A Short Introduction To The Concept Of Symbolic Turiyam Matrix
Mikail Bal, Gaziantep University, Turkey
Prem Kumar Singh1, Department of Computer science and Engineering, Gandhi institute of Technology and Management-Visakhapatnam 530045, India
Katy D. Ahmad2, Islamic university Of Gaza, Palestine
2Email: Katyon765@gmail.com
1Email: premsingh.csjm@gmail.com
Abstract:
Recently, Turiyam set is introduced to deal with data set beyond three-way fuzzy space. In this process a problem is addressed while precise representation of Turiyam attributes in matrix format for knowledge processing tasks. To resolve this issue, current paper defines concept of symbolic Turiyam matrix by using the symbolic Turiyam set concept. In addition, the paper illustrates several examples to clarify the algebraic structure of these matrices such as addition, multiplication, and symmetry of these matrices.
Key words: Symbolic Turiyam matrix, symmetric Turiyam matrix, Hermit-Turiyam matrix, Turiyam Set.
Introduction
Recently, the symbolic turiyam set [91-93] was defined in [78] as a new algebraic generalization of the corresponding neutrosophic one [1-10]. It was used to study Symbolic Turiyam rings structures in [78] with its extensive properties [76-78] for dealing the multi-valued attributes attributes [80-90]. It provides a way to represent the Non-Euclidean data sets [85-91]. One of the suitable example of Turiyam attribute is India-Pakistan match. This match is beyond Win, Draw or Loss. It is not based on refined indeterminacy [92]. In case India is out of the series, Pakistan will lose the match from the lower team also. The Turiyam is to defeat India rather than win, draw or lose the particular match and vice versa. The second example is Israel Philistine conflict we can take as human cognition. It cannot be defined based on acceptation, rejection and uncertain of particular incident. Some country just support Israel (or Philistine) to fulfill their agenda rather than acceptance of incident, rejection of incident, or refined indeterminacy of incident. It is totally beyond the incident. In these cases their cognition support to fulfill their agenda [93]. Another example is recently observed in India where people start voting NOTA(None of The above). The supreme court of India considers it distinct from acceptance, rejection and uncertain vote. This NOTA can be considered as a Turiyam or Liberated state. The people who refused to vote can be found via 1-(Acceptation+Rejection+NOTA+Uncertain or indeterminant).The precise representation of these types of data sets is one of the crucial tasks as they dependent on time. Hence the current paper tried to introduce Turiyam matrix in this paper for knowledge processing tasks.
In the literature, mant authors have contributed to algebraic structures in neutrosophic systems such as neutrosophic rings [11-20], spaces [21-30], modules [31-40] , refined neutrosophic set [41-50], neutrosophic matrices [51-60] and its topology [61-75].
In this work, we extend the previous efforts to the case of the symbolic Turiyam set, where we define the symbolic Turiyam matrix, the symmetric Turiyam matrix, Fuzzy Turiyam matrix, and Hermit-Turiyam matrix.
This work may be very useful in future studied to generalize classical algebraic structures and to study the rerlationships between Turiyam matrices and classical matrices.
On the other hand, many open questions will come to light according to our work. We will list some of them in the conclusion.
Main Discussion
Definition
We define the symbolic Turiyam matrix as follows:
; is a symbolic Turiyam number.
If was real symbolic Turiyam number, then is called real symbolic Turiyam matrix.
If some was complex symbolic Turiyam number, then is called complex Turiyam matrix.
The following example clarifies operations on Turiyam matrices.
Example
Consider the following complex Turiyam matrices:
We have:
It is clear that .
It is clear that .
Example
Take:
Definition
A Turiyam matrix A is called symmetric if and only if .
A is called anti-symmetric if .
We define , if , then is called Hermit -Turiyam matrix.
Example
is Turiyam symmetric matrix.
Thus, is Hermit-Turiyam matrix.
Definition
Let be a Turiyam matrix, we say:
1. is complete Turiyam truth matrix (CTTM) if .
2. is called complete Turiyam Falsity matrix (CTFM) if .
3. is called complete TuriyamIndeterminacy matrix (CTIM) if .
4. is called complete Turiyamawareness matrix (CTAM) if .
Example
is (CTIM)
is (CTAM)
is (CTTM)
Remark
All square (CTTM),(CTFM),(CTIM),(CTAM) are not invertible because the determinant will be a non invertible Turiyam number.
Example
Consider a (CTIM), .
, which is non-invertible.
Remark
We denote to the set of all square (CTTM) by , andby to the (CTFM), and by to the (CTIM),and by to the(CTAM).
Theorem
1) are abelian groups.
2) has a multiplicative identity
3) has a multiplicative identity
4) has a multiplicative identity
5) has a multiplicative identity
Proof.
1) Let ,for each , we have:
, thus , hence is a subgroup of the additive group of Turiyam matrices.
We prove that are additive abelian groups by the same.
2) Let , hence where is an ordinary square matrix.
We have ; is the unitary matrix.
.
3), 4), 5) can be proved by the same.
Remark:
If we take the elements , we get the concept of Fuzzy-Turiyam matrix.
Example:
Consider the matrix .
It is a Fuzzy Turiyam matrix.
Conclusion
In this paper, we have defined for the first time the concept of Turiyam matrix. Also, we have presented many examples about algebraic operations between these matrices.
We suggest many research problems:
1) How can we compute the eigen values of a turiyam matrix?
2) How can we find all eigen vectors of a turiyam matrix?
3) Find an algorithm to diagonalize a Turyiam matrix?
4) What is the algebraic structure of invertible Turiam matrices?. Can we find the invertibility condition depending on the classical parts of a Tutiyam matrix?.
5) Define algebraic operations over Fuzzy-Turiyam Matrices. What are the algebraic properties of these matrices?.
Acknowledgements: Author thanks the editorial team for the valuable time.
Funding :Author declares that, there is no funding for this paper.
Conflicts of Interest: Author declares that, there is no conflict of interest.
Ethics approval: This article does not contain any studies with human or animals participants.
References
[1] Smarandache, F., " A Unifying Field in Logics: Neutrosophic Logic, Neutrosophy, Neutrosophic Set, Neutrosophic Probability", American Research Press. Rehoboth, 2003.
[2] Alhamido, R., and Gharibah, T., "Neutrosophic Crisp Tri-Topological Spaces", Journal of New Theory, Vol. 23 , pp.13-21. 2018.
[3] Edalatpanah. S.A., "Systems of Neutrosophic Linear Equations", Neutrosophic Sets and Systems, Vol. 33, pp. 92-104. 2020.
[4] Sankari, H., and Abobala, M., "Neutrosophic Linear Diophantine Equations With two Variables", Neutrosophic Sets and Systems, Vol. 38, pp. 22-30, 2020.
[5] Sankari, H., and Abobala, M." n-Refined Neutrosophic Modules", Neutrosophic Sets and Systems, Vol. 36, pp. 1-11. 2020.
[6] Alhamido, R., and Abobala, M., "AH-Substructures in Neutrosophic Modules", International Journal of Neutrosophic Science, Vol. 7, pp. 79-86 . 2020.
[7] Abobala, M., "AH-Subspaces in Neutrosophic Vector Spaces", International Journal of Neutrosophic Science, Vol. 6 , pp. 80-86. 2020.
[8] Abobala, M.,. "A Study of AH-Substructures in n-Refined Neutrosophic Vector Spaces", International Journal of Neutrosophic Science", Vol. 9, pp.74-85. 2020.
[9] Hatip, A., Alhamido, R., and Abobala, M., "A Contribution to Neutrosophic Groups", International Journal of Neutrosophic Science", Vol. 0, pp. 67-76 . 2019.
[10] Abobala, M., " n-Refined Neutrosophic Groups I", International Journal of Neutrosophic Science, Vol. 0, pp. 27-34. 2020.
[11] Kandasamy, V.W.B., and Smarandache, F., "Some Neutrosophic Algebraic Structures and Neutrosophic N-Algebraic Structures", Hexis, Phonex, Arizona, 2006.
[12] Agboola, A.A.A., Akinola, A.D., and Oyebola, O.Y., " Neutrosophic Rings I" , International J.Mathcombin, Vol 4,pp 1-14. 2011
[13] Agboola, A.A.A., "On Refined Neutrosophic Algebraic Structures," Neutrosophic Sets and Systems,Vol.10, pp. 99-101. 2015.
[14] Abobala, M., "Classical Homomorphisms Between Refined Neutrosophic Rings and Neutrosophic Rings", International Journal of Neutrosophic Science, Vol. 5, pp. 72-75. 2020.
[15] Smarandache, F., and Abobala, M., n-Refined neutrosophic Rings, International Journal of Neutrosophic Science, Vol. 5 , pp. 83-90, 2020.
[16] Kandasamy, I., Kandasamy, V., and Smarandache, F., "Algebraic structure of Neutrosophic Duplets in Neutrosophic Rings", Neutrosophic Sets and Systems, Vol. 18, pp. 85-95. 2018.
[17] Yingcang, Ma., Xiaohong Zhang ., Smarandache, F., and Juanjuan, Z., "The Structure of Idempotents in Neutrosophic Rings and Neutrosophic Quadruple Rings", Symmetry Journal (MDPI), Vol. 11. 2019.
[18] Kandasamy, V. W. B,. Ilanthenral, K., and Smarandache, F., "Semi-Idempotents in Neutrosophic Rings", Mathematics Journal (MDPI), Vol. 7. 2019.
[19] Abobala, M., On Some Special Substructures of Neutrosophic Rings and Their Properties, International Journal of Neutrosophic Science", Vol. 4 , pp. 72-81, 2020.
[20] Smarandache, F., " An Introduction To neutrosophic Genetics", International Journal of neutrosophic Science, Vol.13, 2021.
[21] Martin, N, Smarandache, F, and Broumi, S., " Covid 19 Decision Making using Extended Plithogenic hypersoft Sets With Dual Dominent Attributes", International Journal of neutrosophic Science, Vol. 13, 2021.
[22]Agboola, A.A., "Introduction To Neutro groups", International Journal of neutrosophic Science, Vol. 6, 2020.
[23] Abobala, M., "On Some Special Substructures of Refined Neutrosophic Rings", International Journal of Neutrosophic Science, Vol. 5, pp. 59-66. 2020.
[24] Smarandache, F., and Ali, M., "Neutrosophic Triplet Group", Neural. Compute. Appl. 2019.
[25] Sankari, H., and Abobala, M.," AH-Homomorphisms In neutrosophic Rings and Refined Neutrosophic Rings", Neutrosophic Sets and Systems, Vol. 38, pp. 101-112, 2020.
[26] Smarandache, F., and Kandasamy, V.W.B., " Finite Neutrosophic Complex Numbers",·Source: arXiv. 2011.
[27]. Abobala, M., " n-Refined Neutrosophic Groups II", International Journal of Neutrosophic Science, Vol. 0, 2020.
[28] Ali, Rozina., " On the Concept of Algebraic Actions In Neutrosophic Groups", Resaechgate.net, 2021.
[29] Ali, Rozina., " Neutrosophic Matrices and Their Properties", researchgat.net, 2021.
[30] Ali, Rozina., " Recent Advantages In Neutrosophic Module Theory", researchgate.net, 2021.
[31] Abobala, M, "n-Cyclic Refined Neutrosophic Algebraic Systems Of Sub-Indeterminacies, An Application To Rings and Modules", International Journal of Neutrosophic Science, Vol. 12, pp. 81-95 . 2020.
[32] Smarandache, F., "Neutrosophic Set a Generalization of the Intuitionistic Fuzzy Sets", Inter. J. Pure Appl. Math., pp. 287-297. 2005.
[33] M. Ali, F. Smarandache, M. Shabir and L. Vladareanu., "Generalization of Neutrosophic Rings and Neutrosophic Fields", Neutrosophic Sets and Systems, vol. 5, pp. 9-14, 2014.
[34] Anuradha V. S., "Neutrosophic Fuzzy Hierarchical Clustering for Dengue Analysis in Sri Lanka", Neutrosophic Sets and Systems, vol. 31, pp. 179-199. 2020.
[35] Olgun, N., and Hatip, A., "The Effect Of The Neutrosophic Logic On The Decision Making, in Quadruple Neutrosophic Theory And Applications", Belgium, EU, Pons Editions Brussels,pp. 238-253. 2020.
[36] Abobala, M., Bal, M., and Hatip, A.," A Review On Recent Advantages In Algebraic Theory Of Neutrosophic Matrices", International Journal of Neutrosophic Science, Vol. 17, 2021.
[37] Turksen, I., "Interval valued fuzzy sets based on normal forms", Fuzzy Sets and Systems, 20, pp.191-210, 1986. 1986.
[38] Chalapathi, T., and Madhavi, L., "Neutrosophic Boolean Rings", Neutrosophic Sets and Systems, Vol. 33, pp. 57-66. 2020.
[39] Abobala, M., "Classical Homomorphisms Between n-refined Neutrosophic Rings", International Journal of Neutrosophic Science", Vol. 7, pp. 74-78. 2020.
[40] Agboola, A.A.A,. Akwu, A.D,. and Oyebo, Y.T., " Neutrosophic Groups and Subgroups", International .J .Math. Combin, Vol. 3, pp. 1-9. 2012.
[41] Smarandache, F., " n-Valued Refined Neutrosophic Logic and Its Applications in Physics", Progress in Physics, 143-146, Vol. 4, 2013.
[42] Adeleke, E.O., Agboola, A.A.A.,and Smarandache, F., "Refined Neutrosophic Rings I", International Journal of Neutrosophic Science, Vol. 2(2), pp. 77-81. 2020.
[43] Hatip, A., and Abobala, M., "AH-Substructures In Strong Refined Neutrosophic Modules", International Journal of Neutrosophic Science, Vol. 9, pp. 110-116 . 2020.
[44] Hatip, A., and Olgun, N., "On Refined Neutrosophic R-Module", International Journal of Neutrosophic Science, Vol. 7, pp.87-96. 2020.
[45] Bal, M., and Abobala, M., "On The Representation Of Winning Strategies Of Finite Games By Groups and Neutrosophic Groups", Journal Of Neutrosophic and Fuzzy Systems, 2022.
[46] Smarandache F., and Abobala, M., "n-Refined Neutrosophic Vector Spaces", International Journal of Neutrosophic Science, Vol. 7, pp. 47-54. 2020.
[47] Sankari, H., and Abobala, M., "Solving Three Conjectures About Neutrosophic Quadruple Vector Spaces", Neutrosophic Sets and Systems, Vol. 38, pp. 70-77. 2020.
[48] Adeleke, E.O., Agboola, A.A.A., and Smarandache, F., "Refined Neutrosophic Rings II", International Journal of Neutrosophic Science, Vol. 2(2), pp. 89-94. 2020.
[49] Abobala, M., On Refined Neutrosophic Matrices and Their Applications In Refined Neutrosophic Algebraic Equations, Journal Of Mathematics, Hindawi, 2021
[50] Abobala, M., A Study of Maximal and Minimal Ideals of n-Refined Neutrosophic Rings, Journal of Fuzzy Extension and Applications, Vol. 2, pp. 16-22, 2021.
[51] Abobala, M., " Semi Homomorphisms and Algebraic Relations Between Strong Refined Neutrosophic Modules and Strong Neutrosophic Modules", Neutrosophic Sets and Systems, Vol. 39, 2021.
[52] Abobala, M., "On Some Neutrosophic Algebraic Equations", Journal of New Theory, Vol. 33, 2020.
[53] Abobala, M., On The Representation of Neutrosophic Matrices by Neutrosophic Linear Transformations, Journal of Mathematics, Hindawi, 2021.
[54] Abobala, M., "On Some Algebraic Properties of n-Refined Neutrosophic Elements and n-Refined Neutrosophic Linear Equations", Mathematical Problems in Engineering, Hindawi, 2021
[55] Kandasamy V, Smarandache F., and Kandasamy I., Special Fuzzy Matrices for Social Scientists . Printed in the United States of America,2007, book, 99 pages.
[56] Khaled, H., and Younus, A., and Mohammad, A., " The Rectangle Neutrosophic Fuzzy Matrices", Faculty of Education Journal Vol. 15, 2019. (Arabic version).
[57] Abobala, M., Partial Foundation of Neutrosophic Number Theory, Neutrosophic Sets and Systems, Vol. 39 , 2021.
[58] F. Smarandache, Neutrosophic Theory and Applications, Le Quy Don Technical University, Faculty of Information technology, Hanoi, Vietnam, 17th May 2016.
[59] Sankari, H, and Abobala, M., " On A New Criterion For The Solvability of non Simple Finite Groups and m-Abelian Solvability, Journal of Mathematics, Hindawi, 2021.
[60] Giorgio, N, Mehmood, A., and Broumi, S.," Single Valued neutrosophic Filter", International Journal of Neutrosophic Science, Vol. 6, 2020.
[61] Abobala, M., "A Study Of Nil Ideals and Kothe's Conjecture In Neutrosophic Rings", International Journal of Mathematics and Mathematical Sciences, hindawi, 2021
[62] Abobala, M., Hatip, A., Olgun, N., Broumi, S., Salama, A,A., and Khaled, E, H., The algebraic creativity In The Neutrosophic Square Matrices, Neutrosophic Sets and Systems, Vol. 40, pp. 1-11, 2021.
[63]Alhamido, K., R., "A New Approach of neutrosophic Topological Spaces", International Journal of neutrosophic Science, Vol.7, 2020.
[64] Abobala, M., "Neutrosophic Real Inner Product Spaces", Neutrosophic Sets and Systems, Vol. 43, 2021.
[65] Abobala, M., "On Some Special Elements In Neutrosophic Rings and Refined Neutrosophic Rings", Journal of New Theory, vol. 33, 2020.
[66] Abobala, M., and Hatip, A., "An Algebraic Approch To Neutrosophic Euclidean Geometry", Neutrosophic Sets and Systems, Vol. 43, 2021.
[67] Sundar, J., Vadivel, A., " New operators Using Neutrosophic Open Sets", Journal Of Neutrosophic and Fuzzy Systems, 2022.
[68] Sankari, H, and Abobala, M, " A Contribution to m-Power Closed Groups", UMM-Alqura University Journal for Applied Sciences, KSA, 2020.
[69] Abobala, M., "On The Characterization of Maximal and Minimal Ideals In Several Neutrosophic Rings", Neutrosophic Sets and Systems, Vol. 45, 2021.
[70] Chellamani, P., and Ajay, D., "Pythagorean neutrosophic Fuzzy Graphs", International Journal of Neutrosophic Science, Vol. 11, 2021.
[71] Milles, S, Barakat, M, and Latrech, A., " Completeness and Compactness In Standard Single Valued neutrosophic Metric Spaces", International Journal of Neutrosophic Science, Vol.12 , 2021.
[72] Es, Haydar, A., "A Note On neutrosophic Soft Menger Topological Spaces", International Journal of Neutrosophic Science, Vol.7, 2020.
[73] Ceven, Y., and Tekin, S., " Some Properties of Neutrosophic Integers", Kırklareli University Journal of Engineering and Science, Vol. 6, pp.50-59, 2020.
[74] Abobala, M., Hatip, A., Bal,M.," A Study Of Some Neutrosophic Clean Rings", International journal of neutrosophic science, 2022.
[75] Ahmad, K., Bal, M., Hajjari, A., Ali, R.," On Imperfect Duplets In Some refined Neutrosophic Rings", Journal of Neutrosophic and Fuzzy Systems, 2022.
[76] Singh, P,K., " Anti-geometry and NeutroGeometry Characterization of Non-Euclidean Data", Journal of Neutrosophic and Fuzzy Systems, Vol 1, Issue 1, pp. 24-33, 2021, DOI: https://doi.org/10.54216/JNFS.0101012
[77] Singh, P,K., " Data With Turiyam Set for Fourth Dimension Quantum Information Processing", Journal of Neutrosophic and Fuzzy Systems, Vol 1, Issue 1, pp. 9-23, DOI: https://doi.org/10.54216/JNFS.010101
[78] Singh, P, K., Ahmad, K., Bal, M., Aswad, M.," On The Symbolic Turiyam Rings", Journal of Neutrosophic and Fuzzy Systems, Vol. 1 , No. 2 , pp. 80-88 , 2021, Doi : https://doi.org/10.54216/JNFS.010204
[79] Ahmad, K., Bal, M., and Aswad, M.," A Short Note on The Solution Of Fermat's Diophantine Equation In Some Neutrosophic Rings", Journal of Neutrosophic and Fuzzy Systems, Vol. 1, 2022.
[80] Ibrahim, M., and Abobala, M., "An Introduction To Refined Neutrosophic Number Theory", Neutrosophic Sets and Systems, Vol. 45, 2021.
[81] Abobala, M., Bal, M., Aswad, M., "A Short Note On Some Novel Applications of Semi Module Homomorphisms", International journal of neutrosophic science, 2022.
[82] Ahmad, K., Bal, M., and Aswad, M.," The kernel of Fuzzy and Anti Fuzzy Groups",Journal of Neutrosophic and Fuzzy Systems, Vol.1, 2022.
[83] Smarandache, F., and Broumi,M., "Neutro-Intelligent Set is a particular case of the refined neutrosophic set", Journal of Neutrosophic and Fuzzy Systems, Vol. 1, 2022.
[84] Olgun, N., Hatip, A., Bal, M., and Abobala, M., " A Novel Approach To Necessary and Sufficient Conditions For The Diagonalization of Refined Neutrosophic Matrices", International Journal of Neutrosophic Science, Vol. 16, pp. 72-79, 2021.
[85] Singh PK, NeutroAlgebra and NeutroGeometry for Dealing Heteroclinic Patterns. In: Theory and Applications of NeutroAlgebras as Generalizations of Classical Algebras, IGI Global Publishers, April 2022, Chapter 6, DOI: 10.4018/978-1-6684-3495-6, ISBN13: 9781668434956
[86] Singh, P.,K., “Data with Non-Euclidean Geometry and its Characterization”, Journal of Artificial Intelligence and Technology, Vol 2, Issue 1, pp. 3-8, 2022, DOI: 10.37965/jait.2021.12001
[87] Singh, P,K, “Three-way n-valued neutrosophic concept lattice at different granulation”, International Journal of Machine Learning and Cybernetics, Vol 9, Issue 11, pp. 1839-1855, 2019.
[88] Bal, M., Ahmad, K., Hajjari, A., Ali, R.," A Short Note On The Kernel Subgroup Of Intuitionistic Fuzzy groups" Journal of Neutrosophic and Fuzzy Systems, 2022.
[89] Bal, M., Ahmad, K., Hajjari, A., Ali, R.," The Structure Of Imperfect Triplets In Several Refined Neutrosophic Rings" Journal of Neutrosophic and Fuzzy Systems, 2022.
[90] Singh, P,K, “Complex Plithogenic Set”, International Journal of Neutrosophic Sciences, Vol 18, Issue 1, pp. 57-72, 2022, Doi : https://doi.org/10.54216/IJNS.180106
[91] Singh, P,K, “Complex multi-–fuzzy context analysis at different granulation”, Granular Computing, Vol. 6, Issue 1, pp. 191-206, Jan 2021, DOI: 10.1007/s41066-019-00180-8
[92] Singh, P,K, “Turiyam set a fourth dimension data representation”, Journal of Applied Mathematics and Physics, Vol. 9, Issue 7, pp. 1821-1828, 2021, DOI: 10.4236/jamp.2021.97116,
[93] Singh, P,K, “Fourth dimension data representation and its analysis using Turiyam Context”, Journal of Computer and Communications, Vol. 9, no. 6, pp. 222-229, 2021, DOI: 10.4236/jcc.2021.96014