EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5914 ISSN 1307-5543 – ejpam.com Published by New York Business Global Investigating Length and Mean-Fuzzy Subalgebras in Sheffer Stroke Hilbert Algebras Neelamegarajan Rajesh1, Tahsin Oner2, Aiyared Iampan3,∗, Akbar Rezaei4 1 Department of Mathematics, Rajah Serfoji Government College, Thanjavur-613005, Tamil Nadu, India 2 Department of Mathematics, Faculty of Science, Ege University, 35100 Izmir, Turkey 3 Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand 4 Department of Mathematics, Faculty of Basic Science, Payame Noor University, P.O. Box 19395-4697, Tehran, Iran Abstract. The aim of this paper is to introduce the notions of the length and the mean of an interval-valued fuzzy structure in Sheffer stroke Hilbert algebras. The notions of length-fuzzy sub- algebras and mean-fuzzy subalgebras of Sheffer stroke Hilbert algebras are introduced, and related properties are investigated. Characterizations of length-fuzzy subalgebras and mean-fuzzy subal- gebras are discussed. Relations between length-fuzzy subalgebras (resp., mean-fuzzy subalgebras) and subalgebras are established. Moreover, we discuss the relationships among length-fuzzy sub- algebras (resp., mean-fuzzy subalgebras) and upper and lower-level subsets of the length (resp., mean) of an interval-valued fuzzy structure in Sheffer stroke Hilbert algebras. 2020 Mathematics Subject Classifications: 20N05, 94D05, 03E72 Key Words and Phrases: Sheffer stroke Hilbert algebra, subalgebra, length-fuzzy subalgebra, mean-fuzzy subalgebra 1. Introduction The Sheffer operation, also known as the Sheffer stroke or NAND operator, was first introduced by Henry Maurice Sheffer [1]. This operation holds significance because it can be used independently, without any other logical operators, to construct a logical system. This means that any axiom of a logical system can be restated using only the Sheffer operation. Because of this property, it becomes easier to control certain properties of the newly constructed logical system. Additionally, it’s worth noting that the axioms of Boolean algebra, which are the algebraic counterpart of classical propositional calculus, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5914 Email addresses: nrajesh topology@yahoo.co.in (N. Rajesh), tahsin.oner@ege.edu.tr (T. Oner), aiyared.ia@up.ac.th (A. Iampan), rezaei@pnu.ac.ir (A. Rezaei) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) N. Rajesh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5914 2 of 21 can be expressed solely using the Sheffer operation. This highlights the fundamental nature and versatility of the Sheffer operation in logical and algebraic systems. In 2002, McCune et al. [2] applied the Sheffer stroke operation for Boolean algebras, and it is shown that there is no shorter axiom in terms of the Sheffer stroke. Algebraic structures play a prominent role in mathematics, with wide-ranging applications in various disciplines, including theoretical physics, computer science, control engineering, information sciences, coding theory, and topological spaces, among others. This provides sufficient motivation for researchers to simplify axioms for various algebraic structures, e.g., see [3–5]. In 1950, Henkin [6] introduced the notion of “implicative model” as a model of positive implicative propositional calculus. In 1960, Monteiro [7] gave the name “Hilbert algebras” to the dual algebras of Henkin’s implicative models. In 1966, Diego [8] intensively studied and developed some properties of Hilbert algebras. In 2021, Oner et al. [9] investigated the relation between Sheffer stroke and Hilbert algebras. Also, see [10]. In 1965, Zadeh [11] proposed a new theory named fuzzy set theory. Then several researchers studied various extensions and generalizations of this theory, e.g., intuitionistic fuzzy sets [12], L-fuzzy sets [13], type-2 fuzzy sets [14], interval-valued fuzzy sets [15], multi fuzzy sets [16], bipolar- valued fuzzy sets [17], m-polar fuzzy sets [18], and neutrosophic sets [19, 20]. Recently, many researchers have studied and applied concepts of fuzzy sets, including fuzzy (weak) filters and deductive systems, to Sheffer stroke Hilbert algebras [21–24]. This paper aims to introduce and explore the concepts of length and mean within interval-valued fuzzy structures in Sheffer stroke Hilbert algebras. Specifically, we define and analyze the notions of length-fuzzy subalgebras and mean-fuzzy subalgebras, inves- tigating their key properties and characterizations. The study establishes relationships between these fuzzy subalgebras and traditional subalgebras, providing a deeper under- standing of their structural interaction. Additionally, we examine the connections between length-fuzzy (resp., mean-fuzzy) subalgebras and their corresponding upper and lower- level subsets within interval-valued fuzzy structures. These findings offer a comprehensive framework for studying gradations of membership and their implications in Sheffer stroke Hilbert algebras, laying the groundwork for further theoretical development and practical applications in fuzzy logic and algebraic systems. 2. Preliminaries Sheffer stroke Hilbert algebras constitute a pivotal framework within the realms of logic and lattice theory, distinguished by the incorporation of the Sheffer stroke (NAND) op- eration—a cornerstone of Boolean algebra. This integration extends the classical Hilbert algebra structure, enabling a more versatile exploration of logical systems and their prop- erties. By bridging algebraic theory and practical applications, Sheffer stroke Hilbert al- gebras provide a robust toolset for analyzing and modeling complex systems characterized by uncertainty, fuzziness, and imprecision. These algebras are particularly relevant in ad- vancing fuzzy logic, decision-making algorithms, and computational frameworks, offering insights that transcend traditional logical paradigms. Moreover, their study contributes to the broader understanding of algebraic hierarchies, enriching both foundational research N. Rajesh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5914 3 of 21 and real-world problem-solving methodologies. This unique combination of theoretical depth and practical utility underscores their significance in contemporary mathematical and computational research. Recall the definitions and results that are taken from [1, 9, 15, 25] for the ready reference of the reader. Definition 1. [1] Let ⟨A, |⟩ be a groupoid. The operation | is said to be a Sheffer stroke operation if it satisfies the following conditions: for all x, y, z ∈ A, (S1) (x|(y|y))|(x|(y|y)) = y|x, (S2) (x|x)|((x|(y|y))|(x|(y|y))) = x, (S3) x|((y|z)|(y|z)) = (((x|(y|y))|(x|(y|y))))|((x|(y|y))|(x|(y|y)))|z, (S4) (x|((x|x)|(y|y)))|(x|((x|x)|(y|y))) = x. Definition 2. [25] An algebra ⟨A,→, 0⟩ of type (2, 0) is called a Hilbert algebra if it satisfies the following axioms: for all x, y, z ∈ A, (H1) x → (y → x) = 0, (H2) (x → (y → z)) → ((x → y) → (x → z)) = 0, (H3) x → y = 0 and y → x = 0 ⇒ x = y. Definition 3. [9] A Sheffer stroke Hilbert algebra (abbreviated SHA) is a structure ⟨A, |, 0⟩ of type (2, 0), in which A is a nonempty set, | is a Sheffer stroke operation on A, and 0 is the fixed element in A such that the following identities are satisfied for all x, y, z ∈ A, (1) (x|(P |P ))|(Q|(R|R))|(Q|(R|R)) = x|(x|x), where P := y|(z|z), Q := x|(y|y) and R := x|(z|z), (2) x|(y|y) = y|(x|x) = x|(x|x) ⇒ x = y. Proposition 1. [9] Let ⟨A, |, 0⟩ be a Sheffer stroke Hilbert algebra. Then the binary relation x ≤ y if and only if x|(y|y) = 0 is a partial order on A. Definition 4. [9] A nonempty subset G of a Sheffer stroke Hilbert algebra ⟨A, |, 0⟩ is called a subalgebra of A if (x|(y|y))|(x|(y|y)) ∈ G for all x, y ∈ G. Definition 5. [15] An interval-valued intuitionistic fuzzy set X over a nonempty set A is an object having the form X = {⟨x, µX(x), γX(x)⟩ : x ∈ A}, where µX(x) : A → D[0, 1] and γX(x) : A → D[0, 1] and D[0, 1] is the set of all intervals of [0, 1]. The intervals µX(x) and γX(x) denote the intervals of the degree of belongingness and non-belongingness of the element x to X, where µX(x) = [µl X(x), µu X(x)] and γX(x) = [γlX(x), γuX(x)] for all x ∈ A with the condition 0 ≤ µl X(x) + γuX(x) ≤ 1. For the sake of simplicity, we shall use the symbol X = (µX , γX) for the interval-valued intuitionistic fuzzy set X = {⟨x, µX(x), γX(x)⟩ : x ∈ A}. Note that µX(x) = [1− µu X(x), 1− µl X(x)] and γX(x) = [1− γuX(x), 1− γlX(x)], where [µX(x), γX(x)] represents the complement of x in X. N. Rajesh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5914 4 of 21 3. Length of an interval-valued fuzzy structure in Sheffer stroke Hilbert algebras In this section, we present the concept of the length of an interval-valued fuzzy structure within the framework of Sheffer stroke Hilbert algebras. We introduce the notion of length- fuzzy subalgebras, which are specific to these algebras, and explore their fundamental properties and interrelationships. This analysis aims to deepen the understanding of how interval-valued fuzziness interacts with the algebraic operations in Sheffer stroke Hilbert algebras, providing new insights into their structural characteristics. Throughout this discussion, we assume A = ⟨A, |, 0⟩ to be a Sheffer stroke Hilbert algebra, serving as the foundational structure for the concepts and results developed herein. Definition 6. Given an interval-valued fuzzy structure (A, f̃) over a nonempty set A, we define two fuzzy structures (A, f̃inf) and (A, f̃sup) in A as follows: f̃inf : A → [0, 1];x 7→ inf{f̃(x)}, and f̃sup : A → [0, 1];x 7→ sup{f̃(x)}. Example 1. [9] Let A = {0, u, v, 1} be a set with the binary operation | given in the following table: | 1 u v 0 1 0 v u 1 u v v 1 1 v u 1 u 1 0 1 1 1 1 Then (A, |) is a Sheffer stroke Hilbert algebra. Define an interval-valued fuzzy structure (A, f̃) over A by the table below: A 1 u v 0 f̃ {0.3, 0.7} [0.2, 0.4] [0.3, 0.7] [0.1, 0.4] Then A 1 u v 0 f̃inf 0.3 0.2 0.3 0.1 f̃sup 0.7 0.4 0.7 0.4 Definition 7. [26] Given an interval-valued fuzzy structure (A, f̃) over A, we define a fuzzy structure (A, f̃ℓ) in A as follows: f̃l : A → [0, 1];x 7→ f̃sup(x)− f̃inf(x), which is called the length of f̃ . N. Rajesh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5914 5 of 21 Example 2. Consider Example 1, we have A 1 u v 0 f̃l 0.4 0.2 0.4 0.3 Definition 8. A fuzzy structure (A, f) in A is called (1) a fuzzy subalgebra of A with type 1 (briefly, 1-fuzzy subalgebra of A) if (∀x, y ∈ A)(f((x|(y|y))|(x|(y|y))) ≥ min{f(x), f(y)}), (2) a fuzzy subalgebra of A with type 2 (briefly, 2-fuzzy subalgebra of A) if (∀x, y ∈ A)(f((x|(y|y))|(x|(y|y))) ≤ min{f(x), f(y)}), (3) a fuzzy subalgebra of A with type 3 (briefly, 3-fuzzy subalgebra of A) if (∀x, y ∈ A)(f((x|(y|y))|(x|(y|y))) ≥ max{f(x), f(y)}), (4) a fuzzy subalgebra of A with type 4 (briefly, 4-fuzzy subalgebra of A) if (∀x, y ∈ A)(f((x|(y|y))|(x|(y|y))) ≤ max{f(x), f(y)}). Example 3. Consider Example 1, we have 3 cases as follows: Case 1: Let x = u and y = v. Then f((x|(y|y))|(x|(y|y))) = f((u|(v|v))|(u|(v|v))) = f((u|u)|(u|u)) = f(v|v) = f(u). Since f(u) = [0.2, 0.4] and f(v) = [0.3, 0.7], it follows that min{f(x), f(y) = min{[0.2, 0.4], [0.3, 0.7]} = [0.2, 0.4]. The condition f((x|(y|y))|(x|(y|y))) ≥ min{f(x), f(y)} is satisfied, as the result f(u) = [0.2, 0.4] is less than or equal to min{f(x), f(y)}. Therefore, (A, f̃) over A forms a 1-fuzzy subalgebra of A. Case 2: Let x = 1 and y = u. Then f((x|(y|y))|(x|(y|y))) = f((1|(u|u))|(1|(u|u))) = f((1|v)|(1|v)) = f(u|u) = f(u). Given that f(1) = [0.3, 0.7] and f(u) = [0.2, 0.4], we can compute the minimum of the two fuzzy sets: min{f(x), f(y) = min{[0.3, 0.7], [0.2, 0.4]} = [0.2, 0.4]. The condition f((x|(y|y))|(x|(y|y))) ≤ min{f(x), f(y)}) is satisfied, as the result f(u) = [0.2, 0.4] is indeed less than or equal to min{f(x), f(y)}. Therefore, (A, f̃) over A forms a 2-fuzzy subalgebra of A. Case 3: Similarly, by choosing x = 1 and y = u, we have (A, f̃) over A forms a 3-fuzzy subalgebra of A, and by selecting x = 0 and y = v, we have (A, f̃) over A forms a 4-fuzzy subalgebra of A. Definition 9. An interval-valued fuzzy structure (A, f̃) over A is called a length 1-fuzzy (resp., 2-fuzzy, 3-fuzzy, 4-fuzzy) subalgebra of A if a fuzzy structure (A, f̃ℓ) is a 1-fuzzy (resp., 2-fuzzy, 3-fuzzy, 4-fuzzy) subalgebra of A. Proposition 2. If (A, f̃) is a length k-fuzzy subalgebra of A for k ∈ {1, 3}, then (∀x ∈ A)(f̃ℓ(0) ≥ f̃ℓ(x)). (1) N. Rajesh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5914 6 of 21 Proof. Let (A, f̃) be a length 1-fuzzy subalgebra of A and x ∈ A. Then f̃ℓ(0) = f̃ℓ(1|1) = f̃ℓ((x|(x|x))|(x|(x|x))) ≥ min{f̃ℓ(x), f̃ℓ(x)} = f̃ℓ(x). Let (A, f̃) be a length 3-fuzzy subalgebra of A. Then f̃ℓ(0) = f̃ℓ(1|1) = f̃ℓ((x|(x|x))|(x|(x|x))) ≥ max{f̃ℓ(x), f̃ℓ(x)} = f̃ℓ(x). Proposition 3. If (A, f̃) is a length k-fuzzy subalgebra of A for k ∈ {2, 4}, then (∀x ∈ A)(f̃ℓ(0) ≤ f̃ℓ(x)). (2) Proof. Let (A, f̃) be a length 2-fuzzy subalgebra of A and x ∈ A. Then f̃ℓ(0) = f̃ℓ(1|1) = f̃ℓ((x|(x|x))|(x|(x|x))) ≤ min{f̃ℓ(x), f̃ℓ(x)} = f̃ℓ(x). Let (A, f̃) be a length 4-fuzzy subalgebra of A. Then f̃ℓ(0) = f̃ℓ(1|1) = f̃ℓ((x|(x|x))|(x|(x|x))) ≤ max{f̃ℓ(x), f̃ℓ(x)} = f̃ℓ(x). Theorem 1. Every length 3-fuzzy subalgebra of A is a length 1-fuzzy subalgebra. Proof. Let (A, f̃) be a length 3-fuzzy subalgebra of A and x, y ∈ A. Then f̃ℓ((x|(y|y))|(x|(y|y))) ≥ max{f̃ℓ(x), f̃ℓ(y)} ≥ min{f̃ℓ(x), f̃ℓ(y)}. Hence, (X, f̃) is a length 1-fuzzy subalgebra of A. Theorem 2. Every length 2-fuzzy subalgebra of A is a length 4-fuzzy subalgebra. Proof. Let (A, f̃) be a length 2-fuzzy subalgebra of A and x, y ∈ A. Then f̃ℓ((x|(y|y))|(x|(y|y))) ≤ min{f̃ℓ(x), f̃ℓ(y)} ≤ max{f̃ℓ(x), f̃ℓ(y)}. Hence, (A, f̃) is a length 4-fuzzy subalgebra of A. Theorem 3. Length 2-fuzzy subalgebra and length 3-fuzzy subalgebra of A coincide. Proof. It is straightforward by Theorems 1 and 2. Theorem 4. Given a subalgebra S of A and B1, B2 ∈ D[0, 1], let (A, f̃) be an interval- valued fuzzy structure over A given by: f̃ : A → D[0, 1];x 7→ { B2 if x ∈ S, B1 otherwise. N. Rajesh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5914 7 of 21 (1) If B1 ⊂ B2, then (A, f̃) is a length 1-fuzzy subalgebra of A. (2) If B2 ⊂ B1, then (A, f̃) is a length 4-fuzzy subalgebra of A. Proof. If x ∈ S, then f̃(x) = B2. Hence, f̃ℓ(x) = f̃sup(x)− f̃inf(x) = sup f̃(x)− inf f̃(x) = supB2 − inf B2. If x /∈ S, then f̃(x) = B1. Hence, f̃ℓ(x) = f̃sup(x)− f̃inf(x) = sup f̃(x)− inf f̃(x) = supB1 − inf B1. (1) Assume that B1 ⊂ B2. Then supB2 − inf B2 ≥ supB1 − inf B1. Case 1: Let x, y ∈ S. Then f̃ℓ(x) = supB2 − inf B2 and f̃ℓ(y) = supB2 − inf B2. Thus, min{f̃ℓ(x), f̃ℓ(y)} = supB2 − inf B2. Since S is a subalgebra of A, we have (x|(y|y))|(x|(y|y)) ∈ S and so f̃ℓ((x|(y|y))|(x|(y|y))) = supB2 − inf B2. Thus, f̃ℓ((x|(y|y))|(x|(y|y))) = supB2 − inf B2 = (≥)min{f̃ℓ(x), f̃ℓ(y)}. Case 2: Let x, y /∈ S. Then f̃ℓ(x) = supB1 − inf B1 and f̃ℓ(y) = supB1 − inf B1, and so min{f̃ℓ(x), f̃ℓ(y)} = supB1 − inf B1. Thus, f̃ℓ((x|(y|y))|(x|(y|y))) ≥ supB1 − inf B1 = min{f̃ℓ(x), f̃ℓ(y)}. Case 3: Let x /∈ S and y ∈ S. Then f̃ℓ(x) = supB1 − inf B1 and f̃ℓ(y) = supB2 − inf B2, and so min{f̃ℓ(x), f̃ℓ(y)} = supB1 − inf B1. Thus, f̃ℓ((x|(y|y))|(x|(y|y))) ≥ supB1 − inf B1 = min{f̃ℓ(x), f̃ℓ(y)}. Case 4: Let x ∈ S and y /∈ S. Then f̃ℓ(x) = supB2 − inf B2 and f̃ℓ(y) = supB1 − inf B1, and so min{f̃ℓ(x), f̃ℓ(y)} = supB1 − inf B1. Thus, f̃ℓ((x|(y|y))|(x|(y|y))) ≥ supB1 − inf B1 = min{f̃ℓ(x), f̃ℓ(y)}. Hence, f̃ℓ is a 1-fuzzy subalgebra of A and so (A, f̃) is a length 1-fuzzy subalgebra of A. (2) Assume that B2 ⊂ B1. Then supB2 − inf B2 ≤ supB1 − inf B1. Case 1: Let x, y ∈ S. Then f̃ℓ(x) = supB2 − inf B2 and f̃ℓ(y) = supB2 − inf B2. Thus, max{f̃ℓ(x), f̃ℓ(y)} = supB2 − inf B2. Since S is a subalgebra of A, we have (x|(y|y))|(x|(y|y)) ∈ S and so f̃ℓ((x|(y|y))|(x|(y|y))) = supB2 − inf B2. Thus, f̃ℓ((x|(y|y))|(x|(y|y))) = supB2 − inf B2 = (≤)max{f̃ℓ(x), f̃ℓ(y)}. Case 2: Let x, y /∈ S. Then f̃ℓ(x) = supB1 − inf B1 and f̃ℓ(y) = supB1 − inf B1, so max{f̃ℓ(x), f̃ℓ(y)} = supB1 − inf B1. Thus, f̃ℓ((x|(y|y))|(x|(y|y))) ≤ supB1 − inf B1 = max{f̃ℓ(x), f̃ℓ(y)}. N. Rajesh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5914 8 of 21 Case 3: Let x /∈ S and y ∈ S. Then f̃ℓ(x) = supB1 − inf B1 and f̃ℓ(y) = supB2 − inf B2, so max{f̃ℓ(x), f̃ℓ(y)} = supB1 − inf B1. Thus, f̃ℓ((x|(y|y))|(x|(y|y))) ≤ supB1 − inf B1 = max{f̃ℓ(x), f̃ℓ(y)}. Case 4: Let x ∈ S and y /∈ S. Then f̃ℓ(x) = supB2 − inf B2 and f̃ℓ(y) = supB1 − inf B1, so max{f̃ℓ(x), f̃ℓ(y)} = supB1 − inf B1. Thus, f̃ℓ((x|(y|y))|(x|(y|y))) ≤ supB1 − inf B1 = max{f̃ℓ(x), f̃ℓ(y)}. Hence, f̃ℓ is a 4-fuzzy subalgebra of A and so (A, f̃) is a length 4-fuzzy subalgebra of A. Definition 10. Let (A, f) be a fuzzy structure in A. For any t ∈ [0, 1], the sets U(f ; t) = {x ∈ A : f(x) ≥ t}, L(f ; t) = {x ∈ A : f(x) ≤ t}, are called an upper t-level subset and a lower t-level subset of f , respectively. Example 4. Consider Example 2, and let t = 0.3. Then U(f̃l; 0.3) = {1, v, 0} and L(f̃l; 0.3) = {u, 0}. If t = 0.5, then U(f̃l; 0.5) = ∅ and L(f̃l; 0.5) = A. If t = 0.1, then U(f̃l; 0.1) = A and L(f̃l; 0.1) = ∅. Theorem 5. An interval-valued fuzzy structure (A, f̃) over A is a length 1-fuzzy subalgebra of A if and only if the set U(f̃ℓ; t) is a subalgebra of A for all t ∈ [0, 1] with U(f̃ℓ; t) ̸= ∅. Proof. Assume that (A, f̃) is a length 1-fuzzy subalgebra of A. Let t ∈ [0, 1] be such that U(f̃ℓ; t) ̸= ∅ and let x, y ∈ U(f̃ℓ; t). Then f̃ℓ(x) ≥ t and f̃ℓ(y) ≥ t. Since (A, f̃) is a length 1-fuzzy subalgebra of A, we have f̃ℓ((x|(y|y))|(x|(y|y))) ≥ min{f̃ℓ(x), f̃ℓ(y)} ≥ t. Thus, (x|(y|y))|(x|(y|y)) ∈ U(f̃ℓ; t). Hence, U(f̃ℓ; t) is a subalgebra of A. Conversely, assume that for all t ∈ [0, 1], the set U(f̃ℓ; t) is a subalgebra of A if U(f̃ℓ; t) ̸= ∅. Let x, y ∈ A. Then f̃ℓ(x), f̃ℓ(y) ∈ [0, 1]. If we take t = min{f̃ℓ(x), f̃ℓ(y)}, then f̃ℓ(x) ≥ t and f̃ℓ(y) ≥ t. Hence, x, y ∈ U(f̃ℓ; t) ̸= ∅. By assumption, we have U(f̃ℓ; t) is a subalgebra of A, and so (x|(y|y))|(x|(y|y)) ∈ U(f̃ℓ; t). Thus, f̃ℓ((x|(y|y))|(x|(y|y))) ≥ t = min{f̃ℓ(x), f̃ℓ(y)}. Hence, (A, f̃ℓ) is a 1-fuzzy subalgebra of A, that is, (A, f̃) is a length 1-fuzzy subalgebra of A. Corollary 1. If (A, f̃) is a length 3-fuzzy subalgebra of A, then the set U(f̃ℓ; t) is a subalgebra of A for all t ∈ [0, 1] with U(f̃ℓ; t) ̸= ∅. N. Rajesh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5914 9 of 21 Proof. It is straightforward by Theorems 1 and 5. Theorem 6. An interval-valued fuzzy structure (A, f̃) over A is a length 4-fuzzy subalgebra of A if and only if the set L(f̃ℓ; t) is a subalgebra of A for all t ∈ [0, 1] with L(f̃ℓ; t) ̸= ∅. Proof. Assume that (A, f̃) is a length 4-fuzzy subalgebra of A. Let t ∈ [0, 1] be such that L(f̃ℓ; t) ̸= ∅ and let x, y ∈ L(f̃ℓ; t). Then f̃ℓ(x) ≤ t and f̃ℓ(y) ≤ t. Since (A, f̃) is a length 4-fuzzy subalgebra of A, we have f̃ℓ((x|(y|y))|(x|(y|y))) ≤ max{f̃ℓ(x), f̃ℓ(y)} ≤ t. Thus, (x|(y|y))|(x|(y|y)) ∈ L(f̃ℓ; t). Hence, L(f̃ℓ; t) is a subalgebra of A. Conversely, assume that for all t ∈ [0, 1], the set L(f̃ℓ; t) is a subalgebra of A if L(f̃ℓ; t) ̸= ∅. Let x, y ∈ A. Then f̃ℓ(x), f̃ℓ(y) ∈ [0, 1]. If we take t = max{f̃ℓ(x), f̃ℓ(y)}. Thus, f̃ℓ(x) ≤ t and f̃ℓ(y) ≤ t, and so x, y ∈ L(f̃ℓ; t) ̸= ∅. By assumption, L(f̃ℓ; t) is a subalgebra of A, and so (x|(y|y))|(x|(y|y)) ∈ L(f̃ℓ; t). Thus, f̃ℓ((x|(y|y))|(x|(y|y))) ≤ t = max{f̃ℓ(x), f̃ℓ(y)}. Hence, (A, f̃ℓ) is a 4-fuzzy subalgebra of A, that is, (A, f̃) is a length 4-fuzzy subalgebra of A. Corollary 2. If (A, f̃) is a length 2-fuzzy subalgebra of A, then the set L(f̃ℓ; t) is a subalgebra of A for all t ∈ [0, 1] with L(f̃ℓ; t) ̸= ∅. Proof. It is straightforward by Theorems 2 and 6. Theorem 7. If (A, f̃) is a length 2-fuzzy subalgebra of A, then Uℓ(f̃ ; t) c is a subalgebra of A for all t ∈ [0, 1] with Uℓ(f̃ ; t) c ̸= ∅. Proof. Assume that (A, f̃) is a length 2-fuzzy subalgebra of A and let x, y ∈ A be such that x ∈ Uℓ(f̃ ; t) c and y ∈ Uℓ(f̃ ; t) c. This shows that f̃ℓ(x) < t and f̃ℓ(y) < t. This implies that f̃ℓ((x|(y|y))|(x|(y|y))) ≤ min{f̃ℓ(x), f̃ℓ(y)} < t, that is, (x|(y|y))|(x|(y|y)) ∈ Uℓ(f̃ ; t) c. Therefore, Uℓ(f̃ ; t) c is a subalgebra of A for all t ∈ [0, 1] with Uℓ(f̃ ; t) c ̸= ∅. Theorem 8. If (A, f̃) is a length 3-fuzzy subalgebra of A, then Lℓ(f̃ ; t) c is a subalgebra of A for all t ∈ [0, 1] with Lℓ(f̃ ; t) c ̸= ∅. Proof. It is similar to the proof of Theorem 7. Theorem 9. If (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃inf) is constant and (A, f̃sup) is a 1-fuzzy subalgebra of A, then (A, f̃) is a length 1-fuzzy subalgebra of A. N. Rajesh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5914 10 of 21 Proof. Assume that (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃inf) is constant and (A, f̃sup) is a 1-fuzzy subalgebra of A. Let x, y ∈ A. Since (A, f̃inf) is constant, we have f̃inf(x) = f̃inf(0) for all x ∈ A. Since (A, f̃sup) is a 1-fuzzy subalgebra of A, we have f̃sup((x|(y|y))|(x|(y|y))) ≥ min{f̃sup(x), f̃sup(y)}. Thus, f̃ℓ((x|(y|y))|(x|(y|y))) = f̃sup((x|(y|y))|(x|(y|y)))− f̃inf((x|(y|y))|(x|(y|y))) = f̃sup((x|(y|y))|(x|(y|y)))− f̃inf(0) ≥ min{f̃sup(x), f̃sup(y)} − f̃inf(0) = min{f̃sup(x)− f̃inf(0), f̃sup(y)− f̃inf(0)} = min{f̃sup(x)− f̃inf(x), f̃sup(y)− f̃inf(y)} = min{f̃ℓ(x), f̃ℓ(y)}. Hence, (A, f̃ℓ) is a 1-fuzzy subalgebra of A, that is, (A, f̃) is a length 1-fuzzy subalgebra of A. Theorem 10. If (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃inf) is constant and (A, f̃sup) is a 4-fuzzy subalgebra of A, then (A, f̃) is a length 4-fuzzy subalgebra of A. Proof. Assume that (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃inf) is constant and (A, f̃sup) is a 4-fuzzy subalgebra of A. Let x, y ∈ A. Since (A, f̃inf) is constant, we have f̃inf(x) = f̃inf(0) for all x ∈ A. Since (A, f̃sup) is a 4-fuzzy subalgebra of A, we have f̃sup((x|(y|y))|(x|(y|y))) ≤ max{f̃sup(x), f̃sup(y)}. Thus, f̃ℓ((x|(y|y))|(x|(y|y))) = f̃sup((x|(y|y))|(x|(y|y)))− f̃inf((x|(y|y))|(x|(y|y))) = f̃sup((x|(y|y))|(x|(y|y)))− f̃inf(0) ≤ max{f̃sup(x), f̃sup(y)} − f̃inf(0) = max{f̃sup(x)− f̃inf(0), f̃sup(y)− f̃inf(0)} = max{f̃sup(x)− f̃inf(x), f̃sup(y)− f̃inf(y)} = max{f̃ℓ(x), f̃ℓ(y)}. Hence, (A, f̃ℓ) is a 4-fuzzy subalgebra of A, that is, (A, f̃) is a length 4-fuzzy subalgebra of A. Theorem 11. If (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃sup) is constant and (A, f̃inf) is a 4-fuzzy subalgebra of A, then (A, f̃) is a length 1-fuzzy subalgebra of A. Proof. Assume that (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃sup) is constant and (A, f̃inf) is a 4-fuzzy subalgebra of A. Let x, y ∈ A. Since (A, f̃sup) is constant, we have f̃sup(x) = f̃sup(0) for all x ∈ A. Since (A, f̃inf) is a 4-fuzzy N. Rajesh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5914 11 of 21 subalgebra of A, we have f̃inf((x|(y|y))|(x|(y|y))) ≤ max{f̃inf(x), f̃inf(y)}. Thus, f̃ℓ((x|(y|y))|(x|(y|y))) = f̃sup((x|(y|y))|(x|(y|y)))− f̃inf((x|(y|y))|(x|(y|y))) = f̃sup(0)− f̃inf((x|(y|y))|(x|(y|y))) ≥ f̃sup(0)−max{f̃inf(x), f̃inf(y)} = min{f̃sup(0)− f̃inf(x), f̃sup(0)− f̃inf(y)} = min{f̃sup(x)− f̃inf(x), f̃sup(y)− f̃inf(y)} = min{f̃ℓ(x), f̃ℓ(y)}. Hence, (A, f̃ℓ) is a 1-fuzzy subalgebra of A, that is, (A, f̃) is a length 1-fuzzy subalgebra of A. Theorem 12. If (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃sup) is constant and (A, f̃inf) is a 1-fuzzy subalgebra of A, then (A, f̃) is a length 4-fuzzy subalgebra of A. Proof. Assume that (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃sup) is constant and (A, f̃inf) is a 1-fuzzy subalgebra of A. Let x, y ∈ A. Since (A, f̃sup) is constant, we have f̃sup(x) = f̃sup(0) for all x ∈ A. Since (A, f̃inf) is a 1-fuzzy subalgebra of A, we have f̃inf((x|(y|y))|(x|(y|y))) ≥ min{f̃inf(x), f̃inf(y)}. Thus, f̃ℓ((x|(y|y))|(x|(y|y))) = f̃sup((x|(y|y))|(x|(y|y)))− f̃inf((x|(y|y))|(x|(y|y))) = f̃sup(0)− f̃inf((x|(y|y))|(x|(y|y))) ≤ f̃sup(0)−min{f̃inf(x), f̃inf(y)} = max{f̃sup(0)− f̃inf(x), f̃sup(0)− f̃inf(y)} = max{f̃sup(x)− f̃inf(x), f̃sup(y)− f̃inf(y)} = max{f̃ℓ(x), f̃ℓ(y)}. Hence, (A, f̃ℓ) is a 4-fuzzy subalgebra of A, that is, (A, f̃) is a length 4-fuzzy subalgebra of A. 4. Mean of an interval-valued fuzzy structure in Sheffer stroke Hilbert algebras This section introduces the concept of the mean of an interval-valued fuzzy structure within Sheffer stroke Hilbert algebras, along with the corresponding notion of mean-fuzzy subalgebras. The fundamental properties of these subalgebras are examined, shedding light on their intrinsic algebraic behavior. We further explore the connections between mean-fuzzy subalgebras and classical subalgebras, offering a comparative perspective on their structural interplay. Additionally, the relationships between mean-fuzzy subalge- bras and various level subsets—namely, upper and lower-level subsets—of the mean of an interval-valued fuzzy structure are analyzed, providing a comprehensive framework for understanding their hierarchical and interval-dependent dynamics in the context of Sheffer stroke Hilbert algebras. N. Rajesh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5914 12 of 21 Definition 11. [26] 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];x 7→ f̃sup(x) + f̃inf(x) 2 , which is called the mean of f̃ . Example 5. Consider Example 1, we have A 1 u v 0 f̃m 0.5 0.3 0.5 0.25 Definition 12. An interval-valued fuzzy structure (A, f̃) over A is called a mean 1-fuzzy (resp., 2-fuzzy, 3-fuzzy and 4-fuzzy) subalgebra of A if a fuzzy structure (A, f̃m) is a 1-fuzzy (resp., 2-fuzzy, 3-fuzzy and 4-fuzzy) subalgebra of A. Proposition 4. If (A, f̃) is a mean k-fuzzy subalgebra of A for k ∈ {1, 3}, then (∀x ∈ A)(f̃m(0) ≥ f̃m(x)). (3) Proof. Let (A, f̃) be a mean k-fuzzy subalgebra of A for k ∈ {1, 3}. Then f̃m(0) = f̃sup(0) + f̃inf(0) 2 ≥ f̃sup(x) + f̃inf(x) 2 = f̃m(x). Proposition 5. If (A, f̃) is a mean k-fuzzy subalgebra of A for k ∈ {2, 4}, then (∀x ∈ A)(f̃m(0) ≤ f̃m(x)). (4) Proof. Let (A, f̃) be a mean k-fuzzy subalgebra of A for k ∈ {2, 4}. Then f̃m(0) = f̃sup(0) + f̃inf(0) 2 ≤ f̃sup(x) + f̃inf(x) 2 = f̃m(x). Theorem 13. Every mean 3-fuzzy subalgebra of A is a mean 1-fuzzy subalgebra. N. Rajesh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5914 13 of 21 Proof. Let (A, f̃) be a mean 3-fuzzy subalgebra of A. Then f̃m((x|(y|y))|(x|(y|y))) = f̃sup((x|(y|y))|(x|(y|y))) + f̃inf((x|(y|y))|(x|(y|y))) 2 = f̃sup((x|(y|y))|(x|(y|y))) 2 + f̃inf((x|(y|y))|(x|(y|y))) 2 ≥ max { f̃sup(x) 2 , f̃sup(y) 2 } +max { f̃inf(x) 2 , f̃inf(y) 2 } ≥ min { f̃sup(x) 2 , f̃sup(y) 2 } +min { f̃inf(x) 2 , f̃inf(y) 2 } = min { f̃sup(x) + f̃inf(x) 2 , f̃sup(y) + f̃inf(y) 2 } = min{f̃m(x), f̃m(y)}. Hence, (A, f̃) is a mean 1-fuzzy subalgebra of A. Theorem 14. Every mean 2-fuzzy subalgebra of A is a mean 4-fuzzy subalgebra. Proof. Let (A, f̃) be a mean 2-fuzzy subalgebra of A. Then f̃m((x|(y|y))|(x|(y|y))) = f̃sup((x|(y|y))|(x|(y|y))) + f̃inf((x|(y|y))|(x|(y|y))) 2 = f̃sup((x|(y|y))|(x|(y|y))) 2 + f̃inf((x|(y|y))|(x|(y|y))) 2 ≤ min { f̃sup(x) 2 , f̃sup(y) 2 } +min { f̃inf(x) 2 , f̃inf(y) 2 } ≤ max { f̃sup(x) 2 , f̃sup(y) 2 } +max { f̃inf(x) 2 , f̃inf(y) 2 } = max { f̃sup(x) + f̃inf(x) 2 , f̃sup(y) + f̃inf(y) 2 } = max{f̃m(x), f̃m(y)}. Hence, (A, f̃) is a mean 4-fuzzy subalgebra of A. Theorem 15. Mean 2-fuzzy subalgebra and mean 3-fuzzy subalgebra of A coincide. Proof. It is straightforward by Theorems 13 and 14. Theorem 16. Given a subalgebra S of A and B1, B2 ∈ D[0, 1], let (A, f̃) be an interval- valued fuzzy structure over A given by f̃ : A → D[0, 1];x 7→ { B2 if x ∈ S, B1 otherwise. N. Rajesh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5914 14 of 21 (1) If supB2 ≥ supB1 and inf B2 ≥ inf B1, then (A, f̃) is a mean 1-fuzzy subalgebra of A. (2) If supB2 ≤ supB1 and inf B2 ≤ inf B1, then (A, f̃) is a mean 4-fuzzy subalgebra of A. Proof. If x ∈ S, then f̃(x) = B2 and so f̃m(x) = f̃sup(x) + f̃inf(x) 2 = sup f̃(x) + inf f̃(x) 2 = supB2 + inf B2 2 . If x /∈ S, then f̃(x) = B1 and so f̃m(x) = f̃sup(x) + f̃inf(x) 2 = sup f̃(x) + inf f̃(x) 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 x, y ∈ S. Then f̃m(x) = supB2 + inf B2 2 and fm(y) = supB2 + inf B2 2 . Thus, min{f̃m(x), f̃m(y)} = supB2 + inf B2 2 . Since S is a subalgebra of A, we have (x|(y|y))|(x|(y|y)) ∈ S and so f̃m((x|(y|y))|(x|(y|y))) = supB2 + inf B2 2 . Thus, f̃m((x|(y|y))|(x|(y|y))) = supB2 + inf B2 2 = (≥)min{f̃m(x), f̃m(y)}. Case 2: Let x, y /∈ S. Then f̃m(x) = supB1 + inf B1 2 and f̃m(y) = supB1 + inf B1 2 , so min{f̃m(x), f̃m(y)} = supB1 + inf B1 2 . Thus, f̃m((x|(y|y))|(x|(y|y))) ≥ supB1 + inf B1 2 = min{f̃m(x), f̃m(y)}. Case 3: Let x /∈ S and y ∈ S. Then f̃m(x) = supB1 + inf B1 2 and f̃m(y) = supB2 + inf B2 2 , so min{f̃m(x), f̃m(y)} = supB1 + inf B1 2 . Thus, f̃m((x|(y|y))|(x|(y|y))) ≥ supB1 + inf B1 2 = min{f̃m(x), f̃m(y)}. Case 4: Let x ∈ S and y /∈ S. Then f̃m(x) = supB2 + inf B2 2 and f̃m(y) = supB1 + inf B1 2 , so min{f̃m(x), f̃m(y)} = supB1 + inf B1 2 . Thus, f̃m((x|(y|y))|(x|(y|y))) ≥ supB1 + inf B1 2 = min{f̃m(x), f̃m(y)}. N. Rajesh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5914 15 of 21 Hence, f̃m is a 1-fuzzy subalgebra of A and so (A, f̃) is a mean 1-fuzzy subalgebra of A. (2) Assume that supB2 ≤ supB1 and inf B2 ≤ inf B1. Then supB2 + inf B2 2 ≤ supB1 + inf B1 2 . Case 1: Let x, y ∈ S. Then f̃m(x) = supB2 + inf B2 2 and f̃m(y) = supB2 + inf B2 2 , so max{f̃m(x), f̃m(y)} = supB2 + inf B2 2 . Since S is a subalgebra ofA, we have (x|(y|y))|(x|(y|y)) ∈ S and so f̃m((x|(y|y))|(x|(y|y))) = supB2 + inf B2 2 . Thus, f̃m((x|(y|y))|(x|(y|y))) = supB2 + inf B2 2 = (≤)max{f̃m(x), f̃m(y)}. Case 2: Let x, y /∈ S. Then f̃m(x) = supB1 + inf B1 2 and f̃m(y) = supB1 + inf B1 2 , so max{f̃m(x), f̃m(y)} = supB1 + inf B1 2 . Thus, f̃m((x|(y|y))|(x|(y|y))) ≤ supB1 + inf B1 2 = max{f̃m(x), f̃m(y)}. Case 3: Let x /∈ S and y ∈ S. Then f̃m(x) = supB1 + inf B1 2 and f̃m(y) = supB2 + inf B2 2 , so max{f̃m(x), f̃m(y)} = supB1 + inf B1 2 . Thus, f̃m((x|(y|y))|(x|(y|y))) ≤ supB1 + inf B1 2 = max{f̃m(x), f̃m(y)}. Case 4: Let x ∈ S and y /∈ S. Then f̃m(x) = supB2 + inf B2 2 and f̃m(y) = supB1 + inf B1 2 , so max{f̃m(x), f̃m(y)} = supB1 + inf B1 2 . Thus, f̃m((x|(y|y))|(x|(y|y))) ≤ supB1 + inf B1 2 = max{f̃m(x), f̃m(y)}. Hence, f̃m is a 4-fuzzy subalgebra of A and so (A, f̃) is a mean 4-fuzzy subalgebra of A. Theorem 17. An interval-valued fuzzy structure (A, f̃) over A is a mean 1-fuzzy sub- algebra of A if and only if the set U(f̃m; t) is a subalgebra of A for all t ∈ [0, 1] with U(f̃m; t) ̸= ∅. N. Rajesh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5914 16 of 21 Proof. Assume that (A, f̃) is a mean 1-fuzzy subalgebra of A. Let t ∈ [0, 1] be such that U(f̃m; t) ̸= ∅ and let x, y ∈ U(f̃m; t). Then f̃m(x) ≥ t and f̃m(y) ≥ t. Since (A, f̃) is a mean 1-fuzzy subalgebra of A, we have f̃m((x|(y|y))|(x|(y|y))) ≥ min{f̃m(x), f̃m(y)} ≥ t. Thus, (x|(y|y))|(x|(y|y)) ∈ U(f̃m; t). Hence, U(f̃m; t) is a subalgebra of A. Conversely, assume that for all t ∈ [0, 1], the set U(f̃m; t) is a subalgebra of A if U(f̃m; t) ̸= ∅. Let x, y ∈ A. Then f̃m(x), f̃m(y) ∈ [0, 1]. Choose t = min{f̃m(x), f̃m(y)}. Thus, f̃m(x) ≥ t and f̃m(y) ≥ t. It follows that x, y ∈ U(f̃m; t) ̸= ∅. By assumption, we have U(f̃m; t) is a subalgebra of A and so (x|(y|y))|(x|(y|y)) ∈ U(f̃m; t). Thus, f̃m((x|(y|y))|(x|(y|y))) ≥ t = min{f̃m(x), f̃m(y)}. Hence, (A, f̃m) is a 1-fuzzy subalgebra of A, that is, (A, f̃) is a mean 1-fuzzy subalgebra of A. Corollary 3. If (A, f̃) is a mean 3-fuzzy subalgebra of A, then U(f̃m; t) is a subalgebra of A for all t ∈ [0, 1] with U(f̃m; t) ̸= ∅. Proof. It is straightforward by Theorems 13 and 17. Theorem 18. An interval-valued fuzzy structure (A, f̃) over A is a mean 4-fuzzy sub- algebra of A if and only if the set L(f̃m; t) is a subalgebra of A for all t ∈ [0, 1] with L(f̃m; t) ̸= ∅. Proof. Assume that (A, f̃) is a mean 4-fuzzy subalgebra of A. Let t ∈ [0, 1] be such that L(f̃m; t) ̸= ∅ and let x, y ∈ L(f̃m; t). Then f̃m(x) ≤ t and f̃m(y) ≤ t. Since (A, f̃) is a mean 4-fuzzy subalgebra of A, we have f̃m((x|(y|y))|(x|(y|y))) ≤ max{f̃m(x), f̃m(y)} ≤ t. Thus, (x|(y|y))|(x|(y|y)) ∈ L(f̃m; t). Hence, L(f̃m; t) is a subalgebra of A. Conversely, assume that for all t ∈ [0, 1], the set L(f̃m; t) is a subalgebra of A if L(f̃m; t) ̸= ∅. Let x, y ∈ A. Then f̃m(x), f̃m(y) ∈ [0, 1]. Choose t = max{f̃m(x), f̃m(y)}. Thus, f̃m(x) ≤ t and f̃m(y) ≤ t, and so x, y ∈ L(f̃m; t) ̸= ∅. By assumption, we have L(f̃m; t) is a subalgebra of A and so (x|(y|y))|(x|(y|y)) ∈ L(f̃m; t). Thus, f̃m((x|(y|y))|(x|(y|y))) ≤ t = max{f̃m(x), f̃m(y)}. Hence, (A, f̃m) is a 4-fuzzy subalgebra of A, that is, (A, f̃) is a mean 4-fuzzy subalgebra of A. Corollary 4. If (A, f̃) is a mean 2-fuzzy subalgebra of A, then L(f̃m; t) is a subalgebra of A for all t ∈ [0, 1] with L(f̃m; t) ̸= ∅. N. Rajesh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5914 17 of 21 Proof. It is straightforward by Theorems 15 and 18. Theorem 19. If (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃inf) is constant and (A, f̃sup) is a 1-fuzzy subalgebra of A, then (A, f̃) is a mean 1-fuzzy subalgebra of A. Proof. Assume that (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃inf) is constant and (A, f̃sup) is a 1-fuzzy subalgebra of A. Let x, y ∈ A. Since (A, f̃inf) is constant, we have f̃inf(x) = f̃inf(0) for all x ∈ A. Since (A, f̃sup) is a 1-fuzzy subalgebra of A, we have f̃sup((x|(y|y))|(x|(y|y))) ≥ min{f̃sup(x), f̃sup(y)}. Thus, f̃m((x|(y|y))|(x|(y|y))) = f̃sup((x|(y|y))|(x|(y|y))) + f̃inf((x|(y|y))|(x|(y|y))) 2 = f̃sup((x|(y|y))|(x|(y|y))) 2 + f̃inf(0) 2 ≥ min { f̃sup(x) 2 + f̃inf(x) 2 } + f̃inf(0) 2 = min { f̃sup(x) 2 + f̃inf(0) 2 , f̃sup(y) 2 + f̃inf(0) 2 } = min { f̃sup(x) + f̃inf(x) 2 , f̃sup(y) + f̃inf(y) 2 } = min{f̃m(x), f̃m(y)}. Hence, (A, f̃m) is a 1-fuzzy subalgebra of A, that is, (A, f̃) is a mean 1-fuzzy subalgebra of A. Theorem 20. If (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃inf) is constant and (A, f̃sup) is a 4-fuzzy subalgebra of A, then (A, f̃) is a mean 4-fuzzy subalgebra of A. Proof. Assume that (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃inf) is constant and (A, f̃sup) is a 4-fuzzy subalgebra of A. Let x, y ∈ A. Since (A, f̃inf) is constant, we have f̃inf(x) = f̃inf(0) for all x ∈ A. Since (A, f̃sup) is a 4-fuzzy subalgebra of A, we have f̃sup((x|(y|y))|(x|(y|y))) ≤ max{f̃sup(x), f̃sup(y)}. N. Rajesh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5914 18 of 21 Thus, f̃m((x|(y|y))|(x|(y|y))) = f̃sup((x|(y|y))|(x|(y|y))) + f̃inf((x|(y|y))|(x|(y|y))) 2 = f̃sup((x|(y|y))|(x|(y|y))) 2 + f̃inf(0) 2 ≥ min { f̃sup(x) 2 + f̃inf(x) 2 } + f̃inf(0) 2 = min { f̃sup(x) 2 + f̃inf(0) 2 , f̃sup(y) 2 + f̃inf(0) 2 } = min { f̃sup(x) + f̃inf(x) 2 , f̃sup(y) + f̃inf(y) 2 } = min{f̃m(x), f̃m(y)}. Hence, (A, f̃m) is a 4-fuzzy subalgebra of A, that is, (A, f̃) is a mean 4-fuzzy subalgebra of A. Theorem 21. If (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃sup) is constant and (A, f̃inf) is a 4-fuzzy subalgebra of A, then (A, f̃) is a mean 4-fuzzy subalgebra of A. Proof. Assume that (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃sup) is constant and (A, f̃inf) is a 4-fuzzy subalgebra of A. Let x, y ∈ A. Since (A, f̃sup) is constant, we have f̃sup(x) = f̃sup(0) for all x ∈ A. Since (A, f̃inf) is a 4-fuzzy subalgebra of A, we have f̃inf((x|(y|y))|(x|(y|y))) ≤ max{f̃inf(x), f̃inf(y)}. Thus, f̃m((x|(y|y))|(x|(y|y))) = f̃sup((x|(y|y))|(x|(y|y))) + f̃inf((x|(y|y))|(x|(y|y))) 2 = f̃sup(0) + f̃inf((x|(y|y))|(x|(y|y))) 2 = f̃sup(0) 2 + f̃inf((x|(y|y))|(x|(y|y))) 2 ≤ f̃sup(0) 2 + max { f̃sup(x) 2 , f̃inf(x) 2 } = max { f̃sup(0) 2 + f̃inf(x) 2 , f̃sup(0) 2 + f̃inf(y) 2 } = max { f̃sup(x) + f̃inf(x) 2 , f̃sup(y) + f̃inf(yx) 2 } = max{f̃m(x), f̃m(y)}. N. Rajesh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5914 19 of 21 Hence, (A, f̃m) is a 4-fuzzy subalgebra of A, that is, (A, f̃) is a mean 4-fuzzy subalgebra of A. Theorem 22. If (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃sup) is constant and (A, f̃inf) is a 1-fuzzy subalgebra of A, then (A, f̃) is a mean 1-fuzzy subalgebra of A. Proof. Assume that (A, f̃) is an interval-valued fuzzy structure over A in which (A, f̃sup) is constant and (A, f̃inf) is a 1-fuzzy subalgebra of A. Let x, y ∈ A. Since (A, f̃sup) is constant, we have f̃sup(x) = f̃sup(0) for all x ∈ A. Since (A, f̃inf) is a 1-fuzzy subalgebra of A, we have f̃inf((x|(y|y))|(x|(y|y))) ≥ min{f̃inf(x), f̃inf(y)}. Thus, f̃m((x|(y|y))|(x|(y|y))) = f̃sup((x|(y|y))|(x|(y|y))) + f̃inf((x|(y|y))|(x|(y|y))) 2 = f̃sup(0) + f̃inf((x|(y|y))|(x|(y|y))) 2 = f̃sup(0) 2 + f̃inf((x|(y|y))|(x|(y|y))) 2 ≥ f̃sup(0) 2 + min { f̃inf(x) 2 , f̃inf(y) 2 } = min { f̃sup(0) 2 + f̃sup(x) 2 , f̃sup(0) 2 , f̃sup(y) 2 } = min { f̃sup(0) + f̃sup(x) 2 , f̃sup(0) + f̃sup(y) 2 } = min{f̃m(x), f̃m(y)}. Hence, (A, f̃m) is a 1-fuzzy subalgebra of A, that is, (A, f̃) is a mean 1-fuzzy subalgebra of A. 5. Conclusion This study advances the theoretical framework of Sheffer stroke Hilbert algebras by introducing the concepts of length-fuzzy subalgebras and mean-fuzzy subalgebras within interval-valued fuzzy structures. These new constructs deepen the understanding of fuzzy logic in algebraic systems, particularly by elucidating the interplay between fuzzy and tra- ditional subalgebras. The investigation reveals key properties and relationships, including their alignment with upper and lower-level subsets, offering a refined perspective on the gradations of membership functions. This framework not only enriches algebraic theory N. Rajesh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5914 20 of 21 but also underscores the practical relevance of fuzzy subalgebras in fields such as logic, computer science, and uncertainty modeling. The findings pave the way for future research to explore these ideas in more complex fuzzy systems or adapt them to other algebraic structures, broadening their applicability and potential impact across diverse domains. Acknowledgements This research was supported by University of Phayao and Thailand Science Research and Innovation Fund (Fundamental Fund 2025, Grant No. 5027/2567). References [1] H. M. Sheffer. A set of five independent postulates for Boolean algebras, with ap- plication to logical constants. 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