New algebraic structure for Diophantine neutrosophic subbisemirings of bisemirings

In this paper, we define the concept of Diophantine neutrosophic subbisemiring (DioNSBS) of
bisemirings (BSs). The DioNSBS is the new approach for fuzzy subbisemiring (FSBS) over a BS. Let Ξ be the
Diophantine neutrosophic subset (DioNSS) in T , we show that Ξ = h(f
T
Ξ , f
I
Ξ, f
F
Ξ ),(ΓΞ, ΛΞ, ΘΞ)i is a DioNSBS
of T if and only if all non-empty level set Ξ(β,γ)
is a subbisemiring (SBS) of T , ∀β, γ ∈ [0, 1]. Let Ξ be the
DioNSBS of a BS T and Z be the strongest Diophantine neutrosophic relation of T . Then Ξ is a DioNSBS
of T if and only if Z is a DioNSBS of T × T . Let Ξ1, Ξ2, . . . , Ξn be the family of DioNSBSs of T1, T2, . . . , Tn,
respectively. We show that Ξ1 × Ξ2 × . . . × Ξn is a DioNSBS of T1 × T2 × . . . × Tn. The homomorphic image
of every DioNSBS is a DioNSBS. Let Ξ be any DioNSBS of T , then pseudo Diophantine neutrosophic coset
(aΞ)p
is a DioNSBS of T , for every a ∈ T . Let (T1, ~1, ~2, ~3) and (T2, ⊗1, ⊗2, ⊗3) be any two BSs. The
homomorphic preimage of every DioNSBS of T2 is a DioNSBS of T1. Let (T1, ~1, ~2, ~3) and (T2, ⊗1, ⊗2, ⊗3)
be any two BSs. Let Ξ and ∆ be any two DioNSBSs of T1 and T2, respectively, then Ξ × ∆ is a DioNSBS of
T1 × T2. If L : T1 → T2 is a homomorphism, then L(Ξ(β,γ)) is a level SBS of DioNSBS Z of T2. Examples are
given to demonstrate our findings.