Characterization of various (♭, ℓ) neutrosophic ideals of an ordered
Γ-semigroups
A. Rajalakshmi1, Nasreen Kausar2, Brikena Vrioni3,∗, K. Lenin Muthu Kumaran1, Nezir Aydin4,5,
Murugan Palanikumar6
1Department of Mathematics, Shanmuga Industries Arts and Science College, Affiliated to Thiruvalluvar
University, Tiruvannamalai, Tamil Nadu, 606603, India
2 Department of Mathematics, Faculty of Arts and Science, Yildiz Technical University, Esenler, 34220,
Istanbul, Turkey
3 School of Arts and Sciences, American International University, Kuwait
4College of Science and Engineering Hamad Bin Khalifa University, 34110 Doha, Qatar
5Department of Industrial Engineering, Yildiz Technical University, Besiktas, 34349, Istanbul, Turkey
6Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical
Sciences, Chennai-602105, India
E-mails: pearlakshmi03@gmail.com; kausar.nasreen57@gmail.com; Brikena.vrioni@yahoo.com;
leninmuthukumaran@gmail.com; naydin@hbku.edu.qa; palanimaths86@gmail.com
Abstract
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 Δ♭ is an subsemigroup
(LI,RI,BI, (1, 2) − ideal) of Z, where Δ♭ = {ϱ ∈ Z|Δ(ϱ) > ♭}, Ψ♭ = {ϱ ∈ Z|Δ(ϱ) > ♭} and
0♭ = {ϱ ∈ Z|Δ(ϱ) < ♭}. A subset ξ = [Δ,Ψ,0] is a (♭, ℓ) − NSS[NLI,NRI,NBI, (1, 2) − ideal]
of Z if and only if each non-empty level subset ξt 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.
Keywords: Ordered Γ-semigroups; neutrosophic ideals; bi-ideals; (♭, ℓ) bi-ideals; (ϵ, ϵ ∨ q) bi-ideals