International Journal of Neutrosophic Science IJNS 2690-6805 2692-6148 10.54216/IJNS https://www.americaspg.com/journals/show/1394 2020 2020 Department of Advanced Mathematical Science, Saveetha School of Engineering, Saveetha University, Saveetha Institute of Medical and Technical Sciences, Chennai-602105, India M. Palanikumar Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand Aiyared Iampan We introduce the notion of Diophantine neutrosophic subbisemiring (DioNSBS), level sets of DioNSBS of a bisemiring. The concept of DioNSBS is a generalization of fuzzy subbisemiring over bisemiring. We interact the theory for (λ, μ)-DioNSBS over bisemiring. Let α be the Diophantine neutrosophic subset in S , we show that α = ⟨(ΞT α , ΞI α , ΞF α ), (Γα, ∆α, Θα)⟩ is a DioNSBS of S if and only if all non empty level set α(t,s) is a subbisemiring of S for t, s ∈ [0, 1]. Let α be the DioNSBS of a bisemiring S and W be the strongest Diophantine neutrosophic relation of S , we observe that α is a DioNSBS of S if and only if W is a DioNSBS of S × S . Let α1, α2, ..., αn be the family of DioN SBSs of S1, S2, ..., Sn respectively. We show that α1× α2 × ... × αn is a DioNSBS of S1 × S2 × ... × Sn. The homomorphic image of DioNSBS is a DioNSBS. The homomorphic preimage of DioNSBS is a DioNSBS. Examples are provided to illustrate our results. 2022 2022 37 48 10.54216/IJNS.190403 https://www.americaspg.com/articleinfo/21/show/1394