EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 3129-3155 ISSN 1307-5543 – ejpam.com Published by New York Business Global Quadri-Polar Fuzzy Fantastic Ideals in BCI-Algebras: A TOPSIS Framework and Application M. Balamurugan1, Khalil H. Hakami2,∗, Moin A. Ansari2,*, Anas Al-Masarwah3, K. Loganathan4 1 Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai, Tamil Nadu, India 2 Department of Mathematics, College of Science, Jazan University, P.O. Box. 114, Jazan 45142, Kingdom of Saudi Arabia 3 Department of Mathematics, Faculty of Science, Ajloun National University, P.O. Box 43, Ajloun 26810, Jordan 4 Department of Mathematics and Statistics, Manipal University Jaipur, Jaipur-303007, India Abstract. A quadri-polar fuzzy (qP-F) set is an extension of a traditional fuzzy set that uses four degrees of membership to represent different aspects of belonging to provide a more detailed framework for handling uncertainty and vagueness. In this paper, we propose the notion of quadri- polar-(ϖ,ϑ)-fuzzy fantastic ideals (qP-(ϖ,ϑ)-FFI(s)) in BCI-algebras based on qP-F set. Also, the notion of quadri-polar-(∈σ̃,∈σ̃ ∨qτ̃ )-fuzzy fantastic ideals (qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFI(s)) is intro- duced, and the characterizations for an ∈σ̃-qP-F set and qτ̃ -qP-F set to be quadri-polar fuzzy ideals (qP-FI) in BCI-algebras are established. Furthermore, we present the qP-F TOPSIS technique for multi-criteria Group decision-making (MCGDM), which is a natural extension of the TOPSIS method and used to rank and choose the best alternatives under qP-F positive and negative ideal solutions. Finally, practical examples interpreting the applicability of our proposed qP-F-TOPSIS are solved. 2020 Mathematics Subject Classifications: 03B47, 03E72, 08A72 Key Words and Phrases: BCK/BCI-algebras, q-polar fuzzy fantastic ideal, q-polar-(ω, ϑ)-fuzzy fantastic ideal, q-polar-(∈σ̃,∈σ̃ ∨qτ̃ )-fuzzy fantastic ideal ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5429 Email addresses: drbalamuruganm@veltech.edu.in (M. Balamurugan), khakami@jazanu.edu.sa (Khalil H. Hakami), maansari@jazanu.edu.sa (Moin A. Ansari), anas.almasarwah@anu.edu.jo (Anas Al-Masarwah), loganathankaruppusamy304@gmail.com (K. Loganathan) https://www.ejpam.com 3129 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3130 1. Introduction Axiom systems, developed by Imai et al. [20, 21] and used in propositional calculi, are collections of axioms and inference guidelines used to derive theorems and prove the cor- rectness of logical arguments. Theorem logic and propositional calculus are other names for propositional logic, which deals with the manipulation and analysis of statements us- ing logical operators like OR, AND, and NOT. Various mathematical systems, including propositional logic, can be modeled and analyzed using algebraic structures, particularly Boolean algebra, which is closely related to propositional logic. Iseki [22, 23] introduced the concept of BCK/BCI-algebras. BCI-algebras, also known as BCK-algebras, generalize Boolean algebras and other related algebraic structures. The idea of fantastic ideals in BCI-algebras is a significant algebraic substructure presented and discussed by Saeid [1]. Zadeh [39] proposed the concept of fuzzy (uncertainty) sets, which address ambiguity and vagueness in real world circumstances. A membership function with a range of [0,1] is used to illustrate an uncertainty structure. Throughout the history of uncertainty set, there are many kinds of uncertainty set extensions, for example bipolar [13] and multi- polar [3] uncertainty sets, etc. The bipolar and multipolar uncertainty sets are in fact a generalization of an uncertainty set with a membership degree range [−1, 1] and [0, 1]q, re- spectively. In [5, 17], the few aspects of the bipolar fuzzy concept are applied to algebraic structures. The qP-FS has an extensive range of implementations to address ambigu- ity and vagueness in real world issues related to the quadri-polar data, quadri-index and quadri-attributes information. Researchers in a lot of different areas are very interested in the multi-polar uncertainty set theory. These areas include Lie algebras [4], ordered semihypergroups [30], subgroups [16] and BCK/BCI-algebras [7, 36]. Rosenfeld [38] introduced fuzzy groups, while Bhakat et al [12] developed a specific type denoted as (∈,∈ ∨q), based on point fuzzy sets within group theory. Jun [27, 28] and Muhiuddin et al. [35] extended this concept to (α, β)-fuzzy subalgebra. Ibrara et al. [19], Dudek et al. [14], and Narayanan et al. [37] furthered this idea with extensions to semigroups, hemirings, and near-rings, respectively. Al-Masarwah et al. [6, 8] explored (α, β) type subalgebras using m-F points within BCK-algebras. Ma et al. [33] intro- duced (∈γ ,∈γ ∨qδ)-fuzzy ideals, while Jana et al. [24] proposed (∈γ ,∈γ ∨qδ) fuzzy soft BCI-algebras. Zulfiqar et al. [42, 43] introduced the idea of (∈γ ,∈γ ∨qδ)-fuzzy subcom- mutative ideals and fuzzy fantastic ideals in BCI/BCH-algebras. Zhan [41] contributed with (∈γ ,∈γ ∨qδ)-fuzzy soft Γ-hyper ideals. Abuhijleh et al. [2] introduced the com- plex fuzzy groups. Fallath et al. [15] introduced cosets and normals of (γ, δ)-fuzzy HX- subgroups. Balamurugan et al. [10, 18] introduced anti-intuitionistic fuzzy soft ideals in BCK/BCI/BG-algebras. Balamurugan et al. [11, 34] introduced tripolar picture fuzzy ideals and bipolar intuitionistic fuzzy soft ideals in BCK/BCI-algebras. Moin et al. [9] introduced and studied a graph associated to UP-algebras. Fuzzy bi-ideals in ternary semirings are studied and explored by Kavikumar [29]. In this work, we combine qP-F sets with BCI-algebras to extend fuzzy set theory and provide new approaches for studying quadri-polar fuzzy BCI-algebras. We introduce a new class of generalized qP-(ϖ,ϑ)-FFI. The properties of qP-(ϖ,ϑ)-FFI(s) are highlighted. K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3131 We then discuss qP-(∈σ̃,∈σ̃ ∨qτ̃ )FFI(s) and explore their properties. Characterization theorems for qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFI(s) are also established. Finally, we present a q-PF TOPSIS methodology, discuss potential applications, compare it with existing TOPSIS methods, and propose future directions. To explain the novelty of this structure, some contributions by several researchers towards qP-FFI(s), qP-(ϖ,ϑ)-FFIs and qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFI(s) in BCI-algebras are presented in Table 1. Table 1: Contributions of several researchers toward ceratin generalizations of qP-FFI(s). Authors Year Contributions Rosenfeld [38] 1971 Creation of fuzzy subgroups. Xi [40] 1991 Creation of fuzzy ideals. Bhakat and Das [12] 1996 Certain extensions of fuzzy subgroups. Jun [26] 2004 Creation of (α, β)-fuzzy ideals. Lee [32] 2009 Creation of bipolar fuzzy ideals. Jana et al. [25] 2017 Extensions of bipolar fuzzy ideals. Al-Masarwah and Ahmad [6–8] 2018 Creation of multi P-FIs. Alqahtani et al. Present Creation of qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFI(s). 2. Preliminaries BCI-algebras are types of algebraic structures used in the study of non-classical log- ics, particularly in the context of certain types of implication algebras. These algebras generalize certain aspects of set theory, logic and have applications in some areas, such as theoretical computer science and mathematical logic. A BCI-algebra is a structure (ℵ̃; ≬, 0) consisting of a non-void set ℵ̃, a binary operation ≬ on ℵ̃, and a constant 0 ∈ ℵ̃, satisfying the following axioms: ∀ς̇ , ϱ̇, κ̇ ∈ ℵ̃ (I1) ((ς̇ ≬ ϱ̇) ≬ (ς̇ ≬ κ̇)) ≬ (κ̇ ≬ ϱ̇) = 0, (I2) (ς̇ ≬ (ς̇ ≬ ϱ̇)) ≬ ϱ̇ = 0, (I3) ς̇ ≬ ς̇ = 0, (I4) ς̇ ≬ ϱ̇ = 0, ϱ̇ ≬ ς̇ = 0 ⇒ ς̇ = ϱ̇. A subset I of ℵ̃ is referred to an ideal of ℵ̃ (see [22, 23]) if it meets: 0 ∈ I and (∀ς̇ , ϱ̇ ∈ I) (ς̇ ≬ ϱ̇ ∈ I, ϱ̇ ∈ I ⇒ ς̇ ∈ I). (1) A subset I of ℵ̃ is referred to a fantastic ideal of ℵ̃ (see [1]) if it meets: 0 ∈ I and (∀ς̇ , ϱ̇, κ̇ ∈ I)((ς̇ ≬ ϱ̇) ≬ κ̇ ∈ I, κ̇ ∈ I ⇒ ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)) ∈ I). (2) Definition 1. [31] A mapping Ã̧ : ℵ̃ → [0, 1] is a fuzzy set FS for the set ℵ̃. K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3132 A FS Ã̧ of ℵ̃ is a FI of ℵ̃ if it meets: (∀ς̇ , ϱ̇ ∈ ℵ̃, Ã̧(0) ≥ Ã̧(ς̇) and Ã̧(ς̇) ≥ Ã̧(ς̇ ≬ ϱ̇) ∧ Ã̧(ϱ̇)). (3) A FS Ã̧ of ℵ̃ is a FFI of ℵ̃ if it meets: (∀ς̇ , ϱ̇ ∈ ℵ̃, Ã̧(0) ≥ Ã̧(ς̇) and Ã̧(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ Ã̧((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ Ã̧(κ̇)). (4) 3. Quadri-Polar Fuzzy Fantastic Ideals Definition 2. A mapping ð̃ : ℵ̃ → [0, 1]4 is a qP-F for the set ℵ̃, where for any ς̇ ∈ ℵ̃, ð̃(ς̇) = (ð̃1(ς̇), ð̃2(ς̇), ð̃3(ς̇), ð̃4(ς̇)) and ð̃q(ς̇) ∈ [0, 1], for q = 1, 2, 3, 4. Definition 3. A qP-F set ð̃ of ℵ̃ is a qP-FFI if, ∀ς̇ , ϱ̇, κ̇ ∈ ℵ̃ and q = 1, 2, 3, 4, ð̃(0) ≥ ð̃(ς̇) and ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇). That is, ð̃q(0) ≥ ð̃q(ς̇) and ð̃q(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ ð̃q((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃q(κ̇). Example 1. Consider ℵ̃ = {0, ς̇ , ϱ̇, κ̇} with the binary operation ≬ defined by Table 2: Table 2. Cayley table representing by “ ≬ ” ≬ 0 ς̇ ϱ̇ κ̇ 0 0 0 0 0 ς̇ ς̇ 0 0 ς̇ ϱ̇ ϱ̇ ς̇ 0 ϱ̇ κ̇ κ̇ κ̇ κ̇ 0 Thus, (ℵ̃; ≬, 0) forms a BCI-algebra. Consider a qP-F set ð̃ defined on ℵ̃ as follows: ð̃(ς̇) =  ⟨0, (.58, .65, .75, .54)⟩, ⟨ς̇ , (.48, .21, .45, .30)⟩, ⟨ϱ̇, (.28, .52, .54, .30)⟩, ⟨κ̇, (.28, .41, .36, .54)⟩.  Thus, ð̃ is a qP-FFI of ℵ̃. Theorem 1. A qP-F set ð̃ is a qP-FFI of ℵ̃ ⇔ for any ρ̃ ∈ (0, 1]4, the ρ̃-cut subset ð̃ρ̃ = {ς̇ ∈ ℵ̃ | ð̃(ς̇) ≥ ρ̃} is a fantastic ideal of ℵ̃. Proof. Let ð̃ be a qP-FFI of ℵ̃ and ρ̃ ∈ (0, 1]4 be such that ð̃ρ̃ = {ς̇ ∈ ℵ̃ | ð̃(ς̇) ≥ ρ̃}. Let ς̇ , ϱ̇, κ̇ ∈ ð̃ρ̃. Then, ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ≥ ρ̃ and ð̃(κ̇) ≥ ρ̃. It follows from Definition 3.2 that, K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3133 ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) = ρ̃ ∧ ρ̃ = ρ̃. Therefore, ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)) ∈ ð̃ρ̃. Hence, ð̃ρ̃ is a fantastic ideal of ℵ̃. Conversely, assume ð̃ρ̃ is a fantastic ideal of ℵ̃. Suppose that ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) < ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇). Then ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) < ρ̃ ≤ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇). But ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ≥ ρ̃ and ð̃(κ̇) ≥ ρ̃. So, ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)) ̸∈ ð̃ρ̃, a contradiction. Therefore, ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇). Hence, ð̃ρ̃ is a FFI of ℵ̃. Consider a qP-F set ð̃ defined on ℵ̃, where ð̃(ς̇) = { ρ̃ ∈ (0, 1]q, if ς̇ ∈ ℵ̃ 0̃, if ς̇ /∈ ℵ̃, then ð̃(ς̇ is a qP-F point with support ℵ̃ and the value ρ̃, and it is symbolized by ς̇ρ̃. Theorem 2. Every fantastic ideal of ℵ̃ is a qP-FFI of ℵ̃. Proof. Suppose ð̃ρ̃ is a fantastic ideal of ℵ̃ and let ð̃ be an qP-FS in ℵ̃ defined by ð̃(ς̇) = { ρ̃ ∈ (0, 1]q, if ς̇ ∈ ℵ̃ 0̃, if ς̇ /∈ ℵ̃ Let ς̇ , ϱ̇ ∈ ℵ̃. To verify that ð̃ is a qP-FFI of ℵ̃. Case 1: If (ς̇ ≬ ϱ̇) ≬ κ̇ ∈ ð̃ and κ̇ ∈ ð̃, then (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∈ ð̃. Thus ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) = ð̃(κ̇) = ρ̃. Hence by Definition 3.2, we have ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) = ρ̃ ∧ ρ̃ = ρ̃. Case 2: If (ς̇ ≬ ϱ̇) ≬ κ̇ ̸∈ ð̃ and κ̇ ̸∈ ð̃, then (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ̸∈ ð̃. Thus, ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) = ð̃(κ̇) = 0̃. Hence by Definition 3.2, we have ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) = 0̃ ∧ 0̃ = 0̃. Case 3: If either (ς̇ ≬ ϱ̇) ≬ κ̇ ∈ ð̃ or κ̇ ∈ ð̃, then either ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) = 0̃ or ð̃(κ̇) = 0̃. So, ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇). Hence, ð̃ is a qP-FFI of ℵ̃. K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3134 4. Quadri-Polar (ϖ,ϑ)-Fuzzy Fantastic Ideals In this section, we introduce the concept of a qP-(ϖ,ϑ)FFI(s) in BCI-algebras and explore various properties associated with it. Here, we use ϖ and ϑ to represent symbols such as ∈σ̃,∈σ̃ ∨qτ̃ , qτ̃ or ∈σ̃ ∧qτ̃ , unless specified otherwise. Consider a qP-F point denoted as ς̇ρ̃ and a qP-F set ð̃ defined on ℵ̃. Then (1) ς̇ρ̃ ∈σ̃ ð̃ if ð̃(ς̇) ≥ ρ̃ > σ̃. (2) ς̇ρ̃qτ̃ ð̃ if ð̃(ς̇) + ρ̃ > 2τ̃ . (3) ς̇ρ̃ ∈σ̃ ∨qτ̃ ð̃ if ς̇ρ̃ ∈σ̃ ð̃ or ς̇ρ̃qτ̃ ð̃. (4) ς̇ρ̃ ∈σ̃ ∧qτ̃ ð̃ if ς̇ρ̃ ∈σ̃ ð̃ and ς̇ρ̃qτ̃ ð̃. (5) ς̇ρ̃ϖð̃ does not hold for ϖ = {∈σ̃, qτ̃ , ∈σ̃ ∨qτ̃ , ∈σ̃ ∧qτ̃},∀σ̃, τ̃ ∈ [0, 1]4, where σ̃ = (σ̃1, σ̃2, σ̃3, σ̃4) < τ̃ = (τ̃1, τ̃2, τ̃3, τ̃4). A qP-F point ς̇ρ̃ ∈ ð̃ if ð̃(ς̇) ≥ ρ̃. That is ð̃q(ς̇) ≥ ρ̃q,∀q = 1, 2, 3, 4. Also, ς̇ρ̃qð̃ if ð̃(ς̇) + ρ̃ > 1̂. That is, ð̃q(ς̇) + ρ̃q > 1̂,∀q = 1, 2, 3, 4. By ς̇ρ̃ ∈ ∨qð̃(resp., ς̇ρ̃ ∈ ∧qð̃) ⇒ ς̇ρ̃ ∈ ð̃ or ς̇ρ̃qð̃(resp., ς̇ρ̃ ∈ ð̃ and ς̇ρ̃qð̃). If ϕ ̸= C̃ ⊆ ℵ̃, then the quadri-polar characteristic fuzzy set (qP-CF) of C̃, say χ̂C̃ , where χ̂C̃ = { 1̂ = (1, 1, 1, 1), if ς̇ ∈ C̃ 0̃ = (0, 0, 0, 0), if ς̇ ̸∈ C̃ Clearly, a qP-CF is a qP-F subset of ℵ̃. Definition 4. A qP-F set ð̃ is a qP-(ϖ,ϑ)-FFI of ℵ̃, if ((ς̇ ≬ ϱ̇) ≬ κ̇)ρ̃ϖð̃, κ̇η̃ϖð̃ ⇒ (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))ρ̃∧η̃ϑð̃, where ϖ ̸=∈σ̃ ∧qτ̃ , ∀σ̃ < ρ̃, η̃ ≤ 1̂ and ((ς̇ ≬ ϱ̇) ≬ κ̇), κ̇ ∈ ℵ̃. Consider a qP-F set ð̃ defined on ℵ̃ such that ð̃(ς̇) ≤ τ̃ , ∀ς̇ ∈ ℵ̃. Let ς̇ ∈ ℵ̃ and σ̃ < ρ̃ ≤ 1̂ be such that ς̇ρ̃ ∈ ∧qτ̃ ð̃. Then, ð̃(ς̇) ≥ ρ̃ > σ̃ and ð̃(ς̇) + ρ̃ > 2τ̃ . Thus, 2τ̃ < ð̃(ς̇) + ρ̃ ≤ ð̃(ς̇) + ð̃(ς̇) = 2ð̃(ς̇) ⇒ ð̃(ς̇) > τ̃. Hence, {ς̇ρ̃ | ς̇ρ̃ ∈σ̃ ∧qτ̃ ð̃} = ϕ. Therefore, we exclude the case ϖ =∈σ̃ ∧qτ̃ in Definition 4.1 is neglected. Theorem 3. Let ð̃ be a qP-(ϖ,ϑ)-FFI and σ̃ + 1̂ = 2τ̃ of ℵ̃. Then, the set ð̃σ̃ = {ς̇ ∈ ℵ̃ | ð̃(ς̇) > σ̃} is a FI of ℵ̃. Proof. Let ς̇ , ϱ̇, κ̇ ∈ ℵ̃ be such that ς̇ , ϱ̇, κ̇ ∈ ð̃σ̃. Then, ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) > σ̃ and ð̃(κ̇) > σ̃. Assume ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≤ σ̃. If ϖ ∈ {∈σ̃, ∈σ̃ ∨qτ̃}, then K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3135 ((ς̇ ≬ ϱ̇) ≬ κ̇)ð̃(ς̇)ϖð̃ and κ̇ð̃(ϱ̇)ϖð̃. But ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≤ σ̃ < ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) and ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) + ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) ≤ σ̃ + 1̂ = 2τ̃ . So, (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))˜̃ð(ς̇)∧ð̃(η̇) ϑð̃,∀ϑ ∈ {∈σ̃, qτ̃ ,∈σ̃ ∨qτ̃ ,∈σ̃ ∧qτ̃}, a contradiction. Hence, ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) > σ̃ ⇒ ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)) ∈ ð̃σ̃. Also, ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) + 1̂ > σ̃ + 1̂ = 2τ̃ ⇒ (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))1̂qτ̃ ð̃. But ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≤ σ̃ ⇒ (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))1̂∈σ̃ð̃ and ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) + 1̂ ≤ σ̃ + 1̂ = 2τ̃ ⇒ (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))1̂qτ̃ ð̃, a contradiction. Thus, ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) > σ̃ ⇒ ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)) ∈ ð̃σ̃. Therefore, ð̃σ̃ is a FI of ℵ̃. Theorem 4. Let ϕ ̸= C̃ ⊆ ℵ̃ and σ̃ + 1̂ = 2τ̃ . Then C̃ is a FI of ℵ̃ if and only if the qP-F subset ð̃ of ℵ̃, which is defined as follows: (1) ð̃(ς̇) ≥ τ̃ , ∀ς̇ ∈ C̃, (2) ð̃(ς̇) ≤ σ̃,∀ς̇ ̸∈ C̃ is a qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFI of ℵ̃. Proof. Let C̃ be a FI of ℵ̃, ς̇ , ϱ̇, κ̇ ∈ ℵ̃ and let σ̃ < ρ̃, η̃ ≤ 1̂ be such that ((ς̇ ≬ ϱ̇) ≬ κ̇)ρ̃ ∈σ̃ ∨qτ̃ ð̃ and κ̇η̃ ∈σ̃ ð̃. Then ð̃(((ς̇ ≬ ϱ̇) ≬ κ̇)) ≥ ρ̃ > σ̃ and ð̃(κ̇) ≥ η̃ > σ̃. Thus, ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)) ∈ C̃ ⇒ ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ τ̃ . If ρ̃ ∧ η̃ ≤ τ̃ , then K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3136 ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ τ̃ ≥ ρ̃ ∧ η̃ > σ̃ ⇒ (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))ρ̃∧η̃ ∈σ̃ ð̃. If ρ̃ ∧ η̃ > τ̃ , then ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) + ρ̃ ∧ η̃ > τ̃ + τ̃ = 2τ̃ ⇒ (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))ρ̃∧η̃qτ̃ ð̃. Thus, (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))ρ̃∧η̃ ∈σ̃ ∨qτ̃ ð̃. Hence, ð̃ is an qP-(∈σ̃,∈σ̃ ∨qτ̃ )FFI of ℵ̃. On the contrary, assume ð̃ is a qP-(∈σ̃,∈σ̃ ∨qτ̃ )FFI of ℵ̃. Then C̃ is equal to ð̃σ̃. Consequently, according to Theorem 4.1, C̃ is a FI of ℵ̃. Corollary 1. Let σ̃ + 1̂ = 2τ̃ and ϕ ̸= C̃ ⊆ ℵ̃. Then, C̃ is a FI of ℵ̃ if and only if the characteristic function χ̂C̃ is a qP-(∈σ̃,∈σ̃ ∨qτ̃ )FFI of ℵ̃. Theorem 5. Let ϕ ̸= C̃ ⊆ ℵ̃ and σ̃ + 1̂ = 2τ̃ . Then, C̃ is a FI of ℵ̃ if and only if the qP-F subset ð̃ of ℵ̃ defined by the following conditions: (1) ð̃(ς̇) ≥ τ̃ , ∀ς̇ ∈ C̃, (2) ð̃(ς̇) ≤ σ̃,∀ς̇ ̸∈ C̃ is a qP-(qτ̃ ,∈σ̃ ∨qτ̃ )-FFI of ℵ̃. Proof. Let C̃ be a FI of ℵ̃, ς̇ , ϱ̇, κ̇ ∈ ℵ̃ and let σ̃ < ρ̃, η̃ ≤ 1̂ be such that ((ς̇ ≬ ϱ̇) ≬ κ̇)ρ̃qτ̃ ð̃ and κ̇η̃qτ̃ ð̃. Then ð̃(((ς̇ ≬ ϱ̇) ≬ κ̇)) + ρ̃ > 2τ̃ ⇒ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) > 2τ̃ − ρ̃ ≥ 2τ̃ − 1̂ = σ̃ and ð̃(κ̇) + η̃ > 2τ̃ ⇒ ð̃(κ̇) > 2τ̃ − η̃ ≥ 2τ̃ − 1̂ = σ̃. Thus, ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)) ∈ C̃ ⇒ ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ τ̃ . Now, if ρ̃ ∧ η̃ ≤ τ̃ , then ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ τ̃ ≥ ρ̃ ∧ η̃ > σ̃. Hence, (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))ρ̃∧η̃ ∈σ̃ ð̃. If ρ̃ ∧ η̃ > τ̃ , then ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) + ρ̃ ∧ ϖ̃ > τ̃ + τ̃ = 2τ̃ ⇒ (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))ρ̃∧η̃qτ̃ ð̃. Therefore, (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))ρ̃∧η̃ ∈ ∨qτ̃ ð̃. Thus, ð̃ is an qP-(qτ̃ ,∈σ̃ ∨qτ̃ )-FFI of ℵ̃. On the contrary, assume ð̃ is a qP-(qτ̃ ,∈σ̃ ∨qτ̃ )-FFI of ℵ̃. Then, C̃ is equal to ð̃σ̃. Consequently, according to Theorem 4.1, C̃ is a FI of ℵ̃. K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3137 Corollary 2. Let ϕ ̸= C̃ ⊆ ℵ̃ and σ̃ + 1̂ = 2τ̃ . Then, C̃ is a FI of ℵ̃ if and only if the characteristic function χ̂C̃ is a qP- (qτ̃ ,∈σ̃ ∨qτ̃ )-FFI of ℵ̃. Theorem 6. Let ϕ ̸= C̃ ⊆ ℵ̃ and σ̃ + 1̂ = 2τ̃ . Then, C̃ is a FI of ℵ̃ if and only if qP-F subset ð̃ of ℵ̃ defined by the following conditions: (1) ð̃(ς̇) ≥ τ̃ , ∀ς̇ ∈ C̃, (2) ð̃(ς̇) ≤ σ̃,∀ς̇ ̸∈ C̃ is a qP-(∈σ̃ ∨qτ̃ ,∈σ̃ ∨qτ̃ )FFI of ℵ̃. Proof. Let C̃ be a FI of ℵ̃, ς̇ , ϱ̇, κ̇ ∈ ℵ̃ and let σ̃ < ρ̃, η̃ ≤ 1̂ be such that ((ς̇ ≬ ϱ̇) ≬ κ̇)ρ̃ ∈σ̃ ∨qτ̃ ð̃ ⇒ ((ς̇ ≬ ϱ̇) ≬ κ̇)ρ̃ ∈σ̃ ð̃ or ((ς̇ ≬ ϱ̇) ≬ κ̇)ρ̃qτ̃ ð̃ and κ̇η̃ ∈σ̃ ∨qτ̃ ð̃ ⇒ κ̇η̃ ∈σ̃ ð̃ or κ̇η̃qτ̃ ð̃. If ((ς̇ ≬ ϱ̇) ≬ κ̇)ρ̃ ∈σ̃ ð̃, and κ̇η̃qτ̃ ð̃, then ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ≥ ρ̃ > σ̃ and ð̃(κ̇) + η̃ > 2τ̃ ⇒ ð̃(κ̇) > 2τ̃ − η̃ ≥ 2τ̃ − 1̂ = σ̃. Thus, ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)) ∈ C̃. Analogous as in Theorems 4.3 and 4.5, (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))ρ̃∧η̃ ∈σ̃ ∨qτ̃ ð̃. Hence, ð̃ is a q-P-(∈σ̃ ∨qτ̃ ,∈σ̃ ∨qτ̃ )-FFI of ℵ̃. The other scenarios can be approached in a similar manner to this one. On the contrary, assume ð̃ is a qP-(∈σ̃ ∨qτ̃ ,∈σ̃ ∨qτ̃ )-FFI of ℵ̃. Then, C̃ is equal to ð̃σ̃. Consequently, according to Theorem 4.1, C̃ is a FI of ℵ̃. Corollary 3. Let ϕ ̸= C̃ ⊆ ℵ̃ and σ̃ + 1̂ = 2τ̃ . Then, C̃ is a FI of ℵ̃ if and only if the characteristic function χ̂C̃ is a qP-(∈σ̃ ∨qτ̃ ,∈σ̃ ∨qτ̃ )-FFI of ℵ̃. Theorem 7. Every qP-(qτ̃ ,∈σ̃ ∨qτ̃ )-FFI of ℵ̃ is a qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFI of ℵ̃. Proof. Let ð̃ be a qP-(qτ̃ ,∈σ̃ ∨qτ̃ )-FFI of ℵ̃, ς̇ , ϱ̇, κ̇ ∈ ℵ̃ and let σ̃ < ρ̃, η̃ ≤ 1̂ be such that ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))ρ̃ ∈σ̃ ð̃ and κ̇η̃ ∈τ̃ ð̃. Then, ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ ρ̃ > σ̃ and ð̃(κ̇) ≥ η̃ > σ̃. Suppose that (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))ρ̃∧η̃∈σ̃ ∨qτ̃ ð̃. Then ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) + ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) < ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) + ρ̃ ∧ ϖ̃ ≤ 2τ̃ . Therefore, ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) < τ̃ . Now, ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∨ σ̃ < ð̃((ς̇ ≬ ϱ̇) ∧ ð̃(κ̇) ∧ τ̃ . Choose σ̃ < r̂ ≤ 1̂ Then K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3138 2τ̃ − ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∨ σ̃ ≥ r̂ > 2τ̃ − ð̃((ς̇ ≬ ϱ̇) ∧ ð̃(κ̇) ∧ τ̃ . Therefore, 2τ̃ − ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∧ (2τ̃ − σ̃) ≥ (2τ̃ − ð̃((ς̇ ≬ ϱ̇)) ∨ (2τ̃ − ð̃(κ̇)) ∨ τ̃ . Thus, r̂ > 2τ̃ − ð̃((ς̇ ≬ ϱ̇), r̂ > 2τ̃ − ð̃(κ̇) and 2τ̃ − ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) > r̂ So, ð̃((ς̇ ≬ ϱ̇) + r̂ > 2τ̃ , ð̃(κ̇) + r̂ > 2τ̃ and ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) + r̂ < 2τ̃ . Thus, ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))r̂qτ̃ ð̃, κ̇r̂qτ̃ ð̃ but (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))r̂ ∈σ̃ ∨qτ̃ ð̃, a contradiction. Hence, ð̃ is a qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFI of ℵ̃. Theorem 8. Every qP-(∈σ̃ ∨qτ̃ ,∈σ̃ ∨qτ̃ )-FFI of ℵ̃ is a qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFI of ℵ̃. Proof. The proof is based on the observation that if ς̇ρ̃ ∈σ̃ ð̃, then ς̇ρ̃ ∈σ̃ ∨qτ̃ ð̃. Theorem 9. Every qP-(∈σ̃,∈σ̃)FFI is a qP-(∈σ̃, ∈σ̃ ∨qτ̃ )FFI of ℵ̃. Proof. Let ð̃ be a qP-(∈σ̃,∈σ̃)-FFI of ℵ̃, ς̇ , ϱ̇, κ̇ ∈ ℵ̃ and let σ̃ < ρ̃, η̃ ≤ 1̂. So the qP-F points ((ς̇ ≬ ϱ̇) ≬ κ̇)ρ̃ ∈σ̃ ð̃ and κ̇η̃ ∈σ̃ ð̃. Then ((ς̇ ≬ ϱ̇) ≬ κ̇)ρ̃ ∈σ̃ ð̃ and κ̇η̃ ∈σ̃ ð̃ ⇒ (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))ρ̃∧η̃ ∈σ̃ ∨qτ̃ ð̃. Thus, ð̃ is a qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFI of ℵ̃. 5. Quadri-Polar (∈σ̃,∈σ̃ ∨qτ̃ )-Fuzzy Fantastic Ideals In this section, we present the notion of a qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFI(s) and explore several of its essential characteristics and properties. Definition 5. A qP-F set ð̃ of ℵ̃ is a qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFI of ℵ̃ if it satisfies condition (1), as follows: (1) ((ς̇ ≬ ϱ̇) ≬ κ̇)ρ̃ ∈σ̃ ð̃, κ̇η̃ ∈σ̃ ð̃ ⇒ (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))ρ̃∧η̃ ∈σ̃ ∨qτ̃ ð̃,∀σ̃ < ρ̃, η̃ ≤ 1̂ and ς̇ , ϱ̇, κ̇ ∈ ℵ̃. Example 2. Consider the BCK-algebra (ℵ̃; ≬, 0) and a qP-F set ð̃ as illustrated in Ex- ample 3.1. It is evident from Definition 5.1, ð̃ is a qP-(∈(0.2,0.1,0.3,0.2),∈(0.2,0.1,0.3,0.2) ∨q(0.61,0.68,0.78,0.57))-FFI of ℵ̃. Theorem 10. For a qP-F set ð̃ of ℵ̃, condition (1) in Definition 5.1 is similar with condition (2), as follows: (2) ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∨ σ̃ ≥ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇), τ̃ , ∀ς̇ , ϱ̇ ∈ ℵ̃. K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3139 Proof. (1) ⇒ (2). Assume that (2) does not hold. Then, ∃ς̇ , ϱ̇, κ̇ ∈ ℵ̃ such that ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∨ σ̃ < ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) ∧ τ̃ . Then, ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∨ σ̃ < ρ̃ ≤ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) ∧ τ̃ . Thus, ((ς̇ ≬ ϱ̇) ≬ κ̇)ρ̃ ∈σ̃ ð̃ and κ̇ρ̃ ∈σ̃ ð̃. But (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))ρ̃∈σ̃ ∨ ∈σ̃ð̃, a contradiction. ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∨ σ̃ ≥ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) ∧ τ̃ . (2) ⇒ (1) Let ((ς̇ ≬ ϱ̇) ≬ κ̇)ρ̃ ∈σ̃ ð̃, κ̇η̃ ∈σ̃ ð̃. Then, ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ≥ ρ̃ > σ̃ and ð̃(κ̇) ≥ η̃ > σ̃. If (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))ρ̃∧η̃ ∈σ̃, then (1) is hold. If (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))ρ̃∧η̃ ∈σ̃ ð̃, then ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) < ρ̃ ∧ η̃. Since ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∨ σ̃ ≥ ð̃(ς̇) ∧ ð̃(ϱ̇) ∧ τ̃ ≥ ρ̃ ∧ ϖ̃ ∧ τ̃ . Therefore, ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ τ̃ and ρ̃ ∧ ϖ̃ > τ̃ . Thus, ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) + ρ̃ ∧ η̃ > τ̃ + τ̃ = 2τ̃ ⇒ (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))ρ̃∧η̃qτ̃ ð̃. Hence, (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))ρ̃∧η̃ ∈σ̃ ∨qτ̃ ð̃. Corollary 4. A qP-F set ð̃ of ℵ̃ is a qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFI of ℵ̃ if it satisfies ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∨ σ̃ ≥ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇), τ̃ , ∀ς̇ , ϱ̇ ∈ ℵ̃. Theorem 11. The intersection of any collection of qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFIs of ℵ̃ is a qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFI of ℵ̃. Proof. Let {ð̃i}i∈I be a collection of qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFIs of ℵ̃ and (ς̇ ≬ ϱ̇) ≬ κ̇, κ̇ ∈ ℵ̃. Then, ð̃i(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∨ σ̃ ≥ ð̃i((ς̇ ≬ ϱ̇)κ̇) ∧ ð̃i(κ̇) ∧ τ̃ . Thus, (∧i∈I ð̃i)(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∨ σ̃ = ∧i∈I ð̃i(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∨ σ̃ ≥ ∧i∈I(ð̃i((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃i(κ̇) ∧ τ̃) ≥ (∧i∈I ð̃j)((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ (∧i∈I ð̃i)(κ̇) ∧ τ̃ . K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3140 Therefore, (∧i∈I ð̃i)(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∨ σ̃ ≥ (∧i∈I ð̃i)((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ (∧i∈I ð̃i)(κ̇) ∧ τ̃ . Hence, ∧i∈I ð̃i is a qP-(∈σ̃,∈σ̃ ∨qτ̃ )FFI of ℵ̃. For any qP-F set ð̃ of ℵ̃ and ρ̃ ∈ [0, 1]q, we define: (1) ð̃σ̃ρ̃ = {ς̇ ∈ ℵ̃ | ς̇ρ̃ ∈σ̃ ð̃}, (2) ⟨ð̃⟩τ̃ρ̃ = {ς̇ ∈ ℵ̃ | ς̇ρ̃qτ̃ ð̃}, (3) [ð̃]τ̃ρ̃ = {ς̇ ∈ ℵ̃ | ς̇ρ̃ ∈σ̃ ∨qτ̃ ð̃}. It is clear that [ð̃]τ̃ρ̃ = ð̃σ̃ρ̃ ∪ ⟨ð̃⟩τ̃ρ̃. The ensuing theorems elucidate the connection between qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFIs and the crisp FIs in ℵ̃. Theorem 12. Let ð̃ be a qP-F set of ℵ̃. Then, ð̃ is a qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFI of ℵ̃ ⇔ ð̃σ̃ρ̃ ̸= ϕ is a FI of ℵ̃, ∀σ̃ < ρ̃ ≤ τ̃ . Proof. Let ð̃ be a qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFI of ℵ̃ and let ς̇ , ϱ̇, κ̇ ∈ ð̃σ̃ρ̃ for σ̃ < ρ̃ ≤ τ̃ . Then ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ≥ ρ̃ > σ̃ and ð̃(κ̇) ≥ ρ̃ > σ̃. Thus, we have ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∨ σ̃ ≥ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) ∧ τ̃ ≥ ρ̃ ∧ ρ̃ ∧ τ̃ = ρ̃ ∧ τ̃ = ρ̃ > σ̃. Therefore, ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ ρ̃ ⇒ ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)) ∈ ð̃σ̃ρ̃ . Thus, ð̃σ̃ρ̃ is a FI of U. On the other hand, suppose that ð̃σ̃ρ̃ is a FI of U,∀σ̃ < ρ̃ ≤ τ̃ . Assume ς̇ , ϱ̇, κ̇ ∈ ℵ̃ such that ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∨ σ̃ < ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) ∧ τ̃ . Select σ̃ < ρ̃ ≤ τ̃ such that ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∨ σ̃ < r̂ = ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) ∧ τ̃ . Then, ((ς̇ ≬ ϱ̇) ≬ κ̇)ρ̃ ∈σ̃ ð̃, κ̇ρ̃ ∈σ̃ ð̃, but (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))r̂∈σ̃ ∨qτ̃ ð̃. Since ð̃σ̃ρ̃ is a FI of ℵ̃, ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)) ∈ ð̃σ̃ρ̃ , a contradiction. Hence, ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∨ σ̃ ≥ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) ∧ τ̃ . Therefore, ð̃ is a qP- (∈σ̃,∈σ̃ ∨qτ̃ )FFI of ℵ̃. By setting σ̃ = 0̃ and τ̃ = 0̂.5 in Theorem 5.3, we can derive the subsequent corollary. K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3141 Corollary 5. Let ð̃ be a qP-F set of ℵ̃. Then ð̃ is a qP-(∈,∈ ∨q)FFI of ℵ̃ ⇔ ð̃ρ̃ = {ς̇ ∈ ℵ̃ | ς̇ρ̃ ∈ ð̃} ≠ ϕ is a FI of ℵ̃,∀ρ̃ ∈ (0, 0.5]q. Theorem 13. Let ð̃ be a qP-F set of ℵ̃. Then (1) ð̃ is a qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFIs of ℵ̃ ⇔ ð̃σ̃ρ̃ ̸= ϕ is a FI of ℵ̃,∀σ̃ < ρ̃ ≤ τ̃ . (2) If 1̂ + σ̃ = 2τ̃ , then ð̃ is a qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFI of ℵ̃ ⇔ ⟨ð̃⟩τ̃ρ̃ ̸= ϕ is a FI of ℵ̃,∀τ̃ < ρ̃ ≤ 1̂. (3) If 1̂ + σ̃ = 2τ̃ , then ð̃ is a qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFI of ℵ̃ ⇔ [ð̃]τ̃ρ̃ ̸= ϕ is a FI of ℵ̃,∀σ̃ < ρ̃ ≤ 1̂. Proof. (1) Let ð̃ be a qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFI of ℵ̃. Let ς̇ ∈ ð̃τ̃ρ̃. Then ð̃(0) ∨ σ̃ ≥ ð̃(ς̇) ∧ τ̃ ≥ ρ̃ ∧ τ̃ > σ̃. Hence, ð̃(0) ≥ ρ̃ ⇒ 0 ∈ ð̃τ̃ρ̃. Let ς̇ , ϱ̇, κ̇ ∈ ð̃τ̃ρ̃. Then, ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ≥ ρ̃ > σ̃ and ð̃(κ̇) ≥ ρ̃ > σ̃. By Theorem 5.1 (2), we have ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∨ σ̃ ≥ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ τ̃ ≥ ρ̃ ∧ ρ̃ ∧ τ̃ = ρ̃ ∧ τ̃ = ρ̃ > σ̃. Therefore, ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ ρ̃ ⇒ ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)) ∈ ð̃σ̃ρ̃ . Hence, ð̃σ̃ρ̃ is a FI of ℵ̃. Conversely, assume ð̃σ̃ρ̃ is a FI of ℵ̃, ∀ρ̃ ∈ (σ, τ ]. Let ς̇ ∈ ℵ̃ be such that ð̃(0)∨σ < ρ̃ = ð̃(ς̇) ∧ τ̃ . Then ς̇ρ̃ ∈σ̃ ð̃, but 0ρ̃∈σ̃ ∨qτ̃ ð̃, a contradiction. Suppose ς̇ , ϱ̇, κ̇ ∈ ℵ̃. Then ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∨ σ̃ < ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) ∧ τ̃ . Select some ρ̃ ∈ (σ̃, τ̃ ] such that ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∨ σ̃ < ρ̃ = ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) ∧ τ̃ . Then ((ς̇ ≬ ϱ̇) ≬ κ̇)ρ̃ ∈σ̃ ð̃, κ̇ρ̃ ∈σ ð̃, but (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))ρ̃∈σ̃ ∨qτ̃ ð̃. Since ð̃σ̃ρ̃ is a FI of ℵ̃, we have ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)) ∈ ð̃σ̃ρ̃ , a contradiction. Hence ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∨ σ̃ ≥ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) ∧ τ̃ . K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3142 Therefore ð̃ be a qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFI of ℵ̃. (2) The proof follows a similar pattern as in (1), and therefore, we omit it for brevity. (3) Let ð̃ is a qP-(∈σ̃,∈σ̃ ∨qτ̃ )FFI of ℵ̃ and ρ̃ ∈ (σ̃, 1̂]. Then ∀ς̇ ∈ [ð̃]τ̃ρ̃, ς̇ρ̃ ∈σ̃ ∨qτ̃ ð̃ ⇒ ð̃(ς̇) ≥ ρ̃ > σ̃ or ð̃(ς̇) > 2τ̃ − ρ̃ > 2τ̃ − 1̂ = σ̃. Since ð̃ is a qP-(∈ σ̃,∈ σ̃ ∨ qτ̃ )FFI of ℵ̃, ð̃(0) ∨ σ̃ ≥ ð̃(ς̇) ∧ τ̃ > σ̃ ∧ τ̃ = σ̃, and so ð̃(0) ≥ σ̃ ⇒ ð̃(0) ≥ ð̃(ς̇) ∧ τ̃ . Case 1: Let ρ̃ ∈ (σ̃, τ̃ ]. Then 2τ̃ − ρ̃ ≥ τ̃ ≥ ρ̃, ð̃(0) ≥ ð̃(ς̇) ∧ τ̃ ≥ ρ̃ ∧ τ̃ = ρ̃ or ð̃(0) ≥ ð̃(ς̇) ∧ τ̃ > (2τ̃ − ρ̃) ∧ τ̃ = ρ̃ ∧ τ̃ = ρ̃. Thus, 0ρ̃ ∈σ̃ ð̃. Case 2: Let ρ̃ ∈ (τ̃ , 1̂]. Then 2τ̃ − ρ̃ < τ̃ < ρ̃, ð̃(0) ≥ ð̃(ς̇) ∧ τ̃ = ρ̃ ∧ τ̃ = τ̃ > 2τ̃ − ρ̃ or ð̃(0) ≥ ð̃(ς̇) ∧ τ̃ > (2τ̃ − ρ̃) ∧ τ̃ = 2τ̃ − ρ̃. Hence, 0ρ̃qτ̃ ð̃ ⇒ 0ρ̃ ∈σ̃ ∨qτ̃ ð̃. Let (ς̇ ≬ ϱ̇) ≬ κ̇, κ̇ ∈ [ð̃]τ̃ρ̃. Then ((ς̇ ≬ ϱ̇) ≬ κ̇)ρ̃, κ̇ρ̃ ∈σ̃ ∨qτ̃ ð̃, ð̃((ς̇≬ϱ̇) ≬ κ̇) ≥ ρ̃ > σ̃ or ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) > 2τ − ρ̃ > 2τ̃ − 1̂ = σ̃ and ð̃(κ̇) ≥ ρ̃ > σ̃ or ≥ (2τ̃ − ρ̃) ∧ τ̃ ð̃(κ̇) > 2τ̃ − ρ̃ = 2τ̃ − ρ̃ > 2τ̃ − 1̂ = σ̃. Since ð̃ is a qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFI of ℵ̃, ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∨ σ̃ ≥ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) ∧ τ̃ > σ̃ ∧ σ̃ ∧ τ̃ > σ̃ ∧ τ̃ = σ̃. Therefore, ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ σ̃ ⇒ ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) ∧ τ̃ . Case 1: Let ρ̃ ∈ (σ̃, τ̃ ]. Then 2τ̃ − ρ̃ ≥ τ̃ ≥ ρ̃, ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) ∧ τ̃ ≥ ρ̃ ∧ ρ̃ ∧ τ̃ = ρ̃ ∧ τ̃ = ρ̃ K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3143 or ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) ∧ τ̃ ≥ ρ̃ ∧ (2τ̃ − ρ̃) ∧ τ̃ = ρ̃ ∧ τ̃ ∧ τ̃ = ρ̃ ∧ τ̃ = ρ̃ or ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) ∧ τ̃ ≥ (2τ̃ − ρ̃) ∧ (2τ̃ − ρ̃) ∧ τ̃ = τ̃ ∧ τ̃ ∧ τ̃ = τ̃ > ρ̃. Hence, (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))ρ̃ ∈σ̃ ð̃. Case 2: Let ρ̃ ∈ (τ̃ , 1̂]. Then 2τ̃ − ρ̃ < τ̃ < ρ̃, ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) ∧ τ̃ ≥ ρ̃ ∧ ρ̃ ∧ τ̃ = ρ̃ ∧ τ̃ = τ̃ > 2τ̃ − ρ̃ or ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) ∧ τ̃ ≥ ρ̃ ∧ (2τ̃ − ρ̃) ∧ τ̃ ≥ τ̃ ∧ (2τ̃ − ρ̃) ∧ τ̃ ≥ τ̃ ∧ (2τ̃ − ρ̃) = (2τ̃ − ρ̃) or ð̃(ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ≥ ð̃((ς̇ ≬ ϱ̇) ≬ κ̇) ∧ ð̃(κ̇) ∧ τ̃ ≥ (2τ̃ − ρ̃) ∧ (2τ̃ − ρ̃) ∧ τ̃ ≥ (2τ̃ − ρ̃) ∧ τ̃ > (2τ̃ − ρ̃). Thus, (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))ρ̃qτ̃ ð̃. Hence, (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇)))ρ̃ ∈σ̃ ∨qτ̃ ð̃ ⇒ (ς̇ ≬ (ϱ̇ ≬ (ϱ̇ ≬ ς̇))) ∈ [ð̃]τ̃ρ̃. K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3144 Therefore [ð̃]τ̃ρ̃ is a FI of ℵ̃. By substituting σ̃ = 0̃ and τ̃ = 0̂.5 into Theorem 5.4, we can derive the ensuing corollary. Corollary 6. Let ð̃ be a qP-F set of ℵ̃. Then (1) ð̃ is a qP-(∈,∈ ∨q)-FFI of ℵ̃ ⇔ ð̃σ̃ρ̃ (̸= ϕ) is a FI of ℵ̃,∀ρ̃ ∈ (0, 0.5]q. (2) ð̃ is a qP-(∈,∈ ∨q)-FFI of ℵ̃ ⇔ ⟨ð̃⟩τ̃ρ̃( ̸= ϕ) is a FI of ℵ̃,∀ρ̃ ∈ (0.5, 1]q. (3) ð̃ is a qP-(∈,∈ ∨q)-FFI of ℵ̃ ⇔ [ð̃]τ̃ρ̃( ̸= ϕ) is a FI of ℵ̃, ∀ρ̃ ∈ (0, 1]q. 6. Quadri-Polar Fuzzy TOPSIS Approach In this section, we present a q-PF TOPSIS approach for multi-criteria group decision- making (MCGDM) problems. For these problems, we use a TOPSIS method based on qPF-sets to address a set of alternatives A̧ = {ς̇1, ς̇2, ς̇3, ς̇4} and a set C = {c1, c2, c3, c4} classified by q. Decision-makers must evaluate the four possibilities based on the q-PF criteria. The possible ratings of alternatives are evaluated in terms of q different attributes among four membership values, represented as (i = 1, 2, 3, 4). Step 1: The degree of each alternative ς̇j ∈ A̧, j = 1, 2, 3, 4) over all the criteria (ck ∈ C, k = 1, 2, 3, 4) may be expressed as q-PFEs. ð̃jk(ς̇) = (ρ1 o ð̃jk(ς̇), ρ2 o ð̃jk(ς̇), ρ3 o ð̃jk(ς̇), ρ4 o ð̃jk(ς̇)), where = (ρ1 o ð̃jk(ς̇) | i = 1, 2, ..., q). The tabular representation of the q-PF decision matrix is given by Table 2, which describes the ratings of alternatives. Table 3. Tablular representation of q-PF decision matrix. Alternatives c1 c2 c3 c4 ς̇1 ð̃11(ς̇1) ð̃12(ς̇1) ð̃13(ς̇1) ð̃14(ς̇1) ς̇2 ð̃21(ς̇2) ð̃22(ς̇2) ð̃23(ς̇2) ð̃24(ς̇2) ς̇3 ð̃31(ς̇3) ð̃32(ς̇3) ð̃33(ς̇3) ð̃34(ς̇3) ς̇4 ð̃41(ς̇4) ð̃42(ς̇4) ð̃43(ς̇4) ð̃44(ς̇4) Step 2: We build the optimistic or pessimistic q-PF decision matrix by adding the max- imal and smallest values to equalize the length of all q-PFEs. Step 3: Weights can be assigned to each q-PF criteria of alternatives by decision-makers based on their choice and importance of each criterion. We assume that the weights assigned by the decision-makers are W̧ = (w1, w2, w3, wq) ∈ (0, 1], K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3145 satisfying the normalized condition∑q k=1wk = 1, q = 1, 2, 3, 4.. Step 4: The weighted q-PF decision matrix is calculated in Table 4. Table 4. Tabular representation of a weighted q-PF decision matrix. Alternatives c1 c2 c3 c4 ς̇1 ð̃(ς̇1)11 ′ ð̃(ς̇1)12 ′ ð̃(ς̇1)13 ′ ð̃(ς̇1)14 ′ ς̇2 ð̃(ς̇2)21 ′ ð̃(ς̇2)22 ′ ð̃(ς̇2)23 ′ ð̃(ς̇2)24 ′ ς̇3 ð̃(ς̇3)31 ′ ð̃(ς̇3)32 ′ ð̃(ς̇3)33 ′ ð̃(ς̇3)34 ′ ς̇4 ð̃(ς̇4)41 ′ ð̃(ς̇4)42 ′ ð̃(ς̇4)43 ′ ð̃(ς̇4)44 ′ For each possible j and k, ð̃jk ′ (ς̇) = (ρ1 o ð̃jk ′ (ς̇), ρ2 o ð̃jk ′ (ς̇), ρ3 o ð̃jk ′ (ς̇), ρ4 o ð̃jk ′ (ς̇)), Step 5: The qP-F positive ideal solution (qP-FPIS) and qP-F negative ideal solution (qP-FNIS) of alternatives under the qP-F environment can be calculated by Equations (5) and (6) as qP − FPIS = {(ð̃1 ′ (ς̇))+, (ð̃2 ′ (ς̇))+, (ð̃3 ′ (ς̇))+, (ð̃4 ′ (ς̇))+}, (5) qP − FNIS = {(ð̃1 ′ (ς̇))−, (ð̃2 ′ (ς̇))−, (ð̃3 ′ (ς̇))−, (ð̃4 ′ (ς̇))−}, (6) where (ð̃k ′ (ς̇))+ = sup j (ð̃k ′ (ς̇)) = sup j (ρ1 o ð̃jk ′ (ς̇), ρ2 o ð̃jk ′ (ς̇), ρ3 o ð̃jk ′ (ς̇), ρ4 o ð̃jk ′ (ς̇)) = ((ρ1 o ð̃k ′ (ς̇))+, (ρ2 o ð̃k ′ (ς̇))+, (ρ3 o ð̃k ′ (ς̇))+, (ρ4 o ð̃k ′ (ς̇))+), and (ð̃k ′ (ς̇))− = inf j (ð̃k ′ (ς̇)) = inf j (ρ1 o ð̃jk ′ (ς̇), ρ2 o ð̃jk ′ (ς̇), ρ3 o ð̃jk ′ (ς̇), ρ4 o ð̃jk ′ (ς̇)) = ((ρ1 o ð̃k ′ (ς̇))−, (ρ2 o ð̃k ′ (ς̇))−, (ρ3 o ð̃k ′ (ς̇))−, (ρ4 o ð̃k ′ (ς̇))−). Step 6: The qP-F Euclidean distance of each alternative ς̇j from qP-FPIS and qP- FNIS can be calculated by Equations (7) and (8). D ′ E(ς̇j , qP − FPIS) = √√√√ 1 16 4∑ k=1 [ 4∑ l=1 { 4∑ i=1 (ρi o ð̃jk′ (ς̇)− ρi o ð̃k′ (ς̇)+}], (7) K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3146 and D ′ E(ς̇j , qP − FNIS) = √√√√ 1 16 4∑ k=1 [ 4∑ l=1 { 4∑ i=1 (ρi o ð̃jk′ (ς̇)− ρi o ð̃k′ (ς̇)−}], (8) Step 7: The relative qP-F closeness coefficient of each alternative ς̇j using the following formula as described by (9), E ′ j = D ′ E(ς̇j , qP − FNIS) D ′ E(ς̇j , qP − FPIS) +D ′ E(ς̇j , qP − FNIS) , j = 1, 2, 3, 4. (9) The alternative with the highest qP-F closeness coefficient is the best one, and we can rank each alternative in order. We present our proposed decision-making method in Algorithm 1. In Section 6.1, we examine the practical usage of our suggested model. Specifically, we Algorithm 1. The algorithm of the proposed approaches for dealing MCGDM problems. Step 1. Input. Step 2. Determine the optimistic or pessimistic q-PF decision matrix. Step 3. Calculate the normalized weights. Step 4. Calculate the weight for the pessimistic qP-F decision matrix. Step 5. Compute the qP-FPIS, and qP-FNIS. Step 6. qP-F Euclidean distance of each alternative ς̇j from qFPIS and qP-FPIS. Step 7. Calculate the relative qP-F closeness coefficients Ej′ . Step 8. Output. Rank the possibilities for the final decision and choose the best one. show how qP-F is useful in the selection of solar power plant stations in a rural region. Selection of Solar Power Plant Station in a Rural Area Assume that the government want to build a solar power plant station in a rural area. The government has four options for the location of a new solar power plant. Each site was appraised by a group of decision makers based on the following criteria as follows: 1. “Solar Irradiance (T1)”, which could have the characteristics shown below: • Solar Irradiance Density: The amount of solar power received per unit area. • Sun Light Duration: The number of sunlight hours per day. • Solar Tracking: The technology used to follow the sun’s path to maximize energy capture. • Temporal Variability: The fluctuation of solar irradiance over time, including daily and seasonal changes. K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3147 2. “Land Availability (T2)”, which could have the characteristics shown below: • Land Size: The total area available for the solar power plant. • Land Ownership: The ownership status of the land, such as government-owned, privately-owned, or leased. • Land Access: The ease of access to the land for construction and maintenance. •Land Cost: The cost of acquiring or leasing the land for the project. 3. “Environmental Impacts (T3)”, which could have the characteristics shown below: • Land Use: The current and previous use of the land and the impact of converting it to a solar power plant. • Water Consumption: The amount of water required for cleaning solar panels and other operations. • Biodiversity: The effect of the solar power plant on local wildlife and plant species. • Materials and Waste: The environmental impact of materials used in the construc- tion and the waste generated. 4. “Proximity to Grid Connections (T4)”,which could have the characteristics shown below: • Grid Connection Cost: The expense associated with connecting the solar power plant to the nearest grid infrastructure. • Electricity Demand: The local demand for electricity and how the new plant will meet or exceed this demand. • Grid Stability: The ability of the existing grid to handle the additional load from the solar power plant. • Grid Capacity: The existing capacity of the grid to integrate the new power supply without significant upgrades. 5. “Local Regulations (T5)”, which could have the characteristics shown below: • Zoning Laws: Regulations that dictate land use in the area. • Permitting Process: The complexity and duration of obtaining the necessary permits for construction and operation. • Incentives and Subsidies: Availability of government incentives, tax credits, and sub- sidies for renewable energy projects. • Compliance Requirements: Environmental and operational regulations that need to be met for the project to proceed. The five criteria and their attributes are shown below: 1. The qP-F initial decision matrix is represented in Table 5. K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3148 Table 5. qP-F initial decision matrix. Alternatives qP-F criteria c1-Solar Radiance ς̇1 {(.18,.73,.43,.67),(.32,.64,.49,.72),(.38,.66,.54,.64)} ς̇2 {(.46,.76,.45,.27),(.56,.91,.36,.48)} ς̇3 {(.85,.37,.45,.59),(.72,.48,.72,.58),(.51,.64,.55,.32)} ς̇4 {(.21,.52,.34,.77),(.41,.61,.43,.78),(.42,.66,.39,.87)} ς̇5 {(.11,.33,.56,.61),(.31,.41,.6,.73)} Alternatives qP-F criteria c2-Local Availability ς̇1 {(.75,.45,.67,.69),(.79,.37,.57,.69) } ς̇2 {(.45,.70,.49,.86),(.56,.72,.66,.74),(.47,.62,.58,.72)} ς̇3 {(.46,.66,.71,.17),(.41,.77,.78,.19),(.48,.80,.83,.15)} ς̇4 {(.57,.64,.38,.57),(.51,.56,.59,.47)} ς̇5 {(.24,.12,.81,.77),(.31,.25,.86,.73),(.23,.31,.80,.75)} ’ Alternatives qP-F criteria c3-Environmental Impacts ς̇1 {(.82,.51,.67,.7),(.8,.64,.69,.8),(.7,.59,.68,.78)} ς̇2 {(.70,.56,.45,.29),(.73,.51,.37,.43),(.83,.63,.65,.54)} ς̇3 {(.74,.46,.61,.8),(.81,.52,.59,.91)} ς̇4 {(.66,.55,.46,.88),(.61,.58,.41,.86),(.71,.61,.48,.93)} ς̇5 {(.11,.76,.42,.61),(.31,.79,.46,.72),(.23,.8,.57,.84)} Alternatives qP-F criteria c4-Local regulations ς̇1 {(.72,.91,.67,.75),(.69,.84,.69,.87),(.73,.87,.60,.81)} ς̇2 {(.60,.44,.76,.80),(.56,.51,.75,.77),(.63 .41,.68,.85)} ς̇3 {(.74,.56,.47,.80),(.81,.56,.53,.91),(.61,.60,.65,.89)} ς̇4 {(.43,.66,.77,.41),(.51,.68,.76,.36),(.53,.60,.87,.27)} ς̇5 {(.21,.65,.52,.61),(.27,.67,.41,.77)} 2. The pessimistic decision matrix qP-F in Table 6. 3. The criteria’s normalised weights are shown below, w1 = .2412, w2 = .2424, w3 = .2548, w4 = .2616, where ∑4 i=1wi = 1. 4. The weight pessimistic of the qP-F decision matrix is calculated in Table 7. K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3149 Table 6. The pessimistic qP-F decision matrix. Alternatives qP-F criteria c1-Solar Radiance ς̇1 {(.18,.73,.43,.67),(.32,.64,.49,.72),(.38,.66,.54,.64)} ς̇2 {(.46,.76,.45,.27),(.46,.76,.45,.27),(.56,.91,.36,.48)} ς̇3 {(.85,.37,.45,.59),(.72,.48,.72,.58),(.51,.64,.55,.32)} ς̇4 {(.21,.52,.34,.77),(.41,.61,.43,.78),(.42,.66,.39,.87)} ς̇5 {(.11,.33,.56,.61),(.11,.33,.56,.61),(.31,.41,.6,.73)} Alternatives qP-F criteria c2-Local Availability ς̇1 {(.75,.45,.67,.69),(.79,.37,.57,.69),(.79,.37,.57,.69) } ς̇2 {(.45,.7,.49,.86),(.56,.72,.66,.74),(.47,.62,.58,.72)} ς̇3 {(.46,.66,.71,.17),(.41,.77,.78,.19),(.48,.80,.83,.15)} ς̇4 {(.57,.64,.38,.57),(.51,.56,.59,.47),(.51,.56,.59,.47)} ς̇5 {(.24,.12,.81,.77),(.31,.25,.86,.73),(.23,.31,.8,.75)} Alternatives qP-F criteria c3-Environmental Impacts ς̇1 {(.82,.51,.67,.7),(.8,.64,.69,.8),(.7,.59,.68,.78)} ς̇2 {(.7,.56,.45,.29),(.73,.51,.37,.43),(.83,.63,.65,.54)} ς̇3 {(.74,.46,.61,.8),(.74,.46,.61,.8),(.81,.52,.59,.91)} ς̇4 {(.66,.55,.46,.88),(.61,.58,.41,.86),(.71,.61,.48,.93)} ς̇5 {(.11,.76,.42,.61),(.31,.79,.46,.72),(.23,.8,.57,.84)} Alternatives qP-F criteria c4-Local regulations ς̇1 {(.72,.91,.67,.75),(.69,.84,.69,.87),(.73,.87,.6,.81)} ς̇2 {(.60,.44,.76,.80),(.56,.51,.75,.77),(.63 .41,.68,.85)} ς̇3 {(.74,.56,.47,.80),(.81,.56,.53,.91),(.61,.60,.65,.89)} ς̇4 {(.43,.66,.77,.41),(.51,.68,.76,.36),(.53,.60,.87,.27)} ς̇5 {(.21,.65,.52,.61),(.21,.65,.54,.61),(.27,.67,.41,.77)} 5. The evaluation of the qP-F positive ideal solution (qP-FPIS) and qP-F negative ideal solution (qP-FNIS) is as follows: qP-FPIS = {{(.2050, .1833, .1351, .1857), (.1737, .1833, .1737, .1881), (.1351, .2195, .1447, .2098)}, {(.1818, .1697, .1963, .2085), (.1915, .1866, .2085, .1794), (.1915, .1939, .2012, .1818)}, {(.2089, .1936, .1707, .2242), (.2038, .2013, .1758, .2191), (.2115, .2038, .1733, .2370)}, {(.1936, .2381, .2014, .2093), (.2119, .2197, .1988, .2381), (.1910, .2276, .2276, .2328)}}, and K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3150 Table 7. The weighted pessimistic qP-F decision matrix. ς̇ qP-F criteria c1-Solar Radiance ς̇1 {(.0434,.1761,.1037,.1568),(.0724,.1544,.1182,.1688),(.0917,.1592,.1302,.1544)} ς̇2 {(.1110,.1833,.1085,.0651),(.1110,.1833,.1085,.0651),(.1351,.2195,.0868,.1158)} ς̇3 {(.2050,.0892,.1085,.1375),(.1737,.1158,.1737,.1399),(.1230,.1544,.1327,.0772)} ς̇4 {(.0507,.1254,.0820,.1857),(.0989,.1471,.1037,.1881),(.1013,.1592,.0941,.2098)} ς̇5 {(.0265,.0796,.1351,.1471),(.0265,.0796,.1351,.1471),(.0748,.0989,.1447,.1761)} ς̇ qP-F criteria c2-Local Availability ς̇1 {(.1818,.1091,.1624,.1673),(.1915,.0897,.1382,.1673),(.1915,.0897,.1382,.1673)} ς̇2 {(.1091,.1697,.1188,.2085),(.1357,.1745,.1600,.1794),(.1139,.1503,.1406,.1745)} ς̇3 {(.1115,.1600,.1721,.0412),(.0994,.1866,.1891,.0461),(.1164,.1939,.2012,.0364)} ς̇4 {(.1382,.1551,.0921,.1382),(.1236,.1357,.1430,.1139),(.1236,.1357,.1430,.1139)} ς̇5 {(.0582,.0291,.1963,.1866),(.0751,.0606,.2085,.1770),(.0558,.0751,.1939,.1818)} ς̇ qP-F criteria c3-Environmental Impacts ς̇1 {(.2089,.1299,.1707,.1784),(.2038,.1631,.1758,.2038),(.1784,.1503,.1733,.1987)} ς̇2 {(.1784,.1427,.1147,.0739),(.1860,.1299,.0943,.1096),(.2115,.1605,.1656,.1376)} ς̇3 {(.1886,.1172,.1554,.2038),(.1886,.1172,.1554,.2038),(.2064,.1325,.1503,.2319)} ς̇4 {(.1682,.1401,.1172,.2242),(.1554,.1478,.1045,.2191),(.1809,.1554,.1223,.2370)} ς̇5 {(.0280,.1936,.1070,.1554),(.0790,.2013,.1172,.1835),(.0586,.2038,.1452,.2140)} ς̇ qP-F criteria c4-Local regulations ς̇1 {(.1884,.2381,.1753,.1962),(.1805,.2197,.1805,.2276),(.1910,.2276,.1570,.2119)} ς̇2 {(.1570,.1151,.1988,.2093),(.1465,.1334,.1962,.2014),(.1648,.1073,.1779,.2224)} ς̇3 {(.1936,.1465,.1230,.2093),(.2119,.1465,.1386,.2381),(.1596,.1570,.1700,.2328)} ς̇4 {(.1125,.1727,.2014,.1073),(.1334,.1779,.1988,.0942),(.1386,.1570,.2276,.0706)} ς̇5 {(.0549,.1700,.1360,.1596),(.0549,.1700,.1413,.1596),(.0706,.1753,.1073,.2014)} qP-FNIS = {{(.0265, .0796, .0820, .0651), (.0265, .0796, .1037, .0651), (.0748, .0989, .0868, .0772)}, {(.0582, .0291, .0921, .0412), (.0751, .0606, .1430, .0461), (.0558, .0751, .1406, .0364)}, {(.0280, .1172, .1070, .0739), (.0790, .1172, .0943, .1096), (.0586, .1325, .1223, .1376)}, {(.0549, .1151, .1230, .1073), (.0549, .1334, .1386, .0942), (.0706, .1073, .1073, .0706)}}, 6. Using (7) and (8), the qP-F Euclidean distance of each alternative ς̇j from qFPIS and qP-FPIS are calculated as: D ′ E(ς̇1, qP-FPIS) = .0951, D ′ E(ς̇2, qP-FPIS) = .1293, D ′ E(ς̇3, qP-FPIS) = .1229, D ′ E(ς̇4, qP-FPIS) = .1311, K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3129-3155 3151 D ′ E(ς̇5, qP-FPIS) = .1786, and D ′ E(ς̇1, qP-FNIS) = .1817, D ′ E(ς̇2, qP-FNIS) = .1627, D ′ E(ς̇3, qP-FNIS) = .1740, D ′ E(ς̇4, qP-FNIS) = .1461, D ′ E(ς̇5, qP-FNIS) = .1218. Using Equation (9), the relative qP-F closeness coefficients Ej′ are calculated as: E1′ = .6565, E2′ = .5572, E3′ = .5860, E4′ = .5270, E5′ = .4055. According to the foregoing computations, the final ranking of power plant selection is as follows: ς̇1 > ς̇3 > ς̇2 > ς̇4 > ς̇5. Hence, solar power plant station ς̇1 is selected in the rural area. As a result of the evaluation among the alternatives, the most risky structure is ς̇1. The positive ideal solution, negative ideal solution and ranking of alternatives based on closeness coefficients is shown in Figure 1. Fig. 1: Positive ideal solution, Negative ideal solution and Ranking of alternatives REFERENCES 3152 7. Conclusions Quadri-polar structures are often preferred over clear-cut circumstances. Various forms of information can be used to manipulate the membership degrees to facilitate the handling of quadri-polar fuzzy information. We have proposed a novel class of generalized qP- FFI(s) of ℵ̃ called, a qP-(ϖ,ϑ)-FFI(s) and compared to the generalizations of present fuzzy sets, it is shown to be a more flexible approach that can be evaluated in quadri- polar ways based on practical interests and requirements. We defined and analyzed the concept of qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFI(s). Moreover, we presented various characterizations of qP-(∈σ̃,∈σ̃ ∨qτ̃ )-FFIs. It is used to manage data that includes quadri-polar information suggested by decision-makers. 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