Characterization of fuzzy algebraic structure based on diophantine Q-neutrosophic subbisemiring of bisemiring

We propose the concept of diophantine Q-neutrosophic subbisemiring(DioQNSBS), level sets of DioQNSBS of a bisemiring. The idea of DioQNSBS is an extension of fuzzy subbisemiring over bisemiring. Exploring the concept for DioQNSBS over bisemiring. Let H be the diophantine Q-neutrosophic subset in D, prove H = ⟨(Γ_H^T,Γ_H^I,Γ_H^F ), (ΛH, ΞH, ΦH )⟩ is a DioQNSBS of D if and only if all non empty level set H(t,s) is a subbisemiring of D for t, s ∈ [0, 1]. Let H be the DioQNSBS of a bisemiring D and M be the strongest diophantine Q-neutrosophic relation (SDioQNSR)of D, we notice H is a DioQNSBS of D if and only if M is a DioQNSBS of D × D. Let H1, H2, ..., Hn be the family of DioQNSBSs of D1, D2, ..., Dn respectively, prove H1 × H2 × ... × Hn is a DioQNSBS of D1 × D2 × ... × Dn. The homomorphic image of DioQNSBS is a DioQNSBS. The homomorphic preimage of DioQNSBS is a DioQNSBS. Illustrations are presented to demonstrate results.