (2021), 170-184.  https://doi.org/10.1016/j.fss.2020.05.017
  9. D. Alghazzawi, U. Shuaib, T. Fatima, A. Razaq & M.A. Binyamin, Algebraic characteristics of anti-intuitionistic fuzzy subgroups over a certain averaging operator, IEEE Access 8 (2020), 205014-205021.  https://doi.org/10.1109/ACCESS.2020.3035590
  10. H. Alolaiyan, U. Shuaib, L. Latif & A. Razaq, T-intuitionistic fuzzification of Lagrange's theorem of t-Intuitionistic fuzzy subgroup, IEEE Access 7 (2019), 158419-158426.  https://doi.org/10.1109/ACCESS.2019.2950167
  11. H.K. Baruah, An introduction to the theory of imprecise sets: The mathematics of partial presence, J. Math. Comput. Sci. 2 (2012), 110-124. 
  12. H.K. Baruah, Fuzzy Membership with respect to a Reference Function, Journal of the Assam Science Society 40 (1999), 65-73. 
  13. H.K. Baruah, The theory of fuzzy sets: beliefs and realities, International Journal of Energy, Information and Communications 2 (2011), 1-22. 
  14. H.K. Baruah, Towards forming a field of fuzzy sets, International Journal of Energy, Information and Communications 2 (2011), 16-20.
  15. I. Masmali, U. Shuaib, A. Razaq, A. Fatima & G. Alhamzi, On Fundamental Algebraic Characterizations of-Fuzzy Normal Subgroups, Journal of Function Spaces (2022). 
  16. J.G. Kim, Fuzzy orders relative to fuzzy subgroups, Information sciences 80 (1994), 341-348.  https://doi.org/10.1016/0020-0255(94)90084-1
  17. J.G. Kim, Orders of fuzzy subgroups and fuzzy p-subgroups, Fuzzy sets and systems 61 (1994), 225-230.  https://doi.org/10.1016/0165-0114(94)90237-2
  18. J.N. Mordeson, K.R. Bhutani & A. Rosenfeld, Lattices of Fuzzy Subgroups, Fuzzy Group Theory (2005), 239-266. 
  19. J. Narzary, S. Borgoyary, J. Basumatary & B. Basumatary, A Short Study on Cosets and Normal Subgroup using Imprecise set definition, Accepted in AIP conference Proceeding 2022. 
  20. K.D. Ahmad, M. Bal & M. Aswad, The kernel of fuzzy and anti-fuzzy groups, Journal of neutrosophic and fuzzy systems (2022), 48-54. 
  21. K.P. Singh & S. Borgoyary, Construction of Normal Imprecise Functions, Advances in Computational Sciences and Technology 10 (2017), 2019-2036. 
  22. K.P. Singh & S. Borgoyary, Rate of Convergence of the Sine Imprecise Functions, International Journal of Intelligent Systems and Applications 8 (2016), 31. 
  23. K.T. Atanassov, Intuitionistic fuzzy sets, Physica-Verlag, HD, 1999, 1-137. 
  24. L.A. Zadeh, Fuzzy sets, Information and control 8 (1965), 338-353.  https://doi.org/10.1016/S0019-9958(65)90241-X
  25. M. Bal, K.D. Ahmad, A.A. Hajjari & R. Ali, A short note on the kernel subgroup of intuitionistic fuzzy groups, Journal of Neutrosophic and Fuzzy Systems 2 (2022), 14-20. 
  26. M. Dhar, A Note on Determinant and Adjoint of Fuzzy Square Matrix, IJ intelligent systems and applications 5 (2013), 58-67.  https://doi.org/10.5815/ijisa.2013.05.07
  27. M. Dhar, Cardinality of fuzzy sets: An overview, International Journal of Energy, Information and Communications 4 (2013), 15-20.  https://doi.org/10.14257/ijeic.2013.4.5.02
  28. M. Dhar, On Geometrical Representation of Fuzzy Numbers, International Journal of Energy Information and Communications 3 (2012), 29-34. 
  29. M. Gulzar, D. Alghazzawi, M.H. Mateen & N. Kausar, A certain class of t-intuitionistic fuzzy subgroups, IEEE access 8 (2020), 163260-163268.  https://doi.org/10.1109/ACCESS.2020.3020366
  30. M. Gulzar, M.H. Mateen, D. Alghazzawi & N. Kausar, A novel applications of complex intuitionistic fuzzy sets in group theory, IEEE Access 8 (2020), 196075-196085.  https://doi.org/10.1109/ACCESS.2020.3034626
  31. N.P. Mukherjee and P. Bhattacharya, Fuzzy groups: some group-theoretic analogs, Information sciences 39 (1986), 247-267.  https://doi.org/10.1016/0020-0255(86)90039-3
  32. P. Bhattacharya, Fuzzy subgroups: some characterizations, Journal of mathematical analysis and applications 128 (1987), 241-252.  https://doi.org/10.1016/0022-247X(87)90228-9
  33. P. Bhattacharya, Fuzzy subgroups: some characterizations II, Information sciences 38 (1986), 293-297.  https://doi.org/10.1016/0020-0255(86)90028-9
  34. P. Bhattacharya & N.P. Mukherjee, Fuzzy relations and fuzzy groups, Information sciences 36 (1985), 267-282.  https://doi.org/10.1016/0020-0255(85)90057-X
  35. P.S. Das, Fuzzy groups and level subgroups, Journal of mathematical analysis and applications 84 (1981), 264-269.  https://doi.org/10.1016/0022-247X(81)90164-5
  36. Q.S. Gao, X.Y. Gao & Y. Hu, A new fuzzy set theory satisfying all classical set formulas, Journal of computer Science and Technology 24 (2009), 798-804.  https://doi.org/10.1007/s11390-009-9250-3
  37. R. Biswas, Intuitionistic fuzzy subgroups, In Mathematical Forum 10 (1989), 37-46. 
  38. R.J. Hussain & S. Palaniyandi, A review on Q-fuzzy subgroups in algebra, International Journal of Applied Engineering Research 14 (2019), 60-63. 
  39. R. Rasuli, Intuitionistic fuzzy complex subgroups with respect to norms (T, S), Journal of fuzzy extension and applications 4 (2023), 92-114. 
  40. R. Rasuli, Norms over Q-intuitionistic fuzzy subgroups of a group, Notes on intuitionistic fuzzy sets 29 (2023), 30-45. 
  41. S. Ardanza-Trevijano, M.J. Chasco & J. Elorza, The annihilator of fuzzy subgroups, Fuzzy Sets and Systems 369 (2019), 122-131.  https://doi.org/10.1016/j.fss.2018.11.001
  42. S. Bhunia, G. Ghorai, Q. Xin & F.I. Torshavn, On the characterization of Pythagorean fuzzy subgroups, AIMS mathematics 6 (2021), 962-978.  https://doi.org/10.3934/math.2021058
  43. S. Borgoyary, A Few Applications of Imprecise Matrices, IJ Intelligent system and Applications 8 (2015), 9-17. 
  44. S. Borgoyary, An introduction of two and three dimensional imprecise numbers, IJ Information Engineering and Electronic Business 7 (2015), 27-38.  https://doi.org/10.5815/ijieeb.2015.05.05
  45. S. Borgoyary & K.P. Singh, Rate of Convergence of the Sine Imprecise Functions, International Journal of Intelligent Systems and Applications 8 (2016), 31. 
  46. T.J. Neog & D.K. Sut, Complement of an extended fuzzy set, IJCA 29 (2011), 0975-8887. 
  47. V.K. Khanna & S.K. Bhamri, A course in abstract algebra, Vikas Publishing House, 2016. 
  48. V. Pushpalatha & R.V. Chandra, Arithmetic mean, geometric mean and harmonic mean of interval valued fuzzy matrices based on reference function, Journal of Data Acquisition and Processing 38 (2023), 1637-1644. 
  49. W. Kosinski, P. Prokopowicz & D. Slezak, On algebraic operations on fuzzy numbers, In Intelligent Information Processing and Web Mining: Proceedings of the International IIS: IIPWM'03 Conference held in Zakopane, Poland, Springer Berlin Heidelberg June 2-5 (2003), 353-362. 
  50. Y.B. Jun, M.A. Ozturk & G. Muhiuddin, A generalization of (∈, ∈ ∧q)-fuzzy subgroups, International Journal of Algebra and statistics 5 (2016), 7-18.  https://doi.org/10.20454/ijas.2016.1041
  51. Y. Li, X. Wang & L. Yang, A Study of (λ, µ)-Fuzzy Subgroups, Journal of Applied Mathematics 2013 (2013).