Volume 25 • Issue 2 • PP: 325-337 • 2025
Characterization of various (b,l) neutrosophic ideals of an ordered Gamma semigroups
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© 2025 The Author(s). Published by ASPG. This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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In this paper, we introduce the notion of $\flat,\ell$-neutrosophic subsemigroup (NSS), neutrosophic left ideal(NLI), neutrosophic right ideal(NRI), neutrosophic ideal (NI), neutrosophic bi-ideal(NBI), $(\epsilon, \epsilon \vee q)$-neutrosophic ideal, neutrosophic bi-ideal of an ordered $\Gamma$-semigroups and discuss some of their properties. The concept of $\flat,\ell$-neutrosophic ideal is a new extension of neutrosophic ideal over ordered $\Gamma$-semigroups $\mathcal{Z}$. A non-empty subset $\xi_{\flat}$ is a $(\flat, \ell)$-NSS (NLI, NRI, NBI, (1,2)-ideal) of $\mathcal{Z}$. Then the lower level set $\Delta_{\flat}$ is an subsemigroup $(LI, RI, BI, (1,2)-ideal)$ of $\mathcal{Z}$, where $\Delta_{\flat}=\{\varrho\in \mathcal{Z}|\Delta(\varrho)> \flat\}$, $\Psi_{\flat}=\{\varrho\in \mathcal{Z} |\Delta(\varrho)> \flat\}$ and $\mho_{\flat}=\{\varrho\in \mathcal{Z}|\Delta(\varrho)< \flat\}$. A subset $\xi=[\Delta,\Psi,\mho]$ is a $(\flat, \ell)- NSS[NLI,NRI,NBI,(1, 2)-ideal]$ of $\mathcal{Z}$ if and only if each non-empty level subset $\xi_{t}$ is a subsemigroup $[LI,RI,BI,(1,2)-ideal]$ of $\mathcal{Z}$ for all $t\in(\flat, \ell]$. Every $(\epsilon, \epsilon \vee q)$NBI of $\mathcal{Z}$ is a $(\flat,\ell)$NBI of $\mathcal{Z}$, but converse need not be true and examples are provided to illustrate our results.
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References
[1] L. A. Zadeh, Fuzzy sets, Information and Control, 8, (1965), 338-353.
[2] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1), (1986) 87-96.
[3] R. R. Yager, Pythagorean membership grades in multi criteria decision-making, IEEE Trans. Fuzzy Systems, 22, (2014), 958-965.
[4] Mahmoud, H. Abdelhafeez, A. ”Spherical Fuzzy Multi-Criteria Decision-Making Approach for Risk Assessment of Natech,” Journal of Neutrosophic and Information Fusion, vol. 2, no. 1, pp. 59-68, 2023.
[5] Ali, O. Mashhadani, S. Alhakam, I. M., S. ”A New Paradigm for Decision Making under Uncertainty in Signature Forensics Applications based on Neutrosophic Rule Engine,” Journal of International Journal of Neutrosophic Science, vol. 24, no. 2, pp. 268-282, 2024.
[6] M Palanikumar, N Kausar, H Garg, A Iampan, S Kadry, M Sharaf, Medical robotic engineering selection based on square root neutrosophic normal interval-valued sets and their aggregated operators, AIMS Mathematics, 8(8), (2023), 17402–17432.
[7] S. Ashraf, S. Abdullah, T. Mahmood, F. Ghani and T. Mahmood, Spherical fuzzy sets and their applications in multi-attribute decision making problems, Journal of Intelligent and Fuzzy Systems, 36, (2019), 2829-284.
[8] Rosenfeld, Fuzzu groups, J.Math. Anal. Appl.35 (1971) 512-517.
[9] N. Kuroki, On fuzzy semigroups, Inform. Sci. 53 (1991), 203-236.
[10] J. N. Mordeson, D. S. Malik, N. Kuroki, Fuzzy semigroups, springer-Verlag Berlin Heidelberg GmbH, 2003.
[11] M. K. Sen, On Γ-semigroups, Proceedings of International conference on Algebra and its Application Decker publication, New yark, (1981), 301.
[12] M. K. Sen and N. K Saha, On Γ-semigroup, I, Bull.Calcutta Math. Soc.,(1986), 78 180-186.
[13] N.Kehayopula, On ordered Γ-semigroups, Scientiae Mathematicae Japonicae Online, e-2010, 37-43.
[14] Somsak Lekkoksung, On Q-fuzzy ideals in ordered semigroups, International Journal of Pure and Applied Mathematics,92(3) (2014), 369–379.
[15] N.Kehayopula and Tsingelis, Fuzzy sets in ordered groupoids, semigroup Forum, 65, (2005) 128-132 .
[16] F. M. Khan, N. H. Sarmin and A. Khan. Some new characterization of ordered semigroups in terms of (λ, θ)-fuzzy bi-ideals, International Journal of Algebra and Statistics, 1(1)(2012), 22–32.
[17] Y. B. Jun, Asghar Khan and Muhammad Shabir, Ordered Semigroups Characterized by Their (ϵ, ϵ,∨q)- Fuzzy Bi-Ideals, Bull. Malays. Math. Sci. Soc. (2) 32(3) (2009), 391-408.
[18] S. K. Bhakat P. DaS. (ϵ, ϵ,∨q)-Fuzzy subsemigroup. Fuzzy Sets and Systems, 80(3)(1996), 359–368.
[19] O. Kazanci and S. Yamak, Generalized fuzzy bi-ideals of semigroup, Soft Computing, 12 (2008), 1119- 1124.
[20] F. Smarandache, A unifying field in logics Neutrosophy Neutrosophic Probability, Set and Logic, Rehoboth American Research Press (1999).
[21] Ozcek, M. ”A Review on the Structure of Fuzzy Regular Proper Mappings in Fuzzy Topological Spaces and Their Properties,” Journal of Pure Mathematics for Theoretical Computer Science, vol. 3, no. 2, pp. 60-71, 2023.
[22] Palanikumar M, Selvi G, Ganeshsree Selvachandran and Tan S.L, New approach to bisemiring theory via the bipolar-valued neutrosophic normal sets, Neutrosophic Sets and Systems, 55, 427-450, 2023.
[23] K. Hila and E. Pisha. On bi-ideals on ordered Γ-semigroups. Hacettepe Journal of Mathematics and Statistics, 40(6), (2011), 793-804.
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