Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) Complex cubic intuitionistic fuzzy set applied to subbisemirings of bisemirings using homomorphism Raed Hatamleh1,∗, Abdallah Al-Husban2,3, N.Sundarakannan4, M. S. Malchijah raj5 1Department of Mathematics, Faculty of Science and Information Technology, Jadara, University, P.O. Box 733, Irbid 21110, Jordan. 2Department of Mathematics, Faculty of Science and Technology, Irbid, National University, P.O. Box: 2600 Irbid, Jordan. 3Jadara Research Center, Jadara University, Irbid 21110, Jordan. 4Department of Mathematics, SRM Valliammai Engineering College, Kattankulathur, Chennai-603203, India. 5Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai-602105, India. E-mails:1raed@jadara.edu.jo, 3dralhosban@inu.edu.jo, 4sundarakannann.maths@srmvalliammai.ac.in, 5malchijahraj@gmail.com ∗Corresponding author: Abdallah Al-Husban. Received: 24-04-2024 Revised: 26-08-2024 Accepted: 05-09-2024. Abstract We develop and analyze the concept of complex cubic intuitionistic fuzzy subbisemiring (ComCIFSBS). We investigate ComCIFSBS its characteristics and homomorphic proper- ties. We suggest the ComCIFSBS level sets of bisemiring. A cubic complex intuitionistic fuzzy set subset Ξ of bisemiring B, if and only if each non-empty level set R(t,s), where R = (µ̂Z · ei2πβ̂Z , ν̂Z · ei2πγ̂Z , µZ · ei2πβZ , νZ · ei2πγZ ) is a ComCIFSBS of B. Let υ be a ComCIFSBS bisemiring B. If Υ is a ComCIFSBS of B ×B, then Z is a ComCIFSBS of bisemiring B. Let Z denote the strongest complex intuitionistic fuzzy relation bisemiring B. It is proved that all ComCIFSBSs have homomorphic images as well as homomorphic pre-images. Examples are presented to demonstrate how our findings are applied. Keywords: CCNSBS, CCNNSBS, SBS, homomorphism. 1 Introduction Fuzzy set (FS) theory was developed by Zadeh1 and is most effective in managing ambiguity and uncertainty. If an element in an FS has a single value inside the interval, it is regarded as a member. The degree of non-membership may not always equal one minus the degree of membership, though, as resistance can occur in real-life situations. An increasing num- ber of hybrid fuzzy models are being developed as FS theory develops significantly. The uncertainties have led to the development of a number of uncertain theories, such as FS,1 intuitionistic FS (IFS),2 Pythagorean FS (PFS),3 and spherical FS (SFS).4 MG sets, or sets with grades between 0 and 1, make up an FS. Despite an assertion made by Atanassov2 that non-membership grades (NMG) can only have a value of 1, IFS is categorized as MG. The https://internationalpubls.com 418 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) total of MGs and NMGs may occasionally exceed 1 throughout a decision-making process. Yager3 used PFS logic to develop the generalized MG and NMG logic, which has a value not exceeding 1 and is determined by the square of the MGs and NMGs. As the neutral state is neither positive nor negative, these theories are unable to describe it. Cuong5 et al. communicated to the picture FS used three grading points: positive, neutral, and negative. The sum of these grades could not be greater than 1. It also outperforms PFS and IFS in several situations. It addresses the truth, indeterminacy, and falsity of FS and IFS and is an autonomous generalization of three models. To handle conflicting and unclear data, Smarandache6 developed the neutrosophic set (NS). The degree to which a proposition is true, ambiguous, or false is established using this logic. Ramot et al. introduce the idea of the complex fuzzy set (CFS)..7 The membership functions of CFS’s deals can have a very broad range of values. While the unit circle of a fuzzy membership function remains fixed, the unit circle of the complex plane is expanded to [0, 1]. Rather than extending exclusively to [0, 1], the membership function xX(x) of the CFS X extends to the unit circle in the complex plane. Hence, xX(x) is a complex-valued function that, for any element x in the discourse universe, provides a grade of membership of the type ηX(x) · ei2πX(x), where i = √ −1. The two real-valued variables, ηX(x) and 2πX(x), where ηX(x), 2πX(x) ∈ [0, 1], define the value of xX(x). Golan8 established the concept of semiring logic and its applications. Hussian et al.9 talked about the concept and use of bisemirings. Lee10 addresses bipolar-valued FSs and associated techniques. Fuzzy semir- ings were studied by Ahsan et al.11 Sen et al.12 introduced the concept of bisemirings. A fuzzy normal subbisemiring of bisemiring that is intuitionistic was presented by Palaniku- mar et al.13 Bisemiring was first proposed by Palanikumar et al.14 utilizing bipolar-valued neutrosophic normal sets. Novel concepts including the fuzzy extension set, neutrosopic set, and specific fuzzy set have been the subject of numerous recent writings15.21 Recently al-husban et al. discussed the new structures such neutrosophic and its applications22.32 We will look at a few aspects of SBS and ComCIFSBS concepts and make some inferences. The article is divided into the following five sections. In Section 1, semirings and SBS are introduced. Section 2 contains information on semiring and SBS preparation. ComCIFSBS properties are enumerated in Section 3. It is recommended to use numerical examples to assess ComCIFSBS. 2 Preliminaries Definition 2.1. An algebraic structure (B,], ,�) is a bisemiring,if (B,], ) and (B, ,�) are semirings, ie.,(B,]), (B, ) and (B,�) are semigroups and 1. κv (κζ ] κη) = (κv κζ) ] (κv κη), 2. (κζ ] κη) κv = (κζ κv) ] (κη κv), 3. κv � (κζ κη) = (κv � κζ) (κv � κη), 4. (κζ κη)� κv = (κζ � κv) (κη � κv), ∀ κv,κζ ,κη ∈ B. Definition 2.2. A non-empty subset Ξ of a bisemiring (B,], ,�) is a subbisemiring if zv ] zζ ∈ Ξ, zv zζ ∈ Ξ and zv � zζ ∈ Ξ for all zv, zζ ∈ Ξ. https://internationalpubls.com 419 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) Definition 2.3. Let (B,+, ·) be semiring. A fuzzy subset Ξ of B is said to be fuzzy sub- bisemiring if (i) πΞ(u1 + u2) ≥ min{πΞ(u1), πΞ(u2)}, (ii) πΞ(u1 · u2) ≥ min{πΞ(u1), πΞ(u2)} for all u1, u2 ∈ B. Definition 2.4. An intuitionistic fuzzy set Ξ in non-empty set U is defined, the form Ξ = {< u, πΞ(u), ωΞ(u)|u ∈ U >} where πΞ : U → [0, 1](define the degree of mem- bership) and ωΞ : U → [0, 1] (degree of non-membership) for every u ∈ U satisfying 0 ≤ πΞ(u) + ωΞ(u) ≤ 1. Definition 2.5. A NS v in the universe U is v = {x, uv(x), uIv(x), uv(x)|x ∈ U},where uv(x),uIv(x) uv(x) represents the TD,ID and FD of v respectively. Consider the mapping uv : U → [0, 1], uIv : U → [0, 1], uv : U → [0, 1] and 0 � uv(x) + uIv(x) + uv(x) � 3. Definition 2.6. Let ∂1 = 〈uT∂1 , uI∂1 , uF∂1 〉,∂2 = 〈uT∂2 , uI∂2 , uF∂2 〉 and ∂3 = 〈uT∂3 , uI∂3 , uF∂3 〉 be the three neutrosophic numbers over U . Then 1. ∂2 t ∂3 = 〈 max(uT∂2 , uT∂3 ),min(uI∂2 , uI∂3 ),min(uF∂2 , uF∂3 ) 〉 , 2. ∂2 u ∂3 = 〈 min(uT∂2 , uT∂3 ),max(uI∂2 , uI∂3 ),max(uF∂2 , uF∂3 ) 〉 , 3. ∂2 � ∂3 iff uT∂2 � uT∂3 and uI∂2 � uI∂3 and uF∂2 � uF∂3 , 4. ∂2 = ∂3 iff uT∂2 = uT∂3 and uI∂2 = uI∂3 and uF∂2 = uF∂3 . Definition 2.7. For any NS ∂ = {x, χTv (x), χIv(x), χFv (x)} of U . Then (t, s)-cut is defined as {x ∈ U |χTv (x) � t, χIv(x) � s, χFv (x) � s}. Definition 2.8. Let V and Y be two NSs of B. Then Cartesian product of V and Y is de- fined as V × Y = {χTV×Y (χ, ∂), χIV×Y (χ, ∂), χFV×Y (χ, ∂)| for all χ, ∂ ∈ B}, where χTV×Y (χ, ∂) = min{χTV (x), χTY (∂)}, χIV×Y (χ, ∂) = χIV (x)+χIY (∂) 2 , χFV×Y (χ, ∂) = max{χFV (x), χFY (∂)}. Definition 2.9. A fuzzy subset v of a bisemiring (B, ‡1, ‡2, ‡3) is represents a fuzzy sub- bisemiring of B if χv(χ‡1ε) � min{χv(x), χv(ε)}, χv(χ‡2ε) � min{χv(x), χv(ε)}, χv(χ‡3 ε) � min{χv(x), χv(ε)},for all χ, ε ∈ B. 3 Complex cubic intuitionistic fuzzy subbisemiring Here B denotes bisemiring unless other stated. Definition 3.1. The complex cubic IFS (ComCIFS) Z in universal set a, Z = {~, µ̂Z(~) · ei2πβ̂Z(~), ν̂Z(~) · ei2πγ̂ (~) Z , µZ(~) · ei2πβZ(~), νZ(~) · ei2πγ (~) Z : ~ ∈ a}, where µ̂Z(~) = [µLZ , µ U Z ], ν̂Z(~) = [νLZ , ν U Z ] and µ, ν : a → D[0, 1], also µ, ν : a → [0, 1] represents the truth degree and false degree respectively. For ComCIF number Z = {~, µ̂Z(~) · ei2πβ̂ (~) Z , ν̂Z(~) · ei2πγ̂ (~) Z , µZ(~) · ei2πβ (~) Z , νZ(~) · ei2πγ (~) Z : ~ ∈ a}. https://internationalpubls.com 420 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) Definition 3.2. Let Z = { ~, µ̂Z(~) · ei2πβ̂ (~) Z , ν̂Z(~) · ei2πγ̂ (~) Z , µZ(~) · ei2πβ (~) Z , νZ(~) · ei2πγ (~) Z } and k = { ~, µ̂k(~)·ei2πβ̂ (~) k , ν̂k(~)·ei2πγ̂ (~) k , µk(~)·ei2πβ (~) k , νk(~)·ei2πγ (~) k } be two ComCIFNs of a. Then we define the intersection and union operation is defined as (i) Zuk = {( ~,min{µ̂Z(~) ·ei2πβ̂ (~) Z , µ̂k(~) ·ei2πβ̂ (~) k },max{ν̂Z(~) ·ei2πγ̂ (~) Z , ν̂k(~) ·ei2πγ̂ (~) k }, min{µZ(~) · ei2πβ (~) Z , µk(~) · ei2πβ (~) k },max{νZ(~) · ei2πγ (~) Z , νk(~) · ei2πγ (~) k } )∣∣∣~ ∈ a } . (ii) Ztk = {( ~,max{µ̂Z(~)·ei2πβ̂ (~) Z , µ̂k(~) ·ei2πβ̂ (~) k },min{ν̂Z(~)·ei2πγ̂ (~) Z , ν̂k(~) ·ei2πγ̂ (~) k }, max{µZ(~) · ei2πβ (~) Z , µk(~) · ei2πβ (~) k },min{νZ(~) · ei2πγ (~) Z , νk(~) · ei2πγ (~) k } )∣∣∣~ ∈ a } . Definition 3.3. For any ComCIFZ = { ~, µ̂Z(~)·ei2πβ̂ (~) Z , ν̂Z(~)·ei2πγ̂ (~) Z , µZ(~)·ei2πβ (~) Z , νZ(~)· ei2πγ (~) Z } of a universal set a. Then (t, s)-cut is defined as { ~ ∈ a|µ̂Z(~) · ei2πβ̂ (~) Z � t, ν̂Z(~) · ei2πγ̂ (~) Z � s, µZ(~) · ei2πβ (~) Z � t, νZ(~) · ei2πγ (~) Z � s } . Definition 3.4. The Cartesian product of Z and k is defined as Z × k = { µ̂Z×k((~,ð))·ei2π ̂ β ((~,ð)) Z×k , ν̂Z×k(~,ð)·ei2π ̂ γ ((~,ð)) Z×k , µZ×k((~,ð))·ei2πβ ((~,ð)) Z×k , νZ×k(~,ð)· ei2πγ ((~,ð)) Z×k | for all ~,ð ∈ S } , where Z and k be the ComCIF of a, where  µ̂Z×k((~,ð)) · ei2π ̂ β ((~,ð)) Z×k = min { µ̂Z(~) · ei2πβ̂ (~) Z , µ̂k(ð) · ei2πβ̂ (ð) k } ν̂Z×k((~,ð)) · ei2π ̂ γ ((~,ð)) Z×k = max { ν̂Z(~) · ei2πγ̂ (~) Z , ν̂k(ð) · ei2πγ̂ (ð) k }  µZ×k((~,ð)) · ei2πβ ((~,ð)) Z×k = min { µZ(~) · ei2πβ (~) Z , µk(ð) · ei2πβ (ð) k } νZ×k((~,ð)) · ei2πγ ((~,ð)) Z×k = max { νZ(~) · ei2πγ (~) Z , νk(ð) · ei2πγ (ð) k }  Definition 3.5. For any ComCIF Z of B is said to be a ComCIFSBS of B if µ̂Z((~�1 ð)) · ei2πβ̂Z((~�1ð)) � min{µ̂Z(~) · ei2πβ̂Z((~), µ̂Z(ð) · ei2πβ̂Z((ð)} µ̂Z((~�2 ð)) · ei2πβ̂Z((~�2ð)) � min{µ̂Z(~) · ei2πβ̂Z((~), µ̂Z(ð) · ei2πβ̂Z((ð)} µ̂Z((~�3 ð)) · ei2πβ̂Z((~�3ð)) � min{µ̂Z(~) · ei2πβ̂Z((~), µ̂Z(ð) · ei2πβ̂Z((ð)}   ν̂Z((~�1 ð)) · ei2πγ̂Z((~�1ð)) � max{ν̂Z(~) · ei2πγ̂Z((~), ν̂Z(ð) · ei2πγ̂Z((ð)} ν̂Z((~�2 ð)) · ei2πγ̂Z((~�2ð)) � max{ν̂Z(~) · ei2πγ̂Z((~), ν̂Z(ð) · ei2πγ̂Z((ð)} ν̂Z((~�3 ð)) · ei2πγ̂Z((~�3ð)) � max{ν̂Z(~) · ei2πγ̂Z((~), ν̂Z(ð) · ei2πγ̂Z((ð)}   µZ((~�1 ð)) · ei2πβZ((~�1ð)) � min{µZ(~) · ei2πβZ((~), µZ(ð) · ei2πβZ((ð)} µZ((~�2 ð)) · ei2πβZ((~�2ð)) � min{µZ(~) · ei2πβZ((~), µZ(ð) · ei2πβZ((ð)} µZ((~�3 ð)) · ei2πβZ((~�3ð)) � min{µZ(~) · ei2πβZ((~), µZ(ð) · ei2πβZ((ð)}  https://internationalpubls.com 421 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025)  νZ((~�1 ð)) · ei2πγZ((~�1ð)) � max{νZ(~) · ei2πγZ((~), νZ(ð) · ei2πγZ((ð)} νZ((~�2 ð)) · ei2πγZ((~�2ð)) � max{νZ(~) · ei2πγZ((~), νZ(ð) · ei2πγZ((ð)} νZ((~�3 ð)) · ei2πγZ((~�3ð)) � max{νZ(~) · ei2πγZ((~), νZ(ð) · ei2πγZ((ð)}  for all ~,ð ∈ B. Example 3.6. Consider the bisemiring B = {x1, x2, x3, x4} with the Cayley table: �1 x1 x2 x3 x4 x1 x1 x1 x1 x1 x2 x1 x2 x1 x2 x3 x1 x1 x3 x3 x4 x1 x2 x3 x4 �2 x1 x2 x3 x4 x1 x1 x2 x3 x4 x2 x2 x2 x4 x4 x3 x3 x4 x3 x4 x4 x4 x4 x4 x4 �3 x1 x2 x3 x4 x1 x1 x1 x1 x1 x2 x1 x2 x3 x4 x3 x4 x4 x4 x4 x4 x4 x4 x4 x4 (w) = x1 (w) = x2 (µ̂Z , β̂Z)(w) [0.45ei2π(0.5), 0.5ei2π(0.55)] [0.35ei2π(0.4), 0.4ei2π(0.45)] (ν̂Z , γ̂Z)(w) [0.4ei2π(0.3), 0.45ei2π(0.35)] [0.45ei2π(0.45), 0.5ei2π(0.55)] (w) = x3 (w) = x4 (µ̂Z , β̂Z)(w) [0.25ei2π(0.3), 0.3ei2π(0.35)] [0.3ei2π(0.35), 0.35ei2π(0.4)] (ν̂Z , γ̂Z)(w) [0.6ei2π(0.55), 0.65ei2π(0.65)] [0.55ei2π(0.5), 0.6ei2π(0.6)] (w) = x1 (w) = x2 (µZ , βZ)(w) 0.5ei2π(0.55) 0.45ei2π(0.5) (νZ , γZ)(w) 0.45ei2π(0.4) 0.5ei2π(0.45) (w) = x3 (w) = x4 (µZ , βZ)(w) 0.3ei2π(0.45) 0.35ei2π(0.4) (νZ , γZ)(w) 0.6ei2π(0.55) 0.55ei2π(0.5) Hence, Z is a ComCIFSBS of B. Theorem 3.7. The intersection of a every ComCIFSBSs is again a ComCIFSBS of B. Proof. Let {Υ̂i : i ∈ I} be the family of ComCIFSBSs of B and Z = ⋂ i∈I Υ̂i. Let ~,ð ∈ B. Now, µ̂Z((~�1 ð)) · ei2πβ̂Z((~�1ð)) = inf i∈I µ̂Υi((~�1 ð)) · ei2πβ̂Z((~�1ð)) � inf i∈I min{µ̂Υi(~) · ei2πβ̂Υi (~), µ̂Υi(ð) · ei2πβ̂Υi (ð)} = min { inf i∈I µ̂Υi(~) · ei2πβ̂Υi (~), inf i∈I µ̂Υi(ð) · ei2πβ̂Υi (ð) } = min{µ̂Z(~) · ei2πβ̂Z(~), µ̂Z(ð) · ei2πβ̂Z(ð)} https://internationalpubls.com 422 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) Similarly, µ̂Z((~�2 ð)) · ei2πβ̂Z((~�2ð)) � min{µ̂Z(~) · ei2πβ̂Z(~), µ̂Z(ð) · ei2πβ̂Z(ð)}, µ̂Z((~�3 ð)) · ei2πβ̂Z((~�3ð)) � min{µ̂Z(~) · ei2πβ̂Z(~), µ̂Z(ð) · ei2πβ̂Z(ð)}. Now, ν̂Z((~�1 ð)) · ei2πγ̂Z((~�1ð)) = sup i∈I ν̂Υi((~�1 ð)) · ei2πγ̂Z((~�1ð)) � sup i∈I max{ν̂Υi(~) · ei2πγ̂Υi (~), ν̂Υi(ð) · ei2πγ̂Υi (ð)} = max { sup i∈I ν̂Υi(~) · ei2πγ̂Υi (~), sup i∈I ν̂Υi(ð) · ei2πγ̂Υi (ð) } = max{ν̂Z(~) · ei2πγ̂Z(~), ν̂Z(ð) · ei2πγ̂Z(ð)} Similarly, ν̂Z((~�2 ð)) · ei2πγ̂Z((~�2ð)) � max{ν̂Z(~) · ei2πγ̂Z(~), ν̂Z(ð) · ei2πγ̂Z(ð)} and ν̂Z((~�3 ð)) · ei2πγ̂Z((~�3ð)) � max{ν̂Z(~) · ei2πγ̂Z(~), ν̂Z(ð) · ei2πγ̂Z(ð)}. Let {Υi : i ∈ I} be the family of ComCIFSBSs of B and Z = ⋂ i∈I Υi. Let ~,ð ∈ B. Now, µZ((~�1 ð)) · ei2πβZ((~�1ð)) = inf i∈I µΥi((~�1 ð)) · ei2πβZ((~�1ð)) � inf i∈I min{µΥi(~) · ei2πβΥi (~), µΥi(ð) · ei2πβΥi (ð)} = min { inf i∈I µΥi(~) · ei2πβΥi (~), inf i∈I µΥi(ð) · ei2πβΥi (ð) } = min{µZ(~) · ei2πβZ(~), µZ(ð) · ei2πβZ(ð)} Similarly, µZ((~�2 ð)) · ei2πβZ((~�2ð)) � min{µZ(~) · ei2πβZ(~), µZ(ð) · ei2πβZ(ð)}, µZ((~�3 ð)) · ei2πβZ((~�3ð)) � min{µZ(~) · ei2πβZ(~), µZ(ð) · ei2πβZ(ð)}. Now, νZ((~�1 ð)) · ei2πγZ((~�1ð)) = sup i∈I νΥi((~�1 ð)) · ei2πγZ((~�1ð)) � sup i∈I max{νΥi(~) · ei2πγΥi (~), νΥi(ð) · ei2πγΥi (ð)} = max { sup i∈I νΥi(~) · ei2πγΥi (~), sup i∈I νΥi(ð) · ei2πγΥi (ð) } = max{νZ(~) · ei2πγZ(~), νZ(ð) · ei2πγZ(ð)} Similarly, νZ((~�2 ð)) · ei2πγZ((~�2ð)) � max{νZ(~) · ei2πγZ(~), νZ(ð) · ei2πγZ(ð)} and νZ((~�3 ð)) · ei2πγZ((~�3ð)) � max{νZ(~) · ei2πγZ(~), νZ(ð) · ei2πγZ(ð)}. Thus,Z is a ComCIFSBS of B. Theorem 3.8. If Z and k be the ComCIFSBSs of B1 and B2 respectively, then Ẑ × kis a ComCIFSBS of B1 ×B2. https://internationalpubls.com 423 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) Proof. Let ~1, ~2 ∈ B1 and ð1,ð2 ∈ B2. Then (~1,ð1) and (~2,ð2) are in B1 ×B2. Now µ̂Z×k[((~1,ð1)�1 (~2,ð2))] · ei2πβ̂Z×k[((~1,ð1)�1(~2,ð2))] = µ̂Z×k((~1 �1 ~2,ð1 �1 ð2)) · ei2πβ̂Z×k((~1�1~2,ð1�1ð2)) = min{µ̂Z((~1 �1 ~2)) · ei2πβ̂Z((~1�1~2)), µ̂k((ð1 �1 ð2)) · ei2πβ̂k((ð1�1ð2))} � min{min{µ̂Z(~1) · ei2πβ̂Z(~1), µ̂Z(~2) · ei2πβ̂Z(~2)},min{µ̂k(ð1) · ei2πβ̂Z(ð1), µ̂k(ð2) · ei2πβ̂Z(ð2)}} = min{min{µ̂Z(~1) · ei2πβ̂Z(~1), µ̂k(ð1) · ei2πβ̂Z(ð1)},min{µ̂Z(~2) · ei2πβ̂Z(~2), µ̂k(ð2) · ei2πβ̂Z(ð2)}} = min{µ̂Z×k((~1,ð1)) · ei2πβ̂Z×k((~1,ð1)), µ̂Z×k((~2, ð2)) · ei2πβ̂Z×k((~2,ð2))} Also µ̂Z×k[((~1,ð1)�2 (~2,ð2))] · ei2πβ̂Z×k[((~1,ð1)�2(~2,ð2))] � min{µ̂Z×k((~1, ð1)) · ei2πβ̂Z×k((~1,ð1)), µ̂Z×k((~2,ð2))ei2πβ̂Z×k((~2,ð2))} and µ̂Z×k[((~1,ð1)�3 (~2,ð2))] · ei2πβ̂Z×k[((~1,ð1)�3(~2,ð2))] � min{µ̂Z×k((~1, ð1)) · ei2πβ̂Z×k((~1,ð1)), µ̂Z×k((~2,ð2)) · ei2πβ̂Z×k((~2,ð2))}. Now, ν̂Z×k[((~1,ð1)�1 (~2,ð2))] · ei2πγ̂Z×k[((~1,ð1)�1(~2,ð2))] = ν̂Z×k((~1 �1 ~2,ð1 �1 ð2)) · ei2πγ̂Z×k((~1�1~2,ð1�1ð2)) = max{ν̂Z((~1 �1 ~2)) · ei2πγ̂Z×k((~1�1~2)), ν̂k((ð1 �1 ð2)) · ei2πγ̂Z×k((ð1�1ð2))} � max{max{ν̂Z(~1) · ei2πγ̂Z(~1), ν̂Z(~2) · ei2πγ̂Z(~2)},max{ν̂k(ð1) · ei2πγ̂k(ð1), ν̂k(ð2) · ei2πγ̂k(ð2)}} = max{max{ν̂Z(~1) · ei2πγ̂Z(~1), ν̂k(ð1) · ei2πγ̂k(ð1)},max{ν̂Z(~2) · ei2πγ̂Z(~2), ν̂k(ð2) · ei2πγ̂k(ð2)}} = max{ν̂Z×k((~1,ð1)) · ei2πβ̂Z×k((~1,ð1)), ν̂Z×k((~2,ð2)) · ei2πβ̂Z×k((~2,ð2))} Also ν̂Z×k[((~1,ð1)�2 (~2, ð2))] · ei2πβ̂Z×k[((~1,ð1)�2(~2,ð2))] � max{ν̂Z×k((~1, ð1)) · ei2πβ̂Z×k((~1,ð1)), ν̂Z×k((~2,ð2)) · ei2πβ̂Z×k((~2,ð2))}, ν̂Z×k[((~1,ð1)�3 (~2,ð2))] · ei2πβ̂Z×k[((~1,ð1)�3(~2,ð2))] � max{ν̂Z×k((~1, ð1)) · ei2πβ̂Z×k((~1,ð1)), ν̂Z×k((~2,ð2)) · ei2πβ̂Z×k((~2,ð2))}. Let ~1, ~2 ∈ B1 and ð1, ð2 ∈ B2. Then (~1,ð1) and (~2,ð2) are in B1 ×B2. Now µZ×k[((~1,ð1)�1 (~2,ð2))] · ei2πβZ×k[((~1,ð1)�1(~2,ð2))] = µZ×k((~1 �1 ~2,ð1 �1 ð2)) · ei2πβZ×k((~1�1~2,ð1�1ð2)) = min{µZ((~1 �1 ~2)) · ei2πβZ((~1�1~2)), µk((ð1 �1 ð2)) · ei2πβk((ð1�1ð2))} � min{min{µZ(~1) · ei2πβZ(~1), µZ(~2) · ei2πβZ(~2)},min{µk(ð1) · ei2πβZ(ð1), µk(ð2) · ei2πβZ(ð2)}} = min{min{µZ(~1) · ei2πβZ(~1), µk(ð1) · ei2πβZ(ð1)},min{µZ(~2) · ei2πβZ(~2), µk(ð2) · ei2πβZ(ð2)}} = min{µZ×k((~1,ð1)) · ei2πβZ×k((~1,ð1)), µZ×k((~2,ð2)) · ei2πβZ×k((~2,ð2))} Also µZ×k[((~1,ð1)�2 (~2, ð2))] · ei2πβZ×k[((~1,ð1)�2(~2,ð2))] � min{µZ×k((~1,ð1)) · ei2πβZ×k((~1,ð1)), µZ×k((~2,ð2))ei2πβZ×k((~2,ð2))} https://internationalpubls.com 424 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) and µZ×k[((~1, ð1)�3 (~2,ð2))] · ei2πβZ×k[((~1,ð1)�3(~2,ð2))] � min{µZ×k((~1,ð1)) · ei2πβZ×k((~1,ð1)), µZ×k((~2,ð2)) · ei2πβZ×k((~2,ð2))}. Now, νZ×k[((~1,ð1)�1 (~2,ð2))] · ei2πγZ×k[((~1,ð1)�1(~2,ð2))] = νZ×k((~1 �1 ~2,ð1 �1 ð2)) · ei2πγZ×k((~1�1~2,ð1�1ð2)) = max{νZ((~1 �1 ~2)) · ei2πγZ×k((~1�1~2)), νk((ð1 �1 ð2)) · ei2πγZ×k((ð1�1ð2))} � max{max{νZ(~1) · ei2πγZ(~1), νZ(~2) · ei2πγZ(~2)},max{νk(ð1) · ei2πγk(ð1), νk(ð2) · ei2πγk(ð2)}} = max{max{νZ(~1) · ei2πγZ(~1), νk(ð1) · ei2πγk(ð1)},max{νZ(~2) · ei2πγZ(~2), νk(ð2) · ei2πγk(ð2)}} = max{νZ×k((~1,ð1)) · ei2πβZ×k((~1,ð1)), νZ×k((~2, ð2)) · ei2πβZ×k((~2,ð2))} Also νZ×k[((~1,ð1)�2 (~2,ð2))] · ei2πβZ×k[((~1,ð1)�2(~2,ð2))] � max{νZ×k((~1,ð1)) · ei2πβZ×k((~1,ð1)), νZ×k((~2,ð2)) · ei2πβZ×k((~2,ð2))}, νZ×k[((~1,ð1)�3 (~2,ð2))] · ei2πβZ×k[((~1,ð1)�3(~2,ð2))] � max{νZ×k((~1,ð1)) · ei2πβZ×k((~1,ð1)), νZ×k((~2,ð2)) · ei2πβZ×k((~2,ð2))}. Thus, Ẑ × k is a ComCIFSBS of B. Corollary 3.9. If Z1, Z2, ..., Zn be the finite collection of ComCIFSBSs of B1,B2, ...,Bn respectively. Then Z1 × Z2 × ...× Zn is a ComCIFSBS of B1 ×B2 × ...×Bn. Definition 3.10. Let Z ⊆ B,the strongest ComCN relation on B is{ µ̂Υ((~,ð)) · ei2πβ̂Υ((~,ð)) = min{µ̂Z(~) · ei2πβ̂Z(~), µ̂Z(ð) · ei2πβ̂Z(ð)} ν̂Υ((~,ð)) · ei2πγ̂Υ((~,ð)) = max{ν̂Z(~) · ei2πγ̂Z(~), ν̂Z(ð) · ei2πγ̂Z(ð)} } Theorem 3.11. Let Z be a ComCIFSBS of B and Υ be a strongest complex cubic neutro- sophic relation of B. Then Z is a ComCIFSBS of B ×B if and only if Υ is a ComCIFSBS of B ×B. Proof. Suppose Z is a ComCIFSBS of B × B and Υ̂ be the strongest complex cubic neutrosophic relation of B. For any ~ = (~1, ~2),ð = (ð1,ð2) ∈ S ×B. Now, µ̂Υ((~�1 ð)) · ei2πβ̂Υ((~�1ð)) = µ̂Υ[((((~1, ~2))�1 ((ð1,ð2))] · ei2πβ̂Υ[((((~1,~2))�1((ð1,ð2))] = µ̂Υ(~1 �1 ð1, ~2 �1 ð2) · ei2πβ̂Υ(~1�1ð1,~2�1ð2) = min{µ̂Z((~1 �1 ð1)) · ei2πβ̂Z((~1�1ð1)), µ̂Z((~2 �1 ð2)) · ei2πβ̂Z((~2�1ð2))} � min{min{µ̂Z(~1) · ei2πβ̂Z(~1), µ̂Z(ð1) · ei2πβ̂Z(ð1)}, min{µ̂Z(~2) · ei2πβ̂Z(~2), µ̂Z(ð2) · ei2πβ̂Z(ð2)}} = min{min{µ̂Z(~1) · ei2πβ̂Z(~1), µ̂Z(~2) · ei2πβ̂Z(~2)}, min{µ̂Z(ð1) · ei2πβ̂Z(ð1), µ̂Z(ð2) · ei2πβ̂Z(ð2)}} = min{µ̂Υ((~1, ~2)) · ei2πβ̂Υ((~1,~2)), µ̂Υ((ð1,ð2)) · ei2πβ̂Υ((ð1,ð2))} = min{µ̂Υ(~) · ei2πβ̂Υ(~), µ̂Υ(ð) · ei2πβ̂Υ(ð)} https://internationalpubls.com 425 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) Also µ̂Υ((~�2 ð)) · ei2πβ̂Υ((~�2ð)) � min{µ̂Υ(~) · ei2πβ̂Υ(~), µ̂Υ(ð) · ei2πβ̂Υ(ð)}, µ̂Υ((~�3 ð)) · ei2πβ̂Υ((~�3ð)) � min{µ̂Υ(~) · ei2πβ̂Υ(~), µ̂Υ(ð) · ei2πβ̂Υ(ð)}. Similarly,ν̂Υ((~�1 ð)) · ei2πγ̂Υ((~�1ð)) � max{ν̂Υ(~) · ei2πγ̂Υ(~)·, ν̂Υ(ð) · ei2πγ̂Υ(ð)}, ν̂Υ((~�2 ð)) · ei2πγ̂Υ((~�2ð)) � max{ν̂Υ(~) · ei2πγ̂Υ(~), ν̂Υ(ð) · ei2πγ̂Υ(ð)} and ν̂Υ((~�3 ð)) · ei2πγ̂Υ((~�3ð)) � max{ν̂Υ(~) · ei2πγ̂Υ(~), ν̂Υ(ð) · ei2πÎIΥ(ð}. For any ~ = (~1, ~2),ð = (ð1,ð2) ∈ S ×B. Now, µΥ((~�1 ð)) · ei2πβΥ((~�1ð)) = µΥ[((((~1, ~2))�1 ((ð1,ð2))] · ei2πβΥ[((((~1,~2))�1((ð1,ð2))] = µΥ(~1 �1 ð1, ~2 �1 ð2) · ei2πβΥ(~1�1ð1,~2�1ð2) = min{µZ((~1 �1 ð1)) · ei2πβZ((~1�1ð1)), µZ((~2 �1 ð2)) · ei2πβZ((~2�1ð2))} � min{min{µZ(~1) · ei2πβZ(~1), µZ(ð1) · ei2πβZ(ð1)}, min{µZ(~2) · ei2πβZ(~2), µZ(ð2) · ei2πβZ(ð2)}} = min{min{µZ(~1) · ei2πβZ(~1), µZ(~2) · ei2πβZ(~2)}, min{µZ(ð1) · ei2πβZ(ð1), µZ(ð2) · ei2πβZ(ð2)}} = min{µΥ((~1, ~2)) · ei2πβΥ((~1,~2)), µΥ((ð1,ð2)) · ei2πβΥ((ð1,ð2))} = min{µΥ(~) · ei2πβΥ(~), µΥ(ð) · ei2πβΥ(ð)} Also µΥ((~�2 ð)) · ei2πβΥ((~�2ð)) � min{µΥ(~) · ei2πβΥ(~), µΥ(ð) · ei2πβΥ(ð)}, µΥ((~�3 ð)) · ei2πβΥ((~�3ð)) � min{µΥ(~) · ei2πβΥ(~), µΥ(ð) · ei2πβΥ(ð)}. Similarly, νΥ((~�1 ð)) · ei2πγΥ((~�1ð)) � max{νΥ(~) · ei2πγΥ(~)·, νΥ(ð) · ei2πγΥ(ð)}, νΥ((~�2 ð)) · ei2πγΥ((~�2ð)) � max{νΥ(~) · ei2πγΥ(~), νΥ(ð) · ei2πγΥ(ð)} and νΥ((~�3 ð)) · ei2πγΥ((~�3ð)) � max{νΥ(~) · ei2πγΥ(~), νΥ(ð) · ei2πIIΥ(ð}. Therefore, Υ is a ComCIFSBS of B ×B. Conversely, suppose that Υ is a ComCIFSBS of B×B. Let ~ = ((~1, ~2)),ð = ((ð1,ð2)) ∈ S ×B. Now, min{µ̂Z((~1 �1 ð1)) · ei2π ̂βZ((~1�1ð1)), µ̂Z((~2 �1 ð2)) · ei2π ̂βZ((~2�1ð2))} = µ̂Υ(~1 �1 ð1, ~2 �1 ð2) · ei2π ̂βΥ(~1�1ð1,~2�1ð2) = µ̂Υ[((~1, ~2))�1 ((ð1,ð2))] · ei2π ̂Iv [((~1,~2))�1((ð1,ð2))] = µ̂Υ(~�1 ð) · ei2π ̂βΥ(~�1ð) � min{µ̂Υ(~) · ei2πβ̂Υ(~), µ̂Υ(ð) · ei2πβ̂Υ(ð)} = min{µ̂Υ((~1, ~2))} · ei2π ̂βΥ((~1,~2))}, µ̂Υ((ð1, ð2)) · ei2πβ̂Υ((ð1,ð2))} = min{min{µ̂Z(~1) · ei2πβ̂Z(~1), µ̂Z(~2) · ei2πβ̂Z(~2)},min{µ̂Z(ð1) · ei2πβ̂Z(ð1), µ̂Z(ð2) · ei2πβ̂Z(ð2)}} If µ̂Z((~1�1ð1))·ei2π ̂βZ((~1�1ð1)) � µ̂Z((~2�1ð2))·ei2π ̂βZ((~2�1ð2)), then µ̂Z(~1)·ei2πβ̂Z(~1) � µ̂Z(~2) · ei2πβ̂Z(~2) and µ̂Z(ð1) · ei2πβ̂Z(ð1) � µ̂Z(ð2) · ei2πβ̂Z(ð2). We get µ̂Z((~1 �1 ð1)) · https://internationalpubls.com 426 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) ei2π ̂βZ((~1�1ð1)) � min{µ̂Z(~1) · ei2πβ̂Z(~1), µ̂Z(ð1) · ei2πβ̂Z(ð1)} for all ~1,ð1 ∈ B,and min{µ̂Z((~1�2ð1)) ·ei2π ̂βZ((~1�2ð1)), µ̂Z((~2�2ð2)) ·ei2π ̂βZ((~2�2ð2))} � min{min{µ̂Z(~1) · ei2πβ̂Z(~1), µ̂Z(~2) · ei2πβ̂Z(~2)},min{µ̂Z(ð1) · ei2πβ̂Z(ð1), µ̂Z(ð2) · ei2πβ̂Z(ð2)}} If µ̂Z((~1 �2 ð1)) · ei2π ̂βZ((~1�2ð1)) � µ̂Z((~2 �2 ð2)) · ei2π ̂βZ((~2�2ð2)),then µ̂Z((~1 �2 ð1)) · ei2π ̂βZ((~1�2ð1)) � min{µ̂Z(~1) · ei2πβ̂Z(~1), µ̂Z(ð1) · ei2πβ̂Z(ð1)}. min{µ̂Z((~1�3ð1)) ·ei2πβ̂Z((~1�3ð1)), µ̂Z((~2�3ð2)) ·ei2πβ̂Z((~2�3ð2))} � min{min{µ̂Z(~1) · ei2πβ̂Z(~1), µ̂Z(~2) · ei2πβ̂Z(~2)},min{µ̂Z(ð1) · ei2πβ̂Z(ð1), µ̂Z(ð2) · ei2πβ̂Z(ð2)}} If µ̂Z((~1 �3 ð1)) · ei2πβ̂Z((~1�3ð1)) � µ̂Z((~2 �3 ð2)) · ei2πβ̂Z((~2�3ð2)),then µ̂Z((~1 �3 ð1)) · ei2πβ̂Z((~1�3ð1)) � min{µ̂Z(~1) · ei2πβ̂Z(~1), µ̂Z(ð1) · ei2πβ̂Z(ð1)}. Similarly to prove that max{ν̂Z((~1�1ð1))·ei2π ̂γZ((~1�1ð1)), ν̂Z((~2�1ð2))·ei2π ̂γZ((~2�1ð2))} � max{max{ν̂Z(~1)·ei2πγ̂Z(~1), ν̂Z(~2)·ei2πγ̂Z(~2)},max{ν̂Z(ð1)·ei2πγ̂Z(ð1), ν̂Z(ð2)·ei2πγ̂Z(ð2)}} If ν̂Z((~1�1ð1)) ·ei2π ̂γZ((~1�1ð1)) � ν̂Z((~2�1ð2)) ·ei2π ̂γZ((~2�1ð2)),then ν̂Z(~1) ·ei2πγ̂Z(~1) � ν̂Z(~2) · ei2πγ̂Z(~2) and ν̂Z(ð1) · ei2πγ̂Z(ð1) � ν̂Z(ð2) · ei2πγ̂Z(ð2). We get ν̂Z((~1 �1 ð1)) · ei2π ̂γZ((~1�1ð1)) � max{ν̂Z(~1) · ei2πγ̂Z(~1), ν̂Z(ð1) · ei2πγ̂Z(ð1)}. max{ν̂Z((~1 �2 ð1)) · ei2π ̂γZ((~1�2ð1)), ν̂Z((~2 �2 ð2)) · ei2π ̂γZ((~2�2ð2))} � max{max{ν̂Z(~1)·ei2πγ̂Z(~1), ν̂Z(~2)·ei2πγ̂Z(~2)},max{ν̂Z(ð1)·ei2πγ̂Z(ð1), ν̂Z(ð2)·ei2πγ̂Z}} If ν̂Z((~1 �2 ð1)) · ei2π ̂γZ((~1�2ð1)) � ν̂Z((~2 �2 ð2)) · ei2π ̂γZ((~2�2ð2)),then ν̂Z((~1 �2 ð1)) · ei2π ̂γZ((~1�2ð1)) � max{ν̂Z(~1) · ei2πγ̂Z(~1), ν̂Z(ð1) · ei2πγ̂Z(ð1)}. max{ν̂Z((~1�3ð1)) ·ei2π ̂γZ((~1�3ð1)), ν̂Z((~2�3ð2)) ·ei2π ̂γZ((~2�3ð2))} � max{max{ν̂Z(~1) · ei2πγ̂Z(~1), ν̂Z(~2) · ei2πγ̂Z(~2)},max{ν̂Z(ð1) · ei2πγ̂Z(ð1), ν̂Z(ð2) · ei2πγ̂Z(ð2)}} If ν̂Z((~1 �3 ð1)) · ei2π ̂βZ((~1�3ð1)) � ν̂Z((~2 �3 ð2)) · ei2π ̂βZ((~2�3ð2)),then ν̂Z((~1 �3 ð1)) · ei2π ̂βZ((~1�3ð1)) � max{ν̂Z(~1) · ei2πγ̂Z(~1), ν̂Z(ð1) · ei2πβ̂Z(ð1)}. Let ~ = ((~1, ~2)),ð = ((ð1,ð2)) ∈ S ×B. Now, min{µZ((~1 �1 ð1)) · ei2πβZ((~1�1ð1)), µZ((~2 �1 ð2)) · ei2πβZ((~2�1ð2))} = µΥ(~1 �1 ð1, ~2 �1 ð2) · ei2πβΥ(~1�1ð1,~2�1ð2) = µΥ[((~1, ~2))�1 ((ð1,ð2))] · ei2πIv [((~1,~2))�1((ð1,ð2))] = µΥ(~�1 ð) · ei2πβΥ(~�1ð) � min{µΥ(~) · ei2πβΥ(~), µΥ(ð) · ei2πβΥ(ð)} = min{µΥ((~1, ~2))} · ei2πβΥ((~1,~2))}, µΥ((ð1,ð2)) · ei2πβΥ((ð1,ð2))} = min{min{µZ(~1) · ei2πβZ(~1), µZ(~2) · ei2πβZ(~2)},min{µZ(ð1) · ei2πβZ(ð1), µZ(ð2) · ei2πβZ(ð2)}} If µZ((~1�1ð1))·ei2πβZ((~1�1ð1)) � µZ((~2�1ð2))·ei2πβZ((~2�1ð2)), then µZ(~1)·ei2πβZ(~1) � µZ(~2) · ei2πβZ(~2) and µZ(ð1) · ei2πβZ(ð1) � µZ(ð2) · ei2πβZ(ð2). We get µZ((~1 �1 ð1)) · ei2πβZ((~1�1ð1)) � min{µZ(~1) · ei2πβZ(~1), µZ(ð1) · ei2πβZ(ð1)} for all ~1,ð1 ∈ B,and min{µZ((~1�2ð1)) ·ei2πβZ((~1�2ð1)), µZ((~2�2ð2)) ·ei2πβZ((~2�2ð2))} � min{min{µZ(~1) · ei2πβZ(~1), µZ(~2) · ei2πβZ(~2)},min{µZ(ð1) · ei2πβZ(ð1), µZ(ð2) · ei2πβZ(ð2)}} If µZ((~1 �2 ð1)) · ei2πβZ((~1�2ð1)) � µZ((~2 �2 ð2)) · ei2πβZ((~2�2ð2)),then µZ((~1 �2 ð1)) · ei2πβZ((~1�2ð1)) � min{µZ(~1) · ei2πβZ(~1), µZ(ð1) · ei2πβZ(ð1)}. https://internationalpubls.com 427 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) min{µZ((~1�3ð1)) ·ei2πβZ((~1�3ð1)), µZ((~2�3ð2)) ·ei2πβZ((~2�3ð2))} � min{min{µZ(~1) · ei2πβZ(~1), µZ(~2) · ei2πβZ(~2)},min{µZ(ð1) · ei2πβZ(ð1), µZ(ð2) · ei2πβZ(ð2)}} If µZ((~1 �3 ð1)) · ei2πβZ((~1�3ð1)) � µZ((~2 �3 ð2)) · ei2πβZ((~2�3ð2)),then µZ((~1 �3 ð1)) · ei2πβZ((~1�3ð1)) � min{µZ(~1) · ei2πβZ(~1), µZ(ð1) · ei2πβZ(ð1)}. Similarly to prove that max{νZ((~1�1ð1)) ·ei2πγZ((~1�1ð1)), νZ((~2�1ð2)) ·ei2πγZ((~2�1ð2))} � max{max{νZ(~1)·ei2πγZ(~1), νZ(~2)·ei2πγZ(~2)},max{νZ(ð1)·ei2πγZ(ð1), νZ(ð2)·ei2πγZ(ð2)}} If νZ((~1�1ð1)) ·ei2πγZ((~1�1ð1)) � νZ((~2�1ð2)) ·ei2πγZ((~2�1ð2)),then νZ(~1) ·ei2πγZ(~1) � νZ(~2) · ei2πγZ(~2) and νZ(ð1) · ei2πγZ(ð1) � νZ(ð2) · ei2πγZ(ð2). We get νZ((~1 �1 ð1)) · ei2πγZ((~1�1ð1)) � max{νZ(~1) · ei2πγZ(~1), νZ(ð1) · ei2πγZ(ð1)}. max{νZ((~1 �2 ð1)) · ei2πγZ((~1�2ð1)), νZ((~2 �2 ð2)) · ei2πγZ((~2�2ð2))} � max{max{νZ(~1)·ei2πγZ(~1), νZ(~2)·ei2πγZ(~2)},max{νZ(ð1)·ei2πγZ(ð1), νZ(ð2)·ei2πγZ}} If νZ((~1 �2 ð1)) · ei2πγZ((~1�2ð1)) � νZ((~2 �2 ð2)) · ei2πγZ((~2�2ð2)),then νZ((~1 �2 ð1)) · ei2πγZ((~1�2ð1)) � max{νZ(~1) · ei2πγZ(~1), νZ(ð1) · ei2πγZ(ð1)}. max{νZ((~1�3ð1)) ·ei2πγZ((~1�3ð1)), νZ((~2�3ð2)) ·ei2πγZ((~2�3ð2))} � max{max{νZ(~1) · ei2πγZ(~1), νZ(~2) · ei2πγZ(~2)},max{νZ(ð1) · ei2πγZ(ð1), νZ(ð2) · ei2πγZ(ð2)}} If νZ((~1 �3 ð1)) · ei2πβZ((~1�3ð1)) � νZ((~2 �3 ð2)) · ei2πβZ((~2�3ð2)),then νZ((~1 �3 ð1)) · ei2πβZ((~1�3ð1)) � max{νZ(~1)·ei2πγZ(~1), νZ(ð1)·ei2πβZ(ð1)}. Therefore, Z is a ComCIFSBS of B. Theorem 3.12. Suppose that Z is a subset of B. Then R = (µ̂Z · ei2πβ̂Z , ν̂Z · ei2πγ̂Z , µZ · ei2πβZ , νZ · ei2πγZ ) is a ComCIFSBS of B if and only ifR(t,s) is a subbisemiring of B for all t, s ∈ D[0, 1]. Proof. Assume that R̂ is a ComCIFSBS of B. For each t, s ∈ D[0, 1] and ~1, ~2 ∈ R̂(t,s). Now, µ̂Z(~1) · ei2πβ̂Z (~1) � t, µ̂Z(~2) · ei2πβ̂Z (~2) � t and ν̂Z(~1) · ei2πγ̂Z (~1) � s, ν̂Z(~2) · ei2πγ̂Z (~2) � s. Now, µ̂Z((~1�1~2))·ei2πβ̂Z ((~1�1~2)) � min{µ̂Z(~1)·ei2πβ̂Z (~1), µ̂Z(~2)· ei2πβ̂Z (~2)} � t and ν̂Z((~1�1 ~2)) · ei2πγ̂Z ((~1�1 ~2)) � max{ν̂Z(~1) · ei2πγ̂Z (~1), ν̂Z(~2) · ei2πγ̂Z (~2)} � s. This implies that ~1 �1 ~2 ∈ R̂(t,s). Similarly, ~1 �2 ~2 ∈ R̂(t,s) and ~1 �3 ~2 ∈ R̂(t,s). Hence, R̂(t,s) is a subbisemiring of B,for all t, s ∈ D[0, 1]. For each t, s ∈ [0, 1] and ~1, ~2 ∈ R(t,s). Now,µZ(~1) ·ei2πβZ (~1) � t, µZ(~2) ·ei2πβZ (~2) � t and νZ(~1) · ei2πγZ (~1) � s, νZ(~2) · ei2πγZ (~2) � s. Now, µZ((~1 �1 ~2)) · ei2πβZ ((~1 �1 ~2)) � min{µZ(~1) ·ei2πβZ (~1), µZ(~2) ·ei2πβZ (~2)} � t and νZ((~1�1 ~2)) ·ei2πγZ ((~1�1 ~2)) � max{νZ(~1)·ei2πγZ (~1), νZ(~2)·ei2πγZ (~2)} � s. This implies that ~1�1~2 ∈ R(t,s). Similarly,~1�2 ~2 ∈ R(t,s) and ~1�3 ~2 ∈ R(t,s). Hence,R(t,s) is a subbisemiring of B, for all t, s ∈ D[0, 1]. Conversely, assume that R̂(t,s) is a subbisemiring of B and t, s ∈ D[0, 1]. Suppose if there exist ~1, ~2 ∈ B such that µ̂Z((~1 �1 ~2)) · ei2πβ̂Z ((~1 �1 ~2)) ≺ min{µ̂Z(~1) · ei2πβ̂Z (~1), µ̂Z(~2) · ei2πβ̂Z (~2)} and ν̂Z((~1 �1 ~2)) · ei2πγ̂Z ((~1 �1 ~2)) � max{ν̂Z(~1) · ei2πγ̂Z (~1), ν̂Z(~2) · ei2πγ̂Z (~2)}. For t, s ∈ D[0, 1] such that µ̂Z((~1 �1 ~2)) · ei2πβ̂Z ((~1 �1 ~2)) ≺ t � min{µ̂Z(~1) ·ei2πβ̂Z (~1), µ̂Z(~2) ·ei2πβ̂Z (~2)} and ν̂Z((~1�1 ~2)) ·ei2πγ̂Z ((~1�1 ~2)) � s � max{ν̂Z(~1)·ei2πγ̂Z (~1), ν̂Z(~2)·ei2πγ̂Z (~2)}. Thus,~1, ~2 ∈ R̂(t,s),but ~1�1~2 /∈ R̂(t,s). This contradicts,R̂(t,s) is a SBS of B. Therefore µ̂Z((~1�1~2))·ei2πβ̂Z ((~1�1~2)) � min{µ̂Z(~1) · ei2πβ̂Z (~1), µ̂Z(~2) · ei2πβ̂Z (~2)} and ν̂Z((~1 �1 ~2)) · ei2πγ̂Z ((~1 �1 ~2)) � max{ν̂Z(~1) · ei2πγ̂Z (~1), ν̂Z(~2) · ei2πγ̂Z (~2)}. Similarly,�2 and �3 cases. Hence R̂ = (µ̂Z · ei2πβ̂Z , R̂I Z · ei2πÎ I Z , ν̂Z · ei2πγ̂Z ) is a ComCIFSBS of B. Let us assume that R(t,s) is a subbisemiring of B and t, s ∈ [0, 1]. Suppose if there exist https://internationalpubls.com 428 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) ~1, ~2 ∈ B such that µZ((~1�1~2)) ·ei2πβZ ((~1�1~2)) ≺ min{µZ(~1) ·ei2πβZ (~1), µZ(~2) · ei2πβZ (~2)} and νZ((~1 �1 ~2)) · ei2πγZ ((~1 �1 ~2)) � max{νZ(~1) · ei2πγZ (~1), νZ(~2) · ei2πγZ (~2)}. For t, s ∈ D[0, 1] such that µZ((~1 �1 ~2)) · ei2πβZ ((~1 �1 ~2)) ≺ t � min{µZ(~1) ·ei2πβZ (~1), µZ(~2) ·ei2πβZ (~2)} andRI Z((~1�1~2)) ·ei2πIIZ ((~1�1~2)) ≺ t � RIZ(~1)·ei2πI I Z (~1)+RIZ(~2)·ei2πI I Z (~2) 2 and νZ((~1�1~2)) ·ei2πγZ ((~1�1~2)) � s � max{νZ(~1) · ei2πγZ (~1), νZ(~2)·ei2πγZ (~2)}. Thus,~1, ~2 ∈ R(t,s),but ~1�1~2 /∈ R(t,s). This contradicts,R(t,s) is a SBS of B. Therefore µZ((~1�1~2))·ei2πβZ ((~1�1~2)) � min{µZ(~1)·ei2πβZ (~1), µZ(~2)· ei2πβZ (~2)} and νZ((~1 �1 ~2)) · ei2πγZ ((~1 �1 ~2)) � max{νZ(~1) · ei2πγZ (~1), νZ(~2) · ei2πγZ (~2)}. Similarly,�2 and �3 cases. Hence R = (µZ · ei2πβZ , νZ · ei2πγZ ) is a Com- CIFSBS of B. Definition 3.13. Let (B1, ∨ 1, ∨ 2, ∨ 3) and (B2, ∧ 1, ∧ 2, ∧ 3) be any two bisemirings. The mapping ` : B1 → B2 and Z be any ComCIFSBS in B1, Υ be any ComCIFSBS in `(B1) = B2. If µZ · ei2πβZ = [µ̂Z · ei2πβ̂Z , ν̂Z · ei2πγ̂Z , µZ · ei2πβZ , µZ · ei2πβZ , νZ · ei2πγZ is a ComCIF in B1, thenRΥ is a ComCIF in B2,defined by µ̂Υ(ð) · ei2πβ̂Z (ð) = { sup µ̂Z(~) · ei2πβ̂Z (~) if ~ ∈ `−1ð 0 otherwise ν̂Υ(ð) · ei2πγ̂Z (ð) = { inf ν̂Z(~) · ei2πγ̂Z (~) if ~ ∈ `−1ð 1 otherwise µΥ(ð) · ei2πβZ (ð) = { supµZ(~) · ei2πβZ (~) if ~ ∈ `−1ð 0 otherwise νΥ(ð) · ei2πγZ (ð) = { inf νZ(~) · ei2πγZ (~) if ~ ∈ `−1ð 1 otherwise for all ~ ∈ B1 and ð ∈ B2 is represents the image of RZ under `. Similarly, If RΥ · ei2πIΥ = [µ̂Υ · ei2πβ̂Z , ν̂Υ · ei2πγ̂Υ ], RΥ · ei2πIΥ , µΥ · ei2πβZ , νΥ · ei2πγΥ is a ComCIF in B2,then ComCIF RZ = ` ◦ RΥ in B1 ie, the ComCIF defined by RZ(~) · ei2πIZ(~),RΥ(`(~)) · ei2πIZ(`(~)), RZ = ` ◦ RΥ in B1 [ie, the ComCIF defined by RZ(~) · ei2πIZ(~) = RΥ(`(~)) · ei2πIZ(`(~)) is represents the preimage ofRΥ under `. Theorem 3.14. The homomorphic image of every ComCIFSBS is a ComCIFSBS. Proof. The mapping ` : B1 → B2 be any homomorphism. Now, `((~ ∨ 1 ð)) = `(~) ∧ 1 `(ð),`((~ ∨ 2 ð)) = `(~) ∧ 2 `(ð) and `((~ ∨ 3 ð)) = `(~) ∧ 3 `(ð) for all ~,ð ∈ B1. Let Υ̂ = `(Z),Z is any ComCIFSBS of B1. Let `(~),`(ð) ∈ B2. Let ~ ∈ u`−1(`(~)) and ð ∈ `−1(`(ð)) be such that µ̂Z(~) · ei2πβ̂Z(~) = sup ~∈`−1(`(~)) µ̂Z(~) · ei2πβ̂Z(~) and µ̂Z(ð) · ei2πβ̂Z(ð) = sup ~∈`−1(`(ð)) µ̂Z(~) · ei2πβ̂Z(~). https://internationalpubls.com 429 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) Now, µ̂Υ((`(~) ∧ 1 `(ð))) · ei2πβ̂Z((`(~) ∧ 1 `(ð))) = sup (~′ )∈`−1(`(~) ∧ 1 `(ð)) µ̂Z(~′) · ei2πβ̂Z(~′ ) = sup (~′ )∈`−1(`((~ ∨ 1 ð)) µ̂Z(~′) · ei2πβ̂Z(~′ ) = µ̂Z((~ ∨ 1 ð)) · ei2πβ̂Z((~ ∨ 1 ð)) � min{µ̂Z(~) · ei2πβ̂Z(~), µ̂Z(ð) · ei2πβ̂Z(ð)} = min{µ̂Υ`(~) · ei2πβ̂Υ`(~), µ̂Υ`(ð) · ei2πβ̂Υ`(ð)}. Thus,µ̂Υ((`(~) ∧ 1 `(ð)))·ei2πβ̂Υ((`(~) ∧ 1 `(ð))) � min{µ̂Υ`(~)·ei2πβ̂Υ`(~), µ̂Υ`(ð)·ei2πβ̂Υ`(ð)}. Similarly,µ̂Υ((`(~) ∧ 2 `(ð)))·ei2πβ̂Υ((`(~) ∧ 2 `(ð))) � min{µ̂Υ`(~)·ei2πβ̂Υ`(~), µ̂Υ`(ð)·ei2πβ̂Υ`(ð)} and µ̂Υ((`(~) ∧ 3 `(ð))) · ei2πβ̂Υ((`(~) ∧ 3 `(ð))) � min{µ̂Υ`(~) · ei2πβ̂Υ`(~), µ̂Υ`(ð) · ei2πβ̂Υ`(ð)}. Let `(~),`(ð) ∈ B2. Let ~ ∈ `−1(`(~)) and ð ∈ `−1(`(ð)) be such that ν̂Z(~) · ei2πγ̂Z(~) = inf ~∈`−1(`(~)) ν̂Z(~) · ei2πγ̂Z(~) and ν̂Z(ð) · ei2πγ̂Z(ð) = inf ~∈`−1(`(ð)) ν̂Z(~) · ei2πγ̂Z(~). Now, ν̂Υ((`(~) ∧ 1 `(ð))) · ei2πγ̂Υ((`(~) ∧ 1 `(ð))) = inf (~′ )∈`−1(`(~) ∧ 1 `(ð)) ν̂Z(~′) · ei2πγ̂Z(~′ ) = inf (~′ )∈`−1(`((~ ∨ 1 ð)) ν̂Z(~′) · ei2πγ̂Z(~′ ) = ν̂Z((~ ∨ 1 ð)) · ei2πγ̂Z((~ ∨ 1 ð)) � max{ν̂Z(~) · ei2πγ̂Z(~), ν̂Z(ð) · ei2πγ̂Z(ð)} = max{ν̂Υ`(~) · ei2πγ̂Υ`(~), ν̂Υ`(ð) · ei2πγ̂Υ`(ð)}. Thus, ν̂Υ((`(~) ∧ 1 `(ð)))·ei2πγ̂Υ((`(~) ∧ 1 `(ð))) � max{ν̂Υ`(~)·ei2πγ̂Υ`(~), ν̂Υ`(ð)·ei2πγ̂Υ`(ð)}. Similarly, ν̂Υ((`(~) ∧ 2 `(ð)))·ei2πγ̂Υ((`(~) ∧ 2 `(ð))) � max{ν̂Υ`(~)·ei2πγ̂Υ`(~), ν̂Υ`(ð)·ei2πγ̂Υ`(ð)} and ν̂Υ((`(~) ∧ 3 `(ð))) · ei2πγ̂Υ((`(~) ∧ 1 `(ð))) � max{ν̂Υ`(~) · ei2πγ̂Υ`(ð), ν̂Υ`(ð) · ei2πγ̂Υ`(ð)}. The mapping ` : B1 → B2 be any homomorphism. Now, `((~ ∨ 1 ð)) = `(~) ∧ 1 `(ð),`((~ ∨ 2 ð)) = `(~) ∧ 2 `(ð) and `((~ ∨ 3 ð)) = `(~) ∧ 3 `(ð) for all ~,ð ∈ B1. Let Υ = `(Z),Z is any ComCIFSBS of B1. Let `(~),`(ð) ∈ B2. Let ~ ∈ `−1(`(~)) and ð ∈ `−1(`(ð)) be such that µZ(~)·ei2πβZ(~) = sup ~∈`−1(`(~)) µZ(~)·ei2πβZ(~) and µZ(ð) · ei2πβZ(ð) = sup ~∈`−1(`(ð)) µZ(~) · ei2πβZ(~). https://internationalpubls.com 430 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) Now, µΥ((`(~) ∧ 1 `(ð))) · ei2πβZ((`(~) ∧ 1 `(ð))) = sup (~′ )∈`−1(`(~) ∧ 1 `(ð)) µZ(~′) · ei2πβZ(~′ ) = sup (~′ )∈`−1(`((~ ∨ 1 ð)) µZ(~′) · ei2πβZ(~′ ) = µZ((~ ∨ 1 ð)) · ei2πβZ((~ ∨ 1 ð)) � min{µZ(~) · ei2πβZ(~), µZ(ð) · ei2πβZ(ð)} = min{µΥ`(~) · ei2πβΥ`(~), µΥ`(ð) · ei2πβΥ`(ð)}. Thus, µΥ((`(~) ∧ 1 `(ð)))·ei2πβΥ((`(~) ∧ 1 `(ð))) � min{µΥ`(~)·ei2πβΥ`(~), µΥ`(ð)·ei2πβΥ`(ð)}. Similarly,µΥ((`(~) ∧ 2 `(ð)))·ei2πβΥ((`(~) ∧ 2 `(ð))) � min{µΥ`(~)·ei2πβΥ`(~), µΥ`(ð)·ei2πβΥ`(ð)} and µΥ((`(~) ∧ 3 `(ð))) · ei2πβΥ((`(~) ∧ 3 `(ð))) � min{µΥ`(~) · ei2πβΥ`(~), µΥ`(ð) · ei2πβΥ`(ð)}. Let `(~),`(ð) ∈ B2. Let ~ ∈ `−1(`(~)) and ð ∈ `−1(`(ð)) be such that νZ(~) · ei2πγZ(~) = inf ~∈`−1(`(~)) νZ(~) · ei2πγZ(~) and νZ(ð) · ei2πγZ(ð) = inf ~∈`−1(`(ð)) νZ(~) · ei2πγZ(~). Now, νΥ((`(~) ∧ 1 `(ð))) · ei2πγΥ((`(~) ∧ 1 `(ð))) = inf (~′ )∈`−1(`(~) ∧ 1 `(ð)) νZ(~′) · ei2πγZ(~′ ) = inf (~′ )∈`−1(`((~ ∨ 1 ð)) νZ(~′) · ei2πγZ(~′ ) = νZ((~ ∨ 1 ð)) · ei2πγZ((~ ∨ 1 ð)) � max{νZ(~) · ei2πγZ(~), νZ(ð) · ei2πγZ(ð)} = max{νΥ`(~) · ei2πγΥ`(~), νΥ`(ð) · ei2πγΥ`(ð)}. Thus, νΥ((`(~) ∧ 1 `(ð)))·ei2πγΥ((`(~) ∧ 1 `(ð))) � max{νΥ`(~)·ei2πγΥ`(~), νΥ`(ð)·ei2πγΥ`(ð)}. Similarly,νΥ((`(~) ∧ 2 `(ð)))·ei2πγΥ((`(~) ∧ 2 `(ð))) � max{νΥ`(~)·ei2πγΥ`(~), νΥ`(ð)·ei2πγΥ`(ð)} and νΥ((`(~) ∧ 3 `(ð))) · ei2πγΥ((`(~) ∧ 1 `(ð))) � max{νΥ`(~) · ei2πγΥ`(ð), νΥ`(ð) · ei2πγΥ`(ð)}. Thus,Υ is a ComCIFSBS of B2. Theorem 3.15. The homomorphic preimage of every ComCIFSBS is a ComCIFSBS. Proof. The mapping ` : B1 → B2 be a homomorphism. Now,`((~ ∨ 1 ð)) = `(~) ∧ 1 `(ð),`((~ ∨ 2 ð)) = `(~) ∧ 2 `(ð) and `((~ ∨ 3 ð)) = `(~) ∧ 3 `(ð) for all ~,ð ∈ B1. Let Υ̂ = `(Z),Υ̂ is a ComCIFSBS of B2. Let ~,ð ∈ B1. Now,µ̂Z((~ ∨ 1 ð))· ei2πβ̂Z((~ ∨ 1 ð)) = µ̂Υ(`((~ ∨ 1 ð)))·ei2πβ̂Υ(`((~ ∨ 1 ð))) = µ̂Υ((`(~) ∧ 1 `(ð)))·ei2πβ̂Υ((`(~) ∧ 1 `(ð))) � min{µ̂Υ`(~) · ei2πβ̂Υ`(~), µ̂Υ`(ð) · ei2πβ̂Υ`(ð)} = min{µ̂Z(~) · ei2πβ̂Z(~), µ̂Z(ð) · ei2πβ̂Z(ð)}. Thus,µ̂Z((~ ∨ 1 ð)) · ei2πβ̂Z((~ ∨ 1 ð)) � min{µ̂Z(~) · ei2πβ̂Z(~), µ̂Z(ð) · ei2πβ̂Z(ð)}. Now,ν̂Z((~ ∨ 1 ð))·ei2πγ̂Z((~ ∨ 1 ð)) = ν̂Υ(`((~ ∨ 1 ð)))·ei2πγ̂Υ(`((~ ∨ 1 ð))) = ν̂Υ((`(~) ∧ 1 `(ð)))· ei2πγ̂Υ((`(~) ∧ 1 `(ð))) � max{ν̂Υ`(~) · ei2πγ̂Υ`(~), ν̂Υ`(ð) · ei2πγ̂Υ`(ð)} = max{ν̂Z(~) · ei2πγ̂Z(~), ν̂Z(ð) · ei2πγ̂Z(ð)}. Thus, ν̂Z((~ ∨ 1 ð)) · ei2πγ̂Z((~ ∨ 1 ð)) � max{ν̂Z(~) · ei2πγ̂Z(~), ν̂Z(ð) · ei2πγ̂Z(ð)}. https://internationalpubls.com 431 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) The mapping ` : B1 → B2 be a homomorphism. Now,`((~ ∨ 1 ð)) = `(~) ∧ 1 `(ð),`((~ ∨ 2 ð)) = `(~) ∧ 2 `(ð) and `((~ ∨ 3 ð)) = `(~) ∧ 3 `(ð) for all ~, ð ∈ B1. Let Υ = `(Z),Υ is a ComCIFSBS of B2. Let ~,ð ∈ B1. Now,µZ((~ ∨ 1 ð))· ei2πβZ((~ ∨ 1 ð)) = µΥ(`((~ ∨ 1 ð)))·ei2πβΥ(`((~ ∨ 1 ð))) = µΥ((`(~) ∧ 1 `(ð)))·ei2πβΥ((`(~) ∧ 1 `(ð))) � min{µΥ`(~)·ei2πβΥ`(~), µΥ`(ð)·ei2πβΥ`(ð)} = min{µZ(~)·ei2πβZ(~), µZ(ð)·ei2πβZ(ð)}. Thus, µZ((~ ∨ 1 ð)) ·ei2πβZ((~ ∨ 1 ð)) � min{µZ(~) ·ei2πβZ(~), µZ(ð) ·ei2πβZ(ð)}. Now,νZ((~ ∨ 1 ð)) · ei2πγZ((~ ∨ 1 ð)) = νΥ(`((~ ∨ 1 ð)))·ei2πγΥ(`((~ ∨ 1 ð))) = νΥ((`(~) ∧ 1 `(ð)))·ei2πγΥ((`(~) ∧ 1 `(ð))) � max{νΥ`(~)·ei2πγΥ`(~), νΥ`(ð)·ei2πγΥ`(ð)} = max{νZ(~)·ei2πγZ(~), νZ(ð)·ei2πγZ(ð)}. Thus, νZ((~ ∨ 1 ð)) · ei2πγZ((~ ∨ 1 ð)) � max{νZ(~) · ei2πγZ(~), νZ(ð) · ei2πγZ(ð)}. Theorem 3.16. If ` : B1 → B2 is a homomorphism,then `(Ẑ(t,s)) is a subbisemiring of ComCIFSBS Υ̂ of B2. Proof. The mapping ` : B1 → B2 be a homomorphism. Now,`((~ ∨ 1 ð)) = `(~) ∧ 1 `(ð),`((~ ∨ 2 ð)) = `(~) ∧ 2 `(ð) and `((~ ∨ 3 ð)) = `(~) ∧ 3 `(ð) for all ~,ð ∈ B1. Let Υ̂ = `(Z),Z is a ComCIFSBS of B1. By Theorem 3.14, Υ̂ is a ComCIFSBS of B2. Let Ẑ(t,s) be any subbisemiring of Z. Suppose that ~, ð ∈ Ẑ(t,s). Then ~ ∨ 1 ð, ~ ∨ 2 ð and ~ ∨ 3 ð ∈ Ẑ(t,s). Now,µ̂Υ(`(~)) · ei2πβ̂Υ(`(~)) = µ̂Z(~) · ei2πβ̂Z(~) � t, µ̂Υ(`(ð)) · ei2πβ̂Υ(`(ð)) = µ̂Z(ð) · ei2πβ̂Z(ð) � t. Thus,µ̂Υ((`(~) ∧ 1 `(ð))) · ei2πβ̂Υ((`(~) ∧ 1 `(ð))) � µ̂Z((~ ∨ 1 ð)) · ei2πβ̂Z((~ ∨ 1 ð)) � t. Now,ν̂Υ(`(~))·ei2πγ̂Υ(`(~)) = ν̂Z(~)·ei2πγ̂Z(~) � s, ν̂Υ(`(ð))·ei2πγ̂Υ(`(ð)) = ν̂Z(ð)·ei2πγ̂Z(ð) � s. Thus, ν̂Υ((`(~) ∧ 1 `(ð))) · ei2πγ̂Υ((`(~) ∧ 1 `(ð))) � ν̂Z((~ ∨ 1 ð)) · ei2πγ̂Z((~ ∨ 1 ð)) � s,for all `(~),`(ð) ∈ B2. Similarly other operations,`(Ẑ(t,s)) is a subbisemiring of ComCIFSBS Υ̂ of B2. The mapping ` : B1 → B2 be a homomorphism. Now,`((~ ∨ 1 ð)) = `(~) ∧ 1 `(ð),`((~ ∨ 2 ð)) = `(~) ∧ 2 `(ð) and `((~ ∨ 3 ð)) = `(~) ∧ 3 `(ð) for all ~,ð ∈ B1. Let Υ = `(Z),Z is a ComCIFSBS of B1. By Theorem 3.14, Υ is a ComCIFSBS of B2. Let Z(t,s) be any subbisemiring of Z. Suppose that ~,ð ∈ Z(t,s). Then ~ ∨ 1 ð, ~ ∨ 2 ð and ~ ∨ 3 ð ∈ Z(t,s). Now,µΥ(`(~)) · ei2πβΥ(`(~)) = µZ(~) · ei2πβZ(~) � t, µΥ(`(ð)) · ei2πβΥ(`(ð)) = µZ(ð) · ei2πβZ(ð) � t. Thus,µΥ((`(~) ∧ 1 `(ð))) · ei2πβΥ((`(~) ∧ 1 `(ð))) � µZ((~ ∨ 1 ð)) · ei2πβZ((~ ∨ 1 ð)) � t. Now,νΥ(`(~)) · ei2πγΥ(`(~)) = νZ(~)·ei2πγZ(~) � s, νΥ(`(ð))·ei2πγΥ(`(ð)) = νZ(ð)·ei2πγZ(ð) � s. Thus,νΥ((`(~) ∧ 1 `(ð)))· ei2πγΥ((`(~) ∧ 1 `(ð))) � νZ((~ ∨ 1 ð)) · ei2πγZ((~ ∨ 1 ð)) � s,for all `(~),`(ð) ∈ B2. Similarly other operations,`(Z(t,s)) is a subbisemiring of ComCIFSBS Υ of B2. Theorem 3.17. If ` : B1 → B2 is any homomorphism,then Ẑ(t,s) is a subbisemiring of ComCIFSBS Z of B1. Proof. The mapping ` : B1 → B2 be any homomorphism. Now `((~ ∨ 1 ð)) = `(~) ∧ 1 `(ð),`((~ ∨ 2 ð)) = `(~) ∧ 2 `(ð) and `((~ ∨ 3 ð)) = `(~) ∧ 3 `(ð) for all ~,ð ∈ B1. Let Υ̂ = `(Z),Υ̂ is a ComCIFSBS of B2. By Theorem 3.15,Z is a Com- CIFSBS of B1. Let `(Ẑ(t,s)) be a subbisemiring of Υ̂. Suppose that `(~),`(ð) ∈ `(Ẑ(t,s)). Now,`((~ ∨ 1 ð)),`((~ ∨ 2 ð)) and `((~ ∨ 3 ð)) ∈ `(Ẑ(t,s)). Now,µ̂Z(~)·ei2πβ̂Z(~) = µ̂Υ(`(~))· https://internationalpubls.com 432 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) ei2πβ̂Υ(`(~)) � t, µ̂Z(ð) · ei2πβ̂Z(ð) = µ̂Υ(`(ð)) · ei2πβ̂Υ(`(ð)) � t. Thus, µ̂Z((~ ∨ 1 ð)) · ei2πβ̂Z((~ ∨ 1 ð)) � min{µ̂Z(~) · ei2πβ̂Z(~), µ̂Z(ð) · ei2πβ̂Z(ð)} � t. Now,ν̂Z(~) · ei2πγ̂Z(~) = ν̂Υ(`(~)) · ei2πγ̂Υ(`(~)) � s, ν̂Z(ð) · ei2πγ̂Z(ð) = ν̂Υ(`(ð)) · ei2πγ̂Υ(`(ð)) � s. Thus, ν̂Z((~ ∨ 1 ð))·ei2πγ̂Z((~ ∨ 1 ð)) = ν̂Υ((`(~) ∧ 1 `(ð)))·ei2πγ̂Υ((`(~) ∧ 1 `(ð))) � max{ν̂Z(~)· ei2πγ̂Z(~), ν̂Z(ð) · ei2πγ̂Z(ð)} � s,for all ~,ð ∈ B1. Similarly other operations,Ẑ(t,s) is a sub- bisemiring of ComCIFSBS Z of B1. The mapping ` : B1 → B2 be any homomorphism. We have `((~ ∨ 1 ð)) = `(~) ∧ 1 `(ð),`((~ ∨ 2 ð)) = `(~) ∧ 2 `(ð) and `((~ ∨ 3 ð)) = `(~) ∧ 3 `(ð) for all ~,ð ∈ B1. Let Υ = `(Z),Υ is a ComCIFSBS of B2. By The- orem 3.15,Z is a ComCIFSBS of B1. Let `(Z(t,s)) be a subbisemiring of Υ. Suppose that `(~),`(ð) ∈ `(Z(t,s)). Now,`((~ ∨ 1 ð)),`((~ ∨ 2 ð)) and `((~ ∨ 3 ð)) ∈ `(Z(t,s)). Now,µZ(~)·ei2πβZ(~) = µΥ(`(~))·ei2πβΥ(`(~)) � t, µZ(ð)·ei2πβZ(ð) = µΥ(`(ð))·ei2πβΥ(`(ð)) � t. Thus,µZ((~ ∨ 1 ð)) · ei2πβZ((~ ∨ 1 ð)) � min{µZ(~) · ei2πβZ(~), µZ(ð) · ei2πβZ(ð)} � t. Now,νZ(~)·ei2πγZ(~) = νΥ(`(~))·ei2πγΥ(`(~)) � s, νZ(ð)·ei2πγZ(ð) = νΥ(`(ð))·ei2πγΥ(`(ð)) � s. Thus, νZ((~ ∨ 1 ð))·ei2πγZ((~ ∨ 1 ð)) = νΥ((`(~) ∧ 1 `(ð)))·ei2πγΥ((`(~) ∧ 1 `(ð))) � max{νZ(~)· ei2πγZ(~), νZ(ð) · ei2πγZ(ð)} � s,for all ~,ð ∈ B1. Similarly other operations,Z(t,s) is a sub- bisemiring of ComCIFSBS Z of B1. References [1] L. A. Zadeh, Fuzzy sets, Information and Control, 8, (1965), 338-353. [2] K. 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[32] Sivakumar, C.. , Omar, Mowafaq. , shihadeh, Abdallah. , Atallah, Ahmed. , Al-Husban, Abdallah. , Maragatha, P.. , Rajesh, N.. , Palanikumar, M.. q-rung square root interval- valued neutrosophic sets with respect to aggregated operators using multiple attribute decision making. International Journal of Neutrosophic Science, 2024, 154-174. https://internationalpubls.com 435 1 Introduction 2 Preliminaries 3 Complex cubic intuitionistic fuzzy subbisemiring