Algebraic Structure for (λ, μ)-Diophantine Neutrosophic Bisemiring

Abstract

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.