Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) New approach towards complex bipolar intuitionistic fuzzy subbisemiring applied to homomorphism R. Raghavendran1, P. Jeyanthi2, M. Palanikumar3, Aiyared Iampan4,∗ 1,3Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai-602105, India. 2Department of Sustainable Engineering, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai-602105, India. 4Department of Mathematics, School of Science, University of Phayao, 19 Moo 2,Tambon Mae Ka, Amphur Mueang, Phayao 56000, Thailand. E-mails:1lakshmiragha1986@gmail.com, 3palanimaths86@gmail.com, 4aiyared.ia@up.ac.th, ∗Corresponding author: Aiyared Iampan. Received: 14-03-2024 Revised: 27-08-2024 Accepted: 19-09-2024. Abstract We establish the concept of complex bipolar intuitionistic fuzzy subbisemiring (CBIFSBS). We analyze the homomorphic features and important properties of CBIFSBS. We construct the CBIFSBS level sets for bisemirings. Suppose thatL is a subset of B. ThenR = (µT − L ·eδχ T− L , µF− L ·eδχ F− L , µT + L ·eδχ T + L , µF+ L · eδχ F+ L ) is a CBIFSBS of B if and only if µ(t,s) is a SBS of B for all t, s ∈ [−1, 0]× [0, 1]. It is shown that there are homomorphic pre-images and homomorphic images for every CBIFSBS. We illustrate the results based on numerical examples. 1 Introduction Zadeh1 developed fuzzy set (FS) theory, which succeeds when dealing with ambiguity and uncertainty. If an element in an FS has a single value inside the interval, it is considered a member. The degree of non- membership does not necessarily equal one minus the degree of membership, however, because resistance might exist in real-world situations. As FS theory advances, a growing number of hybrid fuzzy models appear. Numerous uncertain theories, including FS,1 intuitionistic FS (IFS),2 Pythagorean FS (PFS),3 and spherical FS (SFS),4 have been developed as a result of the uncertainties. An FS is made up of MG sets, or sets with grades ranging from 0 to 1. IFS is classified as MG in spite of Atanassov’s2 assertion that non-membership grades (NMG) may only have a value of 1. Throughout the entirety of a decision-making process, the total of MGs and NMGs may occasionally reach 1. The generalized MG and NMG logic, which has a value not exceeding 1 and is determined by the square of the MGs and NMGs, was developed by Yager3 using PFS logic. These theories are unable to represent the neutral state since it is neither positive nor negative. Cuong5 discussed the photo FS with coworkers. Three grading points were utilized by FS: positive, neutral, and negative. These grades added together could not total more than 1. In a few of scenarios, it also performs better than PFS and IFS. It is an independent generalization of three models that handles the truth, indeterminacy, and falsity of FS and IFS. The concept of bipolar fuzzy sets was introduced in Lee.6 In typical fuzzy sets, the membership degree range is [0, 1]. Bipolar fuzzy sets (FSs) are those whose membership degree range is extended from the interval [0, 1] to the interval [−1, 1]. A bipolar fuzzy set has elements whose membership degree is [−1, 0], which indicates that the implicit contrary property is partially satisfied, and elements whose membership degree is (0, 1], which indicates that the property is not relevant to any extent possible. The neutrosophic set (NS) was developed by Smarandache7 to deal with contradictory and ambiguous data. This logic establishes the degree to which a notion is true, ambiguous, or wrong. In,8 Ramot et al. present the idea of the complex fuzzy set (CFS). The values of the membership functions in CFS transactions might vary greatly. The unit circle of the complex plane has been extended to [0, 1], but the unit circle of a fuzzy membership function stays fixed. The membership function µX(x) of the CFS X extends to the unit circle in the complex plane instead of only to [0, 1]. Thus, µX(x) is a complex-valued function that yields a grade of membership of the kind ηX(x) · eiτX(x), where i = √ −1, for each element x in the discourse universe. The value of µX(x) is defined by the two real-valued variables, ηX(x) and τX(x), where ηX(x) ∈ [0, 1]. Semiring logic and its uses were introduced by Golan.9 Hussian et al.10 discussed about the idea and application of bisemirings. Ahsan et al. researched fuzzy semirings.11 The notion of bisemirings was first presented by https://internationalpubls.com 550 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) Sen et al.12 An intuitionistic fuzzy normal subbisemiring of bisemiring was introduced by Palanikumar et al. (2019).13 Bisemiring was first proposed by Palanikumar et al.14 employing bipolar valued neutrosophic normal sets. We will look at specific components of the SBS and CBIFSBS concepts and make some assumptions. The article is divided into the following three sections. In Section 1 is called an introduced. Section 2 contains basic information. Section 3 has a list of CBIFSBS concepts. Section 4 deals that homomoprhism based on CBIFSBS concepts. 2 Preliminaries Definition 2.1. 12 An algebraic structure (B,u, ,�) is a bisemiring, if (B,u, ) and (B, ,�) are semir- ings, ie.,(B,u), (B, ) and (B,�) are semigroups and 1. a (b u c) = (a b) u (a c), 2. (b u c) a = (b a) u (c a), 3. a� (b c) = (a� b) (a� c), 4. (b c)� a = (b� a) (c� a), ∀ a, b, c ∈ B. Definition 2.2. A bipolar fuzzy set µ in a universe U is an object having the form µ = {〈 x,A+ µ (x), A−µ (x) 〉 : x ∈ U } , where A+ µ : U → [0, 1] and A−µ : U → [−1, 0]. Here A+ µ (x) represents the degree of satisfaction of the element x to the property of A−µ (x) represents the degree of satisfaction of x to some implicit counter property of µ. For simplicity the symbol 〈 A+ µ , A − µ 〉 is used for the bipolar fuzzy set µ = {〈 x,A+ µ (x), A−µ (x) 〉 : x ∈ U } . Definition 2.3. For two bipolar fuzzy subsets µ = (µ+, µ−) and Ψ = (Ψ+,Ψ−), the product of µ and Ψ is denoted by µ ◦Ψ and is defined as (µ+ ◦Ψ+)(x) =  sup (s,t)∈Ax { µ+(s) ∧Ψ+(t) } if Ax 6= 0 0 if Ax = 0 (µ− ◦Ψ−)(x) =  inf (s,t)∈Ax { Ψ−(s) ∨Ψ−(t) } if Ax 6= 0 −1 if Ax = 0 Definition 2.4. 7 A NS µ in the universe U is µ = {x,ATµ (x), AIµ(x), AFµ (x)|x ∈ U},where ATµ (x),AIµ(x) AFµ (x) represents the TD,ID and FD of v respectively. Consider the mapping ATµ : U → [0, 1], AIµ : U → [0, 1], AFµ : U → [0, 1] and 0 ≤ ATµ (x) +AIµ(x) +AFµ (x) ≤ 3. Definition 2.5. 7 Let µ1 = 〈χTµ1 , χIµ1 , χFµ1 〉,µ2 = 〈χTµ2 , χIµ2 , χFµ2 〉 and µ3 = 〈χTµ3 , χIµ3 , χFµ3 〉 be the three neutrosophic numbers over U . Then 1. µ2 � µ3 = 〈 max(χTµ2 , χTµ3 ),min(χIµ2 , χIµ3 ),min(χFµ2 , χFµ3 ) 〉 , 2. µ2 ⊎ µ3 = 〈 min(χTµ2 , χTµ3 ),max(χIµ2 , χIµ3 ),max(χFµ2 , χFµ3 ) 〉 , 3. µ2 ≥ µ3 iff χTµ2 ≥ χTµ3 and χIµ2 ≤ χIµ3 and χFµ2 ≤ χFµ3 , 4. µ2 = µ3 iff χTµ2 = χTµ3 and χIµ2 = χIµ3 and χFµ2 = χFµ3 . Definition 2.6. 7 For any NS µ = {x,ATµ (x), AIµ(x), AFµ (x)} of U . Then (τ, β)-cut is defined as {x ∈ U |ATµ (x) ≥ τ,AIµ(x) ≥ τ,AFµ (x) ≤ β}. https://internationalpubls.com 551 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 3 Complex bipolar intuitionistic fuzzy subbisemiring Here B denotes bisemiring unless other stated, µ stands for real part and χ stands for imaginary part and z = i2π. Definition 3.1. For any complex bipolar intuitionistic fuzzy set(CBIFS) L in universal set U , L = {l, µT − L (l) · eδχT− L (l), µF− L (l) · eδχ F−(l) L , µT + L (l) · eδχT + L (l), µF+ L (l) · eδχ F+(l) L : l ∈ U}, where µT − L (l) · eδχ T−(l) L , µF− L (l) · eδχ F−(l) L : U → [−1, 0]× [0, 1] and µTL(l) · eδχ T (l) L , µFL(l) · eδχ F (l) L : U → [−1, 0]× [0, 1] represents the truth degree,indeterminacy degree and false degree respectively. For simplicity the symbol µT − L , µF− L is CBIFS L = {l, µT − L (l) · eδχ T−(l) L µF− L (l) · eδχ F−(l) L , µT + L (l) · eδχ T +(l) L , µF+ L (l) · eδχ F+(l) L : l ∈ U}. Definition 3.2. Let L = { l, µT − L (l) · eδχ T−(l) L , µF− L (l) · eδχ F−(l) L , µT + L (l) · eδχ T +(l) L , µF+ L (l) · eδχ F+(l) L } and M = { l, µT − M (l) · eδχ T−(l) M , µF− M (l) · eδχ F−(l) M , µT + M (l) · eδχ T +(l) M , µF+ M (l) · eδχ F+(l) M } be two CBIFSs of U . Then we define the intersection and union operation is defined as (i) L∩M = {( l,max{µT − L (l) · eδχ T−(l) L , µT − M (l) · eδχ T−(l) M },min{µF− L (l) · eδχ F−(l) L , µF− M (l) · eδχ F−(l) M }, min{µT + L (l) · eδχ T +(l) L , µT + M (l) · eδχ T +(l) M }, max{µF+ L (l) · eδχ F+(l) L , µF+ M (l) · eδχ F+(l) M } )∣∣∣l ∈ U}. (ii) L∪M = {( l,min{µT − L (l) · eδχ T−(l) L , µT − M (l) · eδχ T−(l) M },max{µF− L (l) · eδχ F−(l) L , µF− M (l) · eδχ F−(l) M }, max{µT + L (l) · eδχ T +(l) L , µT + M (l) · eδχ T +(l) M },max{µI+L (l) · eδχ I+(l) L min{µF+ L (l) · eδχ F+(l) L , µF+ M (l) · eδχ F+(l) M } )∣∣∣l ∈ U}. Definition 3.3. For any CBIFS L = { l, µT − L (l) · eδχ T−(l) L , µF− L (l) · eδχ F−(l) L , µT + L (l) · eδχ T +(l) L , µF+ L (l) · eδχ F+(l) L } of a universal setU . Then (t, s)-cut is defined as { l ∈ U |µT − L (l)·eδχ T−(l) L ≤ t, µF− L (l)·eδχ F−(l) L ≥ s, µT + L (l) · eδχ T +(l) L ≥ t, µF+ L (l) · eδχ F+(l) L ≤ s } . Definition 3.4. The Cartesian product of L and M is defined as L×M = { µT − L×M ((l,m)) · eδχ T−((l,m)) L×M , µF− L×M (l,m) · eδχ F−((l,m)) L×M , µT + L×M ((l,m)) · eδχ T +((l,m)) L×M , µF+ L×M (l,m) · eδχ F+((l,m)) L×M | for all l,m ∈ S } , where L and M be the CBIFS of U µT − L×M ((l,m)) · eδχ T−((l,m)) L×M = max { µT − L (l) · eδχ T−(l) L , µT − M (m) · eδχ T−(m) M } µF− L×M ((l,m)) · eδχ F−((l,m)) L×M = min { µF− L (l) · eδχ F+(l) L , µF− M (m) · eδχ F−(m) M }   µT + L×M ((l,m)) · eδχ T +((l,m)) L×M = min { µT + L (l) · eδχ T +(l) L , µT + M (m) · eδχ T +(m) M } µF+ L×M ((l,m)) · eδχ F+((l,m)) L×M = max { µF+ L (l) · eδχ F+(l) L , µF+ M (m) · eδχ F+(m) M }  Definition 3.5. For any CBIFS L of B is said to be a Q-complex bipolar intuitionistic fuzzy SBS (CBIFSBS) of B if  µT − L ((l ‡1 m)) · eδχT− L ((l‡1m)) ≤ max{µT − L (l) · eδχT− L ((l), µT − L (m) · eδχT− L ((m)} µT − L ((l ‡2 m)) · eδχT− L ((l‡2m)) ≤ max{µT − L (l) · eδχT− L ((l), µT − L (m) · eδχT− L ((m)} µT − L ((l ‡3 m)) · eδχT− L ((l‡3m)) ≤ max{µT − L (l) · eδχT− L ((l), µT − L (m) · eδχT− L ((m)}   µF− L ((l ‡1 m)) · eδχF− L ((l‡1m)) ≥ min{µF− L (l) · eδχF− L ((l), µF− L (m) · eδχF− L ((m)} µF− L ((l ‡2 m)) · eδχF− L ((l‡2m)) ≥ min{µF− L (l) · eδχF− L ((l), µF− L (m) · eδχF− L ((m)} µF− L ((l ‡3 m)) · eδχF− L ((l‡3m)) ≥ min{µF− L (l) · eδχF− L ((l), µF− L (m) · eδχF− L ((m)}  https://internationalpubls.com 552 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025)  µT + L ((l ‡1 m)) · eδχT + L ((l‡1m)) ≥ min{µT + L (l) · eδχT + L ((l), µT + L (m) · eδχT + L ((m)} µT + L ((l ‡2 m)) · eδχT + L ((l‡2m)) ≥ min{µT + L (l) · eδχT + L ((l), µT + L (m) · eδχT + L ((m)} µT + L ((l ‡3 m)) · eδχT + L ((l‡3m)) ≥ min{µT + L (l) · eδχT + L ((l), µT + L (m) · eδχT + L ((m)}  µF+ L ((l ‡1 m)) · eδχF+ L ((l‡1m)) ≤ max{µF+ L (l) · eδχF+ L ((l), µF+ L (m) · eδχF+ L ((m)} µF+ L ((l ‡2 m)) · eδχF+ L ((l‡2m)) ≤ max{µF+ L (l) · eδχF+ L ((l), µF+ L (m) · eδχF+ L ((m)} µF+ L ((l ‡3 m)) · eδχF+ L ((l‡3m)) ≤ max{µF+ L (l) · eδχF+ L ((l), µF+ L (m) · eδχF+ L ((m)}  for all l,m ∈ B. Example 3.6. Consider the bisemiring B = {a, b, c, d} with the Cayley table: ‡1 a b c d a a a a a b a b a b c a a c c d a b c d ‡2 a b c d a a b c d b b b d d c c d c d d d d d d ‡3 a b c d a a a a a b a b c d c d d d d d d d d d ([) = a ([) = b ([) = c ([) = d (µT − L , χT − L )([) −0.8ei2π(−0.65) −0.75ei2π(−0.60) −0.65ei2π(−0.50) −0.7ei2π(0−.55) (µF− L , χF− L )([) −0.7ei2π(−0.55) −0.8ei2π(−0.65) −0.9ei2π(−0.75) −0.85ei2π(−0.7) ([) = a ([) = b ([) = c ([) = d (µT + L , χT + L )([) 0.7ei2π(0.6) 0.6ei2π(0.5) 0.4ei2π(0.3) 0.5ei2π(0.4) (µF+ L , χF+ L )([) 0.6ei2π(0.5) 0.7ei2π(0.6) 0.9ei2π(0.8) 0.8ei2π(0.7) Hence, L is a CBIFSBS of B. Theorem 3.7. The intersection of a every CBIFSBSs is again a CBIFSBS of B. Proof. Let {τi : i ∈ I} be the family of CBIFSBSs of B and L = ei∈Iτi. Let l,m ∈ B. Now, µT − L ((l ‡1 m)) · eδχ T− L ((l‡1m)) = sup i∈I µT − τi ((l ‡1 m)) · eδχ T− L ((l‡1m)) ≤ sup i∈I max{µT − τi (l) · eδχ T− τi (l), µT − τi (m) · eδχ T− τi (m)} = max { sup i∈I µT − τi (l) · eδχ T− τi (l), sup i∈I µT − τi (m) · eδχ T− τi (m) } = max{µT − L (l) · eδχ T− L (l), µT − L (m) · eδχ T− L (m)} Similarly, µT − L ((l ‡2 m)) · eδχT− L ((l‡2m)) ≤ max{µT − L (l) · eδχT− L (l), µT − L (m) · eδχT− L (m)}, µT − L ((l ‡3 m)) · eδχT− L ((l‡3m)) ≤ max{µT − L (l) · eδχT− L (l), µT − L (m) · eδχT− L (m)}. Now, µF− L ((l ‡1 m)) · eδχ F− L ((l‡1m)) = inf i∈I µF− τi ((l ‡1 m)) · eδχ F− L ((l‡1m)) ≥ inf i∈I min{µF− τi (l) · eδχ F− τi (l), µF− τi (m) · eδχ F− τi (m)} = min { inf i∈I µF− τi (l) · eδχ F− τi (l), inf i∈I µF− τi (m) · eδχ F− τi (m) } = min{µF− L (l) · eδχ F− L (l), µF− L (m) · eδχ F− L (m)} https://internationalpubls.com 553 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) Similarly, µF− L ((l ‡2 m)) · eδχF− L ((l‡2m)) ≥ min{µF− L (l) · eδχF− L (l), µF− L (m) · eδχF− L (m)} and µF− L ((l ‡3 m)) · eδχF− L ((l‡3m)) ≥ min{µF− L (l) · eδχF− L (l), µF− L (m) · eδχF− L (m)}. Let {τi : i ∈ I} be the family of CBIFSBSs of B and L = ei∈Iτi. Let l,m ∈ B. Now, µT + L ((l ‡1 m)) · eδχ T + L ((l‡1m)) = inf i∈I µT + τi ((l ‡1 m)) · eδχ T + L ((l‡1m)) ≥ inf i∈I min{µT + τi (l) · eδχ T + τi (l), µT + τi (m) · eδχ T + τi (m)} = min { inf i∈I µT + τi (l) · eδχ T + τi (l), inf i∈I µT + τi (m) · eδχ T + τi (m) } = min{µT + L (l) · eδχ T + L (l), µT + L (m) · eδχ T + L (m)} Similarly, µT + L ((l ‡2 m)) · eδχT + L ((l‡2m)) ≥ min{µT + L (l) · eδχT + L (l), µT + L (m) · eδχT + L (m)}, µT + L ((l ‡3 m)) · eδχT + L ((l‡3m)) ≥ min{µT + L (l) · eδχT + L (l), µT + L (m) · eδχT + L (m)}. Now, µF+ L ((l ‡1 m)) · eδχ F+ L ((l‡1m)) = sup i∈I µF+ τi ((l ‡1 m)) · eδχ F+ L ((l‡1m)) ≤ sup i∈I max{µF+ τi (l) · eδχ F+ τi (l), µF+ τi (m) · eδχ F+ τi (m)} = max { sup i∈I µF+ τi (l) · eδχ F+ τi (l), sup i∈I µF+ τi (m) · eδχ F+ τi (m) } = max{µF+ L (l) · eδχ F+ L (l), µF+ L (m) · eδχ F+ L (m)} Similarly, µF+ L ((l ‡2 m)) · eδχF+ L ((l‡2m)) ≤ max{µF+ L (l) · eδχF+ L (l), µF+ L (m) · eδχF+ L (m)} and µF+ L ((l ‡3 m)) · eδχF+ L ((l‡3m)) ≤ max{µF+ L (l) · eδχF+ L (l), µF+ L (m) · eδχF+ L (m)}. Thus,L is a CBIFSBS of B. Theorem 3.8. IfL andM be the CBIFSBSs of B1 and B2 respectively, thenL×M is a CBIFSBS of B1×B2. Proof. Let l1, l2 ∈ B1 and m1,m2 ∈ B2. Then (l1,m1) and (l2,m2) are in B1 ×B2. Now µT − L×M [((l1,m1) ‡1 (l2,m2))] · eδχ T− L×M [((l1,m1)‡1(l2,m2))] = µT − L×M ((l1 ‡1 l2,m1 ‡1 m2)) · eδχ T− L×M ((l1‡1l2,m1‡1m2)) = max{µT − L ((l1 ‡1 l2)) · eδχ T− L ((l1‡1l2)), µT − M ((m1 ‡1 m2)) · eδχ T− M ((m1‡1m2))} ≤ max{max{µT − L (l1) · eδχ T− L (l1), µT − L (l2) · eδχ T− L (l2)}, max{µT − M (m1) · eδχ T− L (m1), µT − M (m2) · eδχ T− L (m2)}} = max{max{µT − L (l1) · eδχ T− L (l1), µT − M (m1) · eδχ T− L (m1)}, max{µT − L (l2) · eδχ T− L (l2), µT − M (m2) · eδχ T− L (m2)}} = max{µT − L×M ((l1,m1)) · eδχ T− L×M ((l1,m1)), µT − L×M ((l2,m2)) · eδχ T− L×M ((l2,m2))} Also µT − L×M [((l1,m1) ‡2 (l2,m2))] · eδχ T− L×M [((l1,m1)‡2(l2,m2))] ≤ max{µT − L×M ((l1,m1)) · eδχ T− L×M ((l1,m1)), µT − L×M ((l2,m2))eδχ T− L×M ((l2,m2))} https://internationalpubls.com 554 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) and µT − L×M [((l1,m1) ‡3 (l2,m2))] · eδχ T− L×M [((l1,m1)‡3(l2,m2))] ≤ max{µT − L×M ((l1,m1)) · eδχ T− L×M ((l1,m1)), µT − L×M ((l2,m2)) · eδχ T− L×M ((l2,m2))}. Now, µF− L×M [((l1,m1) ‡1 (l2,m2))] · eδχ F− L×M [((l1,m1)‡1(l2,m2))] = µF− L×M ((l1 ‡1 l2,m1 ‡1 m2)) · eδχ F− L×M ((l1‡1l2,m1‡1m2)) = min{µF− L ((l1 ‡1 l2)) · eδχ F− L×M ((l1‡1l2)), µF− M ((m1 ‡1 m2)) · eδχ F− L×M ((m1‡1m2))} ≥ min{min{µF− L (l1) · eδχ F− L (l1), µF− L (l2) · eδχ F− L (l2)}, min{µF− M (m1) · eδχ F− M (m1), µF− M (m2) · eδχ F− M (m2)}} = min{min{µF− L (l1) · eδχ F− L (l1), µF− M (m1) · eδχ F− M (m1)}, min{µF− L (l2) · eδχ F− L (l2), µF− M (m2) · eδχ F− M (m2)}} = min{µF− L×M ((l1,m1)) · eδχ F− L×M ((l1,m1)), µF− L×M ((l2,m2)) · eδχ F− L×M ((l2,m2))} Also µF− L×M [((l1,m1) ‡2 (l2,m2))] · eδχ F− L×M [((l1,m1)‡2(l2,m2))] ≥ min{µF− L×M ((l1,m1)) · eδχ F− L×M ((l1,m1)), µF− L×M ((l2,m2)) · eδχ F− L×M ((l2,m2))}, µF− L×M [((l1,m1) ‡3 (l2,m2))] · eδχ F− L×M [((l1,m1)‡3(l2,m2))] ≥ min{µF− L×M ((l1,m1)) · eδχ F− L×M ((l1,m1)), µF− L×M ((l2,m2)) · eδχ F− L×M ((l2,m2))}. Let l1, l2 ∈ B1 and m1,m2 ∈ B2. Then (l1,m1) and (l2,m2) are in B1 ×B2. Now µT + L×M [((l1,m1) ‡1 (l2,m2))] · eδχ T + L×M [((l1,m1)‡1(l2,m2))] = µT + L×M ((l1 ‡1 l2,m1 ‡1 m2)) · eδχ T + L×M ((l1‡1l2,m1‡1m2)) = min{µT + L ((l1 ‡1 l2)) · eδχ T + L ((l1‡1l2)), µT + M ((m1 ‡1 m2)) · eδχ T + M ((m1‡1m2))} ≥ min{min{µT + L (l1) · eδχ T + L (l1), µT + L (l2) · eδχ T + L (l2)}, min{µT + M (m1) · eδχ T + L (m1), µT + M (m2) · eδχ T + L (m2)}} = min{min{µT + L (l1) · eδχ T + L (l1), µT + M (m1) · eδχ T + L (m1)}, min{µT + L (l2) · eδχ T + L (l2), µT + M (m2) · eδχ T + L (m2)}} = min{µT + L×M ((l1,m1)) · eδχ T + L×M ((l1,m1)), µT + L×M ((l2,m2)) · eδχ T + L×M ((l2,m2))} Also µT + L×M [((l1,m1) ‡2 (l2,m2))] · eδχ T + L×M [((l1,m1)‡2(l2,m2))] ≥ min{µT + L×M ((l1,m1)) · eδχ T + L×M ((l1,m1)), µT + L×M ((l2,m2))eδχ T + L×M ((l2,m2))} and µT + L×M [((l1,m1) ‡3 (l2,m2))] · eδχ T + L×M [((l1,m1)‡3(l2,m2))] ≥ min{µT + L×M ((l1,m1)) · eδχ T + L×M ((l1,m1)), µT + L×M ((l2,m2)) · eδχ T + L×M ((l2,m2))}. https://internationalpubls.com 555 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) Now, µF+ L×M [((l1,m1) ‡1 (l2,m2))] · eδχ F+ L×M [((l1,m1)‡1(l2,m2))] = µF+ L×M ((l1 ‡1 l2,m1 ‡1 m2)) · eδχ F+ L×M ((l1‡1l2,m1‡1m2)) = max{µF+ L ((l1 ‡1 l2)) · eδχ F+ L×M ((l1‡1l2)), µF+ M ((m1 ‡1 m2)) · eδχ F+ L×M ((m1‡1m2))} ≤ max{max{µF+ L (l1) · eδχ F+ L (l1), µF+ L (l2) · eδχ F+ L (l2)}, max{µF+ M (m1) · eδχ F+ M (m1), µF+ M (m2) · eδχ F+ M (m2)}} = max{max{µF+ L (l1) · eδχ F+ L (l1), µF+ M (m1) · eδχ F+ M (m1)}, max{µF+ L (l2) · eδχ F+ L (l2), µF+ M (m2) · eδχ F+ M (m2)}} = max{µF+ L×M ((l1,m1)) · eδχ F+ L×M ((l1,m1)), µF+ L×M ((l2,m2)) · eδχ F+ L×M ((l2,m2))} Also µF+ L×M [((l1,m1) ‡2 (l2,m2))] · eδχ F+ L×M [((l1,m1)‡2(l2,m2))] ≤ max{µF+ L×M ((l1,m1)) · eδχ F+ L×M ((l1,m1)), µF+ L×M ((l2,m2)) · eδχ F+ L×M ((l2,m2))}, µF+ L×M [((l1,m1) ‡3 (l2,m2))] · eδχ F+ L×M [((l1,m1)‡3(l2,m2))] ≤ max{µF+ L×M ((l1,m1)) · eδχ F+ L×M ((l1,m1)), µF+ L×M ((l2,m2)) · eδχ F+ L×M ((l2,m2))}. Thus,L×M is a CBIFSBS of B. Corollary 3.9. If L1, L2, ..., L− be the finite collection of CBIFSBSs of B1,B2, ...,Bn respectively. Then L1 × L2 × ...× Ln is a CBIFSBS of B1 ×B2 × ...×Bn. Definition 3.10. Let L ⊆ B,the strongest CBN relation on B isµT − τ ((l,m)) · eδχT− τ ((l,m)) = min{µT − L (l) · eδχT− L (l), µT − L (m) · eδχT− L (m)} µF− τ ((l,m)) · eδχF− τ ((l,m)) = max{µF− L (l) · eδχF− L (l), µF− L (m) · eδχF− L (m)}  µT + τ ((l,m)) · eδχT + τ ((l,m)) = max{µT + L (l) · eδχT + L (l), µT + L (m) · eδχT + L (m)} µF+ τ ((l,m)) · eδχF+ τ ((l,m)) = min{µF+ L (l) · eδχF+ L (l), µF+ L (m) · eδχF+ L (m)}  Theorem 3.11. Let L be a CBIFSBS of B and τ be a strongest complex bipolar intuitionistic fuzzy relation of B. Then L is a CBIFSBS of B ×B if and only if τ is a CBIFSBS of B ×B. Proof. Suppose L is a CBIFSBS of B×B and τ be the strongest complex bipolar intuitionistic fuzzy relation of B. For any l = (l1, l2),m = (m1,m2) ∈ B ×B. Now, µT − τ ((l ‡1 m)) · eδχ T− τ ((l‡1m)) = µT − τ [((((l1, l2)) ‡1 ((m1,m2))] · eδχ T− τ [((((l1,l2))‡1((m1,m2))] = µT − τ (l1 ‡1 m1, l2 ‡1 m2) · eδχ T− τ (l1‡1m1,l2‡1m2) = max{µT − L ((l1 ‡1 m1)) · eδχ T− L ((l1‡1m1)), µT − L ((l2 ‡1 m2)) · eδχ T− L ((l2‡1m2))} ≤ max{max{µT − L (l1) · eδχ T− L (l1), µT − L (m1) · eδχ T− L (m1)}, max{µT − L (l2) · eδχ T− L (l2), µT − L (m2) · eδχ T− L (m2)}} = max{max{µT − L (l1) · eδχ T− L (l1), µT − L (l2) · eδχ T− L (l2)}, max{µT − L (m1) · eδχ T− L (m1), µT − L (m2) · eδχ T− L (m2)}} = max{µT − τ ((l1, l2)) · eδχ T− τ ((l1,l2)), µT − τ ((m1,m2)) · eδχ T− τ ((m1,m2))} = max{µT − τ (l) · eδχ T− τ (l), µT − τ (m) · eδχ T− τ (m)} https://internationalpubls.com 556 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) Also µT − τ ((l ‡2 m)) · eδχT− τ ((l‡2m)) ≤ max{µT − τ (l) · eδχT− τ (l), µT − τ (m) · eδχT− τ (m)}, µT − τ ((l ‡3 m)) · eδχT− τ ((l‡3m)) ≤ max{µT − τ (l) · eδχT− τ (l), µT − τ (m) · eδχT− τ (m)}. Similarly,µF− τ ((l ‡1 m)) · eδχF− τ ((l‡1m)) ≥ min{µF− τ (l) · eδχF− τ (l)·, µF− τ (m) · eδχF− τ (m)}, µF− τ ((l ‡2 m)) · eδχF− τ ((l‡2m)) ≥ min{µF− τ (l) · eδχF− τ (l), µF− τ (m) · eδχF− τ (m)} and µF− τ ((l ‡3 m)) · eδχF− τ ((l‡3m)) ≥ min{µF− τ (l) · eδχF− τ (l), µF− τ (m) · eδχI − τ (m}. For any l = (l1, l2),m = (m1,m2) ∈ B ×B. Now, µT + τ ((l ‡1 m)) · eδχ T + τ ((l‡1m)) = µT + τ [((((l1, l2)) ‡1 ((m1,m2))] · eδχ T + τ [((((l1,l2))‡1((m1,m2))] = µT + τ (l1 ‡1 m1, l2 ‡1 m2) · eδχ T + τ (l1‡1m1,l2‡1m2) = min{µT + L ((l1 ‡1 m1)) · eδχ T + L ((l1‡1m1)), µT + L ((l2 ‡1 m2)) · eδχ T + L ((l2‡1m2))} ≥ min{min{µT + L (l1) · eδχ T + L (l1), µT + L (m1) · eδχ T + L (m1)}, min{µT + L (l2) · eδχ T + L (l2), µT + L (m2) · eδχ T + L (m2)}} = min{min{µT + L (l1) · eδχ T + L (l1), µT + L (l2) · eδχ T + L (l2)}, min{µT + L (m1) · eδχ T + L (m1), µT + L (m2) · eδχ T + L (m2)}} = min{µT + τ ((l1, l2)) · eδχ T + τ ((l1,l2)), µT + τ ((m1,m2)) · eδχ T + τ ((m1,m2))} = min{µT + τ (l) · eδχ T + τ (l), µT + τ (m) · eδχ T + τ (m)} Also µT + τ ((l ‡2 m)) · eδχT + τ ((l‡2m)) ≥ min{µT + τ (l) · eδχT + τ (l), µT + τ (m) · eδχT + τ (m)}, µT + τ ((l ‡3 m)) · eδχT + τ ((l‡3m)) ≥ min{µT + τ (l) · eδχT + τ (l), µT + τ (m) · eδχT + τ (m)}. Similarly, µF+ τ ((l ‡1 m)) · eδχF+ τ ((l‡1m)) ≤ max{µF+ τ (l) · eδχF+ τ (l)·, µF+ τ (m) · eδχF+ τ (m)}, µF+ τ ((l ‡2 m)) · eδχF+ τ ((l‡2m)) ≤ max{µF+ τ (l) · eδχF+ τ (l), µF+ τ (m) · eδχF+ τ (m)} and µF+ τ ((l ‡3 m)) · eδχF+ τ ((l‡3m)) ≤ max{µF+ τ (l) · eδχF+ τ (l), µF+ τ (m) · eδχF+ τ (m}. Therefore, τ is a CBIFSBS of B ×B. Conversely,suppose that τ is a CBIFSBS of B ×B. Let l = ((l1, l2)),m = ((m1,m2)) ∈ B ×B. Now, max{µT − L ((l1 ‡1 m1)) · eδχ T− L ((l1‡1m1)), µT − L ((l2 ‡1 m2)) · eδχ T− L ((l2‡1m2))} = µT − τ (l1 ‡1 m1, l2 ‡1 m2) · eδχ T− τ (l1‡1m1,l2‡1m2) = µT − τ [((l1, l2)) ‡1 ((m1,m2))] · eδχ T + v [((l1,l2))‡1((m1,m2))] = µT − τ (l ‡1 m) · eδχ T− τ (l‡1m) ≤ max{µT − τ (l) · eδχ T− τ (l), µT − τ (m) · eδχ T− τ (m)} = max{µT − τ ((l1, l2))} · eδχ T− τ ((l1,l2))}, µT − τ ((m1,m2)) · eδχ T− τ ((m1,m2))} = max{max{µT − L (l1) · eδχ T− L (l1), µT − L (l2) · eδχ T− L (l2)}, max{µT − L (m1) · eδχ T− L (m1), µT − L (m2) · eδχ T− L (m2)}} If µT − L ((l1 ‡1 m1)) · eδχT− L ((l1‡1m1)) ≥ µT − L ((l2 ‡1 m2)) · eδχT− L ((l2‡1m2)), then µT − L (l1) · eδχT− L (l1) ≥ µT − L (l2) · eδχT− L (l2) and µT − L (m1) · eδχT− L (m1) ≥ µT − L (m2) · eδχT− L (m2). We get µT − L ((l1 ‡1 m1)) · eδχ T− L ((l1‡1m1)) ≤ max{µT − L (l1) · eδχT− L (l1), µT − L (m1) · eδχT− L (m1)} for all l1,m1 ∈ B,and max{µT − L ((l1‡2m1))·eδχT− L ((l1‡2m1)), µT − L ((l2‡2m2))·eδχT− L ((l2‡2m2))} ≤ max{max{µT − L (l1)·eδχT− L (l1), µT − L (l2)· https://internationalpubls.com 557 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) eδχ T− L (l2)},max{µT − L (m1) · eδχT− L (m1), µT − L (m2) · eδχT− L (m2)}} If µT − L ((l1‡2m1))·eδχT− L ((l1‡2m1)) ≥ µT − L ((l2‡2m2))·eδχT− L ((l2‡2m2)),then µT − L ((l1‡2m1))·eδχT− L ((l1‡2m1)) ≤ max{µT − L (l1) · eδχT− L (l1), µT − L (m1) · eδχT− L (m1)}. max{µT − L ((l1‡3m1))·eδχT− L ((l1‡3m1)), µT − L ((l2‡3m2))·eδχT− L ((l2‡3m2))} ≤ max{max{µT − L (l1)·eδχT− L (l1), µT − L (l2)· eδχ T− L (l2)},max{µT − L (m1) · eδχT− L (m1), µT − L (m2) · eδχT− L (m2)}}. If µT − L ((l1‡3m1))·eδχT− L ((l1‡3m1)) ≥ µT − L ((l2‡3m2))·eδχT− L ((l2‡3m2)),then µT − L ((l1‡3m1))·eδχT− L ((l1‡3m1)) ≤ max{µT − L (l1) · eδχT− L (l1), µT − L (m1) · eδχT− L (m1)}. Similarly to prove that min{µF− L ((l1‡1m1))·eδχF− L ((l1‡1m1)), µF− L ((l2‡1m2))·eδχF− L ((l2‡1m2))} ≥ min{min{µF− L (l1)·eδχF− L (l1), µF− L (l2)· eδχ F− L (l2)},min{µF− L (m1) · eδχF− L (m1), µF− L (m2) · eδχF− L (m2)}} If µF− L ((l1 ‡1 m1)) · eδχF− L ((l1‡1m1)) ≤ µF− L ((l2 ‡1 m2)) · eδχF− L ((l2‡1m2)),then µF− L (l1) · eδχF− L (l1) ≤ µF− L (l2) · eδχF− L (l2) and µF− L (m1) · eδχF− L (m1) ≤ µF− L (m2) · eδχF− L (m2). We get µF− L ((l1 ‡1 m1)) · eδχF− L ((l1‡1m1)) ≥ min{µF− L (l1) · eδχF− L (l1), µF− L (m1) · eδχF− L (m1)}. min{µF− L ((l1 ‡2 m1)) · eδχF− L ((l1‡2m1)), µF− L ((l2 ‡2 m2)) · eδχF− L ((l2‡2m2))} ≥ min{min{µF− L (l1) · eδχF− L (l1), µF− L (l2) · eδχF− L (l2)},min{µF− L (m1) · eδχF− L (m1), µF− L (m2) · eδχF− L }} If µF− L ((l1‡2m1))·eδχF− L ((l1‡2m1)) ≤ µF− L ((l2‡2m2))·eδχF− L ((l2‡2m2)),then µF− L ((l1‡2m1))·eδχF− L ((l1‡2m1)) ≥ min{µF− L (l1) · eδχF− L (l1), µF− L (m1) · eδχF− L (m1)}. min{µF− L ((l1‡3m1))·eδχF− L ((l1‡3m1)), µF− L ((l2‡3m2))·eδχF− L ((l2‡3m2))} ≥ min{min{µF− L (l1)·eδχF− L (l1), µF− L (l2)· eδχ F− L (l2)},min{µF− L (m1) · eδχF− L (m1), µF− L (m2) · eδχF− L (m2)}} If µF− L ((l1‡3m1))·eδχF− L ((l1‡3m1)) ≤ µF− L ((l2‡3m2))·eδχF− L ((l2‡3m2)),then µF− L ((l1‡3m1))·eδχF− L ((l1‡3m1)) ≥ min{µF− L (l1) · eδχF− L (l1), µF− L (m1) · eδχF− L (m1)}. Let l = ((l1, l2)),m = ((m1,m2)) ∈ B ×B. Now, min{µT + L ((l1 ‡1 m1)) · eδχ T + L ((l1‡1m1)), µT + L ((l2 ‡1 m2)) · eδχ T + L ((l2‡1m2))} = µT + τ (l1 ‡1 m1, l2 ‡1 m2) · eδχ T + τ (l1‡1m1,l2‡1m2) = µT + τ [((l1, l2)) ‡1 ((m1,m2))] · eδχ T + v [((l1,l2))‡1((m1,m2))] = µT + τ (l ‡1 m) · eδχ T + τ (l‡1m) ≥ min{µT + τ (l) · eδχ T + τ (l), µT + τ (m) · eδχ T + τ (m)} = min{µT + τ ((l1, l2))} · eδχ T + τ ((l1,l2))}, µT + τ ((m1,m2)) · eδχ T + τ ((m1,m2))} = min{min{µT + L (l1) · eδχ T + L (l1), µT + L (l2) · eδχ T + L (l2)}, min{µT + L (m1) · eδχ T + L (m1), µT + L (m2) · eδχ T + L (m2)}} If µT + L ((l1 ‡1 m1)) · eδχT + L ((l1‡1m1)) ≤ µT + L ((l2 ‡1 m2)) · eδχT + L ((l2‡1m2)), then µT + L (l1) · eδχT + L (l1) ≤ µT + L (l2) · eδχT + L (l2) and µT + L (m1) · eδχT + L (m1) ≤ µT + L (m2) · eδχT + L (m2). We get µT + L ((l1 ‡1 m1)) · eδχ T + L ((l1‡1m1)) ≥ min{µT + L (l1) · eδχT + L (l1), µT + L (m1) · eδχT + L (m1)} for all l1,m1 ∈ B,and min{µT + L ((l1‡2m1))·eδχT + L ((l1‡2m1)), µT + L ((l2‡2m2))·eδχT + L ((l2‡2m2))} ≥ min{min{µT + L (l1)·eδχT + L (l1), µT + L (l2)· eδχ T + L (l2)},min{µT + L (m1) · eδχT + L (m1), µT + L (m2) · eδχT + L (m2)}} If µT + L ((l1‡2m1))·eδχT + L ((l1‡2m1)) ≤ µT + L ((l2‡2m2))·eδχT + L ((l2‡2m2)),then µT + L ((l1‡2m1))·eδχT + L ((l1‡2m1)) ≥ min{µT + L (l1) · eδχT + L (l1), µT + L (m1) · eδχT + L (m1)}. min{µT + L ((l1‡3m1))·eδχT + L ((l1‡3m1)), µT + L ((l2‡3m2))·eδχT + L ((l2‡3m2))} ≥ min{min{µT + L (l1)·eδχT + L (l1), µT + L (l2)· eδχ T + L (l2)},min{µT + L (m1) · eδχT + L (m1), µT + L (m2) · eδχT + L (m2)}} If µT + L ((l1‡3m1))·eδχT + L ((l1‡3m1)) ≤ µT + L ((l2‡3m2))·eδχT + L ((l2‡3m2)),then µT + L ((l1‡3m1))·eδχT + L ((l1‡3m1)) ≥ min{µT + L (l1) · eδχT + L (l1), µT + L (m1) · eδχT + L (m1)}. https://internationalpubls.com 558 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) Similarly to prove that max{µF+ L ((l1 ‡1 m1)) · eδχF+ L ((l1‡1m1)), µF+ L ((l2 ‡1 m2)) · eδχF+ L ((l2‡1m2))} ≤ max{max{µF+ L (l1)·eδχF+ L (l1), µF+ L (l2)·eδχF+ L (l2)},max{µF+ L (m1)·eδχF+ L (m1), µF+ L (m2)·eδχF+ L (m2)}} If µF+ L ((l1 ‡1 m1)) · eδχF+ L ((l1‡1m1)) ≥ µF+ L ((l2 ‡1 m2)) · eδχF+ L ((l2‡1m2)), then µF+ L (l1) · eδχF+ L (l1) ≥ µF+ L (l2) · eδχF+ L (l2) and µF+ L (m1) · eδχF+ L (m1) ≥ µF+ L (m2) · eδχF+ L (m2). We get µF+ L ((l1 ‡1 m1)) · eδχF+ L ((l1‡1m1)) ≤ max{µF+ L (l1) · eδχF+ L (l1), µF+ L (m1) · eδχF+ L (m1)}. max{µF+ L ((l1 ‡2 m1)) · eδχF+ L ((l1‡2m1)), µF+ L ((l2 ‡2 m2)) · eδχF+ L ((l2‡2m2))} ≤ max{max{µF+ L (l1) · eδχF+ L (l1), µF+ L (l2) · eδχF+ L (l2)},max{µF+ L (m1) · eδχF+ L (m1), µF+ L (m2) · eδχF+ L }} If µF+ L ((l1‡2m1))·eδχF+ L ((l1‡2m1)) ≥ µF+ L ((l2‡2m2))·eδχF+ L ((l2‡2m2)),then µF+ L ((l1‡2m1))·eδχF+ L ((l1‡2m1)) ≤ max{µF+ L (l1) · eδχF+ L (l1), µF+ L (m1) · eδχF+ L (m1)}. max{µF+ L ((l1 ‡3 m1)) · eδχF+ L ((l1‡3m1)), µF+ L ((l2 ‡3 m2)) · eδχF+ L ((l2‡3m2))} ≤ max{max{µF+ L (l1)·eδχF+ L (l1), µF+ L (l2)·eδχF+ L (l2)},max{µF+ L (m1)·eδχF+ L (m1), µF+ L (m2)·eδχF+ L (m2)}} If µF+ L ((l1‡3m1))·eδχT + L ((l1‡3m1)) ≥ µF+ L ((l2‡3m2))·eδχT + L ((l2‡3m2)),then µF+ L ((l1‡3m1))·eδχT + L ((l1‡3m1)) ≤ max{µF+ L (l1) · eδχF+ L (l1), µF+ L (m1) · eδχT + L (m1)}. Therefore, L is a CBIFSBS of B. Theorem 3.12. Suppose that L is a subset of B. ThenR = (µT − L ·eδχT− L , µF− L ·eδχF− L , µT + L ·eδχT + L , µF+ L · eδχ F+ L ) is a CBIFSBS of B if and only if µ(t,s) is a SBS of B for all t, s ∈ [−1, 0]× [0, 1]. Proof. Assume that µ is a CBIFSBS of B. For each t, s ∈ [−1, 0]× [0, 1] and l1, l2 ∈ µ(t,s). Now,µT − L (l1) · eδχ T− L (l1) ≤ t, µT − L (l2) · eδχT− L (l2) ≤ t and µI − L (l1) · eδχI − L (l1) ≤ t, µI−L (l2) · eδχI − L (l2) ≤ t and µF− L (l1) · eδχ F− L (l1) ≥ s, µF− L (l2) · eδχF− L (l2) ≥ s. Now,µT − L ((l1 ‡1 l2)) · eδχT− L ((l1 ‡1 l2)) ≤ max{µT − L (l1) · eδχ T− L (l1), µT − L (l2) · eδχT− L (l2)} ≤ t and µF− L ((l1 ‡1 l2)) · eδχF− L ((l1 ‡1 l2)) ≥ min{µF− L (l1) · eδχF− L (l1), µF− L (l2) · eδχF− L (l2)} ≥ s. This implies that l1 ‡1 l2 ∈ µ(t,s). Similarly,l1 ‡2 l2 ∈ µ(t,s) and l1 ‡3 l2 ∈ µ(t,s). Hence,µ(t,s) is a SBS of B,for all t, s ∈ [−1, 0]× [0, 1]. For each t, s ∈ [0, 1] and l1, l2 ∈ µ(t,s). Now,µT + L (l1) · eδχT + L (l1) ≥ t, µT + L (l2) · eδχT + L (l2) ≥ t and µF+ L (l1) · eδχF+ L (l1) ≤ s, µF+ L (l2) · eδχF+ L (l2) ≤ s. Now, µT + L ((l1 ‡1 l2)) · eδχT + L ((l1 ‡1 l2)) ≥ min{µT + L (l1) · eδχT + L (l1), µT + L (l2) · eδχT + L (l2)} ≥ t and µF+ L ((l1 ‡1 l2)) · eδχF+ L ((l1 ‡1 l2)) ≤ max{µF+ L (l1) · eδχF+ L (l1), µF+ L (l2) · eδχF+ L (l2)} ≤ s. This implies that l1 ‡1 l2 ∈ µ(t,s). Similarly,l1 ‡2 l2 ∈ µ(t,s) and l1 ‡3 l2 ∈ µ(t,s). Hence,µ(t,s) is a SBS of B,for all t, s ∈ [−1, 0]× [0, 1]. Conversely, assume that µ(t,s) is a SBS of B and t, s ∈ [−1, 0]× [0, 1]. Suppose if there exist l1, l2 ∈ B such that µT − L ((l1 ‡1 l2)) · eδχT− L ((l1 ‡1 l2)) > max{µT − L (l1) · eδχT− L (l1), µT − L (l2) · eδχT− L (l2)} and µF− L ((l1‡1 l2))·eδχF− L ((l1‡1 l2)) < min{µF− L (l1)·eδχF− L (l1), µF− L (l2)·eδχF− L (l2)}. For t, s ∈ [−1, 0]× [0, 1] such that µT − L ((l1 ‡1 l2)) · eδχT− L ((l1 ‡1 l2)) > t ≥ max{µT − L (l1) · eδχT− L (l1), µT − L (l2) · eδχT− L (l2)} and µF− L ((l1 ‡1 l2)) · eδχF− L ((l1 ‡1 l2)) < s ≤ min{µF− L (l1) · eδχF− L (l1), µF− L (l2) · eδχF− L (l2)}. Thus, l1, l2 ∈ µ(t,s),but l1 ‡1 l2 /∈ µ(t,s). This contradicts, µ(t,s) is a SBS of B. Therefore µT − L ((l1 ‡1 l2)) · eδχT− L ((l1 ‡1 l2)) ≤ max{µT − L (l1) · eδχT− L (l1), µT − L (l2) · eδχT− L (l2)} and µF− L ((l1 ‡1 l2)) · eδχF− L ((l1 ‡1 l2)) ≥ min{µF− L (l1) · eδχF− L (l1), µF− L (l2) · eδχF− L (l2)}. Similarly,‡2 and ‡3 cases. Hence µ = (µT − L · eδχT− L , µF− L · eδχF− L ) is a CBIFSBS of B. Let us assume that µ(t,s) is a SBS of B and t, s ∈ [−1, 0]× [0, 1]. Suppose if there exist l1, l2 ∈ B such that µT + L ((l1 ‡1 l2)) · eδχT + L ((l1 ‡1 l2)) > min{µT + L (l1) · eδχT + L (l1), µT + L (l2) · eδχT + L (l2)} and µF+ L ((l1 ‡1 l2)) · eδχ F+ L ((l1 ‡1 l2)) < max{µF+ L (l1) · eδχF+ L (l1), µF+ L (l2) · eδχF+ L (l2)}. For t, s ∈ [−1, 0] × [0, 1] such that https://internationalpubls.com 559 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) µT + L ((l1 ‡1 l2)) · eδχT + L ((l1 ‡1 l2)) > t ≤ min{µT + L (l1) · eδχT + L (l1), µT + L (l2) · eδχT + L (l2)} and µF+ L ((l1 ‡1 l2)) · eδχF+ L ((l1 ‡1 l2)) < s ≥ max{µF+ L (l1) · eδχF+ L (l1), µF+ L (l2) · eδχF+ L (l2)}. Thus, l1, l2 ∈ µ(t,s),but l1 ‡1 l2 /∈ µ(t,s). This contradicts,µ(t,s) is a SBS of B. Therefore µT + L ((l1 ‡1 l2)) · eδχT + L ((l1 ‡1 l2)) ≥ min{µT + L (l1) · eδχT + L (l1), µT + L (l2) · eδχT + L (l2)} and µF+ L ((l1 ‡1 l2)) · eδχF+ L ((l1 ‡1 l2)) ≤ max{µF+ L (l1) · eδχF+ L (l1), µF+ L (l2) · eδχF+ L (l2)}. Similarly, ‡2 and ‡3 cases. Hence µ = (µT + L · eδχT + L , µF+ L · eδχF+ L ) is a CBIFSBS of B. 4 Homomorphism Definition 4.1. Let (B1,♦1,♦2,♦3) and (B2,>1,>2,>3) be any two bisemirings. The mapping 0 : B1 → B2 and L be any CBIFSBS in B1, τ be any CBIFSBS in 0(B1) = B2. If µL · eδχL = [µT − L · eδχT− L , µF− L · eδχ F− L , µL · eδχL , µT + L · eδχT + L , µF+ L · eδχF+ L is a CBIFS in B1, then µτ is a CBIFS in B2,defined by µT − τ (m) · eδχ T− L (m) = { inf µT − L (l) · eδχT− L (l) if l ∈ 0−1m 0 otherwise µF− τ (m) · eδχ F− L (m) = { supµF− L (l) · eδχF− L (l) if l ∈ 0−1m −1 otherwise µT + τ (m) · eδχ T + L (m) = { supµT + L (l) · eδχT + L (l) if l ∈ 0−1m 0 otherwise µF+ τ (m) · eδχ F+ L (m) = { inf µF+ L (l) · eδχF+ L (l) if l ∈ 0−1m 1 otherwise for all l ∈ B1 and m ∈ B2 is represents the image of RL under 0. Similarly,If µτ · eδχτ = [µT − τ · eδχT− L , µI − τ · eδχ I− τ , µF− τ · eδχF− τ ], µτ · eδχτ , µT + τ · eδχT + L , µF+ τ · eδχF+ τ is a CBIFS in B2,then CBIFS µL = 0 ◦ µτ in B1 ie,the CBIFS defined by µL(l) · eδχL(l), µτ (0(l)) · eδχL(0(l)), µL = 0◦µτ in B1 [ie, the CBIFS defined by µL(l)·eδχL(l) = µτ (0(l))·eδχL(0(l)) is represents the pre-image of µτ under 0. Theorem 4.2. The homomorphic image of every CBIFSBS is a CBIFSBS. Proof. The mapping 0 : B1 → B2 be any homomorphism. Now, 0((l♦1m)) = 0(l) >1 0(m),0((l♦2m)) = 0(l) >2 0(m) and 0((l♦3m)) = 0(l) >3 0(m) for all l,m ∈ B1. Let τ = 0(L),L is any CBIFSBS of B1. Let 0(l),0(m) ∈ B2. Let l ∈ u0−1(0(l)) andm ∈ 0−1(0(m)) be such that µT − L (l)·eδχT− L (l) = inf l∈0−1(0(l)) µT − L (l)·eδχT− L (l) and µT − L (m) · eδχT− L (m) = inf l∈0−1(0(m)) µT − L (l) · eδχT− L (l). Now, µT − τ ((0(l) >1 0(m))) · eδχT− L ((0(l)>10(m))) = inf (l′ )∈0−1(0(l)>10(m)) µT − L (l ′ ) · eδχ T− L (l ′ ) = inf (l′ )∈0−1(0((l♦1m)) µT − L (l ′ ) · eδχ T− L (l ′ ) = µT − L ((l♦1m)) · eδχ T− L ((l♦1m)) ≤ max{µT − L (l) · eδχ T− L (l), µT − L (m) · eδχ T− L (m)} = max{µT − τ 0(l) · eδχ T− τ 0(l), µT − τ 0(m) · eδχ T− τ 0(m)}. https://internationalpubls.com 560 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) Thus,µT − τ ((0(l)>1 0(m))) · eδχT− τ ((0(l)>10(m))) ≤ max{µT − τ 0(l) · eδχT− τ 0(l), µT − τ 0(m) · eδχT− τ 0(m)}. Similarly,µT − τ ((0(l)>20(m)))·eδχT− τ ((0(l)>20(m))) ≤ max{µT − τ 0(l)·eδχT− τ 0(l), µT − τ 0(m)·eδχT− τ 0(m)} and µT − τ ((0(l) >3 0(m))) · eδχT− τ ((0(l)>30(m))) ≤ max{µT − τ 0(l) · eδχT− τ 0(l), µT − τ 0(m) · eδχT− τ 0(m)}. Let 0(l),0(m) ∈ B2. Let l ∈ 0−1(0(l)) and m ∈ 0−1(0(m)) be such that µF− L (l) · eδχF− L (l) = sup l∈0−1(0(l)) µF− L (l)·eδχF− L (l) and µF− L (m)·eδχF− L (m) = sup l∈0−1(0(m)) µF− L (l)·eδχF− L (l). Now, µF− τ ((0(l)>1 0(m))) · eδχF− τ ((0(l)>10(m))) = sup (l′ )∈0−1(0(l)>10(m)) µF− L (l ′ ) · eδχ F− L (l ′ ) = sup (l′ )∈0−1(0((l♦1m)) µF− L (l ′ ) · eδχ F− L (l ′ ) = µF− L ((l♦1m)) · eδχ F− L ((l♦1m)) ≥ min{µF− L (l) · eδχ F− L (l), µF− L (m) · eδχ F− L (m)} = min{µF− τ 0(l) · eδχ F− τ 0(l), µF− τ 0(m) · eδχ F− τ 0(m)}. Thus, µF− τ ((0(l)>1 0(m))) · eδχF− τ ((0(l)>10(m))) ≥ min{µF− τ 0(l) · eδχF− τ 0(l), µF− τ 0(m) · eδχF− τ 0(m)}. Similarly, µF− τ ((0(l) >2 0(m))) · eδχF− τ ((0(l)>20(m))) ≥ min{µF− τ 0(l) · eδχF− τ 0(l), µF− τ 0(m) · eδχF− τ 0(m)} and µF− τ ((0(l) >3 0(m))) · eδχF− τ ((0(l)>10(m))) ≥ min{µF− τ 0(l) · eδχF− τ 0(m), µF− τ 0(m) · eδχF− τ 0(m)}. The mapping 0 : B1 → B2 be any homomorphism. Now, 0((l♦1m)) = 0(l) >1 0(m),0((l♦2m)) = 0(l) >2 0(m) and 0((l♦3m)) = 0(l) >3 0(m) for all l,m ∈ B1. Let τ = 0(L),L is any CBIFSBS of B1. Let 0(l),0(m) ∈ B2. Let l ∈ 0−1(0(l)) and m ∈ 0−1(0(m)) be such that µT + L (l) ·eδχT + L (l) = sup l∈0−1(0(l)) µT + L (l) ·eδχT + L (l) and µT + L (m)·eδχT + L (m) = sup l∈0−1(0(m)) µT + L (l)·eδχT + L (l). Now, µT + τ ((0(l)>10(m)))·eδχT + L ((0(l)>10(m))) = sup (l′ )∈0−1(0(l)>10(m)) µT + L (l ′ ) · eδχ T + L (l ′ ) = sup (l′ )∈0−1(0((l♦1m)) µT + L (l ′ ) · eδχ T + L (l ′ ) = µT + L ((l♦1m)) · eδχ T + L ((l♦1m)) ≥ min{µT + L (l) · eδχ T + L (l), µT + L (m) · eδχ T + L (m)} = min{µT + τ 0(l) · eδχ T + τ 0(l), µT + τ 0(m) · eδχ T + τ 0(m)}. Thus, µT + τ ((0(l) >1 0(m))) · eδχT + τ ((0(l)>10(m))) ≥ min{µT + τ 0(l) · eδχT + τ 0(l), µT + τ 0(m) · eδχT + τ 0(m)}. Similarly, µT + τ ((0(l)>20(m)))·eδχT + τ ((0(l)>20(m))) ≥ min{µT + τ 0(l)·eδχT + τ 0(l), µT + τ 0(m)·eδχT + τ 0(m)} and µT + τ ((0(l) >3 0(m))) · eδχT + τ ((0(l)>30(m))) ≥ min{µT + τ 0(l) · eδχT + τ 0(l), µT + τ 0(m) · eδχT + τ 0(m)}. Let 0(l),0(m) ∈ B2. Let l ∈ 0−1(0(l)) and m ∈ 0−1(0(m)) be such that µF+ L (l) · eδχF+ L (l) = inf l∈0−1(0(l)) µF+ L (l) · eδχF+ L (l) and µF+ L (m) · eδχF+ L (m) = inf l∈0−1(0(m)) µF+ L (l) · eδχF+ L (l). https://internationalpubls.com 561 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) Now, µF+ τ ((0(l) >1 0(m))) · eδχF+ τ ((0(l)>10(m))) = inf (l′ )∈0−1(0(l)>10(m)) µF+ L (l ′ ) · eδχ F+ L (l ′ ) = inf (l′ )∈0−1(0((l♦1m)) µF+ L (l ′ ) · eδχ F+ L (l ′ ) = µF+ L ((l♦1m)) · eδχ F+ L ((l♦1m)) ≤ max{µF+ L (l) · eδχ F+ L (l), µF+ L (m) · eδχ F+ L (m)} = max{µF+ τ 0(l) · eδχ F+ τ 0(l), µF+ τ 0(m) · eδχ F+ τ 0(m)}. Thus, µF+ τ ((0(l)>1 0(m))) · eδχF+ τ ((0(l)>10(m))) ≤ max{µF+ τ 0(l) · eδχF+ τ 0(l), µF+ τ 0(m) · eδχF+ τ 0(m)}. Similarly, µF+ τ ((0(l)>20(m)))·eδχF+ τ ((0(l)>20(m))) ≤ max{µF+ τ 0(l)·eδχF+ τ 0(l), µF+ τ 0(m)·eδχF+ τ 0(m)} and µF+ τ ((0(l)>30(m))) ·eδχF+ τ ((0(l)>10(m))) ≤ max{µF+ τ 0(l) ·eδχF+ τ 0(m), µF+ τ 0(m) ·eδχF+ τ 0(m)}. Thus, τ is a CBIFSBS of B2. Theorem 4.3. The homomorphic preimage of every CBIFSBS is a CBIFSBS. Proof. The mapping 0 : B1 → B2 be a homomorphism. Now,0((l♦1m)) = 0(l) >1 0(m),0((l♦2m)) = 0(l) >2 0(m) and 0((l♦3m)) = 0(l) >3 0(m) for all l,m ∈ B1. Let τ = 0(L), τ is a CBIFSBS of B2. Let l,m ∈ B1. Now,µT − L ((l♦1m)) · eδχT− L ((l♦1m)) = µT − τ (0((l♦1m))) · eδχT− τ (0((l♦1m))) = µT − τ ((0(l) >1 0(m))) · eδχ T− τ ((0(l)>10(m))) ≤ max{µT − τ 0(l)·eδχT− τ 0(l), µT − τ 0(m)·eδχT− τ 0(m)} = max{µT − L (l)·eδχT− L (l), µT − L (m)· eδχ T− L (m)}. Thus,µT − L ((l♦1m)) · eδχT− L ((l♦1m)) ≤ max{µT − L (l) · eδχT− L (l), µT − L (m) · eδχT− L (m)}. Now, µF− L ((l♦1m)) · eδχF− L ((l♦1m)) = µF− τ (0((l♦1m))) · eδχF− τ (0((l♦1m))) = µF− τ ((0(l) >1 0(m))) · eδχF− τ ((0(l)>10(m))) ≥ min{µF− τ 0(l) · eδχF− τ 0(l), µF− τ 0(m) · eδχF− τ 0(m)} = min{µF− L (l) · eδχF− L (l), µF− L (m) · eδχF− L (m)}. Thus,µF− L ((l♦1m)) · eδχF− L ((l♦1m)) ≥ min{µF− L (l) · eδχF− L (l), µF− L (m) · eδχF− L (m)}. The mapping 0 : B1 → B2 be a homomorphism. Now, 0((l♦1m)) = 0(l) >1 0(m),0((l♦2m)) = 0(l) >2 0(m) and 0((l♦3m)) = 0(l) >3 0(m) for all l,m ∈ B1. Let τ = 0(L),τ is a CBIFSBS of B2. Let l,m ∈ B1. Now,µT + L ((l♦1m)) · eδχT + L ((l♦1m)) = µT + τ (0((l♦1m))) · eδχT + τ (0((l♦1m))) = µT + τ ((0(l) >1 0(m))) · eδχ T + τ ((0(l)>10(m))) ≥ min{µT + τ 0(l)·eδχT + τ 0(l), µT + τ 0(m)·eδχT + τ 0(m)} = min{µT + L (l)·eδχT + L (l), µT + L (m)· eδχ T + L (m)}. Thus,µT + L ((l♦1m)) · eδχT + L ((l♦1m)) ≥ min{µT + L (l) · eδχT + L (l), µT + L (m) · eδχT + L (m)}. Now, µF+ L ((l♦1m)) · eδχF+ L ((l♦1m)) = µF+ τ (0((l♦1m))) · eδχF+ τ (0((l♦1m))) = µF+ τ ((0(l) >1 0(m))) · eδχF+ τ ((0(l)>10(m))) ≤ max{µF+ τ 0(l) · eδχF+ τ 0(l), µF+ τ 0(m) · eδχF+ τ 0(m)} = max{µF+ L (l) · eδχF+ L (l), µF+ L (m) · eδχF+ L (m)}. Thus, µF+ L ((l♦1m)) · eδχF+ L ((l♦1m)) ≤ max{µF+ L (l) · eδχF+ L (l), µF+ L (m) · eδχF+ L (m)}. Theorem 4.4. If 0 : B1 → B2 is a homomorphism,then 0(L(t,s)) is a SBS of CBIFSBS τ of B2. Proof. The mapping 0 : B1 → B2 be a homomorphism. Now,0((l♦1m)) = 0(l) >1 0(m),0((l♦2m)) = 0(l) >2 0(m) and 0((l♦3m)) = 0(l)>30(m) for all l,m ∈ B1. Let τ = 0(L),L is a CBIFSBS of B1. By Theorem 4.2, τ is a CBIFSBS of B2. Let L(t,s) be any SBS of L. Suppose that l,m ∈ L(t,s). Then l♦1m, l♦2m and l♦3m ∈ L(t,s). Now,µT − τ (0(l)) · eδχT− τ (0(l)) = µT − L (l) · eδχT− L (l) ≤ t, µT − τ (0(m)) · eδχT− τ (0(m)) = µT − L (m) · eδχ T− L (m) ≤ t. Thus, µT − τ ((0(l) >1 0(m))) · eδχT− τ ((0(l)>10(m))) ≤ µT − L ((l♦1m)) · eδχT− L ((l♦1m)) ≤ t. Now, µF− τ (0(l))·eδχF− τ (0(l)) = µF− L (l)·eδχF− L (l) ≥ s, µF− τ (0(m))·eδχF− τ (0(m)) = µF− L (m)·eδχF− L (m) ≥ s. Thus,µF− τ ((0(l)>10(m)))·eδχF− τ ((0(l)>10(m))) ≥ µF− L ((l♦1m))·eδχF− L ((l♦1m)) ≥ s,for all 0(l),0(m) ∈ B2. Similarly other operations,0(L(t,s)) is a SBS of CBIFSBS τ of B2. https://internationalpubls.com 562 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) The mapping 0 : B1 → B2 be a homomorphism. Now, 0((l♦1m)) = 0(l) >1 0(m),0((l♦2m)) = 0(l) >2 0(m) and 0((l♦3m)) = 0(l)>30(m) for all l,m ∈ B1. Let τ = 0(L),L is a CBIFSBS of B1. By Theorem 4.2, τ is a CBIFSBS of B2. Let L(t,s) be any SBS of L. Suppose that l,m ∈ L(t,s). Then l♦1m, l♦2m and l♦3m ∈ L(t,s). Now,µT + τ (0(l)) · eδχT + τ (0(l)) = µT + L (l) · eδχT + L (l) ≥ t, µT + τ (0(m)) · eδχT + τ (0(m)) = µT + L (m) · eδχ T + L (m) ≥ t. Thus,µT + τ ((0(l) >1 0(m))) · eδχT + τ ((0(l)>10(m))) ≥ µT + L ((l♦1m)) · eδχT + L ((l♦1m)) ≥ t. Now, µF+ τ (0(l))·eδχF+ τ (0(l)) = µF+ L (l)·eδχF+ L (l) ≤ s, µF+ τ (0(m))·eδχF+ τ (0(m)) = µF+ L (m)·eδχF+ L (m) ≤ s. Thus,µF+ τ ((0(l)>10(m)))·eδχF+ τ ((0(l)>10(m))) ≤ µF+ L ((l♦1m))·eδχF+ L ((l♦1m)) ≤ s,for all 0(l),0(m) ∈ B2. Similarly other operations, 0(L(t,s)) is a SBS of CBIFSBS τ of B2. Theorem 4.5. If 0 : B1 → B2 is any homomorphism,then L(t,s) is a SBS of CBIFSBS L of B1. Proof. The mapping 0 : B1 → B2 be any homomorphism. We have 0((l♦1m)) = 0(l)>1 0(m),0((l♦2m)) = 0(l)>2 0(m) and 0((l♦3m)) = 0(l)>3 0(m) for all l,m ∈ B1. Let τ = 0(L),τ is a CBIFSBS of B2. By Theorem 4.3,L is a CBIFSBS of B1. Let 0(L(t,s)) be a SBS of τ . Suppose that 0(l),0(m) ∈ 0(L(t,s)). Now,0((l♦1m)),0((l♦2m)) and 0((l♦3m)) ∈ 0(L(t,s)). Now, µT − L (l)·eδχT− L (l) = µT − τ (0(l))·eδχT− τ (0(l)) ≤ t, µT − L (m)·eδχT− L (m) = µT − τ (0(m))·eδχT− τ (0(m)) ≤ t. Thus,µT − L ((l♦1m)) · eδχT− L ((l♦1m)) ≤ max{µT − L (l) · eδχT− L (l), µT − L (m) · eδχT− L (m)} ≤ t. Now, µF− L (l)·eδχF− L (l) = µF− τ (0(l))·eδχF− τ (0(l)) ≥ s, µF− L (m)·eδχF− L (m) = µF− τ (0(m))·eδχF− τ (0(m)) ≥ s. Thus, µF− L ((l♦1m)) · eδχF− L ((l♦1m)) = µF− τ ((0(l) >1 0(m))) · eδχF− τ ((0(l)>10(m))) ≥ min{µF− L (l) · eδχF− L (l), µF− L (m) · eδχF− L (m)} ≥ s,for all l,m ∈ B1. Similarly other operations,L(t,s) is a SBS of CBIFSBS L of B1. The mapping 0 : B1 → B2 be any homomorphism. We have 0((l♦1m)) = 0(l) >1 0(m),0((l♦2m)) = 0(l) >2 0(m) and 0((l♦3m)) = 0(l) >3 0(m) for all l,m ∈ B1. Let τ = 0(L),τ is a CBIFSBS of B2. By Theo- rem 4.3,L is a CBIFSBS of B1. Let 0(L(t,s)) be a SBS of τ . Suppose that 0(l),0(m) ∈ 0(L(t,s)). Now,0((l♦1m)),0((l♦2m)) and 0((l♦3m)) ∈ 0(L(t,s)). Now,µT + L (l)·eδχT + L (l) = µT + τ (0(l))·eδχT + τ (0(l)) ≥ t, µT + L (m)·eδχT + L (m) = µT + τ (0(m))·eδχT + τ (0(m)) ≥ t. Thus, µT + L ((l♦1m)) · eδχT + L ((l♦1m)) ≥ min{µT + L (l) · eδχT + L (l), µT + L (m) · eδχT + L (m)} ≥ t. Now, µF+ L (l)·eδχF+ L (l) = µF+ τ (0(l))·eδχF+ τ (0(l)) ≤ s, µF+ L (m)·eδχF+ L (m) = µF+ τ (0(m))·eδχF+ τ (0(m)) ≤ s. Thus, µF+ L ((l♦1m)) · eδχF+ L ((l♦1m)) = µF+ τ ((0(l)>1 0(m))) · eδχF+ τ ((0(l)>10(m))) ≤ s, for all l,m ∈ B1. Similarly other operations, L(t,s) is a SBS of CBIFSBS L of B1. Acknowledgment. This research was supported by the University of Phayao and the Thailand Science Re- search and Innovation Fund (Fundamental Fund 2025). 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