EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 1113-1128 ISSN 1307-5543 – ejpam.com Published by New York Business Global Fermatean Neutrosophic INK-Algebras Anas Al-Masarwah 1, M. Kaviyarasu2,∗, Kholood Alnefaie3, M. Rajeshwari 4 1 Department of Mathematics, Faculty of Science, Ajloun National University, P. O. Box 43, Ajloun 26810, Jordan 2 Department of Mathematics, Vel Tech Rangarajan Dr Sagunthala R&D Institute of Science and Technology, Chennai, Tamil Nadu, India-600 062 3 Department of Mathematics, College of Science, Taibah University, Madinah 42353, Saudi Arabia 4 Department of Mathematics, Presidency University, Bangalore, 560064, India. Abstract. This paper introduces the concept of the direct product of sets that involve Fermatean neutrosophic (FN) elements in structures called INK-algebras. It defines terms like the direct product of Fermatean neutrosophic INK-ideals (FNINK-Is) in INK-algebras and Fermatean neu- trosophic sets (FNSs), FNINK-Is, and Fermatean neutrosophic closed INK-ideals (FNCINK-Is). The proof of theorems illustrating the relationships between these ideas is included in the paper. It also defines the INK-subalgebra embedded in an INK-algebra and gives a theorem elucidating the connection between the direct product of FNINK-Is and the images of these subalgebras. In essence, the paper investigates and establishes connections between different mathematical ideas concerning INK-algebras and FNSs. 2020 Mathematics Subject Classifications: 03E72, 03G25, 28E10, 03B52 Key Words and Phrases: INK-algebra, Direct product, Fermatean neutrosophic set, Fermatean neutrosophic INK-ideal, Fermatean neutrosophic closed INK-ideal 1. Introduction Zadeh [1] seminal work on fuzzy sets established the basis for fuzzy logic. Fuzzy sets provide a more flexible representation of uncertainty by assigning degrees of membership to elements. Atanassov [2] paper explores intuitionistic fuzzy sets, which include degrees of membership and degrees of non-membership. This gives a more complete picture of uncertainty in decision-making. Neutrosophic logic has been introduced by Smarandache which involves various disciplines of philosophy and mathematics that studies indetermi- nacy, uncertainty, and contradictions. Neutrosophic logic is a three-valued logic system ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5090 Email addresses: anas.almasarwah@anu.edu.jo (A. Al-Masarwah), kavitamilm@gmail.com (M. Kaviyarasu), knefaie@taibahu.edu.sa (K. Alnefaie), rajeshwari@presidencyuniversity.in (M. Rajeshwari) https://www.ejpam.com 1113 © 2024 EJPAM All rights reserved. M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1113-1128 1114 that includes the truth values ”true” and ”false,” as well as a third value termed indeter- minate, which represents uncertain or ambiguous information. This method is especially beneficial for dealing with difficulties involving inadequate or inconsistent data, which are widespread in domains such as artificial intelligence, decision making, philosophy, and cognitive science. Abdel-Bassset et al. [3] present a decision-making framework for professional selec- tion based on bipolar neutrosophic sets. The approach seeks to address uncertainty and imprecision in decision-making processes, his work expands on intuitionistic fuzzy sets to include neutrosophic sets, allowing for indeterminacy, uncertainty, and contradictory information [4]. Jun [5] research on neutrosophic subalgebras in BCK/BCI-algebras ad- vances our understanding of algebraic structures with neutrosophic elements. Kaviyarasu et al. [6] investigates fuzzy subalgebras and fuzzy INK-ideals in INK-algebras, providing insights into the integration of fuzzy logic in algebraic structures. Additionally, Kaviyarasu and Indhira present a review of BCI/BCK-algebras and discuss their development, con- tributing to the understanding of these specific algebraic structures [7], investigate fuzzy p-ideals in INK-algebras [8], adding to the understanding of fuzzy ideals in the context of specific algebraic structures. Jun et al.[9] collaboration aims to integrate neutrosophic N-structures into BCK/BCI-algebras. Jun et al.[10] investigate neutrosophic positive implicative N-ideals in the context of BCK-algebras, advancing our understanding of neutrosophic structures in algebraic sys- tems. Ozturk and Jun [11] investigates neutrosophic ideals in BCK/BCI algebras based on neutrosophic points, broadening the application of neutrosophic concepts to algebraic structures. Songsaeng and Iampan [12] introduce neutrosophic set theory to UP-algebra and demonstrate its utility in a specific algebraic context.Kaviyarasu, Indhira and Chan- drasekaran [[13], [14], [15]] investigate the direct product of intuitionistic fuzzy INK-ideals, providing insights into the interaction of different algebraic structures; discuss intuitionistic fuzzy translation in INK-Algebra, contributing to the understanding of translation oper- ations in algebraic structures; and apply neutrosophic sets in INK-Algebra, extending the study of neutrosophic concepts to a specific algebraic context. As an extension of partial algebra, Smarandache [16] presents the theory of neutro algebra, which advances the devel- opment of algebraic structures in addition to neutro and anti-algebraic structures, which provide additional insights into mathematical structures. Abdel-Basset et al. proposed a novel plithogenic model for supply chain problem solving, which incorporates neutrosophic and plithogenic sets into optimization theory [17]. Making contributions to the fields of en- vironmental technology and innovation, Mohamed and Abdel [[18], [19], [20]] introduce an integrated plithogenic MCDM approach for assessing the financial performance of manu- facturing industries, utilizing a combination of mathematical decision-making methods, as well as a novel framework for assessing the innovation value proposition for smart product- service systems. Neutrosophic Vague Binary BCK/BCI-algebra, which explores the use of vague and neutrosophic notions in a binary algebraic framework, is covered by Remya and Francina Shalini [21]. Muralikrishna and Manokaran [22] introduce MBJ- neutrosophic B-ideals in B-algebras, which adds to the understanding of neutrosophic ideals in specific algebraic structures. For more results on algebraic structures with uncertainty (see works M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1113-1128 1115 by the authors of [23–27]. This paper presents a new concept based on two distinct sets, called FNSs, and inves- tigates their direct product in the framework of INK-algebra. Specifically, it looks at the relation between FNINK-Ss and FNINK-Is, as well as the conditions that hold for this relation. 1.1. Motivation • It aims to provide a new perspective on FN elements in mathematical structures. • Ensures a systematic and coherent discussion of these mathematical concepts. • Validating the proposed connections rigorously through theorems adds credibility and reliability. • Exploration of Substructures contributes to a more comprehensive understanding of the intricate relationships within the algebraic framework. 1.2. Novelty • The paper aims to present a novel mathematical framework by introducing the con- cept of the direct product, which involves sets with FN elements within the domain of INK-algebras. • The goal is to create a precise mathematical language by defining terms like direct product of FNINK-Is, FN-Ss and FNCINK-Is, which will improve discourse clarity. • The study illuminates complex mathematical relationships through rigorous theorem proofs, delving into interconnected ideas in INK-algebras and FN-Ss. 1.3. Structure of the paper The paper begins with an introductory exploration of the novel concept of the di- rect product within structures known as INK-algebras, which include sets enriched with FN elements. It then defines key terms like the direct product of FNINKs, FN-Ss and FNCINK-Is. The narrative then proceeds to provide a comprehensive exposition, including proofs of theorems that intricately illustrate the relationships between these defined con- cepts. Furthermore, it broadens its scope to define an INK-subalgebra embedded within an INK-algebra, as well as a theorem that explains the relationship between the direct product of FNINKs and the images of these sub algebras. In essence, the paper cul- minates in a thorough investigation, establishing connections and interrelations between diverse mathematical ideas concerning both INK-algebras. M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1113-1128 1116 2. Basic Definitions In the beginning the research, the definition and beneficial properties of INK-algebras will be explained. Definition 1 ([15]). An INK-algebra is a mathematical structure with specific rules; it is represented by the notation (χ, ·, 0). For any elements ϑ, η, z ∈ χ (1) ((ϑ · η) · (ϑ · z)) · (z · η) = 0, (2) ((ϑ · z) · (η · z)) · (ϑ · η) = 0, (3) (ϑ · 0) = 0, (4) (ϑ · η) = 0 and η · ϑ = 0 imply η = ϑ. The operation · denotes a binary operation and 0 is a constant belonging to the set χ. Definition 2 ([6]). A non-empty subset S of a INK-algebra (χ, ·, 0) is considered as an INK-subalgebra of χ, if for every elements ϑ and η ∈ χ, the result of the operation (ϑ · η) is also an element of S. Definition 3 ([6]). Let (χ, ·, 0) be an INK-algebra. An ideal of χ is defined as a nonempty subset ℑ of χ such that it satisfies the following conditions, ∀ϑ, η ∈ χ (1) 0 ∈ ℑ, (2) (ϑ · η) ∈ ℑ and η ∈ ℑ imply ϑ ∈ ℑ. Definition 4 ([6]). Let an INK-algebra χ have a non-empty subset ℑ. If all of the following hold for every ϑ, η, z ∈ χ, then ℑ is called an INK-ideal of χ. (1) 0 ∈ ℑ, (2) (z · ϑ) · (z · η) ∈ ℑ and z ∈ ℑ imply ϑ ∈ ℑ. Definition 5 ([22]). The structure of a FNS M defined on a nonempty set χ can be expressed as: M = {〈 ϑ, ρTM(ϑ), ρIM(ϑ), ρFM(ϑ) 〉 |ϑ ∈ χ } , where ρT : χ → [0, 1] is a mem- bership function ρI : χ → [0, 1] is a indeterminate membership function and ρF : χ → [0, 1] is a non-membership function and these three functions are satisfying the inequalities; 0 ≤ (ρTM(ϑ))3+(ρFM(ϑ))3 ≤ 1, 0 ≤ (ρIM(ϑ))3 ≤ 1 and 0 ≤ (ρTM(ϑ))3+(ρIM(ϑ))3+(ρFM(ϑ))3 ≤ 2. Here, ρTM(ϑ) and ρFM(ϑ) are dependent components and ρIM(ϑ) is an independent com- ponent. Throughout the current research article, we shall use M = 〈 ρTM, ρIM, ρFM 〉 for the FNS M = {〈 ϑ, ρTM(ϑ), ρIM(ϑ), ρFM(ϑ) 〉 |ϑ ∈ χ } . Definition 6 ([22]). If M = {〈 ρTM(ϑ), ρIM(ϑ), ρFM(ϑ) 〉} and N = {〈 ρTN(ϑ), ρ I N(ϑ), ρ F N(ϑ) 〉} be two FNSs, then ∀ϑ ∈ χ M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1113-1128 1117 (i) M = {〈 1− ρTM(ϑ), 1− ρIM(ϑ), 1− ρFM(ϑ) 〉} (ii) M ∩N = {〈 min{ρTM(ϑ), ρTN(ϑ)},max{ρIM(ϑ), ρIN(ϑ)},max{ρFM(ϑ), ρFN(ϑ)} 〉} . Definition 7. A FNS M of χ obtains the title of FNINK-Ss by satisfying the requirements, ∀ ϑ, η ∈ χ (1) ρT(ϑ · η) ≤ min { ρT(ϑ), ρT(η) } , (2) ρI(ϑ · η) ≥ max { ρI(ϑ), ρI(η) } , (3) ρF(ϑ · η) ≥ max { ρF(ϑ), ρF(η) } . Definition 8. A FNS M of χ is considered as a FN-I if it meets the described conditions, ∀ ϑ, η ∈ χ (1) ρT(0) ≤ ρT(ϑ), ρI(0) ≥ ρI(ϑ), ρF(0) ≥ ρF(ϑ), (2) ρT(ϑ) ≤ min { ρT(ϑ · η), ρT(η) } , (3) ρI(ϑ) ≥ max { ρI(ϑ · η), ρI(η) } , (4) ρF(ϑ) ≥ max { ρF(ϑ · η), ρF(η) } . Example 1. If χ = {0, x, y, z} is a set with a binary operation · given by the following Table: Table 1: The operation · · 0 x y z 0 0 x y z x x 0 z y y y z 0 x z z y x 0 Thus, (χ.·, 0) is an INK-algebra. Consider a FNS M in χ, where ρTM(0) = 0.8, ρTM(x) = 0.4, ρTM(y) = ρTM(z) = 0.2, ρIM(0) = 0.7, ρIM(x) = 0.5, ρIM(y) = ρIM(z) = 0.3 and ρFM(0) = 0.1, ρFM(x) = ρFM(y) = 0.4, ρFM(z) = 0.3. Then, M is a FN-I of χ, which is easily verified. Definition 9. A FNs M of χ is considered as a FNINK-I of χ if it meets the described conditions, ∀ ϑ, η, z ∈ χ (1) ρT(0) ≤ ρT(ϑ), ρI(0) ≥ ρI(ϑ), ρF(0) ≥ ρF(ϑ), (2) ρT(ϑ) ≤ min { ρT((z · ϑ) · (z · η)), ρT(η) } , (3) ρI(ϑ) ≥ max { ρI((z · ϑ) · (z · η)), ρI(η) } , (4) ρF(ϑ) ≥ max { ρF((z · ϑ) · (z · η)), ρF(η) } . M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1113-1128 1118 3. Formation of Direct Product: FNINK-Ss and FNINK-Is Definition 10. INK-algebras χ1 and χ2 contain two FNSs, M and N. The structure M×N = 〈 ρT(M×N), ρ I (M×N), ρ F (M×N) 〉 is defined as the direct product of FNSs M and N, specified by, ∀(ϑ, η) ∈ χ1 × χ2 (1) ρT(M×N)(ϑ, η) = min { ρTM(ϑ), ρTN(η) } , (2) ρI(M×N)(ϑ, η) = max { ρIM(ϑ), ρIN(η) } , (3) ρF(M×N)(ϑ, η) = max { ρFM(ϑ), ρFN(η) } . Definition 11. The direct product of FNINK-Ss of χ1 × χ2 is a FNSs M×N = 〈 ρT(M×N), ρ I (M×N), ρ F (M×N) 〉 of χ1 and χ2 if, ∀(ϑ1, η1), (ϑ2, η2) ∈ χ1 × χ2 (1) ρT(M×N)((ϑ1, η1) · (ϑ2, η2)) ≤ min { ρT(M×N)(ϑ1, η1), ρ T (M×N)(ϑ2, η2) } , (2) ρI(M×N)((ϑ1, η1) · (ϑ2, η2)) ≥ max { ρI(M×N)(ϑ1, η1), ρ I (M×N)(ϑ2, η2) } , (3) ρF(M×N)((ϑ1, η1) · (ϑ2, η2)) ≥ max { ρF(M×N)(ϑ1, η1), ρ F (M×N)(ϑ2, η2) } . Definition 12. The direct product of FNINK-I of χ1 × χ2 is a FNSs M×N = 〈 ρT(M×N), ρ I (M×N), ρ F (M×N) 〉 of χ1 and χ2 if ,∀(ϑ1, η1), (ϑ2, η2), (ϑ3, η3) ∈ χ1 × χ2 (1) ρT(M×N)(0, 0) ≤ ρT(M×N)(ϑ, η), ρ I (M×N)(0, 0) ≥ ρI(M×N)(ϑ, η), ρ F (M×N)(0, 0) ≥ ρFM×N(ϑ, η), (2) ρT(M×N)(((ϑ1, η1) ≤ min { ρTM×N(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ T (M×N)(ϑ2, η2) } , (3) ρI(M×N)(ϑ1, η1) ≥ max { ρI(M×N)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ I (M×N)(ϑ2, η2) } , (4) ρF(M×N)(ϑ1, η1) ≥ max { ρF(M×N)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ F (M×N)(ϑ2, η2) } . Definition 13. The direct product of FNCINK-I of χ1 × χ2 is a FNSs M×N = 〈 ρT(M×N), ρ I (M×N), ρ F (M×N) 〉 of χ1 and χ2 if it meets ((2) , (3) and (4) of Definition 12) and the following inequalities, ∀(ϑ, η) ∈ χ1 × χ2 M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1113-1128 1119 (1) ρT(M×N)((0, 0) · (ϑ, η)) ≤ ρT(M×N)(ϑ, η), (2) ρI(M×N)((0, 0) · (ϑ, η)) ≥ ρI(M×N)(ϑ, η), (3) ρF(M×N)((0, 0) · (ϑ, η)) ≥ ρF(M×N)(ϑ, η). Theorem 1. Let M = 〈 ρTM, ρIM, ρFM 〉 and N = 〈 ρTN, ρ I N, ρ F N 〉 be two FNINK-Ss of χ1 and χ2, respectively. Then, the direct product M×N, defined by M×N = 〈 ρT(M×N), ρ I (M×N), ρ F (M×N) 〉 , is a FNINK-S of χ1 × χ2. Proof. Assume that M and N are two FNINK-Ss. Let (ϑ1, η1), (ϑ2, η2) ∈ χ1 × χ2. Then, ρT(M×N)((ϑ1, η1) · (ϑ2, η2)) = { ρTM×N(ϑ1 · ϑ2), (η1 · η2) } = min { ρTM(ϑ1 · ϑ2), ρ T N(η1 · η2) } ≤ min { min { ρTM(ϑ1), ρ T M(ϑ2) } ,min { ρTN(η1), ρ T M(η2) }} = min { min { ρTM(ϑ1), ρ T N(η1) } ,min { ρTM(ϑ2), ρ T N(η2) }} = min { ρT(M×N)(ϑ1, η1), ρ T (M×N)(ϑ2, η2) } , ρI(M×N)((ϑ1, η1) · (ϑ2, η2)) = { ρIM×N(ϑ1 · ϑ2), (η1 · η2) } = max { ρIM(ϑ1 · ϑ2), ρ I N(η1 · η2) } ≥ max { max { ρIM(ϑ1), ρ I M(ϑ2) } ,max { ρIN(η1), ρ I M(η2) }} = max { max { ρIM(ϑ1), ρ I N(η1) } ,max { ρIM(ϑ2), ρ I N(η2) }} = max { ρI(M×N)(ϑ1, η1), ρ I (M×N)(ϑ2, η2) } , and ρF(M×N)((ϑ1η1) · (ϑ2, η2)) = { ρFM×N(ϑ1 · ϑ2), (η1 · η2) } = max { ρFM(ϑ1 · ϑ2), ρ F N(η1 · η2) } ≥ max { max { ρFM(ϑ1), ρ F M(ϑ2) } ,max { ρFN(η1), ρ F M(η2) }} = max { max { ρFM(ϑ1), ρ F N(η1) } ,max { ρFM(ϑ2), ρ F N(η2) }} M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1113-1128 1120 = max { ρF(M×N)(ϑ1, η1), ρ F (M×N)(ϑ2, η2) } . Hence, M×N = 〈 ρT(M×N), ρ I (M×N), ρ F (M×N) 〉 is a FNINK-S of χ1 × χ2. Theorem 2. Let M = 〈 ρTM, ρIM, ρFM 〉 and N = 〈 ρTN, ρ I N, ρ F N 〉 be two FNINK-Is of χ1 and χ2, respectively. Then, the direct product M×N, defined by M×N = 〈 ρT(M×N), ρ I (M×N), ρ F (M×N) 〉 , is a FNINK-I of χ1 × χ2. Proof. For any (ϑ, η) ∈ χ1 × χ2. We have ρT(M×N)(0, 0) = min { ρTM(0), ρTN(0) } ≤ min { ρTM(ϑ), ρTN(η) } = ρI(M×N)(ϑ, η), ρIM×N(0, 0) = max { ρIM(0), ρIN(0) } ≥ max { ρIM(ϑ), ρIN(η) } = ρIM×N(ϑ, η) and ρFM×N(0, 0) = max { ρFM(0), ρFN(0) } ≥ max { ρFM(ϑ), ρFN(η) } = ρFM×N(ϑ, η). Also, for any (ϑ1, η1), (ϑ2, η2), (ϑ3, η3) ∈ χ1 × χ2. We have ρTM×N(ϑ1, η1) = min { ρTM(ϑ1), ρ T N(η1) } ≤ min { min { ρTM((ϑ3 · ϑ1) · (ϑ3 · ϑ2)), ρ T M(ϑ2) } ,min { ρTN((η3 · η1) · (η3 · η2)), ρTN(η2) }} = min { min { ρTM((ϑ3 · ϑ1) · (ϑ3 · ϑ2)), ρ T N((η3 · η1) · (η3 · η2)) } ,min { ρTM(ϑ2), ρ T N(η2) }} = min { ρTM×N(((ϑ3, η3) · (ϑ1, η1)), ((ϑ3, η3) · (ϑ2, η2))), ρ T M×N(ϑ2, η2) } , ρIM×N(ϑ1, η1) = max { ρIM(ϑ1), ρ I N(η1) } M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1113-1128 1121 ≥ max { max { ρIM((ϑ3 · ϑ1) · (ϑ3 · ϑ2)), ρ I M(ϑ2) } ,max { ρIN((η3 · η1) · (η3 · η2)), ρIN(η2) }} = max { max { ρIM((ϑ3 · ϑ1) · (ϑ3 · ϑ2)), ρ I N((η3 · η1) · (η3 · η2)) } ,min { ρIM(ϑ2), ρ I N(η2) }} = max { ρIM×N(((ϑ3, η3) · (ϑ1, η1)), ((ϑ3, η3) · (ϑ2, η2))), ρ I M×N(ϑ2, η2) } and ρFM×N(ϑ1, η1) = max { ρFM(ϑ1), ρ F N(η1) } ≥ max { max { ρFM((ϑ3 · ϑ1) · (ϑ3 · ϑ2)), ρ F M(ϑ2) } ,max { ρFN((η3 · η1) · (η3 · η2)), ρ F N(η2) }} = max { max { ρFM((ϑ3 · ϑ1) · (ϑ3 · ϑ2)), ρ F N((η3 · η1) · (η3 · η2)) } ,min { ρFM(ϑ2), ρ F N(η2) }} = max { ρFM×N(((ϑ3, η3) · (ϑ1, η1)), ((ϑ3, η3) · (ϑ2, η2))), ρ F M×N(ϑ2, η2) } Hence, M×N = 〈 ρT(M×N), ρ I (M×N), ρ F (M×N) 〉 is a FNINK-I of χ1 × χ2. Theorem 3. Let M = 〈 ρTM, ρIM, ρFM 〉 and N = 〈 ρTN, ρ I N, ρ F N 〉 be two FNCINK-Is of χ1 and χ2, respectively. Then, the direct product M×N, defined by M×N = 〈 ρT(M×N), ρ I (M×N), ρ F (M×N) 〉 , is a FNCINK-I of χ1 × χ2. Proof. By applying Theorem 2, the FNS M × N = 〈 ρT(M×N), ρ I (M×N), ρ F (M×N) 〉 is a FNINK-I of χ1 × χ2. Now, ∀(ϑ, η) ∈ χ1 × χ2, we have ρTM×N((0, 0) · (ϑ, η)) ≤ ρTM×N((0 · ϑ), (0 · η)) = min { ρTM(0 · ϑ), ρTN(0 · η) } ≤ min { ρTM(ϑ), ρTN(η) } = ρTM×N(ϑ, η), ρIM×N((0, 0) · (ϑ, η)) ≥ ρIM×N((0 · ϑ), (0 · η)) = max { ρIM(0 · ϑ), ρIN(0 · η) } ≥ max { ρIM(ϑ), ρIN(η) } = ρIM×N(ϑ, η) and ρFM×N((0, 0) · (ϑ, η)) ≥ ρFM×N((0 · ϑ), (0 · η)) M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1113-1128 1122 = max { ρFM(0 · ϑ), ρFN(0 · η) } ≥ max { ρFM(ϑ), ρFN(η) } = ρFM×N(ϑ, η). Hence, M×N = 〈 ρT(M×N), ρ I (M×N), ρ F (M×N) 〉 is a FNCINK-I of χ1 × χ2. Theorem 4. Let M = 〈 ρTM, ρIM, ρFM 〉 and N = 〈 ρTN, ρ I N, ρ F N 〉 be two FNINK-Is of χ1 and χ2, respectively. Then, M × N = 〈 ρT(M×N), ρ I (M×N), ρ T (M×N) 〉 is a FNINK-I of χ1 × χ2, where ρT(M×N) = 1− ρT(M×N). Proof. According to Theorem 2. M×N = 〈 ρT(M×N), ρ I (M×N), ρ F (M×N) 〉 is a FNINK-I of χ1 × χ2. Then, ρT(M×N)(0, 0) ≤ ρT(M×N)(ϑ, η) 1− ρTM×N(0, 0) ≥ 1− ρT(M)×N)(ϑ, η) ρTM×N(0, 0) ≥ ρT(M×N)(ϑ, η). Now, for any (ϑ1, η1), (ϑ2, η2), (ϑ3, η3) ∈ χ1 × χ2. We have ρT(M×N)(ϑ1, η1) ≤ min { ρTM×N(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ T (M×N)(ϑ2, η2) } 1− ρT(M×N)(ϑ1, η1) ≥ 1−min { ρT(M×N)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ T (M×N)(ϑ2, η2) } ρT(M×N)(ϑ1, η1) ≥ max { 1− ρT(M×N)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), 1− ρT(M×N)(ϑ2, η2) } ρT(M×N)(ϑ1, η1) ≥ max { ρT(M×N)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ T (M×N)(ϑ2, η2) } . Hence, M×N = 〈 ρT(M×N), ρ I (M×N), ρ T (M×N) 〉 is a FNINK-I of χ1 × χ2. Theorem 5. Let M = 〈 ρTM, ρIM, ρFM 〉 and N = 〈 ρTN, ρ I N, ρ F N 〉 be two FNINK-Is of χ1 and χ2, respectively. Then, M × N = 〈 ρF(M×N), ρ I (M×N), ρ F (M×N) 〉 is a FNINK-I of χ1 × χ2, where ρF(M×N) = 1− ρF(M×N). Proof. According to Theorem 2. M×N = 〈 ρT(M×N), ρ I (M×N), ρ F (M×N) 〉 is a FNINK-I of χ1 × χ2. Then, ρF(M×N)(0, 0) ≥ ρF(M×N)(ϑ, η) 1− ρFM×N(0, 0) ≤ 1− ρF(M)×N)(ϑ, η) M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1113-1128 1123 ρFM×N(0, 0) ≤ ρF(M×N)(ϑ, η). Now, for any (ϑ1, η1), (ϑ2, η2), (ϑ3, η3) ∈ χ1 × χ2. We have ρF(M×N)(ϑ1, η1) ≥ max { ρFM×N(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ F (M×N)(ϑ2, η2) } 1− ρF(M×N)(ϑ1, η1) ≤ 1−max { ρF(M×N)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ F (M×N)(ϑ2, η2) } ρF(M×N)(ϑ1, η1) ≤ min { 1− ρF(M×N)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), 1− ρF(M×N)(ϑ2, η2) } ρF(M×N)(ϑ1, η1) ≤ min { ρF(M×N)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ F (M×N)(ϑ2, η2) } . Hence, M×N = 〈 ρF(M×N), ρ I (M×N), ρ F (M×N) 〉 is a FNINK-I of χ1 × χ2. Theorem 6. Let M = 〈 ρTM, ρIM, ρFM 〉 and N = 〈 ρTN, ρ I N, ρ F N 〉 be two FNINK-Is of χ1 and χ2, respectively. Then, M × N = 〈 ρF(M×N), ρ I (M×N), ρ T (M×N) 〉 is a FNINK-I of χ1 × χ2, where ρT(M×N) = 1− ρT(M×N) and ρF(M×N) = 1− ρF(M×N). Proof. The proof is produced by using Theorem 4 and Theorem 5 together. Theorem 7. Let M ×N = 〈 ρT(M×N), ρ I (M×N), ρ F (M×N) 〉 be a FNINK-I of χ1 × χ2. Then, (M×N)s = 〈 ρT(M×N)s , ρ I (M×N)s , ρ F (M×N)s 〉 is a FNINK-I of χ1 × χ2. Proof. For any (ϑ, η) ∈ χ1 × χ2. Then, ρT(M×N)(0, 0) ≤ ρT(M×N)(ϑ, η){ ρT(M×N)(0, 0) }s ≤ { ρT(M×N)(ϑ, η) }s { ρT(M×N)(0, 0) s } ≤ { ρT(M×N)(ϑ, η) s } { ρT(M×N)s(0, 0) } ≤ { ρT(M×N)s(ϑ, η) } , ρI(M×N)(0, 0) ≥ ρI(M×N)(ϑ, η){ ρI(M×N)(0, 0) }s ≥ { ρI(M×N)(ϑ, η) }s { ρI(M×N)(0, 0) s } ≥ { ρI(M×N)(ϑ, η) s } { ρI(M×N)s(0, 0) } ≥ { ρI(M×N)s(ϑ, η) } and ρF(M×N)(0, 0) ≥ ρF(M×N)(ϑ, η) M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1113-1128 1124{ ρF(M×N)(0, 0) }s ≥ { ρF(M×N)(ϑ, η) }s { ρF(M×N)(0, 0) s } ≥ { ρF(M×N)(ϑ, η) s } { ρF(M×N)s(0, 0) } ≥ { ρF(M×N)s(ϑ, η) } . If (ϑ1, η1), (ϑ2, η2) and (ϑ3, η3) ∈ χ1 × χ2, then{ ρT(M×N)(ϑ1, η1) }s ≤ min { ρT(M×N)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ T (M×N)(ϑ2, η2) }s { ρT(M×N)(ϑ1, η1) s } ≤ min { ρT(M×N)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))) s, ρT(M×N)(ϑ2, η2) s } { ρT(M×N)s(ϑ1, η1) } ≤ min { ρT(M×N)s(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ T (M×N)s(ϑ2, η2) } , { ρI(M×N)(ϑ1, η1) }s ≥ max { ρI(M×N)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ I (M×N)(ϑ2, η2) }s { ρI(M×N)(ϑ1, η1) s } ≥ max { ρI(M×N)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))) s, ρI(M×N)(ϑ2, η2) s } { ρI(M×N)s(ϑ1, η1) } ≥ max { ρI(M×N)s(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ I (M×N)s(ϑ2, η2) } and{ ρF(M×N)(ϑ1, η1) }s ≥ max { ρF(M×N)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ F (M×N)(ϑ2, η2) }s { ρF(M×N)(ϑ1, η1) s } ≥ max { ρF(M×N)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))) s, ρF(M×N)(ϑ2, η2) s } { ρF(M×N)s(ϑ1, η1) } ≥ max { ρF(M×N)s(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ F (M×N)s(ϑ2, η2) } . Hence, (M×N)s = 〈 ρT(M×N)s , ρ I (M×N)s , ρ F (M×N)s 〉 is a FNINK-I of χ1 × χ2. Theorem 8. Let M×N = 〈 ρT(M×N), ρ I (M×N), ρ F (M×N) 〉 and D× E = 〈 ρT(D×E), ρ I (D×E), ρ F (D×E) 〉 be two FNINK-Is of χ1 × χ2. Then, (M×N) ∩ (D× E) = 〈 ρT(M×N)∩(D×E), ρ I (M×N)∩(D×E), ρ F (M×N)∩(D×E) 〉 is a FNINK-I of χ1 × χ2. Proof. Since M×N and D×E are two FNINK-Is of χ1×χ2. Then, ∀(ϑ, η) ∈ χ1×χ2, we have ρT(M×N)∩(D×E)(0, 0) = min { ρT(M×N)(0, 0), ρ T (D×E)(0, 0) } M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1113-1128 1125 ≤ min { ρT(M×N)(ϑ, η), ρ T (D×E)(ϑ, η) } = ρT(M×N)∩(D×E)(ϑ, η), ρI(M×N)∩(D×E)(0, 0) = max { ρI(M×N)(0, 0), ρ I (D×E)(0, 0) } ≥ max { ρI(M×N)(ϑ, η), ρ I (D×E)(ϑ, η) } = ρI(M×N)∩(D×E)(ϑ, η) and ρF(M×N)∩(D×E)(0, 0) = max { ρF(M×N)(0, 0), ρ I (D×E)(0, 0) } ≥ max { ρF(M×N)(ϑ, η), ρ I (D×E)(ϑ, η) } = ρF(M×N)∩(D×E)(ϑ, η). Now, for any (ϑ1, η1), (ϑ2, η2) and (ϑ3, η3) ∈ χ1 × χ2, we have ρT(M×N)∩(D×E)(ϑ1, η1) = min { ρT(M×N)(ϑ1, η1), ρ T (D×E)(ϑ1, η1) } ≤ min { min { ρT(M×N)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ T (M×N)(ϑ2, η2) } , min { ρT(D×E)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ T (D×E)(ϑ2, η2) }} = min { min { ρT(M×N)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρT(D×E)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))) } ,min { ρT(M×N)(ϑ2, η2), ρ T (D×E)(ϑ2, η2) }} = min { ρT(M×N)∩(D×E)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ T (M×N)∩(D×E)(ϑ2, η2) } , ρI(M×N)∩(D×E)(ϑ1, η1) = max { ρI(M×N)(ϑ1, η1), ρ I (D×E)(ϑ1, η1) } ≥ max { max { ρI(M×N)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ I (M×N)(ϑ2, η2) } , max { ρI(D×E)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ I (D×E)(ϑ2, η2) }} = max { max { ρI(M×N)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1113-1128 1126 ρI(D×E)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))) } ,max { ρT(M×N)(ϑ2, η2), ρ I (D×E)(ϑ2, η2) }} = max { ρI(M×N)∩(D×E)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ I (M×N)∩(D×E)(ϑ2, η2) } , and ρF(M×N)∩(D×E)(ϑ1, η1) = max { ρF(M×N)(ϑ1, η1), ρ F (D×E)(ϑ1, η1) } ≥ max { max { ρF(M×N)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ F (M×N)(ϑ2, η2) } , max { ρF(D×E)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ F (D×E)(ϑ2, η2) }} = max { max { ρF(M×N)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρF(D×E)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))) } ,max { ρT(M×N)(ϑ2, η2), ρ F (D×E)(ϑ2, η2) }} = max { ρF(M×N)∩(D×E)(((ϑ3, η3) · (ϑ1, η1)) · ((ϑ3, η3) · (ϑ2, η2))), ρ F (M×N)∩(D×E)(ϑ2, η2) } . Hence, (M×N)∩ (D×E) = 〈 ρT(M×N)∩(D×E), ρ I (M×N)∩(D×E), ρ F (M×N)∩(D×E) 〉 is a FNINK-I of χ1 × χ2. 4. Comparison Analysis A common ground between FNINK-Algebras and Neutrosophic INK-Algebras is INK- algebra, which emphasizes the integration of non-membership, indeterminacy, and uncer- tainty in algebraic structures. Both approaches are intended for complex system modelling and analysis, where a high prevalence of imprecise and incomplete information exists. Although both approaches provide useful tools for managing uncertainties in algebraic structures, FNINK-Algebras are a better method because of their improved specificity and precision. FNINK-algebras are a more sophisticated and elegant mathematical framework because the incorporation of Fermatean features enables a more detailed representation of indeterminacies and non-memberships. Specialized conditions for FNCINK-Is and direct products add to the robustness and generalizability of the method in different fields. 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