A Novel Study of Fuzzy Bi-Ideals in Ordered Semirings

: In this study, by generalizing the notion of fuzzy bi-ideals of ordered semirings, the notion of ( ∈ , ∈ ∨ ( κ ∗ , q κ )) -fuzzy bi-ideals is established. We prove that ( ∈ , ∈ ∨ ( κ ∗ , q κ )) -fuzzy bi-ideals are fuzzy bi-ideals but that the converse is not true, and an example is provided to support this proof. A condition is given under which fuzzy bi-ideals of ordered semirings coincide with ( ∈ , ∈ ∨ ( κ ∗ , q κ )) -fuzzy bi-ideals. An equivalent condition and certain correspondences between bi-ideals and ( ∈ , ∈ ∨ ( κ ∗ , q κ )) -fuzzy bi-ideals are presented. Moreover, the ( κ ∗ , κ ) -lower part of ( ∈ , ∈ ∨ ( κ ∗ , q κ )) -fuzzy bi-ideals is described and depicted in terms of several classes of ordered semirings. Furthermore, it is shown that the ordered semiring is bi-simple if and only if it is ( ∈ , ∈ ∨ ( κ ∗ , q κ )) -fuzzy bi-simple.


Introduction
The concept of an "ordered semiring" was first used by Gan and Jiang [1] in connection to a semiring with a compatible partial order relation.They also proposed the idea of ideals in ordered semirings.Good et al. [2] developed the concept of bi-ideals in semigroups.Following that, Lajos et al. [3] established bi-ideals in associative rings.Bi-ideals of ordered semirings were described and characterized in terms of regularity, and the relationship between bi-ideals and quasi-ideals was characterized by Palakawong et al. [4].Senarat et al. [5] developed the terms-ordered k-bi-ideal, strong-prime-ordered k-bi-ideal, and prime-ordered k-bi-ideal in ordered semirings.By expanding on the idea of bi-ideals of ordered semirings, Davvaz et al. [6] introduced the concept of bi-hyperideals in ordered semi-hyperrings.The notions of (m, n)-bi-hyperideals and Prime (m, n)-bi-hyperideals were established and inter-related properties were considered by Omidi and Davvaz [7].The characterization of ordered h-regular semirings was considered by Anjum et al. [8].In [9], Patchakhieo and Pibaljommee characterized ordered k-regular semirings using ordered k-ideals.The ordered intra-k-regular semirings have been introduced and defined in different ways by Ayutthaya and Pibaljommee [10].Omidi and Davvaz [7] considered the concepts of (m, n)-bi-hyperideals and Prime (m, n)-bi-hyperideals and established interrelated features.Anjum et al. [8] proposed characterizing ordered h-regular semirings.By using ordered k-ideals, Patchakhieo and Pibaljommee described ordered k-regular semirings in [9].The ordered intra-k-regular semirings have been presented and characterized in various ways by Ayutthaya and Pibaljommee [10].
Fuzzy sets to semirings were initially discussed by Ahsan et al. in [11] and Kuroki [12] applied the idea to semigroups.Mandal [13] pioneered the study of ideals and interior ideals in ordered semirings, as well as their characterizations in the sense of regularity.
He developed the concepts of fuzzy bi-ideals and fuzzy quasi-ideals in ordered semirings in [14].Gao et al. [15] presented semisimple fuzzy ordered semirings and weakly regular fuzzy ordered semirings in terms of different kinds of fuzzy ideals.Saba et al. [16] initiated the study of ordered semirings based on single-valued neutrosophic sets.Several characterizations of regular and intra-regular ordered semigroups in terms of (∈, ∈ ∨q)fuzzy generalized bi-ideals were presented by Jun et al. [17], who also proposed the idea of (α, β)-fuzzy generalized bi-ideal in ordered semigroups.Similar semiring concepts, such as (∈, ∈ ∨q)-fuzzy bi-ideals on semirings, were investigated by Hedayati [18].Additionally, other ideas connected to our research in several domains have been examined in [19][20][21][22][23][24][25].
A subset (∅ =)Σ of Υ is said to be a sub-semiring if ΣΣ ⊆ Σ and Σ + Σ ⊆ Σ.Additionally, Σ refers to the left (resp.right) ideal of Υ if Σ + Σ ⊆ Σ and ΥΣ ⊆ Σ (resp.ΣΥ ⊆ Σ), and (Σ] ⊆ Σ.If it is both the left and right ideals of Υ, it is referred to as an ideal.A sub-semiring P of Υ is called a bi-ideal (in brief, BI) of Υ if PΥP ⊆ P and (P] ⊆ P. A mapping λ f : Υ → [0, 1] is said to be fuzzy set (in brief, FS) of Υ.For the FSs λ f and For Ω ⊆ Υ, the characteristic function χ f Ω is defined as: Define on the set F (Υ) of all FSs of Υ by Fuzzy left (resp.right) ideal (in brief, Fuzzy ideal of Υ if λ f is both fuzzy right and left ideals of Υ.
Then, we have Then, we have Then, by hypothesis, we have Similarly, by hypothesis, We also have as required.