aspg-21-3191-1791472160 20261008150920 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group International Journal of Neutrosophic Science IJNS 2692-6148 2690-6805 2025 25 2 A. Rajalakshmi Department of Mathematics, Shanmuga Industries Arts and Science College, Affiliated to Thiruvalluvar University, Tiruvannamalai, Tamil Nadu, 606603, India Nasreen Kausar Department of Mathematics, Faculty of Arts and Science, Yildiz Technical University, Esenler, 34220, Istanbul, Turkey Brikena Vrioni School of Arts and Sciences, American International University, Kuwait K. Lenin Muthu Kumaran Department of Mathematics, Shanmuga Industries Arts and Science College, Affiliated to Thiruvalluvar University, Tiruvannamalai, Tamil Nadu, 606603, India Nezir Aydin College of Science and Engineering Hamad Bin Khalifa University, 34110 Doha, Qatar; Department of Industrial Engineering, Yildiz Technical University, Besiktas, 34349, Istanbul, Turkey Murugan Palanikumar Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai-602105, India 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. 2025 325 337 3191 https://creativecommons.org/licenses/by/4.0/ 10.54216/IJNS.250228 https://www.americaspg.com/journal/21/article/3191 https://www.americaspg.com/storage/41737827704.pdf