Research Article
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Fuzziness on Hopf Algebraic Structures with Its Application

Year 2021, Issue: 35, 20 - 31, 30.06.2021

Munir Ahmed Tahir Mahmood , Muhammad Munır

https://doi.org/10.53570/jnt.846538

Abstract

The objective of writing this manuscript is to apply the concept of fuzzy set on some basic Hopf algebraic structures. In this manuscript, the novel concepts of fuzzy Hopf subalgebra, fuzzy Hopf ideal, and fuzzy H-submodule are proposed. Some properties of these concepts are discussed, and some significant results are also proved in it. The advantages of the proposed work are also studied in it. The application of the proposed work is also discussed in it.

Keywords

Fuzzy Hopf subalgebra fuzzy Hopf ideal fuzzy H-submodule

References

  • H. Hopf, Uber die Topologie der Gruppen-Mannigfaltigkeiten und ihrer Verallgemeinerungen, Annals of Mathematics 42(1) (1941) 22-52.
  • H. Hopf, Uber den Rang Geschlossener Liescher Gruppen, Commentarii Mathematici Helvetici 13 (1940) 119-143.
  • E. Abe, Hopf Algebras, Cambridge University Press, 1997.
  • M. Cohen, D. Fishman, Hopf Algebra Actions, Journal of Algebra 100 (1986) 363-379.
  • S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Regional Conference Series in Mathematics 82 (1993).
  • S. Majid, Physics for Algebraists: Non-Commutative and Non-Cocomutative Hopf Algebras by a Bicrossproduct Construction, Journal of Algebra 130 (1990) 17-64.
  • L. M. Ionescu, M. Marsalli, A Hopf Algebra Deformation Approach to Renormalization, (2003) arXiv:hep-th/0307112 2003.
  • S. Majid, Hopf Algebra for Physics at the Plank Scale, Classical and Quantum Gravity5(12) (1988) 1587-1606.
  • M. E. Sweedler, Hopf Algebras, Benjamin New-York, 1969.
  • R. J. Blattner, M. Cohen, S. Montgomery, Cross Products and Inner Actions of Hopf Algebras, Transactions of the American Mathematical Society 298(2) (1986) 671-711.
  • S. Montgomery, Indecomposable Coalgebras, Simple Comodules and Pointed Hopf Algebras, Proceedings of the AMS - American Mathematical Society123 (1995) 2343-2351.
  • J. W. Milnor, J. C. Moore, On the Structures of Hopf Algebras, Annals of Mathematics 81 (1965) 211-264.
  • L. A. Zadeh, Fuzzy Sets, Information and Control 8(3) (1965) 338-353.
  • A. Rosenfeld, Fuzzy Groups, Journal of Mathematical Analysis and Applications 35 (1971) 512-517.
  • M. M. Zahedi, R. Ameri, Fuzzy Exact Sequence in Category of Fuzzy Modules, Journal of Fuzzy Mathematics (1994) 409-424.
  • M. Yamin, P. K. Sharma, Group Actions on Fuzzy Modules, Applied Mathematics 7 (2016) 413-421.
  • M. M. Zahedi, Some Results on L-fuzzy Modules, Fuzzy Sets and Systems 55 (1993) 355-361.
  • F. Z. Pan, Fuzzy Finitely Generated Modules, Fuzzy Sets and Systems 21 (1987) 105-113.
  • W. Chen, M. Akram, Fuzzy Subcoalgebras and Fuzzy Subcomodules, Journal of Algebra and Applied Mathematics (10) (2012) 15-32.
  • K. S. Abdulkhalikov, M. S. Tulenbaev, U. U. Umirbaev, On Fuzzy Subalgebras, Fuzzy Sets and Systems 93 (1988) 257-262.
  • W. Chen, Fuzzy Subcoalgebras and Duality, Bulletin of the Malaysian Mathematical Sciences Society 32(3) (2009) 283-294.
  • W. Chen, G. Wenxu, Fuzzy Subbialgebras and Duality, Utilitas Mathematica 87 (2012) 13-20.
  • R. Rasuli, Norms over Fuzzy Lie Algebra, Journal of New Theory 15 (2017) 32-38.
  • S. M. Khalil, New Category of the Fuzzy d-Algebras, Journal of Taibah University for Science 12(2) (2018) 143-149.
  • W. Chen, Y. Guan, The Dual of Generalized Fuzzy Subspaces, Journal of Applied Mathematics Article ID 932014 (2012) 9 pages.

Year 2021, Issue: 35, 20 - 31, 30.06.2021

Munir Ahmed Tahir Mahmood , Muhammad Munır

https://doi.org/10.53570/jnt.846538

Abstract

References

  • H. Hopf, Uber die Topologie der Gruppen-Mannigfaltigkeiten und ihrer Verallgemeinerungen, Annals of Mathematics 42(1) (1941) 22-52.
  • H. Hopf, Uber den Rang Geschlossener Liescher Gruppen, Commentarii Mathematici Helvetici 13 (1940) 119-143.
  • E. Abe, Hopf Algebras, Cambridge University Press, 1997.
  • M. Cohen, D. Fishman, Hopf Algebra Actions, Journal of Algebra 100 (1986) 363-379.
  • S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Regional Conference Series in Mathematics 82 (1993).
  • S. Majid, Physics for Algebraists: Non-Commutative and Non-Cocomutative Hopf Algebras by a Bicrossproduct Construction, Journal of Algebra 130 (1990) 17-64.
  • L. M. Ionescu, M. Marsalli, A Hopf Algebra Deformation Approach to Renormalization, (2003) arXiv:hep-th/0307112 2003.
  • S. Majid, Hopf Algebra for Physics at the Plank Scale, Classical and Quantum Gravity5(12) (1988) 1587-1606.
  • M. E. Sweedler, Hopf Algebras, Benjamin New-York, 1969.
  • R. J. Blattner, M. Cohen, S. Montgomery, Cross Products and Inner Actions of Hopf Algebras, Transactions of the American Mathematical Society 298(2) (1986) 671-711.
  • S. Montgomery, Indecomposable Coalgebras, Simple Comodules and Pointed Hopf Algebras, Proceedings of the AMS - American Mathematical Society123 (1995) 2343-2351.
  • J. W. Milnor, J. C. Moore, On the Structures of Hopf Algebras, Annals of Mathematics 81 (1965) 211-264.
  • L. A. Zadeh, Fuzzy Sets, Information and Control 8(3) (1965) 338-353.
  • A. Rosenfeld, Fuzzy Groups, Journal of Mathematical Analysis and Applications 35 (1971) 512-517.
  • M. M. Zahedi, R. Ameri, Fuzzy Exact Sequence in Category of Fuzzy Modules, Journal of Fuzzy Mathematics (1994) 409-424.
  • M. Yamin, P. K. Sharma, Group Actions on Fuzzy Modules, Applied Mathematics 7 (2016) 413-421.
  • M. M. Zahedi, Some Results on L-fuzzy Modules, Fuzzy Sets and Systems 55 (1993) 355-361.
  • F. Z. Pan, Fuzzy Finitely Generated Modules, Fuzzy Sets and Systems 21 (1987) 105-113.
  • W. Chen, M. Akram, Fuzzy Subcoalgebras and Fuzzy Subcomodules, Journal of Algebra and Applied Mathematics (10) (2012) 15-32.
  • K. S. Abdulkhalikov, M. S. Tulenbaev, U. U. Umirbaev, On Fuzzy Subalgebras, Fuzzy Sets and Systems 93 (1988) 257-262.
  • W. Chen, Fuzzy Subcoalgebras and Duality, Bulletin of the Malaysian Mathematical Sciences Society 32(3) (2009) 283-294.
  • W. Chen, G. Wenxu, Fuzzy Subbialgebras and Duality, Utilitas Mathematica 87 (2012) 13-20.
  • R. Rasuli, Norms over Fuzzy Lie Algebra, Journal of New Theory 15 (2017) 32-38.
  • S. M. Khalil, New Category of the Fuzzy d-Algebras, Journal of Taibah University for Science 12(2) (2018) 143-149.
  • W. Chen, Y. Guan, The Dual of Generalized Fuzzy Subspaces, Journal of Applied Mathematics Article ID 932014 (2012) 9 pages.
There are 25 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Munir Ahmed This is me 0000-0003-4335-6560

Tahir Mahmood 0000-0003-4806-8832

Muhammad Munır This is me 0000-0002-4976-4074

Publication Date June 30, 2021
Submission Date May 1, 2019
Published in Issue Year 2021 Issue: 35

Cite

APA Ahmed, M., Mahmood, T., & Munır, M. (2021). Fuzziness on Hopf Algebraic Structures with Its Application. Journal of New Theory(35), 20-31. https://doi.org/10.53570/jnt.846538
AMA Ahmed M, Mahmood T, Munır M. Fuzziness on Hopf Algebraic Structures with Its Application. JNT. June 2021;(35):20-31. doi:10.53570/jnt.846538
Chicago Ahmed, Munir, Tahir Mahmood, and Muhammad Munır. “Fuzziness on Hopf Algebraic Structures With Its Application”. Journal of New Theory, no. 35 (June 2021): 20-31. https://doi.org/10.53570/jnt.846538.
EndNote Ahmed M, Mahmood T, Munır M (June 1, 2021) Fuzziness on Hopf Algebraic Structures with Its Application. Journal of New Theory 35 20–31.
IEEE M. Ahmed, T. Mahmood, and M. Munır, “Fuzziness on Hopf Algebraic Structures with Its Application”, JNT, no. 35, pp. 20–31, June 2021, doi: 10.53570/jnt.846538.
ISNAD Ahmed, Munir et al. “Fuzziness on Hopf Algebraic Structures With Its Application”. Journal of New Theory 35 (June 2021), 20-31. https://doi.org/10.53570/jnt.846538.
JAMA Ahmed M, Mahmood T, Munır M. Fuzziness on Hopf Algebraic Structures with Its Application. JNT. 2021;:20–31.
MLA Ahmed, Munir et al. “Fuzziness on Hopf Algebraic Structures With Its Application”. Journal of New Theory, no. 35, 2021, pp. 20-31, doi:10.53570/jnt.846538.
Vancouver Ahmed M, Mahmood T, Munır M. Fuzziness on Hopf Algebraic Structures with Its Application. JNT. 2021(35):20-31.
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