Characterization of various (♭, ℓ) neutrosophic ideals of an ordered Γ-semigroups

In this paper, we introduce the notion of ♭, ℓ-neutrosophic subsemigroup (NSS), neutrosophic left ideal(NLI), neutrosophic right ideal(NRI), neutrosophic ideal (NI), neutrosophic bi-ideal(NBI), (ϵ, ϵ ∨ q)-neutrosophic ideal, neutrosophic bi-ideal of an ordered Γ-semigroups and discuss some of their properties. The concept of ♭, ℓ-neutrosophic ideal is a new extension of neutrosophic ideal over ordered Γ-semigroups Z. A non-empty subset ξ_♭ is a (♭, ℓ)-NSS (NLI, NRI, NBI, (1,2)-ideal) of Z. Then the lower level set ∆_♭ is an subsemi-group (LI, RI, BI, (1, 2) − ideal) of Z, where ∆_♭ = {ϱ ∈ Z|∆(ϱ) > ♭}, Ψ_♭ = {ϱ ∈ Z|∆(ϱ) > ♭} and ℧_♭ = {ϱ ∈ Z|∆(ϱ) < ♭}. A subset ξ = [∆, Ψ, ℧] is a (♭, ℓ) − NSS[NLI, NRI, NBI, (1, 2) − ideal] of Z if and only if each non-empty level subset ξ_t is a subsemigroup [LI, RI, BI, (1, 2) − ideal] of Z for all t ∈ (♭, ℓ]. Every (ϵ, ϵ ∨ q)NBI of Z is a (♭, ℓ)NBI of Z, but converse need not be true and examples are provided to illustrate our results.