Volume 25 • Issue 2 • PP: 325-337 • 2025
Characterization of various (b,l) neutrosophic ideals of an ordered Gamma semigroups
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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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