EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 3022-3042 ISSN 1307-5543 – ejpam.com Published by New York Business Global Fermatean Fuzzy Set Theory Applied to IUP-Algebras Kannirun Suayngam1, Rukchart Prasertpong2, Nareupanat Lekkoksung3, Pongpun Julatha4, Aiyared Iampan1,∗ 1 Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand 2 Division of Mathematics and Statistics, Faculty of Science and Technology, Nakhon Sawan Rajabhat University, Nakhon Sawan 60000, Thailand 3 Division of Mathematics, Faculty of Engineering, Rajamangala University of Technology Isan, Khon Kaen Campus, Khon Kaen 40000, Thailand 4 Department of Mathematics, Faculty of Science and Technology, Pibulsongkram Rajabhat University, Phitsanulok 65000, Thailand Abstract. In 1965, Zadeh introduced the foundational concept of fuzzy sets, followed by Atanassov’s introduction of intuitionistic fuzzy sets in 1986. Yager expanded this field with Pythagorean fuzzy sets in 2013, and in 2020, Senapati and Yager further advanced the theory by proposing Fermatean fuzzy sets. This study applies Fermatean fuzzy sets to IUP-algebras, focusing on Fermatean fuzzy IUP-subalgebras, IUP-ideals, IUP-filters, and strong IUP-ideals. We examine their properties, in- cluding characteristic Fermatean fuzzy sets and upper and lower t-(strong) level subsets, offering deeper insights into their structural relationships. 2020 Mathematics Subject Classifications: 03G25, 03E72, 08A72 Key Words and Phrases: IUP-algebra, Fermatean fuzzy set, Fermatean fuzzy IUP-subalgebra, Fermatean fuzzy IUP-ideal, Fermatean fuzzy IUP-filter, Fermatean fuzzy strong IUP-ideal, upper t-(strong) level subset, lower t-(strong) level subset 1. Introduction The concept of fuzzy sets (FSs), introduced by Zadeh [15], revolutionized the handling of uncertainty by allowing elements to have varying degrees of membership. This founda- tional idea was extended by Atanassov [2] with intuitionistic fuzzy sets (IFSs), which added a degree of non-membership. Yager [14] further advanced this field with Pythagorean fuzzy sets (PFSs), where the square sum of membership and non-membership degrees is ≤ 1. The most recent development, Fermatean fuzzy sets (FFSs), was introduced by Senapati ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5367 Email addresses: kannirun.s@gmail.com (K. Suayngam), rukchart.p@nsru.ac.th (R. Prasertpong), nareupanat.le@rmuti.ac.th (N. Lekkoksung), pongpun.j@psru.ac.th (P. Julatha), aiyared.ia@up.ac.th (A. Iampan) https://www.ejpam.com 3022 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) A. Iampan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3022-3042 3023 and Yager [11]. Fermatean fuzzy sets allow the sum of the cubes of membership and non-membership degrees to be ≤ 1, providing even greater flexibility and precision. These progressive enhancements have significantly enriched decision-making, medical diagnosis, and risk assessment, showcasing fuzzy logic’s dynamic evolution and growing sophistica- tion in modelling complex uncertainties. After that, the concept of Fermatean fuzzy sets has been studied extensively and con- tinuously in many spaces, such as Lalitha and Buvaneswari [10] identified and proved various properties, especially those involving the operation A ⇁ B defined as Fermatean fuzzy implication with other operations. Muhammad et al. [8] proposed a new type of fuzzy system known as the Fermatean fuzzy system. More precisely, they presented the notion of Fermatean fuzzy ideal theory and rough Fermatean fuzzy sets in semigroups and initiated the idea of lower and upper approximations in Fermatean fuzzy sets. They extended the study to rough Fermatean fuzzy left (resp., right, interior) ideals in semi- groups. Balamurugan and Nagarajan [3] came up with the idea of a Fermatean fuzzy soft-covered generalized bi-ideal on a semigroup. This extends the idea of a Fermatean fuzzy soft bi-ideal and describes regular semigroups in terms of Fermatean fuzzy soft gen- eralized bi-ideals. They framed the combining of fuzzy relations, composition relations, and compatible relations with Fermatean fuzzy sets. They also introduced the notions of a Fermatean fuzzy soft equivalence relation and a Fermatean fuzzy soft compatible relation on a semigroup. Finally, they provided a Fermatean fuzzy soft inverse relation and a Fermatean fuzzy soft congruence on a semigroup. Balamurugan and Nagarajan [4] first discussed bipolar Fermatean uncertainty subalgebras regarding R-ideals. They also discussed some exciting ideas and examined how bipolar Fermatean uncertainty soft ideals and bipolar Fermatean uncertainty soft R-ideals are related. Adak et al. [1] introduced the concept of Fermatean fuzzy semi-prime ideals and Fermatean fuzzy prime ideals of ordered semigroups. They illustrated some novel concepts to construct Fermatean fuzzy intra-regular and regular ideals and gave several relations for the family of Fermatean fuzzy regular ideals of ordered semigroups. Iampan et al. [7] introduced the groundbreaking concept of IUP-algebras. This innova- tive theory defines four key subsets: IUP-subalgebras, IUP-filters, IUP-ideals, and strong IUP-ideals. Each subset’s fundamental properties were meticulously examined, unveiling new research avenues and applications in the mathematical world. Since its introduction, the mathematical structure of IUP-algebras has captivated numerous researchers, sparking extensive studies that continue to this day. Enthusiastic scholars have delved deep into the intricacies of IUP-algebras and applied its principles to various other concepts. This has led to the creation of many new definitions and theories, significantly expanding the field and demonstrating the far-reaching impact of IUP-algebras on modern mathematics. Chanmanee et al. [6] introduced the concept of the direct product of an infinite family of IUP-algebras. They explored the external direct product of specific subsets and intro- duced the weak direct product. Additionally, they presented fundamental theorems on (anti-)IUP-homomorphisms within this context. Their work significantly advances both the theoretical framework and practical understanding of IUP-algebras. Chanmanee et al. [5] pioneered the concept of the direct product for an infinite family of IUP-algebras, A. Iampan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3022-3042 3024 demonstrating that it forms a DIUP-algebra. Additionally, they introduced the innovative idea of weak direct product DIUP-algebras, further expanding the theoretical framework of IUP-algebras. Kuntama et al. [9] has revolutionized the application of fuzzy set the- ory to IUP-algebras by introducing four groundbreaking concepts: fuzzy IUP-subalgebras, fuzzy IUP-ideals, fuzzy IUP-filters, and fuzzy strong IUP-ideals. Their study delves deep into these innovative ideas, meticulously exploring their unique properties and intricate interrelationships. This work marks a significant advancement in the field, opening new avenues for research and application. Suayngam et al. [13] made significant steps for- ward in the study of IUP-algebras in 2024 by coming up with the ideas of intuitionistic fuzzy IUP-subalgebras, intuitionistic fuzzy IUP-ideals, intuitionistic fuzzy IUP-filters, and intuitionistic fuzzy strong IUP-ideals. This pioneering work expands the theoretical land- scape of IUP-algebras, blending intuitionistic fuzzy set theory with algebraic structures in innovative ways. Building on extensive research into Fermatean fuzzy sets, this paper aims to ex- tend these concepts to IUP-algebras. We introduce and explore Fermatean fuzzy IUP- subalgebras, Fermatean fuzzy IUP-ideals, Fermatean fuzzy IUP-filters, and Fermatean fuzzy strong IUP-ideals. Our study investigates their properties, focusing on characteris- tic Fermatean fuzzy sets, upper t-(strong) level subsets, and lower t-(strong) level subsets. 2. Preliminaries Before delving into our study, let’s review the foundational concepts of IUP-algebras, including their various properties and pertinent definitions crucial to this research. Definition 1. [7] An algebra X = (X; ·, 0) of type (2, 0) is called an IUP-algebra, where X is a non-empty set, · is a binary operation on X, and 0 is a fixed element of X if it satisfies the following axioms: (∀x ∈ X)(0 · x = x) (IUP-1) (∀x ∈ X)(x · x = 0) (IUP-2) (∀x, y, z ∈ X)((x · y) · (x · z) = y · z) (IUP-3) Example 1. [7] Let (G, •, e) be a group such that all elements self-inverse. Then (G, •, e) is an IUP-algebra. Example 2. [7] Let X be a set and P(X) means the power set of X. It follows from Example 1 that (P(X),△, ∅) is an IUP-algebra where the binary operation △ is defined as the symmetric difference of any two sets. Example 3. [7] Let (G, •, e) be a group with the identity element e. Define a binary operation • on G by: (∀x, y ∈ G)(x • y = yx−1) (2.1) Then (G, •, e) is an IUP-algebra. A. Iampan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3022-3042 3025 For convenience, we refer to X as an IUP-algebra X = (X; ·, 0) until otherwise speci- fied. Proposition 1. [7] In X, the following assertions are valid (see [7]). (∀x, y ∈ X)((x · 0) · (x · y) = y) (2.2) (∀x ∈ X)((x · 0) · (x · 0) = 0) (2.3) (∀x, y ∈ X)((x · y) · 0 = y · x) (2.4) (∀x ∈ X)((x · 0) · 0 = x) (2.5) (∀x, y ∈ X)(x · ((x · 0) · y) = y) (2.6) (∀x, y ∈ X)(((x · 0) · y) · x = y · 0) (2.7) (∀x, y, z ∈ X)(x · y = x · z ⇔ y = z) (2.8) (∀x, y ∈ X)(x · y = 0 ⇔ x = y) (2.9) (∀x ∈ X)(x · 0 = 0 ⇔ x = 0) (2.10) (∀x, y, z ∈ X)(y · x = z · x ⇔ y = z) (2.11) (∀x, y ∈ X)(x · y = y ⇒ x = 0) (2.12) (∀x, y, z ∈ X)((x · y) · 0 = (z · y) · (z · x)) (2.13) (∀x, y, z ∈ X)(x · y = 0 ⇔ (z · x) · (z · y) = 0) (2.14) (∀x, y, z ∈ X)(x · y = 0 ⇔ (x · z) · (y · z) = 0) (2.15) the right and the left cancellation laws hold (2.16) In the realm of IUP-algebras, four key subsets are crucial: IUP-subalgebras, IUP- filters, IUP-ideals, and strong IUP-ideals. These subsets provide a nuanced framework essential for understanding and applying IUP-algebras in various mathematical contexts. Definition 2. [7] A non-empty subset S of X is called (i) an IUP-subalgebra of X if it satisfies the following condition: (∀x, y ∈ S)(x · y ∈ S) (2.17) (ii) an IUP-filter of X if it satisfies the following conditions: the constant 0 of X is in S (2.18) (∀x, y ∈ X)(x · y ∈ S and x ∈ S ⇒ y ∈ S) (2.19) (iii) an IUP-ideal of X if it satisfies the condition (2.18) and the following condition: (∀x, y, z ∈ X)(x · (y · z) ∈ S and y ∈ S ⇒ x · z ∈ S) (2.20) (iv) a strong IUP-ideal of X if it satisfies the following condition: (∀x, y ∈ X)(y ∈ S ⇒ x · y ∈ S) (2.21) A. Iampan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3022-3042 3026 According to [7], the concept of IUP-filters serves as a generalization encompassing IUP-ideals and IUP-subalgebras. Both IUP-ideals and IUP-subalgebras, in turn, gen- eralize strong IUP-ideals. In an IUP-algebra X, it is observed that strong IUP-ideals coincide with X itself. This relationship is illustrated in the diagram of special subsets of IUP-algebras, depicted in Figure 1. Figure 1: Special subsets of IUP-algebras 3. Main results Before diving into the definition of Fermatean fuzzy sets, it’s essential to revisit and understand the foundational concepts that underpin them. This background will provide the necessary context and enhance our comprehension of Fermatean fuzzy sets. From now on, we will use abbreviations to represent the following technical terms. Technical terms Abbreviations Fuzzy set FS Fermatean fuzzy set FFS Fermatean fuzzy IUP-subalgebra FFIUP-subalgebra Fermatean fuzzy IUP-ideal FFIUP-ideal Fermatean fuzzy IUP-filter FFIUP-filter Fermatean fuzzy strong IUP-ideal FFSIUP-ideal Definition 3. [2] Let X be a universe of discourse. A Fermatean fuzzy set F (FFS) in X is an object having the form F = {(x, αF (x), βF (x)) : x ∈ X}, where αF (x) : X → [0, 1] and βF (x) : X → [0, 1], including the following condition: (∀x ∈ X)(0 ≤ (αF (x)) 3 + (βF (x)) 3 ≤ 1) (3.1) The numbers αF (x) and βF (x) denote, respectively, the degree of membership and the degree of non-membership of the element x in the set F . For any FFS F and x ∈ X, πF (x) = 3 √ 1− (αF (x))3 − (βF (x))3 is identified as the degree of indeterminacy of x to F . In the interest of simplicity, we shall mention the symbol F = (αF , βF ) for the FFS F = {(x, αF (x), βF (x)) : x ∈ X}. A. Iampan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3022-3042 3027 For a subset G of a non-empty set X, the characteristic functions αFG and βFG are functions of X into {0, 1} defined as follows: αFG (x) = { 1 if x ∈ G 0 otherwise βFG (x) = { 0 if x ∈ G 1 otherwise By the definition of the characteristic function, αFG and βFG are functions of X into {0, 1} ⊂ [0, 1]. Therefore, the FFS FG = (αFG , βFG ) is defined as the characteristic FFS of G in X. Definition 4. Let f be an FS in a non-empty set X. Then the FS f defined by f(x) = 1− f(x) for all x ∈ X is called the complement of f in X. Definition 5. Let F be an FFS in a non-empty set X. Then the FFS F = (αF , βF ) is called the complement of F in X. We extend FFSs to IUP-algebras, introducing four innovative types: Fermatean fuzzy IUP-subalgebras, IUP-ideals, IUP-filters, and strong IUP-ideals. This application opens new dimensions in the study of IUP-algebras, enriching both their theoretical and practical frameworks. Definition 6. An FFS F in X is called a Fermatean fuzzy IUP-subalgebra (FFIUP- subalgebra) of X if it satisfies the following properties: (∀x, y ∈ X)(αF (x · y) ≥ min{αF (x), αF (y)}) (3.2) (∀x, y ∈ X)(βF (x · y) ≤ max{βF (x), βF (y)}) (3.3) Definition 7. An FFS F in X is called a Fermatean fuzzy IUP-ideal (FFIUP-ideal) of X if it satisfies the following properties: (∀x ∈ X)(αF (0) ≥ αF (x)) (3.4) (∀x ∈ X)(βF (0) ≤ βF (x)) (3.5) (∀x, y, z ∈ X)(αF (x · z) ≥ min{αF (x · (y · z)), αF (y)}) (3.6) (∀x, y, z ∈ X)(βF (x · z) ≤ max{βF (x · (y · z)), βF (y)}) (3.7) Definition 8. An FFS F in X is called a Fermatean fuzzy IUP-filter (FFIUP-filter) of X if it satisfies (3.4), (3.5), and the following properties: (∀x, y ∈ X)(αF (y) ≥ min{αF (x · y), αF (x)}) (3.8) (∀x, y ∈ X)(βF (y) ≤ max{βF (x · y), βF (x)}) (3.9) A. Iampan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3022-3042 3028 Definition 9. An FFS F in X is called a Fermatean fuzzy strong IUP-ideal (FFSIUP- ideal) of X if it satisfies the following properties: (∀x, y ∈ X)(αF (x · y) ≥ αF (y)) (3.10) (∀x, y ∈ X)(βF (x · y) ≤ βF (y)) (3.11) Lemma 1. Every FFIUP-subalgebra of X satisfies (3.4) and (3.5). Proof. Assume that F is an FFIUP-subalgebra of X. Let x ∈ X. Then αF (0) = αF (x · x) (by (IUP-2)) ≥ min{αF (x), αF (x)} (by (3.2)) = αF (x), βF (0) = βF (x · x) (by (IUP-2)) ≤ max{βF (x), βF (x)}. (by (3.3)) Hence, F satisfies (3.4) and (3.5). Theorem 1. Every FFSIUP-ideal of X satisfies (3.4) and (3.5). Proof. Assume that F is an FFSIUP-ideal of X. Let x ∈ X. Then αF (0) = αF (x · x) (by (IUP-2)) ≥ αF (x), (by (3.10)) βF (0) = βF (x · x) (by (IUP-2)) ≤ βF (x). (by (3.11)) Hence, F satisfies (3.4) and (3.5). Theorem 2. An FFSIUP-ideal and constant FFS coincide. Proof. Assume that F is an FFSIUP-ideal of X. Let x ∈ X. Then αF (x) = αF ((x · 0) · 0) (by (2.5)) ≥ αF (0), (by (3.10)) βF (x) = βF ((x · 0) · 0) (by (2.5)) ≤ βF (0). (by (3.11)) It follows from Theorem 1 that F is a constant FFS of X. Conversely, it is obviously true that every constant FFS is an FFSIUP-ideal of X. The following theorem is a direct consequence of Theorem 2. Theorem 3. Every FFSIUP-ideal of X is an FFIUP-subalgebra of X. A. Iampan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3022-3042 3029 Example 4. Let X = {0, 1, 2, 3, 4, 5} with the following Cayley table: · 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 0 4 5 2 3 2 3 5 0 4 1 2 3 2 4 5 0 3 1 4 5 3 1 2 0 4 5 4 2 3 1 5 0 Then X is an IUP-algebra. We define an FFS F in X as follows: αF = ( 0 0.9 1 0.1 2 0.1 3 0.1 4 0.5 5 0.5 ) βF = ( 0 0.2 1 0.8 2 0.8 3 0.8 4 0.6 5 0.6 ) Then F is an FFIUP-subalgebra of X. Since αF (1 · 4) = αF (2) = 0.1 ≱ 0.5 = αF (4) and βF (3 · 4) = βF (3) = 0.8 ≰ 0.6 = βF (4). Hence, F is not an FFSIUP-ideal of X. The following theorem is a direct consequence of Theorem 2. Theorem 4. Every FFSIUP-ideal of X is an FFIUP-ideal of X. Example 5. Let X = {0, 1, 2, 3, 4, 5} with the following Cayley table: · 0 1 2 3 4 5 0 0 1 2 3 4 5 1 4 0 3 1 5 2 2 2 5 0 4 3 1 3 5 4 1 0 2 3 4 1 3 5 2 0 4 5 3 2 4 5 1 0 Then X is an IUP-algebra. We define an FFS F in X as follows: αF = ( 0 0.5 1 0.1 2 0.1 3 0.3 4 0.1 5 0.3 ) βF = ( 0 0.6 1 0.9 2 0.9 3 0.7 4 0.9 5 0.7 ) Then F is an FFIUP-ideal of X. Since αF (5 · 0) = αF (3) = 0.3 ≱ 0.5 = αF (0) and βF (1 · 5) = βF (2) = 0.9 ≰ 0.7 = βF (5). Hence, F is not an FFSIUP-ideal of X. Theorem 5. Every FFIUP-ideal of X is an FFIUP-filter of X. A. Iampan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3022-3042 3030 Proof. Assume that F is an FFIUP-ideal of X. By the assumption, it satisfies (3.4) and (3.5). Let x, y ∈ X. Then αF (y) = αF (0 · y) (by (IUP-1)) ≥ min{αF (0 · (x · y)), αF (x)} (by (3.6)) = min{αF (x · y), αF (x)}, (by (IUP-1)) βF (y) = βF (0 · y) (by (IUP-1)) ≤ max{βF (0 · (x · y)), βF (x)} (by (3.7)) = max{βF (x · y), βF (x)}. (by (IUP-1)) Hence, F is an FFIUP-filter of X. Example 6. Let X = {0, 1, 2, 3, 4, 5} with the following Cayley table: · 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 0 5 4 3 2 2 2 4 0 5 1 3 3 3 5 4 0 2 1 4 5 3 1 2 0 4 5 4 2 3 1 5 0 Then X is an IUP-algebra. We define an FFS F in X as follows: αF = ( 0 0.6 1 0.5 2 0.2 3 0.2 4 0.2 5 0.2 ) βF = ( 0 0.1 1 0.7 2 0.9 3 0.9 4 0.9 5 0.9 ) Then F is an FFIUP-filter of X. Since αF (2 · 5) = αF (3) = 0.2 ≱ 0.5 = min{0.6, 0.5} = min{αF (0), αF (1)} = min{αF (2 · 2), αF (1)} = min{αF (2 · (1 · 5)), αF (1)} and βF (3 · 4) = βF (2) = 0.9 ≰ 0.7 = max{0.1, 0.7} = max{βF (0), βF (1)} = max{βF (3 · 3), βF (1)} = max{βF (3 · (1 · 4)), βF (1)}. Hence, F is not an FFIUP-ideal of X. Theorem 6. Every FFIUP-subalgebra of X is an FFIUP-filter of X. Proof. Assume that F is an FFIUP-subalgebra of X. By Lemma 1, we have F satisfies (3.4) and (3.5). Let x, y ∈ X. Then αF (y) = αF (0 · y) (by (IUP-1)) = αF ((x · 0) · (x · y)) (by (IUP-3)) ≥ min{αF (x · 0), αF (x · y)} (by (3.2)) ≥ min{min{αF (x), αF (0)}, αF (x · y)} (by (3.2)) A. Iampan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3022-3042 3031 = min{αF (x), αF (x · y)}, (by (3.4)) βF (y) = βF (0 · y) (by (IUP-1)) = βF ((x · 0) · (x · y)) (by (IUP-3)) ≤ max{βF (x · 0), βF (x · y)} (by (3.3)) ≤ max{max{βF (x), βF (0)}, βF (x · y)} (by (3.3)) = max{βF (x), βF (x · y)}. (by (3.5)) Hence, F is an FFIUP-filter of X. Example 7. [7] Let R∗ be the set of all nonzero real numbers. Define a binary operation · on R∗ by: (∀x, y ∈ R∗)(x · y = y x ). Thus, (R∗, ·, 1) is an IUP-algebra. Example 8. From Example 7, let S = {x ∈ R∗ | x ≥ 1}. Then 1 ∈ S. Next, let x, y, z ∈ R∗ be such that x · (y · z) ≥ 1 and y ≥ 1. Then z yx ≥ 1. Thus, x · z = z x = ( z yx )y ≥ 1, that is, x ·z ∈ S. Hence, S is an IUP-ideal of R∗. Then S is an IUP-filter of R∗. By Theorems 9 and 10, we have FS is an FFIUP-ideal and an FFIUP-filter of R∗. Since 1, 3 ∈ S but 3 · 1 = 1 3 ∈ S, we have S is not an IUP-subalgebra of R∗. By Theorem 8, we have FS is not an FFIUP-subalgebra of R∗. Example 9. Let X = {0, 1, 2, 3, 4, 5} with the following Cayley table: · 0 1 2 3 4 5 0 0 1 2 3 4 5 1 2 0 1 4 5 3 2 1 2 0 5 3 4 3 3 4 5 0 1 2 4 4 5 3 2 0 1 5 5 3 4 1 2 0 Then X is an IUP-algebra. We define an FFS F in X as follows: αF = ( 0 0.8 1 0.2 2 0.2 3 0.7 4 0.2 5 0.2 ) βF = ( 0 0.1 1 0.9 2 0.9 3 0.5 4 0.9 5 0.9 ) Then F is an FFIUP-subalgebra of X. Since αF (1·4) = αF (5) = 0.2 ≱ 0.7 = min{0.8, 0.7} = min{αF (0), αF (3)} = min{αF (1 · (3 · 4)), αF (3)} and βF (1 · 2) = βF (1) = 0.9 ≰ 0.5 = max{0.5, 0.5} = max{βF (3), βF (3)} = max{βF (1 · (3 · 2)), βF (3)}. Hence, F is not an FFIUP-ideal of X. A. Iampan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3022-3042 3032 The study revealed a relationship between the four concepts: FFIUP-ideals and FFIUP- subalgebras are generalizations of FFSIUP-ideals of IUP-algebras, where FFSIUP-ideals of IUP-algebras can only be a constant FFS. FFIUP-filters are a generalization of FFIUP- ideals and FFIUP-subalgebras. We summarize the relationship between these four con- cepts, shown in Figure 2. Figure 2: FFSs in IUP-algebras Theorem 7. If F is an FFIUP-filter of X satisfying the following condition: (∀x, y, z ∈ X) ( αF (y · (x · z)) = αF (x · (y · z)) βF (y · (x · z)) = βF (x · (y · z)) ) (3.12) then F is an FFIUP-ideal of X. Proof. Assume that F is an FFIUP-filter of X satisfying the condition (3.12). By the assumption, it satisfies (3.4) and (3.5). Let x, y, z ∈ X. Then αF (x · z) ≥ min{αF (y · (x · z)), αF (y)} (by (3.8)) = min{αF (x · (y · z)), αF (y)}, (by (3.12)) βF (x · z) ≤ max{βF (y · (x · z)), βF (y)} (by (3.9)) = max{βF (x · (y · z)), βF (y)}. (by (3.12)) Hence, F is an FFIUP-ideal of X. Lemma 2. Let G be a non-empty subset of X. Then the constant 0 is in G if and only if the characteristic FFS FG satisfies (3.4) and (3.5). Proof. Assume that the constant 0 is in G. Then αFG (0) = 1 and βFG (0) = 0. Thus, αFG (0) = 1 ≥ αFG (x) and βFG (0) = 0 ≤ βFG (x) for all x ∈ X, that is, FG satisfies (3.4) and (3.5). Conversely, assume that the characteristic FFS FG satisfies (3.4) and (3.5). Then αFG (0) ≥ αFG (x) for all x ∈ X. Since G is a non-empty subset of X, we let a ∈ G. Then αFG (0) ≥ αFG (a) = 1, so αFG (0) = 1. Hence, the constant 0 is in G. Theorem 8. A non-empty subset G of X is an IUP-subalgebra of X if and only if the characteristic FFS FG is an FFIUP-subalgebra of X. A. Iampan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3022-3042 3033 Proof. Assume that G is an IUP-subalgebra of X. Let x, y ∈ X. Case 1: Suppose x, y ∈ G. Then αFG (x) = 1 and αFG (y) = 1. Since G is an IUP- subalgebra ofX, we have x·y ∈ G. Thus, αFG (x·y) = 1 ≥ min{1, 1} = min{αFG (x), αFG (y)}. Case 2: Suppose x /∈ G or y /∈ G. Then αFG (x) = 0 or αFG (y) = 0. Thus, αFG (x ·y) ≥ 0 = min{αFG (x), αFG (y)}. Case 1’: Suppose x, y ∈ G. Then βFG (x) = 0 and βFG (y) = 0. Since G is an IUP- subalgebra of X, we have x · y ∈ G. Thus, βFG (x · y) = 0 ≤ 0 = max{βFG (x), βFG (y)}. Case 2’ : Suppose x /∈ G or y /∈ G. Then βFG (x) = 1 or βFG (y) = 1. Thus, βFG (x · y) ≤ 1 = max{βFG (x), βFG (y)}. Hence, the characteristic FFS FG is an FFIUP-subalgebra of X. Conversely, assume that the characteristic FFS FG is an FFIUP-subalgebra of X. Let x, y ∈ G. Then αFG (x) = 1 and αFG (y) = 1. By (3.2), we have αFG (x · y) ≥ min{αFG (x), αFG (y)} = min{1, 1} = 1. Thus, αFG (x · y) = 1, that is, x · y ∈ G. Hence, G is an IUP-subalgebra of X. Theorem 9. A non-empty subset G of X is an IUP-ideal of X if and only if the charac- teristic FFS FG is an FFIUP-ideal of X. Proof. Assume that G is an IUP-ideal of X. Since 0 ∈ G, it follows from Lemma 2 that αFG and βFG satisfy (3.4) and (3.5), respectively. Next, let x, y, z ∈ X. Case 1: Suppose x · (y · z) ∈ G and y ∈ G. Since G is an IUP-ideal of X, we have x · z ∈ G. Thus, αFG (x · z) = 1 ≥ 1 = min{1, 1} = min{αFG (x · (y · z)), αFG (y)}. Case 2: Suppose x · (y · z) /∈ G or y /∈ G. Then αFG (x · (y · z)) = 0 or αFG (y) = 0. Thus, αFG (x · z) ≥ 0 = min{αFG (x · (y · z)), αFG (y)}. Case 1’: Suppose x · (y · z) ∈ G and y ∈ G. Since G is an IUP-ideal of X, we have x · z ∈ G. Thus, βFG (x · z) = 0 ≤ 0 = max{0, 0} = max{βFG (x · (y · z)), βFG (y)}. Case 2’: Suppose x · (y · z) /∈ G or y /∈ G. Then βFG (x · (y · z)) = 1 or βFG (y) = 1. Thus, βFG (x · z) ≤ 1 = max{βFG (x · (y · z)), βFG (y)}. Hence, FG is an FFIUP-ideal of X. Conversely, assume that the characteristic FFS FG is an FFIUP-ideal of X. Since αFG satisfies (3.4), it follows from Lemma 2 that 0 ∈ G. Next, let x, y, z ∈ X be such that x · (y · z) ∈ G and y ∈ G. Then αFG (x · (y · z)) = 1 and αFG (y) = 1. Thus, min{αFG (x · (y · z)), αFG (y)} = 1. By (3.6), we have αFG (x · z) ≥ min{αFG (x · (y · z)), αFG (y)} = 1, that is, αFG (x · z) = 1. Hence, x · z ∈ G, so G is an IUP-ideal of X. Theorem 10. A non-empty subset G of X is an IUP-filter of X if and only if the char- acteristic FFS FG is an FFIUP-filter of X. Proof. Assume that G is an IUP-filter of X. Since 0 ∈ G, it follows from Lemma 2 that αFG and βFG satisfy (3.4) and (3.5), respectively. Next, let x, y ∈ X. Case 1: Suppose x · y ∈ G and x ∈ G. Since G is an IUP-filter of X, we have y ∈ G. Thus, αFG (y) = 1 ≥ 1 = min{1, 1} = min{αFG (x · y), αFG (x)}. Case 2: Suppose x · y /∈ G or x /∈ G. Then αFG (x · y) = 0 or αFG (x) = 0. Thus, αFG (y) ≥ 0 = min{αFG (x · y), αFG (x)}. A. Iampan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3022-3042 3034 Case 1’: Suppose x · y ∈ G and x ∈ G. Since G is an IUP-filter of X, we have y ∈ G. Thus, βFG (y) = 0 ≤ 0 = max{0, 0} = max{βFG (x · y), βFG (x)}. Case 2’: Suppose x · y /∈ G or x /∈ G. Then βFG (x · y) = 1 or βFG (x) = 1. Thus, βFG (y) ≤ 1 = max{βFG (x · y), βFG (x)}. Hence, FG is an FFIUP-filter of X. Conversely, assume that the characteristic FFS FG is an FFIUP-filter of X. Since αFG satisfies (3.4), it follows from Lemma 2 that 0 ∈ G. Next, let x, y ∈ G be such that x·y ∈ G and x ∈ G. Then αFG (x · y) = 1 and αFG (x) = 1. Thus, min{αFG (x · y), αFG (x)} = 1. By (3.8), we have αFG (y) = min{αFG (x · y), αFG (x)} = 1, that is, αFG (y) = 1. Hence, y ∈ G, so G is an IUP-filter of X. The following theorem is a direct consequence of Theorem 2. Theorem 11. A non-empty subset G of X is a strong IUP-ideal of X if and only if the characteristic FFS FG is an FFSIUP-ideal of X. Lemma 3. [13] Let f be an FS in a non-empty set X. Then the following statements hold: (∀x, y ∈ X)(1−max{f(x), f(y)} = min{1− f(x), 1− f(y)}) (3.13) (∀x, y ∈ X)(1−min{f(x), f(y)} = max{1− f(x), 1− f(y)}) (3.14) Lemma 4. [13] Let f be an FS in a non-empty set X. Then the following statements hold: (∀x, y, z ∈ X)(f(z) ≥ min{f(x), f(y)} ⇔ f(z) ≤ max{f(x), f(y)}) (3.15) (∀x, y, z ∈ X)(f(z) ≤ max{f(x), f(y)} ⇔ f(z) ≥ min{f(x), f(y)}) (3.16) Before presenting theorems on the relationship between FSSs and their complements, it’s crucial to grasp their basic concept. FSSs extend traditional FSs by incorporating hesitation degrees. The following theorem highlights the key relationship between these sets and their complements. Theorem 12. An FFS F is an FFIUP-subalgebra of X if and only if the FSs αF and βF satisfy (3.2), and the FSs αF and βF satisfy (3.3). Proof. Assume that F is an FFIUP-subalgebra of X. Then αF (x · y) ≥ min{αF (x), αF (y)}, βF (x · y) ≤ max{βF (x), βF (y)}. Thus, αF (x · y) ≤ max{αF (x), αF (y)}, (by (3.15)) βF (x · y) ≥ min{βF (x), βF (y)}, . (by (3.16)) A. Iampan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3022-3042 3035 Hence, the FSs αF and βF satisfy (3.2), and the FSs αF and βF satisfy (3.3). Conversely, assume that the FSs αF and βF satisfy (3.2), and the FSs αF and βF satisfy (3.3). Then αF satisfies (3.2), and βF satisfies (3.3). Hence, F is an FFIUP-subalgebra of X. Theorem 13. An FFS F is an FFIUP-ideal of X if and only if the FSs αF and βF satisfy (3.4) and (3.6), and the FSs αF and βF satisfy (3.5) and (3.7). Proof. Assume that F is an FFIUP-ideal of X. Then αF (0) ≥ αF (x), βF (0) ≤ βF (x), αF (x · z) ≥ min{αF (x · (y · z)), αF (y)}, βF (x · z) ≤ max{βF (x · (y · z)), βF (y)}. Thus, αF (0) ≤ αF (x), βF (0) ≥ βF (x), αF (x · z) ≤ max{αF (x · (y · z)), αF (y)}, βF (x · z) ≥ min{βF (x · (y · z)), βF (y)}. Hence, the FSs αF and βF satisfy (3.4) and (3.6), and the FSs αF and βF satisfy (3.5) and (3.7). Conversely, assume that the FSs αF and βF satisfy (3.4) and (3.6), and the FSs αF and βF satisfy (3.5) and (3.7). Then αF satisfies (3.4) and (3.6), and βF satisfies (3.5) and (3.7). Hence, F is an FFIUP-ideal of X. Theorem 14. An FFS F is an FFIUP-filter of X if and only if the FSs αF and βF satisfy (3.4) and (3.8), and the FSs αF and βF satisfy (3.5) and (3.9). Proof. Assume that F is an FFIUP-ideal of X. Then αF (0) ≥ αF (x), βF (0) ≤ βF (x), αF (y) ≥ min{αF (x · y), αF (x)}, βF (y) ≤ max{βF (x · y), βF (x)}. Thus, αF (0) ≤ αF (x), βF (0) ≥ βF (x), αF (y) ≤ max{αF (x · y), αF (x)}, A. Iampan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3022-3042 3036 βF (y) ≥ min{βF (x · y), βF (x)}. Hence, the FSs αF and βF satisfy (3.4) and (3.8), and the FSs αF and βF satisfy (3.5) and (3.9). Conversely, assume that the FSs αF and βF satisfy (3.4) and (3.8), and the FSs αF and βF satisfy (3.5) and (3.9). Then αF satisfies (3.4) and (3.8), and βF satisfies (3.5) and (3.9). Hence, F is an FFIUP-filter of X. The following theorem is a direct consequence of Theorem 2. Theorem 15. An FFS F is an FFSIUP-ideal of X if and only if the FSs αF and βF satisfy (3.10), and the FSs αF and βF satisfy (3.11). Theorem 16. An FFS F is an FFIUP-subalgebra of X if and only if FFS ∗F = (αF , αF ) and △F = (βF , βF ) are FFIUP-subalgebras of X. Proof. It is straightforward by Theorem 12. Theorem 17. An FFS F is an FFIUP-ideal of X if and only if FFS ∗F = (αF , αF ) and △F = (βFβF ) are FFIUP-ideals of X. Proof. It is straightforward by Theorem 13. Theorem 18. An FFS F is an FFIUP-filter of X if and only if FFS ∗F = (αF , αF ) and △F = (βFβF ) are FFIUP-filters of X. Proof. It is straightforward by Theorem 14. Theorem 19. An FFS F is an FFSIUP-ideal of X if and only if FFS ∗F = (αF , αF ) and △F = (βFβF ) are FFSIUP-ideals of X. Proof. It is straightforward by Theorem 15. Definition 10. [12] Let f be an FS in a non-empty set X. For any t ∈ [0, 1], the sets U(f ; t) = {x ∈ X | f(x) ≥ t}, (3.17) L(f ; t) = {x ∈ X | f(x) ≤ t} (3.18) are called an upper t-level subset and a lower t-level subset of f , respectively. The sets U + (f ; t) = {x ∈ X | f(x) > t}, (3.19) L − (f ; t) = {x ∈ X | f(x) < t} (3.20) are called an upper t-strong level subset and a lower t-strong level subset of f , respectively. A. Iampan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3022-3042 3037 Before presenting theorems on the relationship between level subsets and their corre- sponding FFSs, it’s essential to grasp the key concepts. Level subsets characterize FFSs by detailing the distribution of membership degrees. The following theorem formalizes this relationship, providing insights into the structure of FFSs. Theorem 20. An FFS F is an FFIUP-subalgebra of X if and only if for all t, s ∈ [0, 1], the sets U(αF ; t) and L(βF ; s) are either empty or IUP-subalgebras of X. Proof. Assume that F is an FFIUP-subalgebra of X. Let t ∈ [0, 1] be such that U(αF ; t) ̸= ∅. Let x, y ∈ U(αF ; t). Then αF (x) ≥ t and αF (y) ≥ t. Thus, min{αF (x), αF (y)} ≥ t. By (3.2), we have αF (x · y) ≥ min{αF (x), αF (y)} ≥ t, that is, αF (x · y) ≥ t. Thus, x · y ∈ U(αF ; t). Hence, U(αF ; t) is an IUP-subalgebra of X. Let s ∈ [0, 1] be such that L(βF ; s) ̸= ∅. Let x, y ∈ L(βF ; s). Then βF (x) ≤ s and βF (y) ≤ s. Thus, max{βF (x), βF (y)} ≤ s. By (3.3), we have βF (x · y) ≤ max{βF (x), βF (y)} ≤ s, that is, βF (x · y) ≤ s. Thus, x · y ∈ L(βF ; s). Hence, L(βF ; s) is an IUP-subalgebra of X. Conversely, assume that for all t, s ∈ [0, 1], the sets U(αF ; t) and L(βF ; s) are either empty or IUP-subalgebras of X. Let x, y ∈ X. Let t = min{αF (x), αF (y)}. Then αF (x) ≥ t and αF (y) ≥ t. Thus, x, y ∈ U(αF ; t) ̸= ∅. By the assumption, we have U(αF ; t) is an IUP-subalgebra of X. By (2.17), we have x ·y ∈ U(αF ; t). Thus, αF (x ·y) ≥ t = min{αF (x), αF (y)}. Let x, y ∈ X. Let s = max{βF (x), βF (y)}. Then βF (x) ≤ s and βF (y) ≤ s. Thus, x, y ∈ L(βF ; s) ̸= ∅. By the assumption, we have L(βF ; s) is an IUP-subalgebra of X. By (2.17), we have x · y ∈ L(βF ; s). Thus, βF (x · y) ≤ s = max{βF (x), βF (y)}. Hence, F is an FFIUP-subalgebra of X. Theorem 21. An FFS F in X is an FFIUP-ideal of X if and only if for all t, s ∈ [0, 1], the sets U(αF ; t) and L(βF ; s) are either empty or IUP-ideals of X. Proof. Assume that F is an FFIUP-ideal of X. Let t ∈ [0, 1] be such that U(αF ; t) ̸= ∅. Let r ∈ U(αF ; t). Then αF (r) ≥ t. By (3.4), we have αF (0) ≥ αF (r) ≥ t. Thus, 0 ∈ U(αF ; t). Let x, y, z ∈ X be such that x · (y · z) ∈ U(αF ; t) and y ∈ U(αF ; t). Then αF (x · (y · z)) ≥ t and αF (y) ≥ t. Thus, min{αF (x · (y · z)), αF (y)} ≥ t. By (3.6), we have αF (x · z) ≥ min{αF (x · (y · z)), αF (y)} ≥ t. Thus, x · z ∈ U(αF ; t). Hence, U(αF ; t) is an IUP-ideal of X. Let s ∈ [0, 1] be such that L(βF ; s) ̸= ∅. Let β ∈ L(βF ; s). Then βF (l) ≤ s. By (3.5), we have βF (0) ≤ βF (l) ≤ s. Thus, 0 ∈ L(βF ; s). Let x, y, z ∈ X be such that x · (y · z) ∈ L(βF ; s) and y ∈ L(βF ; s). Then βF (x · (y · z)) ≤ s and βF (y) ≤ s. Thus, max{βF (x·(y·z)), βF (y)} ≤ s. By (3.7), we have βF (x·z) ≤ max{βF (x·(y·z)), βF (y)} ≤ s. Thus, x · z ∈ L(βF ; s). Hence, L(βF ; s) is an IUP-ideal of X. Conversely, assume that for all t, s ∈ [0, 1], the sets U(αF ; t) and L(βF ; s) are either empty or IUP-ideals of X. Let x ∈ X. Let t = αF (x). Then αF (x) ≥ t. Thus, x ∈ U(αF ; t) ̸= ∅. By the assumption, we have U(αF ; t) is an IUP-ideal of X. By (2.18), we have 0 ∈ U(αF ; t). Then αF (0) ≥ t = αF (x). Let x, y, z ∈ X. Let t = min{αF (x · (y · A. Iampan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3022-3042 3038 z)), αF (y)}. Then αF (x · (y · z)) ≥ t and αF (y) ≥ t. Thus, x · (y · z), y ∈ U(αF ; t) ̸= ∅. By the assumption, we have U(αF ; t) is an IUP-ideal of X. By (2.20), we have x·z ∈ U(αF ; t). Thus, αF (x · z) ≥ t = min{αF (x · (y · z)), αF (y)}. Let x ∈ X. Let s = βF (x). Then βF (x) ≤ s. Thus, x ∈ L(βF ; s) ̸= ∅. By the assumption, we have L(βF ; s) is an IUP-ideal of X. By (2.18), we have 0 ∈ L(βF ; s). Then βF (0) ≤ s = βF (x). Let x, y, z ∈ X. Let s = max{βF (x · (y · z)), βF (y)}. Then βF (x · (y · z)) ≤ s and βF (y) ≤ s. Thus, x · (y · z), y ∈ L(βF ; s) ̸= ∅. By the assumption, we have U(βF ; s) is an IUP-ideal of X. By (2.20), we have x · z ∈ L(βF ; s). Thus, βF (x · z) ≤ s = max{βF (x · (y · z)), βF (y)}. Hence, F is an FFIUP-ideal of X. Theorem 22. An FFS F in X is an FFIUP-filter of X if and only if for all t, s ∈ [0, 1], the sets U(αF ; t) and L(βF ; s) are either empty or IUP-filters of X. Proof. Assume that F is an FFIUP-filter of X. Let t ∈ [0, 1] be such that U(αF ; t) ̸= ∅. Let r ∈ U(αF ; t). Then αF (r) ≥ t. By (3.4), we have αF (0) ≥ αF (r) ≥ t. Thus, 0 ∈ U(αF ; t). Let x, y ∈ X be such that x · y ∈ U(αF ; t) and x ∈ U(αF ; t). Then αF (x · y) ≥ t and αF (x) ≥ t. Thus, min{αF (x · y), αF (x)} ≥ t. By (3.8), we have αF (y) ≥ min{αF (x · y), αF (x)} ≥ t. Thus, y ∈ U(αF ; t). Hence, U(αF ; t) is an IUP-filter of X. Let s ∈ [0, 1] be such that L(βF ; s) ̸= ∅. Let l ∈ L(βF ; s). Then βF (l) ≤ s. By (3.5), we have βF (0) ≤ βF (l) ≤ s. Thus, 0 ∈ L(βF ; s). Let x, y ∈ X be such that x ·y ∈ L(βF ; s) and x ∈ L(βF ; s). Then βF (x·y) ≤ s and βF (x) ≤ s. Thus, max{βF (x·y), βF (x)} ≤ s. By (3.9), we have βF (y) ≤ max{βF (x · y), βF (x)} ≤ s. Thus, y ∈ L(βF ; s). Hence, L(βF ; s) is an IUP-ideal of X. Conversely, assume that for all t, s ∈ [0, 1], the sets U(αF ; t) and L(βF ; s) are either empty or IUP-filters of X. Let x ∈ X. Let t = αF (x). Then αF (x) ≥ t. Thus, x ∈ U(αF ; t) ̸= ∅. By the assumption, we have U(αF ; t) is an IUP-filter ofX. By (2.18), we have 0 ∈ U(αF ; t). Then αF (0) ≥ t = αF (x). Let x, y ∈ X. Let t = min{αF (x·y), αF (x)}. Then αF (x · y) ≥ t and αF (x) ≥ t. Thus, x · y, x ∈ U(αF ; t) ̸= ∅. By the assumption, we have U(αF ; t) is an IUP-filter of X. By (2.19), we have y ∈ U(αF ; t). Thus, αF (y) ≥ t = min{αF (x · y), αF (x)}. Let x ∈ X. Let s = βF (x). Then βF (x) ≤ s. Thus, x ∈ L(βF ; s) ̸= ∅. By the assumption, we have L(βF ; s) is an IUP-filter of X. By (2.18), we have 0 ∈ L(βF ; s). Then βF (0) ≤ s = βF (x). Let x, y ∈ X. Let s = max{βF (x · y), βF (x)}. Then βF (x · y) ≤ s and βF (x) ≤ s. Thus, x · y, x ∈ L(βF ; s) ̸= ∅. By the assumption, we have L(βF ; s) is an IUP- filter of X. By (2.19), we have y ∈ L(βF ; s). Thus, βF (y) ≤ s = max{βF (x · y), βF (x)}. Hence, F is an FFIUP-filter of X. The following theorem is a direct consequence of Theorem 2. Theorem 23. An FFS F in X is an FFSIUP-ideal of X if and only if for all t, s ∈ [0, 1], the sets U(αF ; t) and L(βF ; s) are either empty or strong IUP-ideal of X. Theorem 24. An FFS F in X is an FFIUP-subalgebra of X if and only if for all t, s ∈ [0, 1], the sets U + (αF ; t) and L − (βF ; s) are either empty or IUP-subalgebras of X. A. Iampan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3022-3042 3039 Proof. Assume that F is an FFIUP-subalgebra of X. Let t ∈ [0, 1] be such that U + (αF ; t) ̸= ∅. Let x, y ∈ U + (αF ; t). Then αF (x) > t and αF (y) > t. Thus, min{αF (x), αF (y)} > t. By (3.2), we have αF (x · y) ≥ min{αF (x), αF (y)} > t. Thus, x · y ∈ U + (αF ; t). Hence, U + (αF ; t) is an IUP-subalgebra of X. Let s ∈ [0, 1] be such that L − (βF ; s) ̸= ∅. Let x, y ∈ L − (βF ; s). Then βF (x) < s and βF (y) < s. Thus, max{βF (x), βF (y)} < s. By (3.3), we have βF (x · y) ≤ max{βF (x), βF (y)} < s. Thus, x · y ∈ L − (βF ; s). Hence, L − (βF ; s) is an IUP-subalgebra of X. Conversely, assume that for all t, s ∈ [0, 1], the sets U + (αF ; t) and L − (βF ; s) are either empty or IUP-subalgebras ofX. Let x, y ∈ X. Assume that αF (x·y) < min{αF (x), αF (y)}. Let t = αF (x·y). Then αF (x) > t and αF (y) > t. Thus, x, y ∈ U + (αF ; t). By the assump- tion, we have U + (αF ; t) is an IUP-subalgebra. By (2.17), we have x · y ∈ U + (αF ; t). So αF (x · y) > t = αF (x · y), which is a contradiction. Thus, αF (x · y) ≥ min{αF (x), αF (y)}. Let x, y ∈ X. Assume that βF (x · y) > max{βF (x), βF (y)}. Let s = βF (x · y). Then βF (x) < s and βF (y) < s. Thus, x, y ∈ L − (βF ; s). By the assumption, we have L − (βF ; s) is an IUP-subalgebra. By (2.17), we have x · y ∈ L − (βF ; s). So βF (x · y) < s = βF (x · y), which is a contradiction. Thus, βF (x · y) ≤ max{βF (x), βF (y)}. Hence, F is an FFIUP-subalgebra of X. Theorem 25. An FFS F in X is an FFIUP-ideal of X if and only if for all t, s ∈ [0, 1], the sets U + (αF ; t) and L − (βF ; s) are either empty or IUP-ideals of X. Proof. Assume that F is an FFIUP-ideal of X. Let t ∈ [0, 1] be such that U + (αF ; t) ̸= ∅. Let a ∈ U + (αF ; t). Then αF (a) > t. By (3.4), we have αF (0) ≥ αF (a) > t. Thus, 0 ∈ U + (αF ; t). Let x, y, z ∈ U + (αF ; t) be such that x · (y · z), y ∈ U + (αF ; t). Then αF (x · (y · z)) > t and αF (y) > t. Thus, min{αF (x · (y · z)), αF (y)} > t. By (3.6). we have αF (x · z) ≥ min{αF (x · (y · z)), αF (y)} > t. Thus, x · z ∈ U + (αF ; t). Hence, U + (αF ; t) is an IUP-ideal of X. Let s ∈ [0, 1] be such that L − (βF ; s) ̸= ∅. Let β ∈ L − (βF ; s). Then βF (β) < s. By (3.5), we have βF (0) ≤ βF (β) < s. Thus, 0 ∈ L − (βF ; s). Let x, y, z ∈ L − (βF ; s) be such that x · (y · z), y ∈ L − (βF ; s). Then βF (x · (y · z)) < s and βF (y) < s. Thus, max{βF (x·(y·z)), βF (y)} < s. By (3.7). we have βF (x·z) ≤ max{βF (x·(y·z)), βF (y)} > s. Thus, x · z ∈ L − (βF ; s). Hence, L − (βF ; s) is an IUP-ideal of X. Conversely, assume that for all t, s ∈ [0, 1], the sets U + (αF ; t) and L − (βF ; s) are either empty or IUP-ideals of X. Let x ∈ X. Assume that αF (0) < αF (x). Let t = αF (0). Then x ∈ U + (αF ; t) ̸= ∅. By the assumption, we have U + (αF ; t) is an IUP-ideal of X. By (2.18), we have 0 ∈ U + (αF ; t). So αF (0) > t = αF (0), which is a contradiction. Thus, αF (0) ≥ αF (x). Let x, y, z ∈ X. Assume that αF (x · z) < min{αF (x · (y · z)), αF (y)}. Let t = αF (x · z). Then x · (y · z), y ∈ U + (αF ; t) ̸= ∅. By the assumption, we have U + (αF ; t) is an IUP-ideal of X. By (2.20), we have x · z ∈ U + (αF ; t). So αF (x · z) > t = αF (x · z), which is a contradiction. Thus, αF (x · z) ≥ min{αF (x · (y · z)), αF (y)}. Let x ∈ X. Assume that βF (0) > βF (x). Let s = βF (0). Then x ∈ L − (βF ; s) ̸= ∅. By the assumption, we have L − (βF ; s) is an IUP-ideal of X. By (2.18), we have 0 ∈ A. Iampan et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3022-3042 3040 L − (βF ; s). So βF (0) < s = βF (0), which is a contradiction. Thus, βF (0) ≤ βF (x). Let x, y, z ∈ X. Assume that βF (x · z) > max{βF (x · (y · z)), βF (y)}. Let s = βF (x · z). Then x · (y · z), y ∈ L − (βF ; s) ̸= ∅. By the assumption, we have L − (βF ; s) is an IUP-ideal of X. By (2.20), we have x ·z ∈ L − (βF ; s). So βF (x ·z) < s = βF (x ·z), which is a contradiction. Thus, βF (x · z) ≤ max{βF (x · (y · z)), βF (y)}. Hence, F is an FFIUP-ideal of X. Theorem 26. An FFS A in X is an FFIUP-filter of X if and only if for all t, s ∈ [0, 1], the sets U + (αF ; t) and L − (βF ; s) are either empty or IUP-filters of X. Proof. Assume that F is an FFIUP-filter of X. Let t ∈ [0, 1] be such that U + (αF ; t) ̸= ∅. Let a ∈ U + (αF ; t). Then αF (a) > t. By (3.4), we have αF (0) ≥ αF (a) > t. Thus, 0 ∈ U + (αF ; t). Let x, y ∈ U + (αF ; t) be such that x · y, x ∈ U + (αF ; t). Then αF (x · y) > t and αF (x) > t. Thus, min{αF (x · y), αF (x)} > t. By (3.8). we have αF (y) ≥ min{αF (x · y), αF (x)} > t. Thus, y ∈ U + (αF ; t). Hence, U + (αF ; t) is an IUP-filter of X. Let s ∈ [0, 1] be such that L − (βF ; s) ̸= ∅. Let β ∈ L − (βF ; s). Then βF (β) < s. By (3.5), we have βF (0) ≤ βF (β) < s. Thus, 0 ∈ L − (βF ; s). Let x, y ∈ L − (βF ; s) be such that x · y, x ∈ L − (βF ; s). Then βF (x · y) < s and βF (x) < s. Thus, max{βF (x · y), βF (x)} < s. By (3.9). we have βF (y) ≤ max{βF (x · y), βF (x)} > s. Thus, y ∈ L − (βF ; s). Hence, L − (βF ; s) is an IUP-ideal of X. Conversely, assume that for all t, s ∈ [0, 1], the sets U + (αF ; t) and L − (βF ; s) are either empty or IUP-filters of X. Let x ∈ X. Assume that αF (0) < αF (x). Let t = αF (0). Then x ∈ U + (αF ; t) ̸= ∅. By the assumption, we have U + (αF ; t) is an IUP-ideal of X. By (2.18), we have 0 ∈ U + (αF ; t). So αF (0) > t = αF (0), which is a contradiction. Thus, αF (0) ≥ αF (x). Let x, y ∈ X. Assume that αF (y) < min{αF (x · y), αF (x)}. Let t = αF (y). Then x · y, x ∈ U + (αF ; t) ̸= ∅. By the assumption, we have U + (αF ; t) is an IUP-filter of X. By (2.19), we have y ∈ U + (αF ; t). So αF (y) > t = αF (y), which is a contradiction. Thus, αF (y) ≥ min{αF (x · y), αF (x)}. Let x ∈ X. Assume that βF (0) > βF (x). Let s = βF (0). Then x ∈ L − (βF ; s) ̸= ∅. By the assumption, we have L − (βF ; s) is an IUP-filter of X. By (2.18), we have 0 ∈ L − (βF ; s). So βF (0) < s = βF (0), which is a contradiction. Thus, βF (0) ≤ βF (x). Let x, y ∈ X. Assume that βF (y) > max{βF (x · y), βF (x)}. Let s = βF (y). Then x · y, x ∈ L − (βF ; s) ̸= ∅. By the assumption, we have L − (βF ; s) is an IUP-filter of X. By (2.19), we have y ∈ L − (βF ; s). So βF (y) < s = βF (y), which is a contradiction. Thus, βF (y) ≤ max{βF (x · y), βF (x)}. Hence, F is an FFIUP-filter of X. The following theorem is a direct consequence of Theorem 2. Theorem 27. An FFS F in X is an FFSIUP-ideal of X if and only if for all t, s ∈ [0, 1], the sets U + (αF ; t) and L − (βF ; s) are either empty or strong IUP-ideals of X. REFERENCES 3041 4. Conclusions and future work Our paper introduces pioneering concepts like FFIUP-subalgebras, FFIUP-ideals, FFIUP- filters, and FFSIUP-ideals. 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