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Characterization of Associative PU-algebras by the Notion of Derivations

Received: 18 January 2021     Accepted: 2 February 2021     Published: 4 March 2021
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Abstract

In this manuscript we insert the concept of derivations in associative PU-algebras and discuss some of its important results such that we prove that for a mapping being a (Left, Right) or (Right, Left)-derivation of an associative PU-algebra then such a mapping is one-one. If a mapping is regular then it is identity. If any element of an associative PU-algebra satisfying the criteria of identity function then such a map is identity. We also prove some useful properties for a mapping being (Left, Right)-regular derivation of an associative PU-algebra and (Right, Left)-regular derivation of an associative PU-algebra. Moreover we prove that if a mapping is regular (Left, Right)-derivation of an associative PU-algebra then its Kernel is a subalgebra. We have no doubt that the research along this line can be kept up, and indeed, some results in this manuscript have already made up a foundation for further exploration concerning the further progression of PU-algebras. These definitions and main results can be similarly extended to some other algebraic systems such as BCH-algebras, Hilbert algebras, BF-algebras, J-algebras, WS-algebras, CI-algebras, SU-algebras, BCL-algebras, BP-algebras and BO-algebras, Z- algebras and so forth. The main purpose of our future work is to investigate the fuzzy derivations ideals in PU-algebras, which may have a lot of applications in different branches of theoretical physics and computer science.

Published in American Journal of Mathematical and Computer Modelling (Volume 6, Issue 1)
DOI 10.11648/j.ajmcm.20210601.13
Page(s) 14-18
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

PU-Algebras, (Left, Right)-derivations of PU-algebras, (Right, Left)-derivations of PU-algebras, Regular Derivations of PU-algebras.

References
[1] Iséki, K. (1966). An algebra related with a propositional calculus. Proceedings of the Japan Academy, 42 (1), 26-29.
[2] Iséki, K. (1978). An introduction to the theory of BCK- algebras. Math. Japonica, 23, 1-26.
[3] Imai, Y., & Iseki, K. (1966). On axioms of Proportional calculi xiv proc. Japan Acad, 42, 19-22.
[4] Neggers, J., Ahn, S. S., & Kim, H. S. (2001). On Q- algebras. International Journal of Mathematics and Mathematical Sciences, 27 (12), 749-757.
[5] Megalai, K., & Tamilarasi, A. (2010). Classification of TM- algebra. IJCA Special Issue on “Computer Aided Soft Computing Techniques for Imaging and Biomedical Applications” CASCT.
[6] Mostafa, Samy M., MA Abdel Naby, and A. I. Elkabany. "New view of ideals on PU-algebra." International Journal of Computer Applications 111.4, 18 pages, (2015). View at: Publisher Site | Google Scholar.
[7] Hvala, B. (1998). Generalized derivations in rings. Communications in Algebra, 26 (4), 1147-1166.
[8] Bell, H. E., & Kappe, L. C. (1989). Rings in which derivations satisfy certain algebraic conditions. Acta Mathematica Hungarica, 53 (3-4), 339-346.
[9] Bell, H. E., & Mason, G. (1987). On derivations in near- rings. In North-Holland Mathematics Studies (Vol. 137, pp. 31-35). North-Holland.
[10] Brešar, M., & Vukman, J. (1990). On left derivations and related mappings. Proceedings of the American Mathematical Society, 110 (1), 7-16.
[11] Bresar, M. (1991). On the distance of the composition of two derivations to the generalized derivations. Glasgow Mathematical Journal, 33 (1), 89-93.
[12] Muhiuddin, G., & Al-Roqi, A. M. (2012). On-Derivations in BCI-Algebras. Discrete Dynamics in Nature and Society, 2012.
[13] Ilbira, S., Firat, A., & Jun, Y. B. (2011). On symmetric bi- derivations of BCI-algebras. Applied Mathematical Sciences, 5 (57-60), 2957-2966.
[14] Al-Kadi, D. (2017). On (f, g)-Derivations of G-Algebras.
[15] Zhao, W. (2018). Some open problems on locally finite or locally nilpotent derivations and ε-derivations. Communications in Contemporary Mathematics, 20 (04), 1750056.
[16] Alekseev, A. V., & Arutyunov, A. A. (2020). Derivations in semigroup algebras. Eurasian Mathematical Journal, 11 (2), 9-18.
[17] Muhiuddin, G., & Al-Roqi, A. M. (2012). On (α, β)- derivations in BCI-algebras. Discrete Dynamics in Nature and Society, 2012.
[18] Abujabal, H. A., & Al-Shehri, N. O. (2017). On left derivations of BCI-algebras. Soochow Journal of Mathematics, 33 (3), 435.
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  • APA Style

    Mehmood Khan, Dawood Khan, Khalida Mir Aalm. (2021). Characterization of Associative PU-algebras by the Notion of Derivations. American Journal of Mathematical and Computer Modelling, 6(1), 14-18. https://doi.org/10.11648/j.ajmcm.20210601.13

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    ACS Style

    Mehmood Khan; Dawood Khan; Khalida Mir Aalm. Characterization of Associative PU-algebras by the Notion of Derivations. Am. J. Math. Comput. Model. 2021, 6(1), 14-18. doi: 10.11648/j.ajmcm.20210601.13

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    AMA Style

    Mehmood Khan, Dawood Khan, Khalida Mir Aalm. Characterization of Associative PU-algebras by the Notion of Derivations. Am J Math Comput Model. 2021;6(1):14-18. doi: 10.11648/j.ajmcm.20210601.13

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  • @article{10.11648/j.ajmcm.20210601.13,
      author = {Mehmood Khan and Dawood Khan and Khalida Mir Aalm},
      title = {Characterization of Associative PU-algebras by the Notion of Derivations},
      journal = {American Journal of Mathematical and Computer Modelling},
      volume = {6},
      number = {1},
      pages = {14-18},
      doi = {10.11648/j.ajmcm.20210601.13},
      url = {https://doi.org/10.11648/j.ajmcm.20210601.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20210601.13},
      abstract = {In this manuscript we insert the concept of derivations in associative PU-algebras and discuss some of its important results such that we prove that for a mapping being a (Left, Right) or (Right, Left)-derivation of an associative PU-algebra then such a mapping is one-one. If a mapping is regular then it is identity. If any element of an associative PU-algebra satisfying the criteria of identity function then such a map is identity. We also prove some useful properties for a mapping being (Left, Right)-regular derivation of an associative PU-algebra and (Right, Left)-regular derivation of an associative PU-algebra. Moreover we prove that if a mapping is regular (Left, Right)-derivation of an associative PU-algebra then its Kernel is a subalgebra. We have no doubt that the research along this line can be kept up, and indeed, some results in this manuscript have already made up a foundation for further exploration concerning the further progression of PU-algebras. These definitions and main results can be similarly extended to some other algebraic systems such as BCH-algebras, Hilbert algebras, BF-algebras, J-algebras, WS-algebras, CI-algebras, SU-algebras, BCL-algebras, BP-algebras and BO-algebras, Z- algebras and so forth. The main purpose of our future work is to investigate the fuzzy derivations ideals in PU-algebras, which may have a lot of applications in different branches of theoretical physics and computer science.},
     year = {2021}
    }
    

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  • TY  - JOUR
    T1  - Characterization of Associative PU-algebras by the Notion of Derivations
    AU  - Mehmood Khan
    AU  - Dawood Khan
    AU  - Khalida Mir Aalm
    Y1  - 2021/03/04
    PY  - 2021
    N1  - https://doi.org/10.11648/j.ajmcm.20210601.13
    DO  - 10.11648/j.ajmcm.20210601.13
    T2  - American Journal of Mathematical and Computer Modelling
    JF  - American Journal of Mathematical and Computer Modelling
    JO  - American Journal of Mathematical and Computer Modelling
    SP  - 14
    EP  - 18
    PB  - Science Publishing Group
    SN  - 2578-8280
    UR  - https://doi.org/10.11648/j.ajmcm.20210601.13
    AB  - In this manuscript we insert the concept of derivations in associative PU-algebras and discuss some of its important results such that we prove that for a mapping being a (Left, Right) or (Right, Left)-derivation of an associative PU-algebra then such a mapping is one-one. If a mapping is regular then it is identity. If any element of an associative PU-algebra satisfying the criteria of identity function then such a map is identity. We also prove some useful properties for a mapping being (Left, Right)-regular derivation of an associative PU-algebra and (Right, Left)-regular derivation of an associative PU-algebra. Moreover we prove that if a mapping is regular (Left, Right)-derivation of an associative PU-algebra then its Kernel is a subalgebra. We have no doubt that the research along this line can be kept up, and indeed, some results in this manuscript have already made up a foundation for further exploration concerning the further progression of PU-algebras. These definitions and main results can be similarly extended to some other algebraic systems such as BCH-algebras, Hilbert algebras, BF-algebras, J-algebras, WS-algebras, CI-algebras, SU-algebras, BCL-algebras, BP-algebras and BO-algebras, Z- algebras and so forth. The main purpose of our future work is to investigate the fuzzy derivations ideals in PU-algebras, which may have a lot of applications in different branches of theoretical physics and computer science.
    VL  - 6
    IS  - 1
    ER  - 

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Author Information
  • Department of Mathematics, University of Balochistan, Quetta, Pakistan

  • Department of Mathematics, University of Balochistan, Quetta, Pakistan

  • Department of Mathematics, University of Balochistan, Quetta, Pakistan

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