On The Symbolic Turiyam Rings
Prem Kumar Singh, Department of Computer science and Engineering, Gandhi institute of Technology and Management, India, premsingh.csjm@gmail.com
Katy D. Ahmad, Islamic University Of Gaza, Palestine, katyon765@gmail.com
Mikail Bal, Gaziantep University, Turkey, mikailbal46@hotmail.com
Malath Aswad, Al-baath University, Syria, malaz.aswad@yahoo.com
Abstract: In this paper, we define for the first time the concept of symbolic Turiyam ring as a direct application of Turiyam symbolic set and as a new generalization of neutrosophic rings. Also, we study many of essential properties and related concepts of these rings such as AH-ideals and subrings.
On the other hand, we illustrate many examples to clarify the validity of our work.
Key words: turiyam symbolic set, turyiam symbolic ring, AH-ideal, AH-subring.
1. Introduction
After the work of Zadeh [54], many authors tend to study algebraic structures of fuzzy sets [2], and their generalizations such as neutrosophic sets [1].
In the literature, we find many algebraic studies of neutrosophic algebraic structures such as rings [12-17], modules [4-5], spaces [6-7, 20], matrices [31,35,41-48], and groups [8,9,16].
The concept of Turiyam set was supposed in [55], as a new generalization of neutrosophic set was studied in [38].
In this work, we aim to study the algebraic structure related with turiyam concepts in a similar way of neutrosophic algebraic structures. From this point of view, we present the concept of symbolic Turyiam set, and we use this concept to define symbolic tyriam ring as a new generalization of neutrosophic rings. Also, we generalize neutrosophic AH-ideals to turyiam AH-ideals and AH-subrings.
This work may be used to define many turyiam symbolic algebraic structures.
Main Concepts and Discussion
Definition 1:
We define the Turiyam symbolic set as follows:
, Where denotes to falsity, denotes to truth, denotes to Indeteminacy and denotes to fourth case of awareness or (Turiyamic) property.
We define an algebraic operation between symbols in as follows:
, , , , ,
, , ,
Definition 2
Let be a ring, we define the symbolic Turiyam ring as follows:
.
We denote it by .
If a field, then is called symbolic Turiyam field .
Example 3
Let be the ring of integers module 2.
The corresponding is:
.
Definition 4: (Algebraic operations)
Let be a , , be two arbitrary elements in , we define:
Addition:
.
Multiplication:
.
.
Example 5:
Consider the , take:
.
Remark 6
is a ring in the algebraic ordinary meaning i.e. is abelian group.
The multiplication is associative and distributive with respect to addition.
Example 7
Consider the ring of integers ,Let be the corresponding , take:
.
.
It is clear that .
On other hand:
5T F 14 .
, easily we see that
Definition 8:
Let be a , we define the Turyiam AHS-ideal as follows:
.
is called AH-ideal if , where is an ideal of for all
Example 9:
Let be the ring og integers module 5, consider the following ideals:
The set is a Turiyam AH-ideal.
Definition 10:
If is a subring of , hence is called an AH-subring.
If , then is called an AHS-subring.
Theorem 11
Let be a , then it contains a neutrosophic ring .
Proof.
It is clear that , this means that the is a generalization of neutrosophic rings.
Remark 12
Every neutrosophic AH-ideal can be considered as a Turiyam AH-ideal.
Theorem 13
Let be a finite ring with , hence .
The proof is obvious.
Example14
a). If , hence .
b). If , hence .
Conclusion
In this paper, we have defined the concept of symbolic turiyam ring (STR) as a new generalization of neutrosophic rings and an application of turiyam sets with four dimensions.
In The future, we aim to study symbolic Turiyam modules and spaces by using (STR) concept.
References
[1] Smarandache, F., " A Unifying Field in Logics: Neutrosophic Logic, Neutrosophy, Neutrosophic Set, Neutrosophic Probability", American Research Press. Rehoboth, 2003.
[2] P.K. Sharma , “( α , β ) – Cut of Intuitionistic fuzzy Groups” International
Mathematical Forum ,Vol. 6, 2011 , no. 53 , 2605-2614
[3] Sankari, H., and Abobala, M., "Neutrosophic Linear Diophantine Equations With two Variables", Neutrosophic Sets and Systems, Vol. 38, pp. 22-30, 2020.
[4] Sankari, H., and Abobala, M." n-Refined Neutrosophic Modules", Neutrosophic Sets and Systems, Vol. 36, pp. 1-11. 2020.
[5] Alhamido, R., and Abobala, M., "AH-Substructures in Neutrosophic Modules", International Journal of Neutrosophic Science, Vol. 7, pp. 79-86 . 2020.
[6] Abobala, M., "AH-Subspaces in Neutrosophic Vector Spaces", International Journal of Neutrosophic Science, Vol. 6 , pp. 80-86. 2020.
[7] Abobala, M.,. "A Study of AH-Substructures in n-Refined Neutrosophic Vector Spaces", International Journal of Neutrosophic Science", Vol. 9, pp.74-85. 2020.
[8] Hatip, A., Alhamido, R., and Abobala, M., "A Contribution to Neutrosophic Groups", International Journal of Neutrosophic Science", Vol. 0, pp. 67-76 . 2019.
[9] Abobala, M., " n-Refined Neutrosophic Groups I", International Journal of Neutrosophic Science, Vol. 0, pp. 27-34. 2020.
[10] Kandasamy, V.W.B., and Smarandache, F., "Some Neutrosophic Algebraic Structures and Neutrosophic N-Algebraic Structures", Hexis, Phonex, Arizona, 2006.
[11] Agboola, A.A.A., Akinola, A.D., and Oyebola, O.Y., " Neutrosophic Rings I" , International J.Mathcombin, Vol 4,pp 1-14. 2011
[12] Abobala, M., "Classical Homomorphisms Between Refined Neutrosophic Rings and Neutrosophic Rings", International Journal of Neutrosophic Science, Vol. 5, pp. 72-75. 2020.
[13] Smarandache, F., and Abobala, M., n-Refined neutrosophic Rings, International Journal of Neutrosophic Science, Vol. 5 , pp. 83-90, 2020.
[14] Abobala, M., On Some Special Substructures of Neutrosophic Rings and Their Properties, International Journal of Neutrosophic Science", Vol. 4 , pp. 72-81, 2020.
[15] Abobala, M., "On Some Special Substructures of Refined Neutrosophic Rings", International Journal of Neutrosophic Science, Vol. 5, pp. 59-66. 2020.
[16] Smarandache, F., and Ali, M., "Neutrosophic Triplet Group", Neural. Compute. Appl. 2019.
[17] Sankari, H., and Abobala, M.," AH-Homomorphisms In neutrosophic Rings and Refined Neutrosophic Rings", Neutrosophic Sets and Systems, Vol. 38, pp. 101-112, 2020.
[18] Smarandache, F., and Kandasamy, V.W.B., " Finite Neutrosophic Complex Numbers",·Source: arXiv. 2011.
[19] Ali, Rozina., " Recent Advantages In Neutrosophic Module Theory", researchgate.net, 2021.
[20] 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.
[21] Smarandache, F., "Neutrosophic Set a Generalization of the Intuitionistic Fuzzy Sets", Inter. J. Pure Appl. Math., pp. 287-297. 2005.
[22] 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.
[23] Abobala, M., "On The Characterization of Maximal and Minimal Ideals In Several Neutrosophic Rings", Neutrosophic Sets and Systems, Vol. 45, 2021.
[24] Chalapathi, T., and Madhavi, L., "Neutrosophic Boolean Rings", Neutrosophic Sets and Systems, Vol. 33, pp. 57-66. 2020.
[25] Abobala, M., "Classical Homomorphisms Between n-refined Neutrosophic Rings", International Journal of Neutrosophic Science", Vol. 7, pp. 74-78. 2020.
[26] 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.
[27] Hatip, A., and Abobala, M., "AH-Substructures In Strong Refined Neutrosophic Modules", International Journal of Neutrosophic Science, Vol. 9, pp. 110-116 . 2020.
[28] Smarandache F., and Abobala, M., "n-Refined Neutrosophic Vector Spaces", International Journal of Neutrosophic Science, Vol. 7, pp. 47-54. 2020.
[29] Sankari, H., and Abobala, M., "Solving Three Conjectures About Neutrosophic Quadruple Vector Spaces", Neutrosophic Sets and Systems, Vol. 38, pp. 70-77. 2020.
[30] 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.
[31] Abobala, M., On Refined Neutrosophic Matrices and Their Applications In Refined Neutrosophic Algebraic Equations, Journal Of Mathematics, Hindawi, 2021
[32] 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.
[33] Abobala, M., " Semi Homomorphisms and Algebraic Relations Between Strong Refined Neutrosophic Modules and Strong Neutrosophic Modules", Neutrosophic Sets and Systems, Vol. 39, 2021.
[34] Abobala, M., "On Some Neutrosophic Algebraic Equations", Journal of New Theory, Vol. 33, 2020.
[35] Abobala, M., On The Representation of Neutrosophic Matrices by Neutrosophic Linear Transformations, Journal of Mathematics, Hindawi, 2021.
[36] Ahmad, K., Bal, M., and Aswad, M.," The kernel of Fuzzy and Anti Fuzzy Groups",Journal of Neutrosophic and Fuzzy Systems, Vol.1, 2022.
[37] 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.
[38] Singh, P,K., " Data With Turiyam Set for Fourth Dimension Quantum Information Processing", Journal of Neutrosophic and Fuzzy Systems, vol.1, 2022.
[39] Singh, P,K., " Anti-geometry and NeutroGeometry Characterization of Non-Euclidean Data", Journal of Neutrosophic and Fuzzy Systems, vol. 1, 2022.
[40] 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.
[41] Abobala, M., "On Some Algebraic Properties of n-Refined Neutrosophic Elements and n-Refined Neutrosophic Linear Equations", Mathematical Problems in Engineering, Hindawi, 2021
[42] Abobala, M., "A Study Of Nil Ideals and Kothe's Conjecture In Neutrosophic Rings", International Journal of Mathematics and Mathematical Sciences, hindawi, 2021
[43] 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.
[44]Alhamido, K., R., "A New Approach of neutrosophic Topological Spaces", International Journal of neutrosophic Science, Vol.7, 2020.
[45] Abobala, M., "Neutrosophic Real Inner Product Spaces", Neutrosophic Sets and Systems, Vol. 43, 2021.
[46] Abobala, M., "On Some Special Elements In Neutrosophic Rings and Refined Neutrosophic Rings", Journal of New Theory, vol. 33, 2020.
[47] Abobala, M., and Hatip, A., "An Algebraic Approch To Neutrosophic Euclidean Geometry", Neutrosophic Sets and Systems, Vol. 43, 2021.
[48] 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.
[49] Sankari, H, and Abobala, M, " A Contribution to m-Power Closed Groups", UMM-Alqura University Journal for Applied Sciences, KSA, 2020.
[50] Hajjari, A., and Ali, R., " A Contribution To Kothe's Conjecture In Refined Neutrosophic Rings", International Journal of Neutrosophic Science", Vol. 16, 2021.
[51] Ali, R., "A Short Note On The Solution of n-Refined Neutrosophic Linear Diophantine Equations", International Journal Of Neutrosophic Science, Vol. 15, 2021.
[52] 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.
[53] Ibrahim, M., and Abobala, M., "An Introduction To Refined Neutrosophic Number Theory", Neutrosophic Sets and Systems, Vol. 45, 2021.
[54] Zadeh, L., "Fuzzy Sets", Information. Control, 1965.
[55] Singh, P.K., " Turiyam Set a Fourth Dimension Data representation", Journal of Applied Mathematics and Physics, 2021.