Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 5s (2025) Complex tangent trigonometric approach applied to (γ, τ)-rung fuzzy set using weighted averaging, geometric operators and its extension Raed Hatamleh1, Abdallah Al-Husban2,3, K. Sundareswari4,∗, G.Balaj5, M.Palanikumar6 1Department of Mathematics, Faculty of Science, Jadara University, P.O. Box 733, Irbid 21110, Jordan. 2Department of Mathematics, Faculty of Science, Irbid National University, P.O. Box: 2600 Irbid, Jordan. 3Jadara Research Center, Jadara University, Irbid 21110, Jordan. 4,5Department of Mathematics, Al- Ameen Engineering College, Erode. 6Department 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, 4sundarimaths@gmail.com, 5balajivaithesh@gmail.com, 6palanimaths86@gmail.com, ∗Corresponding author: K. Sundareswari. Received: 04-10-2024 Revised: 26-11-2024 Accepted: 04-12-2024. Abstract This paper presents a new method that generates complex tangent trigonometric (γ, τ)-rung fuzzy sets. This article will deal with averaging, geometric, generalized weighted averaging, generalized weighted geometric using complex tangent trigonometric (γ, τ)-rung fuzzy set. We used an aggregating model to get the weighted average and geometric. Several sets with significant characteristics will be further studied using the algebraic approaches. Keywords: (γ, τ)-rung, WA, WG, GWA, GWG. 1 Introduction Numerous ideas have been put out to explain uncertainty, including fuzzy sets (FS),1 which have membership grades (MG) ranging from zero to one. Atanassov.2 For $, κ ∈ [0, 1] created an intuitionistic FS (IFS) in which each element has two MGs: positive $ and negative κ, and 0 ≤ $ + κ ≤ 1. The Pythagorean FSs (PFS) idea was developed by Yager3 and is distinguished by its MG and non-MG (NMG) with $ + κ ≥ 1 to$2+κ2 ≤ 1. The three main concepts of the picture FS are positive MG ($), neutral MG (γ), and negative MG (κ), as stated by Cuong et al.4 It also provides more advantages than PFS and IFS with 0 ≤ $+γ+κ ≤ 1 since$, γ, κ ∈ [0, 1]. Expert comments such as ”yes,” ”abstain,” ”no,” and ”refusal” will be sent, in accordance with the image FS description. Shahzaib et al.5 used MADM to define the SFS for certain AOs. Instead of 0 ≤ $ + γ + κ ≤ 1, SFS demands that 0 ≤ $2 + γ2 + κ2 ≤ 1. The idea of an intelligent decision support system for SFS was initially put out by Hussain et al.6 Both the MG and the NMG have power q in the q-rung orthogonal pair FS (q-ROFS), but their sum can never be more than one. Xu et al. developed geometric operators, including weighted, ordered weighted, and hybrid operators, that were derived from IFSs.7 Generalized ordered weighted averaging operators (GOWs) were suggested by Li et al.8 in 2002. Al-husband et al.9-14 and20-30 discussed the concept of various FS and its extension. Zeng et al.15 explained how to compute ordered weighted distances using AOs and distance measurements. Based on the features of AOs, Peng et al. investigated a simple PFS.16 Various algebraic structures and aggregation techniques with applications were studied by Palanikumar et al.17–19 For the rest of my work, I will keep to the format provided here. In Section 2 deals that PFS and NS were discussed. Section 3 describes numerous methods on (γ, τ)-rung FNs. In Section 4, the AOs based on CT (γ, τ)-rung FN are discussed. 2 Background Many important definitions that we should review for future learning are included in this section. https://internationalpubls.com 133 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 5s (2025) Definition 2 .1. Let A be a universal. The PFS Θ = { δ, 〈 M ᵀ(δ),M `(δ) 〉∣∣δ ∈ A } , M ᵀ : A → (0, 1) and M ` : A → (0, 1) called the MG and NMG of δ ∈ A to Θ, respectively and 0 � (M ᵀ(δ))2+(M `(δ))2 � 1. For Θ = 〈 M ᵀ,M `〉 is called a Pythagorean fuzzy number (PFN). Definition 2.2. The NS Θ = { δ, 〈 M ᵀ(δ),Mi(δ),M `(δ) 〉∣∣δ ∈ A } , where M ᵀ,Mi,M ` : A → (0, 1) is denote the MG, IMG and NMG of δ ∈ A , respectively and 0 ≤ (M ᵀ(δ)) + (Mi(δ)) + (M `(δ)) ≤ 2. For M = 〈 M ᵀ,Mi,M `〉 is called a neutrosophic number (-rung FN). Definition 2.3. The Pythagorean NS Θ = { δ, 〈 M ᵀ(δ),Mi(δ),M `(δ) 〉∣∣δ ∈ A } , where M ᵀ,Mi,M ` : A → (0, 1) is called the MG, IMG and NMG of δ ∈ A , respectively and 0 ≤ (M ᵀ(δ))2 + (Mi(δ))2 + (M `(δ))2 ≤ 2. For M = 〈 M ᵀ,Mi,M `〉 is called a Pythagorean neutrosophic number (Py-rung FN). Definition 2.4. Let Θ1 = (a1, b1) ∈ Nand Θ2 = (a2, b2) ∈ N . Then the distance between Θ1 and Θ2 is defined as D(Θ1,Θ2) = √ (a1 − a2)2 + 1 2 (b1 − b2)2, where N is a natural number. 3 Operations for CT (γ, τ)-rung FN We introduce the notion of a complex tangent trigonometric, the (γ, τ)-rung FN. Consequently, tanπ/2 = a and the CT (γ, τ)-rung FN and its operations were established. Definition 3.1. The (γ, τ) NS Θ = { δ, 〈( (a · Rᵀ)(δ) · e(a·I ᵀ)(δ), (a · R`)(δ) · e(a·I `)(δ) )〉∣∣∣δ ∈ A } , where (a · Rᵀ), (a · R`) : A → (0, 1) denote the MG and NMG of δ ∈ A to Θ, respectively and 0 ≤ ((a ·Rᵀ)(δ))γ + ((a ·R`)(δ))τ ≤ 1 and 0 ≤ ((a ·I ᵀ)(δ))γ + ((a ·I `)(δ))τ ≤ 1. For, Θ = 〈( (a ·Rᵀ) · e(a·I ᵀ), (a ·R`) · e(a·I `) )〉 is represent a CT (γ, τ)-rung FN. Definition 3.2. Let Θ = 〈((a ·Rᵀ) · e(a·I ᵀ), ((a ·R`)) · e(a·I `))〉,Θ1 = 〈((a ·Rᵀ 1 ) · e(a·I ᵀ 1 ), (a ·R` 1 ) · e(a·I ` 1 ))〉,Θ2 = 〈((a ·Rᵀ 2 ) · e(a·I ᵀ 2 ), (a ·R` 2 ) · e(a·I ` 2 ))〉 be any three CT (γ, τ)-rung FNs, and (γ, τ) > 0. Then 1. Θ1 ⊕ Θ2 =  γ √ ((a ·Rᵀ 1 ))γ + ((a ·Rᵀ 2 ))γ −((a ·Rᵀ 1 ))γ · ((a ·Rᵀ 2 ))γ · e γ √√√√√ ((a ·I ᵀ 1 ))γ + ((a ·I ᵀ 2 ))γ −((a ·I ᵀ 1 ))γ · ((a ·I ᵀ 2 ))γ , ((a ·R` 1 ))τ ((a ·R` 2 ))τ · e((a·I ` 1 ))τ ((a·I ` 2 ))τ  , 2. Θ1 �Θ2 =  ((a ·Rᵀ 1 ))γ((a ·Rᵀ 2 ))γ · e((a·I ᵀ 1 ))γ((a·I ᵀ 2 ))γ , τ √ ((a ·R` 1 ))τ + ((a ·R` 2 ))τ −((a ·R` 1 ))τ · ((a ·R` 2 ))τ · e τ √√√√√ ((a ·R` 1 ))τ + ((a ·I ` 2 ))τ −((a ·R` 1 ))τ · ((a ·I ` 2 ))τ  3. ∂ ·Θ =  γ √ 1− ( 1− (a · (Rᵀ)γ )∂ · e γ√1− ( 1−(a·(I ᵀ)γ )∂ , ((a · (R`)τ )∂ · e((a·(I `)τ )∂  4. Θ∂ =  ((a · (Rᵀ)γ)∂ · e((a·(I ᵀ)γ)∂ , τ √ 1− ( 1− (a · (R`)τ )∂ · e τ√1− ( 1−(a·(I `)τ )∂  . Definition 3.3. For any two CT (γ, τ)-rung FNs Θ1 = 〈(((a ·Rᵀ 1 ), (a ·R` 1 )))〉 and Θ2 = 〈(((a ·Rᵀ 2 ), (a · R` 2 )))〉. Then DE(Θ1,Θ2) = √ 1 2 [ [ 1 + ((a ·Rᵀ 1 ))2 − ((a ·R` 1 ))2 − ( 1 + ((a ·Rᵀ 2 ))2 − ((a ·R` 2 ))2 )]2 + [ ((a ·I ᵀ 1 ))2 − ((a ·I ` 1 ))2 − ( ((a ·I ᵀ 2 ))2 − ((a ·I ` 2 ))2 )]2] https://internationalpubls.com 134 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 5s (2025) where DE(Θ1, Θ2) is called the ED between Θ1 and Θ2. DH(Θ1,Θ2) = 1 2 [∣∣∣∣∣ 1 + ((a ·Rᵀ 1 ))2 − ((a ·R` 1 ))2 − ( 1 + ((a ·Rᵀ 2 ))2 − ((a ·R` 2 ))2 )∣∣∣∣∣+ ∣∣∣∣∣ ((a ·I ᵀ 1 ))2 − ((a ·I ` 1 ))2( −((a ·I ᵀ 2 ))2 − ((a ·I ` 2 ))2 )∣∣∣∣∣ ] where DH(Θ1,Θ2) is called the HD between Θ1 and Θ2. 4 AOs based on CT (γ, τ)-rung FN We use CT (γ, τ)-rung FNWA, CT (γ, τ)-rung FNWG, GCT (γ, τ)-rung FNWA, and GCT (γ, τ)-rung FNWG to describe the AOs. 4.1 CT (γ, τ)NWA Definition 4.1. Let Θi = 〈(((a · Rᵀ i ) · e(a·I ᵀ i ), (a · R` i ) · e(a·I ` i )))〉 be the CT (γ, τ)-rung FNs, W = (ω1, ω2, ..., ω`) be the weight of Θi, ωi ≥ 0 and ⊕` i=1 ωi = 1. Then CT (γ, τ)-rung FNWA (Θ1,Θ2, ...,Θ`) =⊕` i=1 ωiΘi. Theorem 4.2. Let Θi = 〈 (((a ·Rᵀ i ) · e(a·I ᵀ i ), (a ·R` i ) · e(a·I ` i ))) 〉 be the CT (γ, τ)-rung FNs. Then CT (γ, τ)NWA(Θ1,Θ2, ...,Θ`) =  γ √ 1− ⊗` i=1 ( 1− ((a ·Rᵀ i ))γ )ωi · e γ √ 1− ⊗` i=1 ( 1−((a·I ᵀ i ))γ )ωi ,⊗` i=1(((a ·R` i ))τ )ωi · e ⊗` i=1(((a·I ` i ))τ )ωi  . Proof. If ` = 2, then CT (γ, τ)-rung FNWA(Θ1,Θ2) = ω1Θ1 ⊕ ω2Θ2, where ω1Θ1 =  γ √ 1− ( 1− ((a ·Rᵀ 1 ))γ )ω1 · e γ √ 1− ( 1−((a·I ᵀ 1 ))γ )ω1 , (((a ·R` 1 ))τ )ω1 · e(((a·I ` 1 ))τ )ω1  ω2Θ2 =  γ √ 1− ( 1− ((a ·Rᵀ 2 ))γ )ω2 · e γ √ 1− ( 1−((a·I ᵀ 2 ))γ )ω2 , (((a ·R` 2 ))τ )ω2 · e(((a·I ` 2 ))τ )ω2  . Now, ω1Θ1 ⊕ ω2Θ2 =  γ √√√√√√√√√√√√√ ( 1− ( 1− ((a ·Rᵀ 1 ))γ )ω1 ) +( 1− ( 1− ((a ·Rᵀ 2 ))γ )ω2 ) − ( 1− ( 1− ((a ·Rᵀ 1 ))γ )ω1 ) ·( 1− ( 1− ((a ·Rᵀ 2 ))γ )ω2 ) , · e γ √√√√√√√√√√√√√√√√√√ ( 1− ( 1− ((a ·I ᵀ 1 ))γ )ω1 ) +( 1− ( 1− ((a ·I ᵀ 2 ))γ )ω2 ) − ( 1− ( 1− ((a ·I ᵀ 1 ))γ )ω1 ) ·( 1− ( 1− ((a ·I ᵀ 2 ))γ )ω2 ) , (((a ·R` 1 ))τ )ω1 · (((a ·R` 2 ))τ )ω2 · e(((a·I ` 1 ))τ )ω1 ·(((a·I ` 2 ))τ )ω2  https://internationalpubls.com 135 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 5s (2025) =  γ √ 1− ( 1− ((a ·Rᵀ 1 ))γ )ω1 ( 1− ((a ·Rᵀ 2 ))γ )ω2 · e γ √ 1− ( 1− ((a ·I ᵀ 1 ))γ )ω1 ( 1− ((a ·I ᵀ 2 ))γ )ω2 , (((a ·R` 1 ))τ )ω1 · (((a ·R` 2 ))τ )ω2 · e(((a·I ` 1 ))τ )ω1 ·(((a·I ` 2 ))τ )ω2  Hence, CT (γ, τ)NWA(Θ1,Θ2) =  γ √ 1− ⊗2 i=1 ( 1− ((a ·Rᵀ i ))γ )ωi · e γ √ 1− ⊗2 i=1 ( 1−((a·I ᵀ i ))γ )ωi ,⊗2 i=1(((a ·R` i ))τ )ωi · e ⊗2 i=1(((a·I ` i ))τ )ωi  . It valid for ` ≥ 3, Thus, CT (γ, τ)NWA(Θ1,Θ2, ...,Θ`) =  γ √ 1− ⊗` i=1 ( 1− ((a ·Rᵀ i ))γ )ωi · e γ √ 1− ⊗` i=1 ( 1−((a·I ᵀ i ))γ )ωi ,⊗` i=1(((a ·R` i ))τ )ωi · e ⊗` i=1(((a·I ` i ))τ )ωi  . If ` = `+ 1, then CT (γ, τ)-rung FNWA (Θ1,Θ2, ...,Θ`,Θ`+1) =  γ √√√√√√√√√√ ⊕̀ i=1 ( 1− ( 1− ((a ·Rᵀ i ))γ )ωi) + ( 1− ( 1− (Rᵀ `+1)γ )ω`+1 ) − ⊗̀ i=1 ( 1− ( 1− ((a ·Rᵀ i ))γ )ωi) · ( 1− ( 1− (Rᵀ `+1)γ )ω`+1 )· e γ √√√√√√√√√√√√√√ ⊕̀ i=1 ( 1− ( 1− ((a ·I ᵀ i ))γ )ωi) + ( 1− ( 1− (I ᵀ `+1)γ )ω`+1 ) − ⊗̀ i=1 ( 1− ( 1− ((a ·I ᵀ i ))γ )ωi) · ( 1− ( 1− (I ᵀ `+1)γ )ω`+1 ) , ⊗` i=1(((a ·R` i ))τ )ωi · ((R` `+1)τ )ω`+1 · e ⊗` i=1(((a·I ` i ))τ )ωi ·((I ` `+1) τ )ω`+1  =  γ √√√√ 1− `+1⊗ i=1 ( 1− ((a ·Rᵀ i ))γ )ωi · e γ √√√√√√1− `+1⊗ i=1 ( 1− ((a ·I ᵀ i ))γ )ωi ,⊗`+1 i=1(((a ·R` i ))τ )ωi · e ⊗`+1 i=1(((a·I ` i ))τ )ωi  . Theorem 4.3. Let Θi = 〈 (((a ·Rᵀ i ) · e(a·I ᵀ i ), (a ·R` i ) · e(a·I ` i ))) 〉 be the CT (γ, τ)-rung FNs. Then CT (γ, τ)-rung FNWA (Θ1,Θ2, ...,Θ`) = Θ (idempotency property). https://internationalpubls.com 136 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 5s (2025) Proof. Since (a ·Rᵀ i ) = (a ·Rᵀ) , (a ·R` i ) = (a ·R`) and (a ·I ᵀ i ) = (a ·I ᵀ) , (a ·I ` i ) = (a ·I `) and ⊕` i=1 ωi = 1. Now, CT (γ, τ)NWA(Θ1,Θ2, ...,Θ`) =  γ √√√√ 1− ⊗̀ i=1 ( 1− ((a ·Rᵀ i ))γ )ωi · e γ √√√√√√1− ⊗̀ i=1 ( 1− ((a ·I ᵀ i ))γ )ωi ,⊗` i=1(((a ·R` i ))τ )ωi · e ⊗` i=1(((a·I ` i ))τ )ωi  =  γ √ 1− ( 1− (a · (Rᵀ)γ )⊕` i=1 ωi · e γ √√√√1− ( 1− (a · (I ᵀ)γ )⊕` i=1 ωi , ((a · (R`)τ ) ⊕` i=1 ωi · e((a·(I `)τ ) ⊕` i=1 ωi  =  γ √ 1− ( 1− (a · (Rᵀ)γ ) · e γ √ 1− ( 1− (a · (I ᵀ)γ ) , (a · (R`)τ · e(a·(I `)τ  = Θ. Theorem 4.4. Let Θi = 〈 (((a ·Rᵀ i ) · e(a·I ᵀ i ), (a ·R` i ) · e(a·I ` i ))) 〉 be the CT (γ, τ)-rung FNs. Then CT (γ, τ)-rung FNWA(Θ1,Θ2, ...,Θ`), where ←−−−−− (a ·Rᵀ) = min(a ·Rᵀ ij), ̂(a ·Rᵀ) = max(a ·Rᵀ ij), ←−−−−− (a ·R`) = min(a ·R` ij), ̂(a ·R`) = max(a ·R` ij) and ←−−−−− (a ·I ᵀ) = min(a ·I ᵀ ij), ̂(a ·I ᵀ) = max(a ·I ᵀ ij), ←−−−−− (a ·I `) = min(a ·I ` ij), ̂(a ·I `) = max(a ·I ` ij) and where 1 ≤ i ≤ n, j = 1, 2, ..., ij . Then,〈←−−−−−−−−−−−− (a ·Rᵀ) · e(a·I ᵀ), ̂(a ·R`) · e(a·I `) 〉 ≤ CT (γ, τ)NWA(Θ1,Θ2, ...,Θ`) ≤ 〈 ̂(a ·Rᵀ) · e(a·I ᵀ), ←−−−−−−−−−−−− (a ·R`) · e(a·I `) 〉 . (Boundedness property). Proof. Since, ←−−−−− (a ·Rᵀ) = min(a ·Rᵀ ij), ̂(a ·Rᵀ) = max(a ·Rᵀ ij) and ←−−−−− (a ·Rᵀ) ≤ (a ·Rᵀ ij) ≤ ̂(a ·Rᵀ) and ←−−−−− (a ·I ᵀ) = min(a ·I ᵀ ij), ̂(a ·I ᵀ) = max(a ·I ᵀ ij) and ←−−−−− (a ·I ᵀ) ≤ (a ·I ᵀ ij) ≤ ̂(a ·I ᵀ). Now ←−−−−− (a ·Rᵀ) · e ←−−−−− (a·I ᵀ) = γ √√√√1− ⊗̀ i=1 ( 1− ( ←−−−−− (a ·Rᵀ))γ )ωi · e γ √ 1− ⊗` i=1 ( 1−( ←−−−−− (a·I ᵀ))γ )ωi ≤ γ √√√√1− ⊗̀ i=1 ( 1− (((a ·Rᵀ ij))) γ )ωi · e γ √ 1− ⊗` i=1 ( 1−(((a·I ᵀ ij))) γ )ωi ≤ γ √√√√1− ⊗̀ i=1 ( 1− ( ̂(a ·Rᵀ))γ )ωi · e γ √ 1− ⊗` i=1 ( 1−( ̂(a·I ᵀ))γ )ωi = ̂(a ·Rᵀ). Since, ←−−−−−−− (a · (R`)τ ) = min((a ·R` ij)) τ , ̂(a · (R`)τ ) = max((a ·R` ij)) τ and ←−−−−−−− (a · (R`)τ ) ≤ ((a ·R` ij)) τ ≤ ̂(a · (R`)τ ) and ←−−−−−−−− (a · (I `)τ ) = min((a · I ` ij)) τ , ̂(a · (I `)τ ) = max((a · I ` ij)) τ and ←−−−−−−−− (a · (I `)τ ) ≤ ((a · I ` ij)) τ ≤ ̂(a · (I `)τ ). https://internationalpubls.com 137 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 5s (2025) We have, ←−−−−−−− (a · (R`)τ = ⊗̀ i=1 ←−−−−−−− (a · (R`)τ )ωi · e ⊗` i=1 ←−−−−−−− (a·(I `)τ )ωi ≤ ⊗̀ i=1 (((a ·R` ij)) τ )ωi · e ⊗` i=1(((a·I ` ij)) τ )ωi ≤ ⊗̀ i=1 ̂(a · (R`)τ ) ωi · e ⊗` i=1 ̂(a·(I `)τ ) ωi = ̂(a · (R`)τ ) · e ̂(a·(I `)τ ). Therefore, 1 2 ×   γ √√√√ 1− ⊗̀ i=1 ( 1− ( ←−−−−− (a ·Rᵀ))γ )ωi2 + 1− (⊗` i=1(( ̂(a ·R`))τ )ωi )2 +  γ √√√√ 1− ⊗̀ i=1 ( 1− ( ←−−−−− (a ·I ᵀ))γ )ωi2 − (⊗` i=1(( ̂(a ·I `))τ )ωi )2  ≤ 1 2 ×   γ √√√√ 1− ⊗̀ i=1 ( 1− ((a · (a ·Rᵀ ij))) γ )ωi2 + 1− (⊗` i=1(((a ·R` ij)) τ )ωi )2 +  γ √√√√ 1− ⊗̀ i=1 ( 1− ((a · (a ·I ᵀ ij))) γ )ωi2 − (⊗` i=1(((a ·I ` ij)) τ )ωi )2  ≤ 1 2 ×   γ √√√√ 1− ⊗̀ i=1 ( 1− ( ̂(a ·Rᵀ))γ )ωi2 + 1− (⊗` i=1( ←−−−−− (a ·R`))τ )ωi )2 +  γ √√√√ 1− ⊗̀ i=1 ( 1− ( ̂(a ·I ᵀ))γ )ωi2 − (⊗` i=1( ←−−−−− (a ·I `))τ )ωi )2  . Hence, 〈←−−−−−−−−−−−− (a ·Rᵀ) · e(a·I ᵀ), ̂(a ·R`) · e(a·I `) 〉 ≤ CT (γ, τ)NWA(Θ1,Θ2, ...,Θ`) ≤ 〈 ̂(a ·Rᵀ) · e(a·I ᵀ), ←−−−−−−−−−−−− (a ·R`) · e(a·I `)〉. Theorem 4.5. Let Θi = 〈((a ·Rᵀ tij ) · e (a·I ᵀ tij ) , (a ·R` tij ) · e (a·I ` tij ) )〉 and Wi = 〈((a ·Rᵀ hij ) · e(a·I ᵀ hij ) , (a ·R` hij ) · e(a·I ` hij ) )〉, be the CT (γ, τ)-rung FNWAs. For any i, if there is (a ·Rᵀ tij ) 2 ≤ (a ·Rᵀ hij )2 and (a ·R` tij ) 2 ≥ (a ·R` hij )2 and (a ·I ᵀ tij ) 2 ≤ (a ·I ᵀ hij )2 and (a ·I ` tij ) 2 ≥ (a · I ` hij )2or Θi ≤ Wi. Prove that CT (γ, τ)NWA(Θ1,Θ2, ...,Θ`) ≤ CT (γ, τ)NWA(W1,W2, ...,W`), where (i = 1, 2, ..., `); (j = 1, 2, ..., ij) (monotonicity property). Proof. For any i, (a ·Rᵀ tij ) 2 ≤ (a ·Rᵀ hij )2. Therefore, 1− ((a ·Rᵀ ti)) 2 ≥ 1− ((a ·Rᵀ hi ))2. Hence, ⊗` i=1 ( 1− ( (a ·Rᵀ ti) )2)ωi ≥⊗` i=1 ( 1− ( (a ·Rᵀ hi ) )2)ωi and γ √ 1− ⊗` i=1 ( 1− ( (a ·Rᵀ ti) )γ)ωi ≤ γ √ 1− ⊗` i=1 ( 1− ( (a ·Rᵀ hi ) )γ)ωi . Similarly, (a ·I ᵀ tij ) 2 ≤ (a ·I ᵀ hij )2. Therefore, 1− ((a ·I ᵀ ti )) 2 ≥ 1− ((a ·I ᵀ hi ))2. Hence, ⊗` i=1 ( 1− ( (a ·I ᵀ ti ) )2)ωi ≥⊗` i=1 ( 1− ( (a ·I ᵀ hi ) )2)ωi https://internationalpubls.com 138 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 5s (2025) and γ √ 1− ⊗` i=1 ( 1− ( (a ·I ᵀ ti ) )γ)ωi ≤ γ √ 1− ⊗` i=1 ( 1− ( (a ·I ᵀ hi ) )γ)ωi . For any i, ( (a ·R` tij ) )2 ≥ ( (a ·R` hij ) )2 and ( (a ·R` tij ) )τ ≥ ( (a ·R` hij ) )τ . Therefore, 1− (⊗` i=1(a ·R` tij ) )τ ≤ 1− (⊗` i=1(a ·R` hij ) )τ . Similarly, for any i,( (a ·I ` tij ) )2 ≥ ( (a ·I ` hij ) )2 and ( (a ·I ` tij ) )τ ≥ ( (a ·I ` hij ) )τ . Therefore, − (⊗` i=1(a ·I ` tij ) )τ ≤ − (⊗` i=1(a ·I ` hij ) )τ . Hence, 1 2 ×    γ √√√√ 1− ⊗̀ i=1 ( 1− ((a ·Rᵀ ti)) γ )ωi2 +1− (⊗` i=1((a ·R` ti)) τ )2 +   γ √√√√ 1− ⊗̀ i=1 ( 1− ((a ·I ᵀ ti)) γ )ωi2 − (⊗` i=1((a ·I ` ti )) τ )2   ≤ 1 2 ×    γ √√√√ 1− ⊗̀ i=1 ( 1− ((a ·Rᵀ hi)) γ )ωi2 +1− (⊗` i=1((a ·R` hi ))τ )2 +   γ √√√√ 1− ⊗̀ i=1 ( 1− ((a ·I ᵀ hi)) γ )ωi2 − (⊗` i=1((a ·I ` hi ))τ )2   . Hence, CT (γ, τ)NWA (Θ1,Θ2, ...,Θ`) ≤ CT (γ, τ)NWA (W1,W2, ...,W`). 4.2 CT (γ, τ)-rung FNWG Definition 4.6. Let Θi = 〈( ((a · Rᵀ i ) · e(a·I ᵀ i ), (a · R` i ) · e(a·I ` i )) )〉 be the CT (γ, τ)-rung FNs. Then (γ, τ)-rung FNWG (Θ1,Θ2, ...,Θ`) = ⊗` i=1 Θωi i . Corollary 4.7. Let Θi = 〈( ((a ·Rᵀ i ) · e(a·I ᵀ i ), (a ·R` i ) · e(a·I ` i )) )〉 be the CT (γ, τ)-rung FNs. Then CT (γ, τ)-rung FNWG (Θ1,Θ2, ...,Θ`) =  ⊗` i=1(((a ·Rᵀ i ))γ)ωi · e ⊗` i=1(((a·I ᵀ i ))γ)ωi τ √ 1− ⊗` i=1 ( 1− ((a ·R` i ))τ )ωi · e τ √ 1− ⊗` i=1 ( 1−((a·I ` i ))τ )ωi . Corollary 4.8. (i) Let Θi = 〈( ((a ·Rᵀ i ) · e(a·I ᵀ i ), (a ·R` i ) · e(a·I ` i )) )〉 be the CT (γ, τ)-rung FNs and all are equal. Then (γ, τ)-rung FNWG(Θ1,Θ2, ...,Θ`) = Θ. (ii) It has other properties, including boundedness and monotonicity, as well as having (γ, τ)-rung FNWG. 4.3 Generalized CT (γ, τ)-rung FNWA (GCT(γ, τ)-rung FNWA) Definition 4.9. Let Θi = 〈 ( ((a ·Rᵀ i ), (a ·R` i )) ) 〉 be the CT (γ, τ)-rung FN. Then GCT (γ, τ)-rung FNWA (Θ1,Θ2, ...,Θ`) = (⊕` i=1 ωiΘ ∂ i )1/∂ . https://internationalpubls.com 139 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 5s (2025) Theorem 4.10. Let Θi = 〈 ( ((a ·Rᵀ i ), (a ·R` i )) ) 〉 be the CT (γ, τ)-rung FNs. Then GCT (γ, τ)-rung FNWA (Θ1,Θ2, ...,Θ`) =  ( γ √√√√ 1− ⊗̀ i=1 ( 1− ( ((a ·Rᵀ i ))γ )γ)ωi )1/γ · e ( γ √√√√√√1− ⊗̀ i=1 ( 1− ( ((a ·I ᵀ i ))γ )γ)ωi )1/γ , τ √√√√1− ( 1− (⊗̀ i=1 ( τ √ 1− ( 1− ((a ·R` i ))τ )τ)ωi )τ)1/τ · e τ √√√√√√1− ( 1− (⊗̀ i=1 ( τ √ 1− ( 1− ((a ·I ` i ))τ )τ)ωi )τ)1/τ  . Proof. To illustrate this, we may first show that, ⊕` i=1 ωiΘ γ i =  γ √√√√ 1− ⊗̀ i=1 ( 1− ( ((a ·Rᵀ i ))γ )γ)ωi · e γ √√√√√√√√√√ 1− ⊗̀ i=1 ( 1− ( ((a ·I ᵀ i ))γ )γ)ωi ⊗` i=1 ( τ √ 1− ( 1− ((a ·R` i ))τ )τ)ωi · e ⊗̀ i=1 ( τ √ 1− ( 1− ((a ·I ` i ))τ )τ)ωi  . Put ` = 2, ω1Θ1 ⊕ ω2Θ2 =  γ √√√√√√√√√√ ( γ √ 1− ( 1− ( ((a ·Rᵀ 1 ))γ )γ)ω1 )γ + ( γ √ 1− ( 1− ( ((a ·Rᵀ 2 ))γ )γ)ω1 )γ − ( γ √ 1− ( 1− ( ((a ·Rᵀ 1 ))γ )γ)ω1 )γ · ( γ √ 1− ( 1− ( ((a ·Rᵀ 2 ))γ )γ)ω1 )γ ·e γ √√√√√√√√√√√√√√ ( γ √ 1− ( 1− ( ((a ·I ᵀ 1 ))γ )γ)ω1 )γ + ( γ √ 1− ( 1− ( ((a ·I ᵀ 2 ))γ )γ)ω1 )γ − ( γ √ 1− ( 1− ( ((a ·I ᵀ 1 ))γ )γ)ω1 )γ · ( γ √ 1− ( 1− ( ((a ·I ᵀ 2 ))γ )γ)ω1 )γ ,( τ √ 1− ( 1− ((a ·R` 1 ))τ )τ)ω1 · ( τ √ 1− ( 1− ((a ·R` 2 ))τ )τ)ω1 ·e ( τ √ 1− ( 1− ((a ·I ` 1 ))τ )τ)ω1 · ( τ √ 1− ( 1− ((a ·I ` 2 ))τ )τ)ω1  =  γ √ 1− ⊗2 i=1 ( 1− ( ((a ·Rᵀ 1 ))γ )γ)ωi · e γ √√√√1− ⊗2 i=1 ( 1− ( ((a·I ᵀ 1 ))γ )γ)ωi ⊗2 i=1 ( τ √ 1− ( 1− ((a ·R` i ))τ )τ)ωi · ⊗2 i=1 ( τ √ 1− ( 1− ((a ·I ` i ))τ )τ)ωi  . Hence, https://internationalpubls.com 140 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 5s (2025) ⊕̀ i=1 ωiΘ ∂ i =  γ √ 1− ⊗` i=1 ( 1− ( ((a ·Rᵀ 1 ))γ )γ)ωi · e γ √√√√1− ⊗` i=1 ( 1− ( ((a·I ᵀ 1 ))γ )γ)ωi ⊗` i=1 ( τ √ 1− ( 1− ((a ·R` i ))τ )τ)ωi · e ⊗` i=1 ( τ √ 1− ( 1−((a·I ` i ))τ )τ)ωi  . If ` = `+ 1, then ⊕` i=1 ωiΘ ∂ i + ω`+1Θ∂ `+1 = ⊕`+1 i=1 ωiΘ ∂ i . Now, ⊕` i=1 ωiΘ ∂ i + ω`+1Θ∂ `+1 = ω1Θ∂ 1 ⊕ ω2Θ∂ 2 ⊕ ... ⊕ ω`Θ ∂ ` ⊕ ω`+1Θ∂ `+1 =  γ √√√√√√√√√√√ ( γ √√√√ 1 − ⊗̀ i=1 ( 1 − ( ((a · Rᵀ i ))γ )γ)ωi )γ + ( γ √ 1 − ( 1 − ( (Rᵀ `+1)γ )γ)ω1 )γ − ( γ √√√√ 1 − ⊗̀ i=1 ( 1 − ( ((a · Rᵀ i ))γ )γ)ωi )γ · ( γ √ 1 − ( 1 − ( (Rᵀ `+1)γ )γ)ω1 )γ ·e γ √√√√√√√√√√√√√√√√√√ ( γ √√√√ 1 − ⊗̀ i=1 ( 1 − ( ((a · I ᵀ i ))γ )γ)ωi )γ + ( γ √ 1 − ( 1 − ( (I ᵀ `+1)γ )γ)ω1 )γ − ( γ √√√√ 1 − ⊗̀ i=1 ( 1 − ( ((a · I ᵀ i ))γ )γ)ωi )γ · ( γ √ 1 − ( 1 − ( (I ᵀ `+1)γ )γ)ω1 )γ ⊗̀ i=1 ( τ √ 1 − ( 1 − ((a · R` i ))τ )τ)ωi · ( τ √ 1 − ( 1 − (R` `+1)τ )τ)ω1 ·e ⊗̀ i=1 ( τ √ 1 − ( 1 − ((a · I ` i ))τ )τ)ωi · ( τ √ 1 − ( 1 − (I ` `+1)τ )τ)ω1  `+1⊕ i=1 ωiΘ γ i =  γ √ 1 − ⊗`+1 i=1 ( 1 − ( ((a · Rᵀ 1 ))γ )γ)ωi · e γ √√√√√1− ⊗`+1 i=1 ( 1− ( ((a·Iᵀ 1 ))γ )γ)ωi ⊗`+1 i=1 ( τ √ 1 − ( 1 − ((a · R` i ))τ )τ)ωi · e ⊗`+1 i=1 ( τ √ 1− ( 1−((a·I` i ))τ )τ)ωi  . ( `+1⊕ i=1 ωiΘ ∂ i )1/∂ =  ( γ √√√√ 1 − `+1⊗ i=1 ( 1 − ( ((a · Rᵀ i ))γ )γ)ωi )1/γ · e ( γ √√√√√√1 − `+1⊗ i=1 ( 1 − ( ((a · I ᵀ i ))γ )γ)ωi )1/γ τ √√√√1 − ( 1 − ( `+1⊗ i=1 ( τ √ 1 − ( 1 − ((a · R` i ))τ )τ)ωi )2)1/τ · e τ √√√√√√√1− ( 1 − ( `+1⊗ i=1 ( τ √ 1 − ( 1 − ((a · I ` i ))τ )τ)ωi )2)1/τ  Corollary 4.11. (i) If (γ, τ) = (1, 1), then CT (γ, τ)-rung FNWA operator is used instead of the GCT (γ, τ)- rung FNWA operator. (ii) If all Θi = 〈( ((a · Rᵀ i ) · e(a·I ᵀ i ), (a · R` i ) · e(a·I ` i )) )〉 and all are equal. Then GCT (γ, τ)-rung FNWA(Θ1,Θ2, ...,Θ`) = Θ. (iii) The GCT (γ, τ)-rung FNWA operator meets both boundedness and monotonicity constraints. https://internationalpubls.com 141 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 5s (2025) 4.4 Generalized CT (γ, τ)-rung FNWG ( GCT (γ, τ)-rung FNWG) Definition 4.12. Let Θi = 〈( ((a ·Rᵀ i ) · e(a·I ᵀ i ), (a ·R` i ) · e(a·I ` i )) )〉 be the CT (γ, τ)-rung FNs. Then GCT(γ, τ)-rung FNWG (Θ1,Θ2, ...,Θ`) = 1 ∂ (⊗` i=1(∂Θi) ωi ) . Corollary 4.13. Let Θi = 〈( ((a · Rᵀ i ) · e(a·I ᵀ i ), (a · R` i ) · e(a·I ` i )) )〉 be the CT (γ, τ)-rung FNs. Then GCT(γ, τ)-rung FNWG(Θ1,Θ2, ...,Θ`) =  γ √√√√1− ( 1− (⊗̀ i=1 ( γ √ 1− ( 1− ((a ·Rᵀ i ))γ )γ)ωi )γ)1/γ · e γ √√√√√√1− ( 1− (⊗̀ i=1 ( γ √ 1− ( 1− ((a ·I ᵀ i ))γ )γ)ωi )γ)1/γ ( τ √√√√ 1− ⊗̀ i=1 ( 1− ( ((a ·R` i ))τ )τ)ωi )1/τ · e ( τ √√√√√√1− ⊗̀ i=1 ( 1− ( ((a ·I ` i ))τ )τ)ωi )1/τ  . Corollary 4.14. (i) When ∂ = 1, the GCT (γ, τ)-rung FNWG is converted to the (γ, τ)-rung FNWG. (ii) GCT(γ, τ)-rung FNWG operators satisfy the boundedness and monotonicity characteristics. (iii) If all Θi = 〈( ((a ·Rᵀ i ) · e(a·I ᵀ i ), (a ·R` i ) · e(a·I ` i )) )〉 are equal. Then GCT(γ, τ)-rung FNWG(Θ1,Θ2, ...,Θ`) = Θ. References [1] L. A. Zadeh, Fuzzy sets, Information and control, 8(3), (1965), 338-353. [2] K. 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Hatamleh, On a Novel Topological Space Based on Partially Ordered Ring of Weak Fuzzy Complex Numbers and its Relation with the Partially ordered Neutrosophic Ring of Real Numbers, Neutrosophic Sets and Systems, 78, (2025), 578-590. https://internationalpubls.com 144 1 Introduction 2 Background 3 Operations for CT (, )-rung FN 4 AOs based on CT (, )-rung FN 4.1 CT (, ) NWA 4.2 CT (, )-rung FNWG 4.3 Generalized CT (, )-rung FNWA (GCT(, )-rung FNWA) 4.4 Generalized CT (, )-rung FNWG ( GCT (, )-rung FNWG)