International Journal of Neutrosophic Science

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https://doi.org/10.54216/IJNS

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Volume 24 , Issue 1 , PP: 94-103, 2024 | Cite this article as | XML | Html | PDF | Full Length Article

On The Real Inner Products and Orthogonality for Plithogenic Vector Spaces of Orders 4 and 5

Mayada Abualhomos 1 , Ahmed Atallah Alsaraireh 2 , Malik Bataineh 3 , Nabeela Abu-Alkishik 4 , Isra Al-Shbeil 5 , Abdallah Al-Husban 6 *

  • 1 Applied Science Private University Amman,11931, Jordan - (abuhomos@asu.edu.jo)
  • 2 Department of Computer Information Systems, The University of Jordan – Aqaba, Jordan - (a.alsarairah@ju.edu.jo)
  • 3 Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan - (msbataineh@just.edu.jo)
  • 4 Department of Mathematics, Jerash University, Jerash, Jordan - (Nabeelakishik@yahoo.com)
  • 5 Department of Mathematics, Faculty of Sciences, University of Jordan, Amman 11942, Jordan - (i.shbeil@ju.edu.jo)
  • 6 Department of Mathematics, Faculty of Science and Technology, Irbid National University, P.O. Box: 2600 Irbid, Jordan - (dralhosban@inu.edu.jo)
  • Doi: https://doi.org/10.54216/IJNS.240109

    Received: August 26, 2023 Revised: December 12, 2023 Accepted: March 28, 2024
    Abstract

    This paper will study the necessary conditions and sufficient conditions for the problem of orthogonality in 4-plithogenic and 5-plithogenic vector spaces, where we provided a definition of the real scalar products defined above these spaces, and we used the new concept of scalar product that was defined in determining the necessary and sufficient condition for the orthogonality of symbolic 4-plithogenic and 5-plithogenic vectors, in addition to calculating various norms and their metric properties. We have also provided some examples that aim to explain the scientific contribution of this work.

    Keywords :

    plithogenic vector space , inner product , plithogenic norm , orthogonal basis , ortho-normed vectors.

    References

    [1] Merkepci, H., and Abobala, M., " On The Symbolic 2-Plithogenic Rings", International Journal of Neutrosophic Science, 2023.

    [2] Smarandache, F., " Introduction to the Symbolic Plithogenic Algebraic Structures (revisited)", Neutrosophic Sets and Systems, vol. 53, 2023.

    [3] Taffach, N., and Ben Othman, K., " An Introduction to Symbolic 2-Plithogenic Modules Over Symbolic 2-Plithogenic Rings", Neutrosophic Sets and Systems, Vol 54, 2023.

    [4] Albasheer, O., Hajjari., A., and Dalla., R., " On The Symbolic 3-Plithogenic Rings and TheirAlgebraic Properties", Neutrosophic Sets and Systems, Vol 54, 2023.

    [5] Ben Othman, K., "On Some Algorithms For Solving Symbolic 3-Plithogenic Equations", Neoma Journal Of Mathematics and Computer Science, 2023.

    [6] Rawashdeh, A., "An Introduction To The Symbolic 3-plithogenic Number Theory", Neoma Journal Of Mathematics and Computer Science, 2023.

    [7] Alfahal, A.; Alhasan, Y.; Abdulfatah, R.; Mehmood, A.; Kadhim, M. On Symbolic 2-Plithogenic Real Matrices and Their Algebraic Properties. Int. J. Neutrosophic Sci. 2023, 21.

    [8] Merkepci, H., "On Novel Results about the Algebraic Properties of Symbolic 3-Plithogenic and 4-Plithogenic Real Square Matrices", Symmetry, MDPI, 2023.

    [9] Soueycatt, M., Charchekhandra, B., Abu Hakmeh, R., " On The Algebraic Properties Of Symbolic 6-Plithogenic Integers", Neutrosophic Sets and Systems, vol. 59, 2023.

    [10] Ben Othman, K., Von Shtawzen, O., Khaldi, A., and Ali, R., "On The Concept Of Symbolic 7-Plithogenic Real Matrices", Pure Mathematics For Theoretical Computer Science, Vol.1, 2023.

    [11] Ben Othman, K., Von Shtawzen, O., Khaldi, A., and Ali, R., "On The Symbolic 8-Plithogenic Matrices", Pure Mathematics For Theoretical Computer Science, Vol.1, 2023.

    [12] Abobala, M., " n-Cyclic Refined Neutrosophic Algebraic Systems of Sub-Indeterminacies, An Application to Rings and Modules", International Journal of Neutrosophic Science, 2020.

    [13] Abobala, M., Hatip, A., and Bal, M., " A Review On Recent Advantages In Algebraic Theory Of Neutrosophic Matrices", International Journal of Neutrosophic Science, Vol.17, 2021.

    [14] Abobala, M. On Refined Neutrosophic Matrices and Their Applications in Refined Neutrosophic Algebraic Equations. J. Math. 2021, 2021, 5531093.

    [15] Abobala, M., On Refined Neutrosophic Matrices and Their Applications In Refined Neutrosophic Algebraic Equations, Journal Of Mathematics, Hindawi, 2021

    [16] Abobala, M., " A Study of Maximal and Minimal Ideals of n-Refined Neutrosophic Rings", Journal of Fuzzy Extension and Applications, 2021.

    [17] Soueycatt, M., Charchekhandra, B., Abu Hakmeh, R., " On The Foundations Of Symbolic 5-Plithogenic Number Theory", Neutrosophic Sets and Systems, vol. 59, 2023.

    [18] Merkepci, H., and Rawashdeh, A., " On The Symbolic 2-Plithogenic Number Theory and Integers ", Neutrosophic Sets and Systems, Vol 54, 2023.

    [19] Sankari, H., and Abobala, M., " On The Classification of The Group of Units of Rational and Real

    2-Cyclic Refined Neutrosophic Rings", Neutrosophic Sets and Systems, 2023.

    [20] Sankari, H., and Abobala, M., " On The Algebraic Homomorphisms Between Symbolic 2-plithogenic Rings And 2-cyclic Refined Rings", Neutrosophic Sets and Systems, 2023.

    [21] Salman, N, K., " On The Symbolic 2-Plithogenic Real Analysis By Using Algebraic Functions", Neutrosophic Sets and Systems, 2023.

    [22] Salman, N, K., " On The Computing Of Symbolic 2-Plithogenic And 3-Plithogenic Complex Roots Of Unity", Neutrosophic Sets and Systems, 2023.

    [23] Merkepci, M., Abobala, M., and Allouf, A., " The Applications of Fusion Neutrosophic Number Theory in Public Key Cryptography and the Improvement of RSA Algorithm ", Fusion: Practice and Applications, 2023.

    [24] Merkepci, M., and Abobala, M., " Security Model for Encrypting Uncertain Rational Data Units Based on Refined Neutrosophic Integers Fusion and El Gamal Algorithm ", Fusion: Practice and Applications, 2023.

    [25] Nadweh, R., Von Shtawzen, O., Khaldi, A., and Ali, R., " A Study of Symbolic 2-Plithogenic Split-Complex Linear Diophantine Equations in Two Variables", Neutrosophic Sets and Systems, 2023.

    [26] Von Shtawzen, O., Rawashdeh, A., and Zayood, K., " On The Symbolic 2-Plithogenic Split-Complex Real Square Matrices and Their Algebraic Properties", Neutrosophic Sets and Systems, 2023.

    [27] Nadweh, R., Von Shtawzen, O., Khaldi, A., "An Introduction To The Split-Complex Symbolic 2-Plithogenic Numbers", Neutrosophic Sets and Systems, 2023.

    [28] Ali, R., and Hasan, Z., "An Introduction To The Symbolic 3-Plithogenic Vector Spaces", Galoitica Journal Of Mathematical Structures and Applications, vol. 6, 2023.

    [29] Ali, R., and Hasan, Z., " An Introduction To The Symbolic 3-Plithogenic Modules ", Galoitica Journal Of Mathematical Structures and Applications, vol. 6, 2023.

    [30] Hatip, A., Alsheikh, M., and Alhamamde, I., " On The Orthogonality in Real Symbolic 2-Plithogenic

    and 3-Plithogenic Vector Spaces", Neutrosophic Sets and Systems, Vol. 59, 2023.

     

    Cite This Article As :
    Abualhomos, Mayada. , Atallah, Ahmed. , Bataineh, Malik. , Abu-Alkishik, Nabeela. , Al-Shbeil, Isra. , Al-Husban, Abdallah. On The Real Inner Products and Orthogonality for Plithogenic Vector Spaces of Orders 4 and 5. International Journal of Neutrosophic Science, vol. , no. , 2024, pp. 94-103. DOI: https://doi.org/10.54216/IJNS.240109
    Abualhomos, M. Atallah, A. Bataineh, M. Abu-Alkishik, N. Al-Shbeil, I. Al-Husban, A. (2024). On The Real Inner Products and Orthogonality for Plithogenic Vector Spaces of Orders 4 and 5. International Journal of Neutrosophic Science, (), 94-103. DOI: https://doi.org/10.54216/IJNS.240109
    Abualhomos, Mayada. Atallah, Ahmed. Bataineh, Malik. Abu-Alkishik, Nabeela. Al-Shbeil, Isra. Al-Husban, Abdallah. On The Real Inner Products and Orthogonality for Plithogenic Vector Spaces of Orders 4 and 5. International Journal of Neutrosophic Science , no. (2024): 94-103. DOI: https://doi.org/10.54216/IJNS.240109
    Abualhomos, M. , Atallah, A. , Bataineh, M. , Abu-Alkishik, N. , Al-Shbeil, I. , Al-Husban, A. (2024) . On The Real Inner Products and Orthogonality for Plithogenic Vector Spaces of Orders 4 and 5. International Journal of Neutrosophic Science , () , 94-103 . DOI: https://doi.org/10.54216/IJNS.240109
    Abualhomos M. , Atallah A. , Bataineh M. , Abu-Alkishik N. , Al-Shbeil I. , Al-Husban A. [2024]. On The Real Inner Products and Orthogonality for Plithogenic Vector Spaces of Orders 4 and 5. International Journal of Neutrosophic Science. (): 94-103. DOI: https://doi.org/10.54216/IJNS.240109
    Abualhomos, M. Atallah, A. Bataineh, M. Abu-Alkishik, N. Al-Shbeil, I. Al-Husban, A. "On The Real Inner Products and Orthogonality for Plithogenic Vector Spaces of Orders 4 and 5," International Journal of Neutrosophic Science, vol. , no. , pp. 94-103, 2024. DOI: https://doi.org/10.54216/IJNS.240109