Generated Fuzzy Quasi-ideals in Ternary Semigroups

Ravi Srivastava

doi:doi:10.11648/j.ijmfs.20251103.12

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Generated Fuzzy Quasi-ideals in Ternary Semigroups

Ravi Srivastava^*

Published in International Journal of Management and Fuzzy Systems (Volume 11, Issue 3)

Received: 8 June 2025 Accepted: 1 July 2025 Published: 5 August 2025

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Abstract

Here in this paper, we provide characterizations of fuzzy quasi-ideal in terms of level and strong level subsets. Along with it, we provide expression for the generated fuzzy quasi-ideal generated by level subsets and strong level subsets of the given fuzzy set in explicit manner. We also establish the existence for a generated fuzzy quasi-ideal in ternary semigroup by showing that the non-empty intersection of an arbitrary family of fuzzy quasi-ideals is again a fuzzy quasi-ideal. This paper introduces and explores the concept of generated fuzzy quasi-ideals in ternary semigroups, extending classical algebraic notions into the fuzzy domain. A fuzzy quasi-ideal is defined as a fuzzy set that satisfies specific conditions analogous to those of crisp quasi-ideals under the ternary operation. A foundational result established in this work is that the non-empty intersection of any family of fuzzy quasi-ideals in a ternary semigroup remains a fuzzy quasi-ideal, reinforcing the internal consistency of the structure. Furthermore, we explore key properties of these generated fuzzy quasi-ideals, including their relationships with level sets and strong level subsets. The central focus of the paper is on how fuzzy quasi-ideals can be generated by arbitrary fuzzy sets within a ternary semigroup. We establish methods for constructing the smallest fuzzy quasi-ideal containing a given fuzzy set, along with expressions for this generated structure in terms of the quasi-ideals generated by its level and strong level subsets. Through constructive proofs, we demonstrate the existence by showing the non-empty intersection of an arbitrary family of fuzzy quasi-ideals in a ternary semigroup is itself a fuzzy quasi-ideal and uniqueness of such generated fuzzy quasi-ideals. The findings contribute to a deeper understanding of the internal structure of ternary semigroups and provide a foundational framework for further research in fuzzy algebraic systems. In summary, this work establishes a comprehensive framework for generated fuzzy quasi-ideals in ternary semigroups, revealing their structural properties, generation mechanisms, and theoretical importance. These results contribute meaningfully to the study of fuzzy algebraic systems and open new avenues for further research in fuzzy ternary algebra.

Published in	International Journal of Management and Fuzzy Systems (Volume 11, Issue 3)
DOI	10.11648/j.ijmfs.20251103.12
Page(s)	102-108
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), 2025. Published by Science Publishing Group

Keywords

Ternary Semigroups, Arbitrary Intersection, Level Subsets, Strong Level Subsets, Generated Fuzzy Ideals

References

[1]	V. N. Dixit, and S. Dewan (1995). A Note on Quasi and Bi-Ideals of Ternary Semigroups. Internat. J. Math. & Math. Science 3: 501-508. https://doi.org/10.1155/S0161171295000640
[2]	T. K. Dutta, S. Kar, and B. K. Maity (2008). On Ideals in Regular Ternary Semigroups. General Algebra and Applications 28: 147-159. https://doi.org/10.7151/dmgaa.1140
[3]	E. Hewitt, and H. S. Zuckerman (1955). Finite Dimensional Convolution Algebras. Acta Mathematics 93: 67-119. https://doi.org/10.1007/BF02392520
[4]	A. Iampan (2007). Lateral Ideals of Ternary Semigroups. Ukrainian Math. Bulletin 4: 323-334. http://dspace.nbuv.gov.ua/handle/123456789/124531
[5]	S. Kar, and P. Sarkar (2012). On Fuzzy Ideals of Ternary Semigroups. Fuzzy Information and Engineering 4(2): 407-423. https://doi.org/10.1007/s12543-012-0110-4
[6]	S. Kar, and P. Sarkar (2012). On Fuzzy Quasi-Ideals and Fuzzy Bi-Ideals of Ternary Semigroups. Annals of Fuzzy Mathematics and Informatics 2: 181-193. https://api.semanticscholar.org/CorpusID:126201623
[7]	R. Kumar (1982). Fuzzy Subgroups, Fuzzy Ideals and Fuzzy Cosets, Some Properties. Fuzzy Sets and Systems 48: 133-139. https://doi.org/10.1016/0165-0114(92)90341-Z
[8]	D. H. Lehmer (1932). A Ternary Analoue of Abelian Groups. Amer. J. Mathematics 59: 329-338. https://doi.org/10.2307/2370997
[9]	J. Los (1955). On the Extending of Models I, Fundam. Math. 42: 38-54. https://doi.org/10.2307/2963709
[10]	A. S. Prajapati, and R. Srivatava (2006). Metatheorem and Fuzzy Quasi-Ideals in Semigroups. The Journal of Fuzzy Mathematics 14(4): 851-865.
[11]	M. L. Santiago, and S. S. Bala (2010). Ternary semigroups. Semigroup Forum 81: 380-388. https://doi.org/10.1007/s00233-010-9254-x
[12]	F. M.Sioson(1965).Ideal Theory in Ternary Semigroups. Mathematica Japonica 10:63-84.
[13]	R. Srivastava (2024). New Way for Extending Ideals of Ternary Semigroup to Fuzzy Setting. Journal of Computational Analysis and Applications 36(6): 1291-1306. https://eudoxuspress.com/index.php/pub/article/view/1741
[14]	L. A. Zadeh (1965). Fuzzy Sets, Inform. and Control. 8: 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X

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Srivastava, R. (2025). Generated Fuzzy Quasi-ideals in Ternary Semigroups. International Journal of Management and Fuzzy Systems, 11(3), 102-108. https://doi.org/10.11648/j.ijmfs.20251103.12

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ACS Style

Srivastava, R. Generated Fuzzy Quasi-ideals in Ternary Semigroups. Int. J. Manag. Fuzzy Syst. 2025, 11(3), 102-108. doi: 10.11648/j.ijmfs.20251103.12

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AMA Style

Srivastava R. Generated Fuzzy Quasi-ideals in Ternary Semigroups. Int J Manag Fuzzy Syst. 2025;11(3):102-108. doi: 10.11648/j.ijmfs.20251103.12

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@article{10.11648/j.ijmfs.20251103.12,
  author = {Ravi Srivastava},
  title = {Generated Fuzzy Quasi-ideals in Ternary Semigroups
},
  journal = {International Journal of Management and Fuzzy Systems},
  volume = {11},
  number = {3},
  pages = {102-108},
  doi = {10.11648/j.ijmfs.20251103.12},
  url = {https://doi.org/10.11648/j.ijmfs.20251103.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijmfs.20251103.12},
  abstract = {Here in this paper, we provide characterizations of fuzzy quasi-ideal in terms of level and strong level subsets. Along with it, we provide expression for the generated fuzzy quasi-ideal generated by level subsets and strong level subsets of the given fuzzy set in explicit manner. We also establish the existence for a generated fuzzy quasi-ideal in ternary semigroup by showing that the non-empty intersection of an arbitrary family of fuzzy quasi-ideals is again a fuzzy quasi-ideal. This paper introduces and explores the concept of generated fuzzy quasi-ideals in ternary semigroups, extending classical algebraic notions into the fuzzy domain. A fuzzy quasi-ideal is defined as a fuzzy set that satisfies specific conditions analogous to those of crisp quasi-ideals under the ternary operation. A foundational result established in this work is that the non-empty intersection of any family of fuzzy quasi-ideals in a ternary semigroup remains a fuzzy quasi-ideal, reinforcing the internal consistency of the structure. Furthermore, we explore key properties of these generated fuzzy quasi-ideals, including their relationships with level sets and strong level subsets. The central focus of the paper is on how fuzzy quasi-ideals can be generated by arbitrary fuzzy sets within a ternary semigroup. We establish methods for constructing the smallest fuzzy quasi-ideal containing a given fuzzy set, along with expressions for this generated structure in terms of the quasi-ideals generated by its level and strong level subsets. Through constructive proofs, we demonstrate the existence by showing the non-empty intersection of an arbitrary family of fuzzy quasi-ideals in a ternary semigroup is itself a fuzzy quasi-ideal and uniqueness of such generated fuzzy quasi-ideals. The findings contribute to a deeper understanding of the internal structure of ternary semigroups and provide a foundational framework for further research in fuzzy algebraic systems. In summary, this work establishes a comprehensive framework for generated fuzzy quasi-ideals in ternary semigroups, revealing their structural properties, generation mechanisms, and theoretical importance. These results contribute meaningfully to the study of fuzzy algebraic systems and open new avenues for further research in fuzzy ternary algebra.
},
 year = {2025}
}

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TY  - JOUR
T1  - Generated Fuzzy Quasi-ideals in Ternary Semigroups

AU  - Ravi Srivastava
Y1  - 2025/08/05
PY  - 2025
N1  - https://doi.org/10.11648/j.ijmfs.20251103.12
DO  - 10.11648/j.ijmfs.20251103.12
T2  - International Journal of Management and Fuzzy Systems
JF  - International Journal of Management and Fuzzy Systems
JO  - International Journal of Management and Fuzzy Systems
SP  - 102
EP  - 108
PB  - Science Publishing Group
SN  - 2575-4947
UR  - https://doi.org/10.11648/j.ijmfs.20251103.12
AB  - Here in this paper, we provide characterizations of fuzzy quasi-ideal in terms of level and strong level subsets. Along with it, we provide expression for the generated fuzzy quasi-ideal generated by level subsets and strong level subsets of the given fuzzy set in explicit manner. We also establish the existence for a generated fuzzy quasi-ideal in ternary semigroup by showing that the non-empty intersection of an arbitrary family of fuzzy quasi-ideals is again a fuzzy quasi-ideal. This paper introduces and explores the concept of generated fuzzy quasi-ideals in ternary semigroups, extending classical algebraic notions into the fuzzy domain. A fuzzy quasi-ideal is defined as a fuzzy set that satisfies specific conditions analogous to those of crisp quasi-ideals under the ternary operation. A foundational result established in this work is that the non-empty intersection of any family of fuzzy quasi-ideals in a ternary semigroup remains a fuzzy quasi-ideal, reinforcing the internal consistency of the structure. Furthermore, we explore key properties of these generated fuzzy quasi-ideals, including their relationships with level sets and strong level subsets. The central focus of the paper is on how fuzzy quasi-ideals can be generated by arbitrary fuzzy sets within a ternary semigroup. We establish methods for constructing the smallest fuzzy quasi-ideal containing a given fuzzy set, along with expressions for this generated structure in terms of the quasi-ideals generated by its level and strong level subsets. Through constructive proofs, we demonstrate the existence by showing the non-empty intersection of an arbitrary family of fuzzy quasi-ideals in a ternary semigroup is itself a fuzzy quasi-ideal and uniqueness of such generated fuzzy quasi-ideals. The findings contribute to a deeper understanding of the internal structure of ternary semigroups and provide a foundational framework for further research in fuzzy algebraic systems. In summary, this work establishes a comprehensive framework for generated fuzzy quasi-ideals in ternary semigroups, revealing their structural properties, generation mechanisms, and theoretical importance. These results contribute meaningfully to the study of fuzzy algebraic systems and open new avenues for further research in fuzzy ternary algebra.

VL  - 11
IS  - 3
ER  -

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Author Information

Ravi Srivastava

Department of Mathematics, University of Delhi, Delhi, India

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http://orcid.org/0009-0003-6661-4646

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APA Style

Srivastava, R. (2025). Generated Fuzzy Quasi-ideals in Ternary Semigroups. International Journal of Management and Fuzzy Systems, 11(3), 102-108. https://doi.org/10.11648/j.ijmfs.20251103.12

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ACS Style

Srivastava, R. Generated Fuzzy Quasi-ideals in Ternary Semigroups. Int. J. Manag. Fuzzy Syst. 2025, 11(3), 102-108. doi: 10.11648/j.ijmfs.20251103.12

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AMA Style

Srivastava R. Generated Fuzzy Quasi-ideals in Ternary Semigroups. Int J Manag Fuzzy Syst. 2025;11(3):102-108. doi: 10.11648/j.ijmfs.20251103.12

Copy | Download

@article{10.11648/j.ijmfs.20251103.12,
  author = {Ravi Srivastava},
  title = {Generated Fuzzy Quasi-ideals in Ternary Semigroups
},
  journal = {International Journal of Management and Fuzzy Systems},
  volume = {11},
  number = {3},
  pages = {102-108},
  doi = {10.11648/j.ijmfs.20251103.12},
  url = {https://doi.org/10.11648/j.ijmfs.20251103.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijmfs.20251103.12},
  abstract = {Here in this paper, we provide characterizations of fuzzy quasi-ideal in terms of level and strong level subsets. Along with it, we provide expression for the generated fuzzy quasi-ideal generated by level subsets and strong level subsets of the given fuzzy set in explicit manner. We also establish the existence for a generated fuzzy quasi-ideal in ternary semigroup by showing that the non-empty intersection of an arbitrary family of fuzzy quasi-ideals is again a fuzzy quasi-ideal. This paper introduces and explores the concept of generated fuzzy quasi-ideals in ternary semigroups, extending classical algebraic notions into the fuzzy domain. A fuzzy quasi-ideal is defined as a fuzzy set that satisfies specific conditions analogous to those of crisp quasi-ideals under the ternary operation. A foundational result established in this work is that the non-empty intersection of any family of fuzzy quasi-ideals in a ternary semigroup remains a fuzzy quasi-ideal, reinforcing the internal consistency of the structure. Furthermore, we explore key properties of these generated fuzzy quasi-ideals, including their relationships with level sets and strong level subsets. The central focus of the paper is on how fuzzy quasi-ideals can be generated by arbitrary fuzzy sets within a ternary semigroup. We establish methods for constructing the smallest fuzzy quasi-ideal containing a given fuzzy set, along with expressions for this generated structure in terms of the quasi-ideals generated by its level and strong level subsets. Through constructive proofs, we demonstrate the existence by showing the non-empty intersection of an arbitrary family of fuzzy quasi-ideals in a ternary semigroup is itself a fuzzy quasi-ideal and uniqueness of such generated fuzzy quasi-ideals. The findings contribute to a deeper understanding of the internal structure of ternary semigroups and provide a foundational framework for further research in fuzzy algebraic systems. In summary, this work establishes a comprehensive framework for generated fuzzy quasi-ideals in ternary semigroups, revealing their structural properties, generation mechanisms, and theoretical importance. These results contribute meaningfully to the study of fuzzy algebraic systems and open new avenues for further research in fuzzy ternary algebra.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Generated Fuzzy Quasi-ideals in Ternary Semigroups

AU  - Ravi Srivastava
Y1  - 2025/08/05
PY  - 2025
N1  - https://doi.org/10.11648/j.ijmfs.20251103.12
DO  - 10.11648/j.ijmfs.20251103.12
T2  - International Journal of Management and Fuzzy Systems
JF  - International Journal of Management and Fuzzy Systems
JO  - International Journal of Management and Fuzzy Systems
SP  - 102
EP  - 108
PB  - Science Publishing Group
SN  - 2575-4947
UR  - https://doi.org/10.11648/j.ijmfs.20251103.12
AB  - Here in this paper, we provide characterizations of fuzzy quasi-ideal in terms of level and strong level subsets. Along with it, we provide expression for the generated fuzzy quasi-ideal generated by level subsets and strong level subsets of the given fuzzy set in explicit manner. We also establish the existence for a generated fuzzy quasi-ideal in ternary semigroup by showing that the non-empty intersection of an arbitrary family of fuzzy quasi-ideals is again a fuzzy quasi-ideal. This paper introduces and explores the concept of generated fuzzy quasi-ideals in ternary semigroups, extending classical algebraic notions into the fuzzy domain. A fuzzy quasi-ideal is defined as a fuzzy set that satisfies specific conditions analogous to those of crisp quasi-ideals under the ternary operation. A foundational result established in this work is that the non-empty intersection of any family of fuzzy quasi-ideals in a ternary semigroup remains a fuzzy quasi-ideal, reinforcing the internal consistency of the structure. Furthermore, we explore key properties of these generated fuzzy quasi-ideals, including their relationships with level sets and strong level subsets. The central focus of the paper is on how fuzzy quasi-ideals can be generated by arbitrary fuzzy sets within a ternary semigroup. We establish methods for constructing the smallest fuzzy quasi-ideal containing a given fuzzy set, along with expressions for this generated structure in terms of the quasi-ideals generated by its level and strong level subsets. Through constructive proofs, we demonstrate the existence by showing the non-empty intersection of an arbitrary family of fuzzy quasi-ideals in a ternary semigroup is itself a fuzzy quasi-ideal and uniqueness of such generated fuzzy quasi-ideals. The findings contribute to a deeper understanding of the internal structure of ternary semigroups and provide a foundational framework for further research in fuzzy algebraic systems. In summary, this work establishes a comprehensive framework for generated fuzzy quasi-ideals in ternary semigroups, revealing their structural properties, generation mechanisms, and theoretical importance. These results contribute meaningfully to the study of fuzzy algebraic systems and open new avenues for further research in fuzzy ternary algebra.

VL  - 11
IS  - 3
ER  -

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