EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5779 ISSN 1307-5543 – ejpam.com Published by New York Business Global 1 On Length and Mean Fuzzy Ideals of Sheffer Stroke2 Hilbert Algebras3 Neelamegarajan Rajesh1, Tahsin Oner2, Aiyared Iampan3,∗, Ibrahim Senturk24 1 Department of Mathematics, Rajah Serfoji Government College, Thanjavur-613005, Tamil5 Nadu, India6 2 Department of Mathematics, Faculty of Science, Ege University, 35100 Izmir, Turkey7 3 Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang,8 Phayao 56000, Thailand9 10 Abstract. This paper presents a detailed exploration of Sheffer stroke Hilbert algebras, introduc- ing the innovative concepts of length fuzzy ideals and mean fuzzy ideals within an interval-valued fuzzy framework. These new constructs extend classical ideal theory by incorporating fuzzy logic, providing precise mathematical tools to analyze and measure membership gradations. Specifically, the study establishes critical relationships between length fuzzy ideals and mean fuzzy ideals, their hierarchical subsets, and their implications for algebraic consistency and computational logic. Key findings demonstrate that length fuzzy ideals align closely with interval-valued fuzzy subsets, while mean fuzzy ideals offer a unique averaging perspective for understanding ideal structures. These contributions significantly advance the field of fuzzy algebra, offering theoretical insights and po- tential applications in computational logic, uncertainty modeling, and algorithmic design. 2020 Mathematics Subject Classifications: 20N05, 94D05, 03E7211 Key Words and Phrases: Sheffer stroke Hilbert algebra (SHA), ideal, length fuzzy ideal, mean12 fuzzy ideal13 14 1. Introduction15 Hilbert algebras, often referred to as implicative algebras, are algebraic structures16 that extend the classical operations of logic. These algebras are typically defined by a17 set of axioms involving a binary operation, the Sheffer stroke, which is a generalization18 of the NAND operation in propositional logic. The study of Hilbert algebras is integral19 to understanding non-classical logics, modal logics, and lattice theory, offering essential20 insights into the foundational structure of logical systems [3].21 The Sheffer stroke is a fundamental element in both classical and non-classical logic22 due to its property as a functionally complete operation [13]. This means it can operate by23 ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5779 Email addresses: nrajesh topology@yahoo.co.in (N. Rajesh), tahsin.oner@ege.edu.tr (T. Oner), aiyared.ia@up.ac.th (A. Iampan), ibrahim.senturk@ege.edu.tr (I. Senturk) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) N. Rajesh, T. Oner, A. Iampan, I. Senturk / Eur. J. Pure Appl. Math, 18 (1) (2025), 5779 2 of 18 itself without requiring any other logical operators to form a comprehensive logical system.24 In simpler terms, every logical axiom can be restated using just the Sheffer stroke. This25 capability greatly simplifies the manipulation and control of various properties within26 the logical system it creates. Moreover, it is noteworthy that the axioms of Boolean27 algebra, which correspond to classical propositional logic, can be entirely represented28 using the Sheffer stroke. This highlights the Sheffer stroke’s foundational importance and29 its versatility within both logical and algebraic systems.30 The Sheffer stroke has been utilized in various algebraic structures, such as Boolean31 algebras, basic algebras, MV-algebras, BCK-algebras, MTL-algebras and ortholattices,32 among others, and is also explored within fuzzy contexts (see [1, 4–7, 9–12]). In 2021,33 Oner et al. [6] extended the Sheffer stroke to Hilbert algebras, defining the Sheffer stroke34 Hilbert algebra and studying its various properties. In [5], they introduced the concepts35 of a deductive system and filter for Sheffer stroke Hilbert algebras and explored their36 fuzzification. Additionally, Oner et al. [6] presented the idea of an ideal in Sheffer stroke37 Hilbert algebras and analyzed its properties.38 The field of fuzzy logic, introduced by [15], broadens classical logic by incorporating39 truth values that range continuously between 0 and 1, rather than being restricted to40 binary true/false values. This flexibility makes fuzzy logic particularly useful in scenarios41 involving uncertainty and imprecision. Integrating fuzzy logic with Hilbert algebras results42 in the concept of fuzzy ideals, where the elements of an ideal can have varying degrees of43 membership rather than being limited to crisp values. This extension offers a more refined44 approach to analyzing the algebraic properties of Hilbert algebras [2].45 A recent innovation in the theory of fuzzy ideals is the introduction of length-fuzzy46 ideals. This concept enhances the classical definition of an ideal in Sheffer stroke Hilbert al-47 gebras by associating a fuzzy function that measures the “length” or degree of membership48 of elements within an ideal. This new perspective provides a more nuanced understanding49 of the structure and behavior of these algebras, enriching the classical theory with elements50 of fuzzy logic [8]. The application of length-fuzzy ideals allows for a more refined analysis51 of ideals with fuzzy characteristics, enabling better modeling of systems with inherent52 uncertainty or imprecision. By using fuzzy functions to measure membership degrees, this53 approach is applicable in decision-making processes under ambiguity, the design of algo-54 rithms for complex computations, and the study of structures in systems with incomplete55 or vague data. Integrating fuzzy logic into classical theory not only deepens its theoretical56 base but also extends its applicability to fields such as computer science, engineering, and57 areas involving uncertain or imprecise information processing.58 This paper examines the properties and characteristics of length-fuzzy ideals in Sheffer59 stroke Hilbert algebras. By investigating these properties, the goal is to provide fresh per-60 spectives on the theoretical foundation of Hilbert algebras and their potential applications61 in fields such as logic, computer science, and beyond. The concepts of length fuzzy ideals62 and mean fuzzy ideals are introduced in the context of Sheffer stroke Hilbert algebras, and63 their properties are analyzed. The paper further explores the relationships between length64 fuzzy ideals (and mean fuzzy ideals) and traditional ideals. Additionally, it discusses how65 length fuzzy ideals (and mean fuzzy ideals) are related to upper and lower level subsets66 N. Rajesh, T. Oner, A. Iampan, I. Senturk / Eur. J. Pure Appl. Math, 18 (1) (2025), 5779 3 of 18 based on the length (or mean) of a fuzzy structure within Sheffer stroke Hilbert algebras.67 2. Preliminaries68 Sheffer stroke Hilbert algebras represent an important algebraic system in the study69 of logic and lattice theory. These algebras are characterized by the inclusion of the Sheffer70 stroke (NAND) operation, a fundamental logical connectives in Boolean algebra. By71 extending classical Hilbert algebras with this operation, Sheffer stroke Hilbert algebras72 provide a powerful framework for investigating logical structures, with applications in73 fuzzy logic, decision-making, and computational theory. Their study enhances both the74 theoretical understanding of algebraic systems and their practical applications in modeling75 uncertainty and imprecision.76 Definition 1. [13] The operation | in a groupoid A = (A, |) is referred to as the Sheffer77 stroke or Sheffer operation if it satisfies the following condition: for all c, b, d ∈ A,78 (S1) c|b = b|c, (S2) (c|c)|(c|b) = b, (S3) c|((b|d)|(b|d)) = ((c|b)|(c|b))|b, (S4) (c|((c|c)|(b|b)))|(c|((c|c)|(b|b))) = c. To improve the clarity of this manuscript on Sheffer stroke Hilbert algebras, we will79 adopt the following notation throughout:80 p|(q|q) = pq. Definition 2. [6] A Sheffer stroke Hilbert algebra (abbreviated SHA) refers to a groupoid81 A = (A, |, 0) equipped with a Sheffer stroke operation | and 0 is the fixed element in A,82 and it must satisfy the following conditions: for all p, q, r ∈ A,83 (1) (p|(qr|pq))|((pq)(pr)|(pq)(pr)) = pp,84 (2) pq = qp ⇒ p = q.85 Proposition 1. [6] Let A = (A, |, 0) be an SHA. Then the binary relation p ≤ q if and86 only if pq = 1 is a partial order on A.87 Definition 3. [6] Let A = (A, |, 0) be an SHA. A nonempty subset G of A is called an88 ideal of A if for all p, q ∈ A,89 (1) 0 ∈ G,90 (2) pq ∈ G and q ∈ G ⇒ p ∈ G.91 N. Rajesh, T. Oner, A. Iampan, I. Senturk / Eur. J. Pure Appl. Math, 18 (1) (2025), 5779 4 of 18 3. Length fuzzy ideals of Sheffer stroke Hilbert algebras92 This paper introduces the concept of length fuzzy ideals in Sheffer stroke Hilbert93 algebras and examines their associated properties. It establishes the connections between94 length fuzzy ideals and conventional ideals. Furthermore, it explores the relationships95 between length fuzzy ideals and the upper and lower level subsets of the length in an96 interval-valued fuzzy structure within Sheffer stroke Hilbert algebras.97 From now on, unless stated otherwise, we denote an SHA by A = (A, |, 0).98 Definition 4. A fuzzy structure (A, f) of A is defined as:99 (1) a fuzzy ideal of A of type 1 (simply a 1-fuzzy ideal of A) if100 (∀p ∈ A)(f(0) ≥ f(p)), (1) (∀p, q ∈ A)(f(p) ≥ min{f(pq), f(q)}). (2) (2) a fuzzy ideal of A of type 2 (simply a 2-fuzzy ideal of A) if101 (∀p ∈ A)(f(0) ≤ f(p)), (3) (∀p, q ∈ A)(f(p) ≤ min{f(pq), f(q)}). (4) (3) a fuzzy ideal of A of type 3 (simply a 3-fuzzy ideal of A) if102 (∀p ∈ A)(f(0) ≥ f(p)), (5) (∀p, q ∈ A)(f(p) ≥ max{f(pq), f(q)}). (6) (4) a fuzzy ideal of A of type 4 (simply a 4-fuzzy ideal of A) if103 (∀p ∈ A)(f(0) ≤ f(p)), (7) (∀p, q ∈ A)(f(p) ≤ max{f(pq), f(q)}). (8) Definition 5. [14] Given an interval-valued fuzzy structure (A, f̃) over A, we define a fuzzy structure (A, f̃l) on A as follows: f̃l : A → [0, 1]; p 7→ f̃sup(p)− f̃inf(p), which is referred to as the length of f̃ .104 Definition 6. An interval-valued fuzzy structure (A, f̃) over A is referred to as a length105 1-fuzzy (resp., 2-fuzzy, 3-fuzzy, 4-fuzzy) ideal of A if the fuzzy structure (A, f̃l) is a 1-fuzzy106 (resp., 2-fuzzy, 3-fuzzy, 4-fuzzy) ideal of A.107 Proposition 2. Given an interval-valued fuzzy structure (A, f̃) on A, the following state-108 ments hold.109 N. Rajesh, T. Oner, A. Iampan, I. Senturk / Eur. J. Pure Appl. Math, 18 (1) (2025), 5779 5 of 18 (1) If (A, f̃) is a length k-fuzzy ideal of A for k ∈ {1, 3}, then (∀p, q ∈ A)(p ≤ p ⇒ f̃l(p) ≥ f̃l(p)). (2) If (A, f̃) is a length k-fuzzy ideal of A for k ∈ {2, 4}, then (∀p, q ∈ A)(p ≤ q ⇒ f̃l(p) ≤ f̃l(q)). Proof. Let p, q ∈ A be such that p ≤ q. If (A, f̃) is a length k-fuzzy ideal of A for110 k ∈ {1, 3}, then111 f̃l(p) ≥ min{f̃l(pq), f̃l(q)} = min{f̃l(0), f̃l(q)} = f̃l(q) and112 f̃l(p) ≤ max{f̃l(pq), f̃l(q)} = max{f̃l(0), f̃l(q)} = f̃l(q). If (A, f̃) is a length k-fuzzy ideal of A for k ∈ {2, 4}, then113 f̃l(p) ≥ min{f̃l(pq), f̃l(q)} = min{f̃l(0), f̃l(q)} = f̃l(q) and114 f̃l(p) ≤ max{f̃l(pq), f̃l(q)} = max{f̃l(0), f̃l(q)} = f̃l(q). Theorem 1. For any interval-valued fuzzy structure (A, f̃) on A, the following assertions115 are true:116 (1) Every length 3-fuzzy ideal of A is also a length 1-fuzzy ideal of A.117 (2) Every length 2-fuzzy ideal of A is also a length 4-fuzzy ideal of A.118 Proof. (1) Let (A, f̃) be a length 3-fuzzy ideal of A and p, q ∈ A. Then119 f̃l(p) ≥ max{f̃l(pq), f̃l(q)} ≥ min{f̃l(pq), f̃l(q)}. Hence, (A, f̃) is a length 1-fuzzy ideal of A.120 (2) Let (A, f̃) be a length 2-fuzzy ideal of A and p, q ∈ A. Then121 f̃l(p) ≤ min{f̃l(pq), f̃l(q)} ≤ max{f̃l(pq), f̃l(q)}. Hence, (A, f̃) is a length 4-fuzzy ideal of A.122 N. Rajesh, T. Oner, A. Iampan, I. Senturk / Eur. J. Pure Appl. Math, 18 (1) (2025), 5779 6 of 18 Theorem 2. Given an ideal S of A and B1, B2 ∈ P ([0, 1]), let (A, f̃) be an interval-valued fuzzy structure over A given by f̃ : A → P ([0, 1]); p 7→ { B2 if p ∈ S, B1 otherwise. (1) If B1 ⊂ B2, then (A, f̃) is a length 1-fuzzy ideal of A.123 (2) If B2 ⊂ B1, then (A, f̃) is a length 4-fuzzy ideal of A.124 Proof. If p ∈ S, then f̃(p) = B2 and so f̃l(p) = f̃sup(p)− f̃inf(p) = sup f̃(p)− inf f̃(p) = supB2 − inf B2. If p /∈ S, then f̃(p) = B1 and so f̃l(p) = f̃sup(p)− f̃inf(p) = sup f̃(p)− inf f̃(p) = supB1 − inf B1. (1) Assume that B1 ⊂ B2. Then supB2 − inf B2 ≥ supB1 − inf B1. Since 0 ∈ I,125 f̃l(0) = f̃sup(0)− f̃inf(0) = supB2 − inf B2 ≥ f̃l(p) for all p ∈ A.126 Case 1: Let pq, q ∈ S. Then f̃l(p q) = supB2 − inf B2 and f̃l(q) = supB2 − inf B2.127 Thus, min{f̃l(pq), f̃l(q)} = supB2 − inf B2. Since S is an ideal of A, p ∈ S and so128 f̃l(p) = supB2 − inf B2. Thus, f̃l(p) = supB2 − inf B2 = min{f̃l(pq), f̃l(q)}.129 Case 2: Let pq, q /∈ S. Then f̃l(p q) = supB1 − inf B1 and f̃l(q) = supB1 − inf B1, so130 min{f̃l(pq), f̃l(q)} = supB1 − inf B1. Thus, f̃l(p) ≥ supB1 − inf B1 = min{f̃l(pq), f̃l(q)}.131 Case 3: Let pq /∈ S and q ∈ S. Then f̃l(p q) = supB1 − inf B1 and f̃l(q) = supB2 −132 inf B2, so min{f̃l(pq), f̃l(q)} = supB1−inf B1. Thus, f̃l(p) ≥ supB1−inf B1 = min{f̃l(pq), f̃l(q)}.133 Case 4: Let pq ∈ S and q /∈ S. Then f̃l(p q) = supB2 − inf B2 and f̃l(q) = supB1 −134 inf B1, so min{f̃l(pq), f̃l(q)} = supB1−inf B1. Thus, f̃l(p) ≥ supB1−inf B1 = min{f̃l(pq), f̃l(q)}.135 Hence, f̃l is a 1-fuzzy ideal of A and so (A, f̃) is a length 1-fuzzy ideal of A.136 (2) Assume that B2 ⊂ B1. Then supB2 − inf B2 ≤ supB1 − inf B1. Since 0 ∈ I,137 f̃l(0) = f̃sup(0)− f̃inf(0) = supB2 − inf B2 ≤ f̃l(p) for all p ∈ A.138 Case 1: Let pq, q ∈ S. Then f̃l(p q) = supB2 − inf B2 and f̃l(q) = supB2 − inf B2.139 Thus, max{f̃l(pq), f̃l(q)} = supB2 − inf B2. Since S is an ideal of A, x ∈ S and so140 f̃l(p) = supB2 − inf B2. Thus, f̃l(p) = supB2 − inf B2 = max{f̃l(pq), f̃l(q)}.141 Case 2: Let pq, q /∈ S. Then f̃l(p q) = supB1 − inf B1 and f̃l(q) = supB1 − inf B1, so142 max{f̃l(pq), f̃l(q)} = supB1 − inf B1. Thus, f̃l(p) ≤ supB1 − inf B1 = max{f̃l(pq), f̃l(q)}.143 Case 3: Let pq /∈ S and q ∈ S. Then f̃l(p q)) = supB1 − inf B1 and f̃l(q) =144 supB2 − inf B2, so max{f̃l(pq), f̃l(q)} = supB1 − inf B1. Thus, f̃l(p) ≤ supB1 − inf B1 =145 max{f̃l(pq), f̃l(q)}.146 Case 4: Let pq ∈ S and q /∈ S. Then f̃l(p q) = supB2 − inf B2 and f̃l(q) = supB1 −147 inf B1, so max{f̃l(pq), f̃l(q)} = supB1−inf B1. Thus, f̃l(p) ≤ supB1−inf B1 = max{f̃l(pq), f̃l(q)}.148 Hence, f̃l is a 4-fuzzy ideal of A and so (A, f̃) is a length 4-fuzzy ideal of A.149 N. Rajesh, T. Oner, A. Iampan, I. Senturk / Eur. J. Pure Appl. Math, 18 (1) (2025), 5779 7 of 18 Definition 7. Let (A, f) be a fuzzy structure in A. For any t ∈ [0, 1], the sets U(f, t) = {p ∈ A : f(p) ≥ t}, L(f, t) = {p ∈ A : f(p) ≤ t}, are called upper t-level subset and lower t-level subset of f , respectively.150 Theorem 3. An interval-valued fuzzy structure (A, f̃) over A is a length 1-fuzzy ideal of151 A if and only if the set U(f̃l, t) is an ideal of A for all t ∈ [0, 1] with U(f̃l, t) ̸= ∅.152 Proof. Assume that an interval-valued fuzzy structure (A, f̃) over A is a length 1-fuzzy153 ideal of A and let t ∈ [0, 1] be such that U(f̃ , t) is nonempty. Obviously, 0 ∈ U(f̃ , t). Let154 p, q ∈ A be such that pq ∈ U(f̃ , t) and q ∈ U(f̃ , t). Then f̃l(p q) ≥ t and f̃l(q) ≥ t, which155 imply from (2) that f̃l(p) ≥ min{f̃l(pq), f̃l(q)} ≥ t. Hence, p ∈ U(f̃ , t), and therefore156 U(f̃ , t) is an ideal of A.157 Conversely, suppose that U(f̃l, t) is an ideal of A for all t ∈ [0, 1] with U(f̃l, t) ̸= ∅. If158 f̃l(0) < f̃l(k) for some k ∈ A, then k ∈ U(f̃l, f̃l(k)) and hence U(f̃l, f̃l(k)) is an ideal of A.159 Thus, 0 ∈ U(f̃l, f̃l(k)), and so f̃l(0) ≥ f̃l(k). This is a contradiction, and thus f̃l(0) ≥ f̃l(p)160 for all p ∈ A. Assume that there exist k, l ∈ A such that f̃l(k) < min{f̃l(kl), f̃l(l)}.161 Taking t = min{f̃l(kl), f̃l(l)} implies that k ∈ U(f̃l, t). Since U(f̃l, t) is an ideal of A,162 a ∈ U(f̃l, t). Hence, f̃l(k) ≥ t = min{f̃l(kl), f̃l(l)}, which is a contradiction. Hence,163 f̃l(p) ≥ min{f̃l(pq), f̃l(q)} for all p, q ∈ A. Therefore, (A, f̃) is a length 1-fuzzy ideal of A.164 Corollary 1. If (A, f̃) is a length 3-fuzzy ideal of A, then the set U(f̃l, t) is an ideal of A165 for all t ∈ [0, 1] with U(f̃l, t) ̸= ∅.166 Proof. It is straightforward by Theorems 1 and 3.167 Theorem 4. An interval-valued fuzzy structure (A, f̃) over A is a length 4-fuzzy ideal of168 A if and only if the set L(f̃l, t) is an ideal of A for all t ∈ [0, 1] with L(f̃l, t) ̸= ∅.169 Proof. Assume that an interval-valued fuzzy structure (A, f̃) over A is a length 4-fuzzy170 ideal of A and let t ∈ [0, 1] be such that L(f̃ , t) is nonempty. Obviously, 0 ∈ L(f̃ , t). Let171 p, q ∈ A be such that pq ∈ L(f̃ , t) and q ∈ L(f̃ , t). Then f̃l(p q) ≤ t and f̃l(q) ≤ t, which172 imply from (8) that f̃l(p) ≤ min{f̃l(pq), f̃l(q)} ≤ t. Hence, p ∈ L(f̃ , t), and therefore173 L(f̃ , t) is an ideal of A.174 Conversely, suppose that L(f̃l, t) is an ideal of A for all t ∈ [0, 1] with L(f̃l, t) ̸= ∅. If175 f̃l(0) > f̃l(k) for some k ∈ A, then k ∈ L(f̃l, f̃l(k)) and hence L(f̃l, f̃l(k)) is an ideal of A.176 Thus, 0 ∈ L(f̃l, f̃l(k)), and so f̃l(0) ≤ f̃l(k). This is a contradiction, and thus f̃l(0) ≤ f̃l(p)177 for all p ∈ A. Assume that there exist k, l ∈ A such that f̃l(k) > max{f̃l(kl), f̃l(l)}.178 Taking t = max{f̃l(kl), f̃l(l)} implies that k ∈ L(f̃l, t). Since L(f̃l, t) is an ideal of A,179 k ∈ L(f̃l, t). Hence, f̃l(k) ≤ t = max{f̃l(kl), f̃l(l)}, which is a contradiction. Hence,180 f̃l(p) ≤ max{f̃l(pq), f̃l(q)} for all p, q ∈ A. Therefore, (A, f̃) is a length 4-fuzzy ideal of A.181 N. Rajesh, T. Oner, A. Iampan, I. Senturk / Eur. J. Pure Appl. Math, 18 (1) (2025), 5779 8 of 18 Corollary 2. If (A, f̃) is a length 2-fuzzy ideal of A, then the set L(f̃l, t) is an ideal of A182 for all t ∈ [0, 1] with L(f̃l, t) ̸= ∅.183 Proof. It is straightforward by Theorems 1 and 4.184 Theorem 5. If (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃inf) is185 constant and (A, f̃sup) is a 1-fuzzy ideal of A, then (A, f̃) is a length 1-fuzzy ideal of A.186 Proof. Assume that (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃inf)187 is constant and (A, f̃sup) is a 1-fuzzy ideal of A. Let p, q ∈ A. Since (A, f̃inf) is constant,188 f̃inf(p) = f̃inf(0) for all p ∈ A. Since (A, f̃sup) is a 1-fuzzy ideal of A,189 (∀p ∈ A)(f̃sup(0) ≥ f̃sup(p)), (9) (∀p, q ∈ A)(f̃sup(p) ≥ min{f̃sup(pq), f̃sup(q)}). (10) Let p ∈ A. Then190 f̃l(0) = f̃sup(0)− f̃inf(0) ≥ f̃sup(p)− f̃inf(0) = f̃sup(p)− f̃inf(p) = f̃l(p). Let p, q ∈ A. Then191 f̃l(p) = f̃sup(p)− f̃inf(p) = f̃sup(p)− f̃inf(0) ≥ min{f̃sup(pq), f̃sup(q)} − f̃inf(0) = min{f̃sup(pq)− f̃inf(0), f̃sup(q)− f̃inf(0)} = min{f̃sup(pq)− f̃inf(p q), f̃sup(q)− f̃inf(q)} = min{f̃l(pq), f̃l(q)}. Hence, (A, f̃l) is a 1-fuzzy ideal of A, that is, (A, f̃) is a length 1-fuzzy ideal of A.192 Theorem 6. If (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃inf) is193 constant and (A, f̃sup) is a 4-fuzzy ideal of A, then (A, f̃) is a length 4-fuzzy ideal of A.194 Proof. Assume that (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃inf)195 is constant and (A, f̃sup) is a 4-fuzzy ideal of A. Let p, q ∈ A. Since (A, f̃inf) is constant,196 we have f̃inf(p) = f̃inf(0) for all p ∈ A. Since (A, f̃sup) is a 4-fuzzy ideal of A, we have197 (∀p ∈ A)(f̃sup(0) ≤ f̃sup(p)), (11) (∀p, q ∈ A)(f̃sup(p) ≤ max{f̃sup(pq), f̃sup(q)}). (12) N. Rajesh, T. Oner, A. Iampan, I. Senturk / Eur. J. Pure Appl. Math, 18 (1) (2025), 5779 9 of 18 Let p ∈ A. Then198 f̃l(0) = f̃sup(0)− f̃inf(0) ≤ f̃sup(p)− f̃inf(0) = f̃sup(p)− f̃inf(p) = f̃l(p). Let p, q ∈ A. Then199 f̃l(p) = f̃sup(p)− f̃inf(p) = f̃sup(p)− f̃inf(0) ≤ max{f̃sup(pq), f̃sup(p)} − f̃inf(0) = max{f̃sup(pq)− f̃inf(0), f̃sup(q)− f̃inf(0)} = max{f̃sup(pq)− f̃inf(p q), f̃sup(q)− f̃inf(q)} = max{f̃l(pq), f̃l(q)}. Hence, (A, f̃l) is a 4-fuzzy ideal of A, that is, (A, f̃) is a length 4-fuzzy ideal of A.200 Corollary 3. If (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃inf) is201 constant and (A, f̃sup) is a 2-fuzzy ideal of A, then (A, f̃) is a length 4-fuzzy ideal of A.202 Proof. It is straightforward by Theorems 1 and 6.203 Corollary 4. For j ∈ {2, 4}, every (2(3), j)-hyperfuzzy ideal of A is a length 4-fuzzy ideal.204 Proof. It is straightforward by Theorem 6 and Corollary 3.205 Theorem 7. If (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃sup) is206 constant and (A, f̃inf) is a 4-fuzzy ideal of A, then (A, f̃) is a length 1-fuzzy ideal of A.207 Proof. Assume that (A, f̃) is an interval-valued fuzzy structure over A in which208 (A, f̃sup) is constant and (A, f̃inf) is a 4-fuzzy ideal of A. Let p, q ∈ A. Since (A, f̃sup) is209 constant, we have f̃sup(p) = f̃sup(0) for all p ∈ A. Since (A, f̃inf) is a 4-fuzzy ideal of A,210 we have211 (∀p ∈ A)(f̃inf(0) ≤ f̃inf(p)), (13) (∀p, q ∈ A)(f̃inf(p) ≤ max{f̃inf(pq), f̃inf(q)}). (14) Let p ∈ A. Then212 f̃l(0) = f̃sup(0)− f̃inf(0) ≥ f̃sup(0)− f̃inf(p) = f̃sup(p)− f̃inf(p) = f̃l(p). N. Rajesh, T. Oner, A. Iampan, I. Senturk / Eur. J. Pure Appl. Math, 18 (1) (2025), 5779 10 of 18 Let p, q ∈ A. Then213 f̃l(p) = f̃sup(p)− f̃inf(p) = f̃sup(0)− f̃inf(p) ≥ f̃sup(0)−max{f̃inf(pq), f̃inf(q)} = min{f̃sup(0)− f̃inf(p q), f̃sup(0)− f̃inf(q)} = min{f̃sup(pq)− f̃inf(p q), f̃sup(q)− f̃inf(q)} = min{f̃l(pq), f̃l(q)}. Hence, (A, f̃l) is a 1-fuzzy ideal of A, that is, (A, f̃) is a length 1-fuzzy ideal of A.214 Corollary 5. If (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃sup) is215 constant and (A, f̃inf) is a 2-fuzzy ideal of A, then (A, f̃) is a length 1-fuzzy ideal of A.216 Proof. It is straightforward by Theorems 1 and 7.217 Corollary 6. For i ∈ {2, 4}, every (i, 2(3))-hyperfuzzy ideal of A is a length 1-fuzzy ideal.218 Proof. It is straightforward by Theorem 7 and Corollary 5.219 Theorem 8. If (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃sup) is220 constant and (A, f̃inf) is a 1-fuzzy ideal of A, then (A, f̃) is a length 4-fuzzy ideal of A.221 Proof. Assume that (A, f̃) is an interval-valued fuzzy structure over A in which222 (A, f̃sup) is constant and (A, f̃inf) is a 1-fuzzy ideal of A. Let p, q ∈ A. Since (A, f̃sup) is223 constant, we have f̃sup(p) = f̃sup(0) for all p ∈ A. Since (A, f̃inf) is a 1-fuzzy ideal of A,224 we have225 (∀p ∈ A)(f̃inf(0) ≥ f̃inf(p)), (15) (∀p, q ∈ A)(f̃inf(p) ≥ min{f̃inf(pq), f̃inf(q)}). (16) Let p ∈ A. Then226 f̃l(0) = f̃sup(0)− f̃inf(0) ≤ f̃sup(0)− f̃inf(p) = f̃sup(p)− f̃inf(p) = f̃l(p). Let p, q ∈ A. Then227 f̃l(p) = f̃sup(p)− f̃inf(p) = f̃sup(0)− f̃inf(p) ≤ f̃sup(0)−min{f̃pinf(pq), f̃inf(q)} = max{f̃sup(0)− f̃inf(p q), f̃sup(0)− f̃inf(q)} = max{f̃sup(pq)− f̃inf(p q), f̃sup(qq)− f̃inf(q)} = max{f̃l(pq), f̃l(q)}. Hence, (A, f̃l) is a 4-fuzzy ideal of A, that is, (A, f̃) is a length 4-fuzzy ideal of A.228 N. Rajesh, T. Oner, A. Iampan, I. Senturk / Eur. J. Pure Appl. Math, 18 (1) (2025), 5779 11 of 18 4. Mean fuzzy ideals of Sheffer stroke Hilbert algebras229 In this section, we introduce the concept of the mean of an interval-valued fuzzy230 structure within Sheffer stroke Hilbert algebras. We also define the notion of mean fuzzy231 ideals in these algebras and investigate their related properties. Furthermore, we establish232 the relationships between mean fuzzy ideals and traditional fuzzy ideals.233 Definition 8. [14] Given an interval-valued fuzzy structure (A, f̃) over A, we define a fuzzy structure (A, f̃m) in A as follows: f̃m : A → [0, 1]; p 7→ f̃sup(p) + f̃inf(p) 2 , which is called the mean of f̃ .234 Definition 9. An interval-valued fuzzy structure (A, f̃) over A is called a mean 1-fuzzy235 (resp., 2-fuzzy, 3-fuzzy and 4-fuzzy) ideal of A if the fuzzy structure (A, f̃m) is a 1-fuzzy236 (resp., 2-fuzzy, 3-fuzzy and 4-fuzzy) ideal of A.237 Proposition 3. If (A, f̃) is a mean k-fuzzy ideal of A for k = 1, 3, then (∀p ∈ A)(f̃m(0) ≥ f̃m(p)). Proof. Let (A, f̃) be a mean k-fuzzy ideal of A for k = 1, 3 and p ∈ A. Then238 f̃m(0) = f̃sup(0) + f̃inf(0) 2 ≥ f̃sup(p) + f̃inf(p) 2 = f̃m(p). Proposition 4. If (A, f̃) is a mean k-fuzzy ideal of A for k = 2, 4, then (∀p ∈ A)(f̃m(0) ≤ f̃m(p)). Proof. Let (A, f̃) be a mean k-fuzzy ideal of A for k = 2, 4 and p ∈ A. Then239 f̃m(0) = f̃sup(0) + f̃inf(0) 2 ≤ f̃sup(p) + f̃inf(p) 2 = f̃m(p). Theorem 9. Every mean 3-fuzzy ideal of A is a mean 1-fuzzy ideal of A.240 N. Rajesh, T. Oner, A. Iampan, I. Senturk / Eur. J. Pure Appl. Math, 18 (1) (2025), 5779 12 of 18 Proof. Let (A, f̃) be a mean 3-fuzzy ideal of A and p, q ∈ A. Then241 f̃m(p) = f̃sup(p) + f̃inf(p) 2 = f̃sup(p) 2 + f̃inf(p) 2 ≥ max { f̃sup(p q) 2 , f̃sup(q) 2 } +max { f̃inf(p q) 2 , f̃inf(q) 2 } ≥ min { f̃sup(p q) 2 , f̃sup(q) 2 } +min { f̃inf(p q) 2 , f̃inf(q) 2 } = min { f̃sup(p q) + f̃inf(p q) 2 , f̃sup(q) + f̃inf(q) 2 } = min{f̃m(pq), f̃m(q)}. Hence, (A, f̃) is a mean 1-fuzzy ideal of A.242 Theorem 10. Every mean 2-fuzzy ideal of A is a mean 4-fuzzy ideal of A.243 Proof. Let (A, f̃) be a mean 2-fuzzy ideal of A and p, q ∈ A. Then244 f̃m(p) = f̃sup(p) + f̃inf(p) 2 = f̃sup(p) 2 + f̃inf(p) 2 ≤ min { f̃sup(p q) 2 , f̃sup(q) 2 } +min { f̃inf(p q) 2 , f̃inf(q) 2 } ≤ max { f̃sup(p q) 2 , f̃sup(q) 2 } +max { f̃inf(p q) 2 , f̃inf(q) 2 } = max { f̃sup(p q) + f̃inf(p q) 2 , f̃sup(q) + f̃inf(q) 2 } = max{f̃m(pq), f̃m(q)}. Hence, (A, f̃) is a mean 4-fuzzy ideal of A.245 Theorem 11. Mean 2-fuzzy ideal and mean 3-fuzzy ideal of A coincide.246 Proof. It is straightforward by Theorems 9 and 10.247 Theorem 12. Given an ideal S of A and B1, B2 ∈ P ([0, 1]), let (A, f̃) be an interval- valued fuzzy structure over A given by f̃ : A → P ([0, 1]); p 7→ { B2, if p ∈ S B1, otherwise. N. Rajesh, T. Oner, A. Iampan, I. Senturk / Eur. J. Pure Appl. Math, 18 (1) (2025), 5779 13 of 18 (1) If supB2 ≥ supB1 and inf B2 ≥ inf B1, then (A, f̃) is a mean 1-fuzzy ideal of A.248 (2) If supB2 ≤ supB1 and inf B2 ≤ inf B1, then (A, f̃) is a mean 4-fuzzy ideal of A.249 Proof. If p ∈ S, then f̃(p) = B2 and so f̃m(p) = f̃sup(p) + f̃inf(p) 2 = sup f̃(p) + inf f̃(p) 2 = supB2 + inf B2 2 . If p /∈ S, then f̃(p) = B1 and so f̃m(p) = f̃sup(p) + f̃inf(p) 2 = sup f̃(p) + inf f̃(p) 2 = supB1 + inf B1 2 . (1) Assume that supB2 ≥ supB1 and inf B2 ≥ inf B1. Then supB2 + inf B2 2 ≥ supB1 + inf B1 2 . Case 1: Let pq, q ∈ S. Then f̃m(pq) = supB2 + inf B2 2 and fm(q) = supB2 + inf B2 2 .250 Thus, min{f̃m(pq), f̃m(q)} = supB2 + inf B2 2 . Since S is an ideal of A, we have p ∈ S and251 so f̃m(p) = supB2 + inf B2 2 . Thus, f̃m(p) = supB2 + inf B2 2 = min{f̃m(pq), f̃m(q)}.252 Case 2: Let pq, q /∈ S. Then f̃m(pq) = supB1 + inf B1 2 and f̃m(q) = supB1 + inf B1 2 , so253 min{f̃m(pq), f̃m(q)} = supB1 + inf B1 2 . Thus, f̃m(p) ≥ supB1 + inf B1 2 = min{f̃m(pq), f̃m(q)}.254 255 Case 3: Let pq /∈ S and q ∈ S. Then f̃m(pq) = supB1 + inf B1 2 and f̃m(q) =256 supB2 + inf B2 2 , so min{f̃m(pq), f̃m(q)} = supB1 + inf B1 2 . Thus, f̃m(p) ≥ supB1 + inf B1 2 =257 min{f̃m(pq), f̃m(q)}.258 259 Case 4: Let pq ∈ S and q /∈ S. Then f̃m(pq) = supB2 + inf B2 2 and f̃m(q) =260 supB1 + inf B1 2 , so min{f̃m(pq), f̃m(q)} = supB1 + inf B1 2 . Thus, f̃m(p) ≥ supB1 + inf B1 2 =261 min{f̃m(pq), f̃m(q)}.262 Hence, f̃m is a 1-fuzzy ideal of A and so (A, f̃) is a mean 1-fuzzy ideal of A.263 (2) Assume that supB2 ≤ supB1 and inf B2 ≤ inf B1. Then supB2 + inf B2 2 ≤ supB1 + inf B1 2 . Case 1: Let pq, q ∈ S. Then f̃m(pq) = supB2 + inf B2 2 and f̃m(q) = supB2 + inf B2 2 ,264 so max{f̃m(pq), f̃m(q)} = supB2 + inf B2 2 . Since S is an ideal of A, we have p ∈ S and so265 f̃m(p) = supB2 + inf B2 2 . Thus, f̃m(p) = supB2 + inf B2 2 = max{f̃m(pq), f̃m(q)}.266 N. Rajesh, T. Oner, A. Iampan, I. Senturk / Eur. J. Pure Appl. Math, 18 (1) (2025), 5779 14 of 18 Case 2: Let pq, q /∈ S. Then f̃m(pq) = supB1 + inf B1 2 and f̃m(q) = supB1 + inf B1 2 , so267 max{f̃m(pq), f̃m(q)} = supB1 + inf B1 2 . Thus, f̃m(p) ≤ supB1 + inf B1 2 = max{f̃m(pq), f̃m(q)}.268 Case 3: Let pq /∈ S and q ∈ S. Then f̃m(pq) = supB1 + inf B1 2 and f̃m(q) =269 supB2 + inf B2 2 , so max{f̃m(pq), f̃m(q)} = supB1 + inf B1 2 . Thus, f̃m(p) ≤ supB1 + inf B1 2 =270 max{f̃m(pq)), f̃m(q)}.271 Case 4: Let pq ∈ S and q /∈ S. Then f̃m(pq) = supB2 + inf B2 2 and f̃m(q) =272 supB1 + inf B1 2 , so max{f̃m(pq), f̃m(q)} = supB1 + inf B1 2 . Thus, f̃m(p) ≤ supB1 + inf B1 2 =273 max{f̃m(pq), f̃m(q)}.274 Hence, f̃m is a 4-fuzzy ideal of A and so (A, f̃) is a mean 4-fuzzy ideal of A.275 Theorem 13. An interval-valued fuzzy structure (A, f̃m) over A is a mean 1-fuzzy ideal276 of A if and only if the set U(f̃m, t) is an ideal of A for all t ∈ [0, 1] with U(f̃m, t) ̸= ∅.277 Proof. Assume that an interval-valued fuzzy structure (A, f̃m) over A is a mean 1-fuzzy278 ideal of A and let t ∈ [0, 1] be such that U(f̃m, t) is nonempty. Obviously, 0 ∈ U(f̃m, t).279 Let p, q ∈ A be such that pq ∈ U(f̃m, t) and q ∈ U(f̃m, t). Then f̃m(pq) ≥ t and f̃m(q) ≥ t,280 which imply from (2) that f̃m(p) ≥ min{f̃m(pq), f̃m(q)} ≥ t. Hence, p ∈ U(f̃m, t), and281 therefore U(f̃m, t) is an ideal of A.282 Conversely, suppose that U(f̃m, t) is an ideal of A for all t ∈ [0, 1] with U(f̃m, t) ̸= ∅.283 If f̃m(0) < f̃m(k) for some k ∈ A, then k ∈ U(f̃m, f̃m(k)) and hence U(f̃m, f̃m(k)) is an284 ideal of A. Thus, 0 ∈ U(f̃m, f̃m(k)), and so f̃m(0) ≥ f̃m(k). This is a contradiction,285 and thus f̃m(0) ≥ f̃m(p) for all p ∈ A. Assume that there exist k, l ∈ A such that286 f̃m(k) < min{f̃m(kl), f̃m(l)}. Taking t = min{f̃m(kl), f̃m(l)} implies that k ∈ U(f̃ , t). Since287 U(f̃m, t) is an ideal of A, we have k ∈ U(f̃ , t). Hence, f̃m(k) ≥ t = min{f̃m(kl), f̃m(k)},288 which is a contradiction. Hence, f̃m(p) ≥ min{f̃m(pq), f̃m(q)} for all p, q ∈ A. Therefore,289 (A, f̃m) is a mean 1-fuzzy ideal of A.290 Corollary 7. If (A, f̃) is a mean 3-fuzzy ideal of A, then U(f̃m, t) is an ideal of A for all291 t ∈ [0, 1] with U(f̃m, t) ̸= ∅.292 Proof. It is straightforward by Theorems 9 and 13.293 Theorem 14. An interval-valued fuzzy structure (A, f̃) over A is a mean 4-fuzzy ideal of294 A if and only if the set L(f̃m, t) is an ideal of A for all t ∈ [0, 1] with L(f̃m, t) ̸= ∅.295 Proof. Assume that an interval-valued fuzzy structure (A, f̃m) over A is a mean 4-fuzzy296 ideal of A and let t ∈ [0, 1] be such that L(f̃m, t) is nonempty. Obviously, 0 ∈ L(f̃m, t).297 Let p, q ∈ A be such that pq ∈ L(f̃m, t) and q ∈ L(f̃m, t). Then f̃m(pq) ≤ t and f̃m(q) ≤ t,298 which imply from (8) that f̃m(p) ≤ max{f̃m(pq), f̃m(q)} ≤ t. Hence, p ∈ L(f̃m, t), and299 therefore L(f̃m, t) is an ideal of A.300 N. Rajesh, T. Oner, A. Iampan, I. Senturk / Eur. J. Pure Appl. Math, 18 (1) (2025), 5779 15 of 18 Conversely, suppose that L(f̃m, t) is an ideal of A for all t ∈ [0, 1] with L(f̃m, t) ̸= ∅.301 If f̃m(0) > f̃m(k) for some k ∈ A, then k ∈ L(f̃m, f̃m(k)) and hence L(f̃m, f̃m(k)) is an302 ideal of A. Thus, 0 ∈ L(f̃m, f̃m(k)), and so f̃m(0) ≤ f̃m(k). This is a contradiction, and303 thus f̃m(0) ≤ f̃m(p) for all p ∈ A. Assume that there exist k, l ∈ A such that f̃m(k) >304 max{f̃m(kl)), f̃m(l)}. Taking t = max{f̃m(kl), f̃m(l)} implies that k ∈ L(f̃m, t). Since305 L(f̃m, t) is an ideal of A, we have k ∈ L(f̃m, t). Hence, f̃m(k) ≤ t = max{f̃m(kl), f̃m(l)},306 which is a contradiction. Hence, f̃m(p) ≤ max{f̃m(pq), f̃m(q)} for all p, q ∈ A. Therefore,307 (A, f̃m) is a mean 4-fuzzy ideal of A.308 Corollary 8. If (A, f̃) is a mean 2-fuzzy ideal of A, then L(f̃m, t) is an ideal of A for all309 t ∈ [0, 1] with L(f̃m, t) ̸= ∅.310 Proof. It is straightforward by Theorems 1 and 14.311 Theorem 15. If (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃inf) is312 constant and (A, f̃sup) is a 1-fuzzy ideal of A, then (A, f̃) is a mean 1-fuzzy ideal of A.313 Proof. Assume that (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃inf)314 is constant and (A, f̃sup) is a 1-fuzzy ideal of A. Let p, q ∈ A. Since (A, f̃inf) is constant,315 we have f̃inf(p) = f̃inf(0) for all p ∈ A. Since (A, f̃sup) is a 1-fuzzy ideal of A, we have316 f̃sup(p) ≥ min{f̃sup(p), f̃sup(q)}. Thus,317 f̃m(p) = f̃sup(p) + f̃inf(p) 2 = f̃sup(p) 2 + f̃inf(0) 2 ≥ min { f̃sup(p q) 2 + f̃inf(q) 2 } + f̃inf(0) 2 = min { f̃sup(p q) 2 + f̃inf(0) 2 , f̃sup(q) 2 + f̃inf(0) 2 } = min { f̃sup(p q)) + f̃inf(p) 2 , f̃sup(q) + f̃inf(q) 2 } = min{f̃m(pq), f̃m(q)}. Hence, (A, f̃m) is a 1-fuzzy ideal of A, that is, (A, f̃) is a mean 1-fuzzy ideal of A.318 Theorem 16. If (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃inf) is319 constant and (A, f̃sup) is a 4-fuzzy ideal of A, then (A, f̃) is a mean 4-fuzzy ideal of A.320 Proof. Assume that (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃inf)321 is constant and (A, f̃sup) is a 4-fuzzy ideal of A. Let p, q ∈ A. Since (A, f̃inf) is constant,322 N. Rajesh, T. Oner, A. Iampan, I. Senturk / Eur. J. Pure Appl. Math, 18 (1) (2025), 5779 16 of 18 we have f̃inf(p) = f̃inf(0) for all p ∈ A. Since (A, f̃sup) is a 4-fuzzy ideal of A, we have323 f̃sup(p) ≤ max{f̃sup(p), f̃sup(q)}. Thus,324 f̃m(p) = f̃sup(p) + f̃inf(p) 2 = f̃sup(p) 2 + f̃inf(0) 2 ≥ min { f̃sup(p q) 2 + f̃inf(q) 2 } + f̃inf(0) 2 = min { f̃sup(p q) 2 + f̃inf(0) 2 , f̃sup(q) 2 + f̃inf(0) 2 } = min { f̃sup(p q) + f̃inf(p q) 2 , f̃sup(q) + f̃inf(q) 2 } = min{f̃m(pq), f̃m(q)}. Hence, (A, f̃m) is a 4-fuzzy ideal of A, that is, (A, f̃) is a mean 4-fuzzy ideal of A.325 Theorem 17. If (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃sup) is326 constant and (A, f̃inf) is a 4-fuzzy ideal of A, then (A, f̃) is a mean 4-fuzzy ideal of A.327 Proof. Assume that (A, f̃) is an interval-valued fuzzy structure over A in which328 (A, f̃sup) is constant and (A, f̃inf) is a 4-fuzzy ideal of A. Let p, q ∈ A. Since (A, f̃sup) is329 constant, we have f̃sup(p) = f̃sup(0) for all p ∈ A. Since (A, f̃inf) is a 4-fuzzy ideal of A,330 we have f̃inf(p) ≤ max{f̃inf(p), f̃inf(q)}. Thus,331 f̃m(p) = f̃sup(p) + f̃inf(p) 2 = f̃sup(0) + f̃inf(p) 2 = f̃sup(0) 2 + f̃inf(p) 2 ≤ f̃sup(0) 2 + max { f̃sup(p q) 2 , f̃inf(q) 2 } = max { f̃sup(0) 2 + f̃inf(p q) 2 , f̃sup(0) 2 + f̃inf(q) 2 } = max { f̃sup(p q)) + f̃inf(p q) 2 , f̃sup(q) + f̃inf(q) 2 } = max{f̃m(pq), f̃m(q)}. Hence, (A, f̃m) is a 4-fuzzy ideal of A, that is, (A, f̃) is a mean 4-fuzzy ideal of A.332 N. Rajesh, T. Oner, A. Iampan, I. Senturk / Eur. J. Pure Appl. Math, 18 (1) (2025), 5779 17 of 18 Theorem 18. If (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃sup) is333 constant and (A, f̃inf) is a 1-fuzzy ideal of A, then (A, f̃) is a mean 1-fuzzy ideal of A.334 Proof. Assume that (A, f̃) is an interval-valued fuzzy structure over A in which335 (A, f̃sup) is constant and (A, f̃inf) is a 1-fuzzy ideal of A. Let p, q ∈ A. Since (A, f̃sup) is336 constant, we have f̃sup(p) = f̃sup(0) for all p ∈ A. Since (A, f̃inf) is a 1-fuzzy ideal of A,337 we obtain f̃inf(p) ≥ min{f̃inf(p), f̃inf(q)}. Thus,338 f̃m(p) = f̃sup(p) + f̃inf(p) 2 = f̃sup(0) + f̃inf(p) 2 = f̃sup(0) 2 + f̃inf(p) 2 ≥ f̃sup(0) 2 + min { f̃inf(p q) 2 , f̃inf(q) 2 } = min { f̃sup(0) 2 + f̃sup(p q) 2 , f̃sup(0) 2 , f̃sup(q) 2 } = min{ f̃sup(0) + f̃sup(p q) 2 , f̃sup(0) + f̃sup(q) 2 } = min{f̃m(pq), f̃m(q)}. Hence, (A, f̃m) is a 1-fuzzy ideal of A, that is, (A, f̃) is a mean 1-fuzzy ideal of A.339 5. Conclusion340 This study extends the theoretical foundation of Sheffer stroke Hilbert algebras by341 introducing and analyzing the notions of length fuzzy ideals and mean fuzzy ideals within342 an interval-valued fuzzy structure. By defining these concepts, the research provides a343 more nuanced understanding of fuzzy logic applications in algebraic structures, empha-344 sizing the relationships between fuzzy ideals and traditional ideals. The characterizations345 and properties of length fuzzy ideals and mean fuzzy ideals demonstrate their alignment346 with upper and lower level subsets, offering a framework to explore the gradations of347 membership functions. Furthermore, the findings highlight the potential of these fuzzy348 constructs in bridging algebraic theory with practical applications in logic, computer sci-349 ence, and uncertainty modeling. 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