EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6500 ISSN 1307-5543 – ejpam.com Published by New York Business Global Generalized Fuzzy Subalgebras of Sheffer Stroke Hilbert Algebras Neelamegarajan Rajesh1, Aiyared Iampan2,∗, Tahsin Oner3, Arsham Borumand Saeid1 1 Department of Mathematics, Rajah Serfoji Government College, Thanjavur-613005, Tamil Nadu, India 2 Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand 3 Department of Mathematics, Faculty of Science, Ege University, 35100 Izmir, Turkey 4 Department of Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran Abstract. This paper introduces new generalized fuzzy subalgebras and investigates their impor- tant properties within the framework of Sheffer stroke Hilbert algebras. We characterize these generalized subalgebras through their level subsets and establish key properties that define their structure. The Sheffer stroke operation, known for its ability to construct logical systems indepen- dently of other operators, plays a central role in our study. Using fuzzy set theory, we adapt the traditional ideas of subalgebras to fit fuzzy contexts, giving a detailed look at (∈,∈ ∨qm)-fuzzy subalgebras. Our results include necessary and sufficient conditions for a fuzzy set to qualify as such a subalgebra, along with theorems addressing their intersections, unions, and homomorphic invariance. This work contributes to the broader understanding of algebraic structures in fuzzy logic and their applications in logical systems. 2020 Mathematics Subject Classifications: 03G25, 03E72 Key Words and Phrases: Sheffer stroke Hilbert algebra, subalgebra, fuzzy set, fuzzy subalgebra 1. Introduction The study of algebraic structures in logic has been profoundly influenced by the dis- covery of universal operations, among which the Sheffer stroke (also known as the NAND operator) stands out as a cornerstone. Introduced by Sheffer in 1913 [1], this operation possesses the remarkable property of functional completeness: it can express all other log- ical connectives independently, thereby simplifying the axiomatization of logical systems. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6500 Email addresses: nrajesh topology@yahoo.co.in (N. Rajesh), aiyared.ia@up.ac.th (A. Iampan), tahsin.oner@ege.edu.tr (T. Oner), arsham@uk.ac.ir (A. B. Saeid) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) T. Oner et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6500 2 of 13 This universality extends to Boolean algebras, where the Sheffer stroke alone suffices to formulate their axioms, offering a minimalist yet powerful framework for algebraic and logical investigations [2]. Inspired by its efficiency and expressive power, recent studies by Oner et al. [3] introduced Sheffer stroke Hilbert algebras, merging the conceptual simplic- ity of the NAND operation with the structured implications of Hilbert algebras. Their work revealed new algebraic properties and paved the way for exploring substructures within these algebras, particularly in fuzzy settings. In parallel, Hilbert algebras—initially studied in the 1930s as algebraic models for the implicative fragment of intuitionistic logic [4–7]—emerged as pivotal objects in algebraic logic due to their ability to model logical implication in a purely algebraic setting. Over time, researchers have extended Hilbert algebras to enriched systems, incorporating oper- ations such as infimum and supremum to enhance their expressive power [7]. However, it was not until the synthesis with Sheffer stroke operations that a minimalist yet expressive representation of logical structures was achieved [3]. The introduction of fuzzy set theory by Zadeh in 1965 [8] revolutionized the modeling of vagueness and imprecision in mathematical systems. This extension inspired the fuzzi- fication of various algebraic structures, beginning with fuzzy subgroups [9] and evolving into fuzzy ideals in rings [10]. Building on these foundations, Dudek and Jun [11] extended the concept of ideals in Hilbert algebras to fuzzy ideals, introducing foundational results on closure properties and their connections to deductive systems. Borzooei et al. [12] introduced fuzzy weak filters in Sheffer stroke Hilbert algebras, exploring their definitions and key properties. The recent incorporation of fuzzy principles into Sheffer stroke Hilbert algebras by Oner et al. [3] marks a critical step forward, demonstrating that the Sheffer stroke’s functional completeness persists even in fuzzy settings. Despite these advancements, the study of fuzzy subalgebras within Sheffer stroke Hilbert algebras, particularly those characterized by (∈,∈ ∨qm)-fuzzy substructures, re- mains largely uncharted. The generalization of subalgebras to fuzzy contexts introduces graded membership, which allows for varying degrees of inclusion, thereby reflecting real- world imprecision more effectively. This gap presents a promising opportunity for deep- ening the theoretical understanding of fuzzy algebraic systems and expanding their appli- cations in logic and information processing. The exploration of Sheffer stroke Hilbert algebras has witnessed increasing depth and diversity, highlighting their central role in algebraic logic and fuzzy systems. Foundational work has addressed the algebraic basis of these structures, particularly the interplay be- tween Sheffer stroke and Hilbert algebras [3], and has been extended through the study of fuzzy ideals [13] and fuzzy filters [14]. The introduction of fuzzy weak filters [12] and bipolar-valued fuzzy deductive systems [15] further enriched the theoretical framework, allowing nuanced treatment of uncertainty and graded reasoning. More recent develop- ments have focused on structural generalizations, such as the incorporation of N -based soft subalgebras and ideals [16], and the formulation of length and mean-fuzzy ideals [17] and subalgebras [18], revealing new perspectives on the quantitative dimensions of fuzzy membership. Building upon this progression, the present paper introduces a new class of fuzzy sub- T. Oner et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6500 3 of 13 structures—namely, (∈,∈ ∨qm)-fuzzy subalgebras—within Sheffer stroke Hilbert algebras. These subalgebras extend existing frameworks by incorporating graded membership with tolerance for ambiguity, providing refined algebraic tools to model fuzzy logic under Shef- fer stroke operations. Our investigation establishes necessary and sufficient conditions, level set characterizations, and structural invariance under algebraic operations and ho- momorphisms, thereby contributing to the ongoing evolution of fuzzy algebraic theory. In this paper, we leverage this interaction to introduce generalized fuzzy subalgebras within Sheffer stroke Hilbert algebras. Specifically, we define and analyze (∈,∈ ∨qm)- fuzzy subalgebras, a class of fuzzy substructures that extend traditional subalgebras by incorporating graded membership and tolerance for ambiguity. These subalgebras are characterized by their interaction with level subsets and their behavior under algebraic operations, offering a nuanced perspective on the interplay between logic and fuzziness. The primary contributions of this paper include: • The definition and characterization of (∈,∈ ∨qm)-fuzzy subalgebras in Sheffer stroke Hilbert algebras. • The establishment of necessary and sufficient conditions for a fuzzy set to be an (∈,∈ ∨qm)-fuzzy subalgebra. • The investigation of level subsets and their role in characterizing these subalgebras. • The study of algebraic properties such as intersections, unions, and homomorphic invariance in the context of fuzzy subalgebras. Our results not only extend the theoretical understanding of Sheffer stroke Hilbert algebras but also provide a foundation for future applications in fuzzy logic and algebraic structures. 2. Preliminaries Sheffer stroke Hilbert algebras form a significant algebraic system bridging logic and lattice theory. These structures incorporate the Sheffer stroke (NAND) operation—an es- sential logical connective in Boolean algebra—into the classical Hilbert algebra framework. By extending Hilbert algebras with this operation, they provide a robust tool for analyzing logical structures and addressing applications in fuzzy logic, decision-making, and compu- tational frameworks. Their study deepens the theoretical foundation of algebraic systems and offers practical models for uncertainty and vagueness. Definition 1. [1] Let H = ⟨H, |⟩ be a groupoid. The operation | is said to be a Sheffer stroke operation if it satisfies the following conditions: (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. T. Oner et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6500 4 of 13 Definition 2. [3] A Sheffer stroke Hilbert algebra is a structure H = ⟨H, |, 0⟩ of type (2, 0), in which H is a non-empty set, | is a Sheffer stroke operation on H, and 0 is the fixed element in H such that the following identities are satisfied for all x, y, z ∈ H: (1) (x|((y|(z|z))|(y|(z|z))))|(((x|(y|y))|((x|(z|z))|(x|(z|z))))| ((x|(y|y))|((x|(z|z))|(x|(z|z))))) = x|(x|x) (2) x|(y|y) = y|(x|x) ⇒ x = y. Proposition 1. [3] Let H = ⟨H, |, 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 H. Definition 3. [3] A nonempty subset G of a Sheffer stroke Hilbert algebra H = ⟨H, |, 0⟩ is called a subalgebra of H if (x|(y|y))|(x|(y|y)) ∈ G for all x, y ∈ G. Definition 4. A fuzzy set µ in a non-empty set X of the form µ = { t ∈ (0, 1] if y = x 0 if y ̸= x is said to be a fuzzy point with support x and value t and is denoted by xt. The general form of the symbol xt q λ as follows: for an arbitrary element k of [0, 1), we say that • xt qk λ if λ(x) + t+ k > 1. • xt ∈ ∨qkλ if xt ∈ λ or xtqkλ. 3. New fuzzy subalgebras of Sheffer stroke Hilbert algebras In this section, let H = ⟨H, |, 0⟩ denote the Sheffer stroke Hilbert algebra unless oth- erwise specified. Definition 5. A fuzzy set µ in H is called an (∈,∈ ∨q)-fuzzy subalgebra of H = ⟨H, |, 0⟩, if it satisfies (∀x, y ∈ H, t1, t2 ∈ (0, 1])(xt1 , yt2 ∈ µ ⇒ ((x|(y|y))|(x|(y|y)))min{t1,t2} ∈ ∨qµ). (1) Remark 1. Let m be an element of [0, 1) unless otherwise specified. By xtqmµ, we mean µ(x) + t+m > 1, t ∈ (0, 1−m 2 ]. The notation xt ∈ ∨qmµ means that xt ∈ µ or xtqmµ. Definition 6. A fuzzy set µ in H is called an (∈,∈ ∨qm)-fuzzy subalgebra of H if (∀x, y ∈ H, t1, t2 ∈ (0, 1])(xt1 , yt2 ∈ µ ⇒ ((x|(y|y))|(x|(y|y)))min{t1,t2} ∈ ∨qmµ). (2) We note that different types of fuzzy subalgebras can be constructed for different values of m ∈ [0, 1). Hence, an (∈,∈ ∨qm)-fuzzy subalgebra with m = 0 is called an (∈,∈ ∨q)-fuzzy subalgebra. T. Oner et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6500 5 of 13 Proposition 2. Every (∈,∈)-fuzzy subalgebra is an (∈,∈ ∨qm)-fuzzy subalgebra. Proof. Straightforward. Theorem 1. A fuzzy set µ in H is an (∈,∈ ∨qm)-fuzzy subalgebra of H if and only if µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1−m 2 } holds for all x, y ∈ H. Proof. Let µ be an (∈,∈ ∨qm)-fuzzy subalgebra of H. Assume that µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1−m 2 } is not true. Then there exist x′, y′ ∈ H such that µ((x′|(y′|y′))|(x′|(y′|y′))) < min{µ(x′), µ(y′), 1−m 2 }. If min{µ(x′), µ(y′)} < 1−m 2 , then µ((x′|(y′|y′))|(x′|(y′|y′))) < min{µ(x′), µ(y′)}. Thus, µ((x′|(y′|y′))|(x′|(y′|y′))) < t ≤ min{µ(x′), µ(y′)} for some t ∈ (0, 1]. It follows that x′t ∈ µ and y′t ∈ µ, but ((x′|(y′|y′))|(x′|(y′|y′)))∈µ, a contradiction. Moreover, µ((x′|(y′|y′))|(x′|(y′|y′))) + t < 2t < 1−m, and so ((x′|(y′|y′))|(x′|(y′|y′)))tqmµ. Hence, ((x′|(y′|y′))|(x′|(y′|y′)))t∈ ∨qmµ, a contradic- tion. On the other hand, if min{µ(x′), µ(y′)} ≥ 1−m 2 , then µ(x′) ≥ 1−m 2 , µ(y′) ≥ 1−m 2 and µ((x′|(y′|y′))|(x′|(y′|y′))) < 1−m 2 . Thus, x′1−m 2 ∈ µ and y′1−m 2 ∈ µ, but ((x′|(y′|y′))|(x′|(y′|y′))) 1−m 2 ∈µ. Also, µ((x′|(y′|y′))|(x′|(y′|y′))) + 1−m 2 < 1−m 2 + 1−m 2 = 1−m, that is, ((x′|(y′|y′))|(x′|(y′|y′))) 1−m 2 qmµ. Hence, ((x′|(y′|y′))|(x′|(y′|y′))) 1−m 2 ∈ ∨qmµ, a contradiction. Hence, µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1−m 2 } holds for all x, y ∈ H. Conversely, assume that µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1−m 2 } holds for all x, y ∈ H. Let x, y ∈ H and t1, t2 ∈ (0, 1] be such that xt1 ∈ µ and yt2 ∈ µ. Then µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1−m 2 } ≥ min{t1, t2, 1−m 2 }. Assume that t1 ≤ 1−m 2 or t2 ≤ 1−m 2 . Then µ((x|(y|y))|(x|(y|y))) ≥ min{t1, t2}, which implies that ((x|(y|y))|(x|(y|y)))min{t1,t2} ∈ µ. Now, suppose that t1 > 1−m 2 and t2 > 1−m 2 . Then µ((x|(y|y))|(x|(y|y))) ≥ 1−m 2 , and thus µ((x|(y|y))|(x|(y|y))) + min{t1, t2} > 1−m 2 + 1−m 2 = 1−m, that is, ((x|(y|y))|(x|(y|y)))min{t1,t2}qmµ. Hence, ((x|(y|y))|(x|(y|y)))min{t1,t2} ∈ ∨qmµ, and consequently, µ is an (∈,∈ ∨qm)-fuzzy subalgebra of H. T. Oner et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6500 6 of 13 Theorem 2. A fuzzy set µ of H is an (∈,∈ ∨qm)-fuzzy subalgebra of H if and only if each nonempty level set U(µ, t) = {x ∈ H : µ(x) ≥ t}, t ∈ (0, 1−m 2 ], is a subalgebra of X. Proof. Assume that a fuzzy set µ is an (∈,∈ ∨q)-fuzzy subalgebra of H. Let t ∈ (0, 1−m 2 ] and x, y ∈ U(µ, t). Then µ(x) ≥ t and µ(y) ≥ t. It follows from µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1−m 2 } holds for all x, y ∈ H that µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1−m 2 } ≥ min{t, 1−m 2 } = t, so that (x|(y|y))|(x|(y|y)) ∈ U(µ, t). Hence, U(µ, t) is an (∈,∈ ∨qm)-fuzzy subalgebra of H. Conversely, suppose that the nonempty set U(µ, t) is a subalgebra of H for all t ∈ (0, 1−m 2 ]. If the condition µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1−m 2 } holds for all x, y ∈ H is not true, then there exist a, b ∈ H such that µ((a|(b|b))|(a|(b|b))) < min{µ(a), µ(b), 1−m 2 }. Hence, we can take t ∈ (0, 1] such that µ((a|(b|b))|(a|(b|b))) < t1 < min{µ(a), µ(b), 1−m 2 }. Then t ∈ (0, 1−m 2 ] and a, b ∈ U(µ, t). Since U(µ, t) is a subalgebra of H, we have (a|(b|b))|(a|(b|b)) ∈ U(µ, t), so µ((a|(b|b))|(a|(b|b))) ≥ t. This is a contradiction. Therefore, µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1−m 2 } holds for all x, y ∈ H, and so µ is an (∈,∈ ∨qm)-fuzzy subalgebra of H. Theorem 3. Let µ be a fuzzy set of H. Then the nonempty level set U(µ, t) is a subalgebra of H for all t ∈ (1−m 2 , 1] if and only if max{µ((x|(y|y))|(x|(y|y))), 1−m 2 } ≥ min{µ(x), µ(y)} for all x, y ∈ H. Proof. Suppose that U(µ, t) ̸= ∅ is a subalgebra of H. Assume that max{µ((x|(y|y))|(x|(y|y))), 1−m 2 } < min{µ(x), µ(y)} = t for some x, y ∈ H. Then t ∈ (1−m 2 , 1], µ((x|(y|y))|(x|(y|y))) < t, x ∈ U(µ, t) and y ∈ U(µ, t). Since x, y ∈ U(µ, t), we have U(µ, t) is a subalgebra of H, so (x|(y|y))|(x|(y|y)) ∈ U(µ, t), a contradiction. The proof of the second part of the theorem is straightforward. Theorem 4. Let µ be an (∈,∈ ∨qm)-fuzzy subalgebra of H. If it satisfies µ(x) < 1−m 2 for all x ∈ H, then it is a fuzzy subalgebra of H. Proof. Let x, y ∈ H and t1, t2 ∈ (0, 1] be such that xt1 ∈ µ and yt2 ∈ µ. Then µ(x) ≥ t1 and µ(y) ≥ t2. It follows from Theorem 1 that µ((x|(y|y))|(x|(y|y))) > min{µ(x), µ(y), 1−m 2 } = min{µ(x), µ(y)} = min{t1, t2}, so ((x|(y|y))|(x|(y|y)))min{t1,t2} ∈ µ. Hence, µ is a fuzzy subalgebra of H. T. Oner et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6500 7 of 13 Theorem 5. If 0 ≤ m < n < 1, then each (∈,∈ ∨qm)-fuzzy subalgebra of H is an (∈,∈ ∨qn)-fuzzy subalgebra of H. Proof. Let µ be an (∈,∈ ∨qm)-fuzzy subalgebra of H and let x, y ∈ H. Then µ((x|(y|y))|(x|(y|y))) > min{µ(x), µ(y), 1−m 2 } ≥ min{µ(x), µ(y), 1−n 2 }. Thus, from Theo- rem 1, we have µ is an (∈,∈ ∨qn)-fuzzy subalgebra of H. Note that an (∈,∈ ∨qn)-fuzzy subalgebra may not be an (∈,∈ ∨qm)-fuzzy subalgebra for 0 ≤ m < n < 1. Theorem 6. A nonempty subset M of H is a subalgebra of H if and only if its charac- teristic function is an (∈,∈ ∨qm)-fuzzy subalgebra of H. Proof. Let M be a subalgebra of H. Then χM (x) = 1 for x ∈ M and χM (x) = 0 for x /∈ M . Thus, U(µM , t) = M for all t ∈ (0, 1−m 2 ]. Hence, by Theorem 2, we have χM is an (∈,∈ ∨qm)-fuzzy subalgebra of H. Conversely, suppose that µM is an (∈,∈ ∨qm)-fuzzy subalgebra of H. Then µ((x|(y|y))|(x|(y|y))) > min{χM (x), χM (y), 1−m 2 } = min{1, 1−m 2 } = 1−m 2 for all x, y ∈ H. Since m ∈ [0, 1), χM ((x|(y|y))|(x|(y|y))) = 1, so (x|(y|y))|(x|(y|y)) ∈ M . Hence, M is a subalgebra of H. Theorem 7. For every subalgebra M of H and every t ∈ (0, 1−m 2 ] there exists an (∈,∈ ∨qm)-fuzzy subalgebra µ of H such that U(µ, t) = M . Proof. Let µ be a fuzzy set in H defined by µ(x) = { t if x ∈ M 0 otherwise, where t ∈ (0, 1−m 2 ]. Obviously, U(µ, t) = M . Assume that µ((x|(y|y))|(x|(y|y))) < min{µ(x), µ(y), 1−m 2 } for some x, y ∈ H. Since |Im(µ)| = 2, µ((x|(y|y))|(x|(y|y))) = 0 and min{µ(x), µ(y), 1−m 2 } = t. Hence, µ(x) = µ(y) = t, and so x, y ∈ M . Since M is a subalgebra of H, (x|(y|y))|(x|(y|y)) ∈ M . Thus, µ((x|(y|y))|(x|(y|y))) = t, which is a contradiction. Therefore, µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1−m 2 } for all x, y ∈ H. By Theorem 1, we have µ is an (∈,∈ ∨qm)-fuzzy subalgebra of H. Theorem 8. The intersection of any family of (∈,∈ ∨qm)-fuzzy subalgebras of H is an (∈,∈ ∨qm)-fuzzy subalgebra of H. T. Oner et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6500 8 of 13 Proof. Let µ = ⋂ i∈∆ µi, where µi is an (∈,∈ ∨qm)-fuzzy subalgebras of H. Then µ((x|(y|y))|(x|(y|y))) = inf i∈∆ µi((x|(y|y))|(x|(y|y))) ≥ inf i∈∆ min{µi(x), µi(y), 1−m 2 } ≥ min { inf i∈∆ µi(x), inf i∈∆ µi(y), 1−m 2 } = min { ⋂ i∈∆ µi(x), ⋂ i∈∆ µi(y), 1−m 2 } = min{µ(x), µ(y), 1−m 2 }. Hence, by Theorem 1, we have µ is an (∈,∈ ∨qm)-fuzzy subalgebra of H. Theorem 9. For any finite strictly increasing chain of subalgebras of H there exists an (∈,∈ ∨qm)-fuzzy subalgebra µ of H whose level subalgebras are precisely the members of the chain with µ 1−m 2 = H0 ⊂ H1 ⊂ . . . ⊂ Hn = H. Proof. Let {ti : ti ∈ (0, 1−m 2 ], i = 1, 2, . . . , n} be such that 1−m 2 > t1 > t2 > t3 > . . . > tn. Consider the fuzzy set µ defined by µ(x) = { 1−m 2 if x ∈ H0 tk if x ∈ Hk\Hk−1, k = 1, 2, . . . , n. Let x, y ∈ H be such that x ∈ Hi\Hi−1 and y ∈ Hj\Hj−1, where 1 ≤ i, j ≤ n. If i ≥ j, then x ∈ Hi and y ∈ Hi, so (x|(y|y))|(x|(y|y)) ∈ Hi. Thus, µ((x|(y|y))|(x|(y|y))) ≥ ti = min{ti, tj} = min{µ(x), µ(y), 1−m 2 }. If i < j, then x ∈ Hj and y ∈ Hj , so (x|(y|y))|(x|(y|y)) ∈ Hj . Thus, µ((x|(y|y))|(x|(y|y))) ≤ tj = min{ti, tj} = min{µ(x), µ(y), 1−m 2 }. Hence, µ is an (∈,∈ ∨qm)-fuzzy subalgebra of H. Definition 7. For any fuzzy set µ in H and t ∈ (0, 1], we define the sets [µ]t = {x ∈ H : xt ∈ ∨qmµ} and Q(µ, t) = {x ∈ H : xtqmµ}. It is clear that [µ]t = U(µ, t) ∪Q(µ, t). Theorem 10. Let µ be a fuzzy set in H. Then µ is an (∈,∈ ∨qm)-fuzzy subalgebra of H if and only if [µ]t is a subalgebra of H for all t ∈ (0, 1]. We call [µ]t an (∈ ∨qm)-level subalgebra of µ. Proof. Assume that µ is an (∈,∈ ∨qm)-fuzzy subalgebra of Hand let x, y ∈ [µ]t for t ∈ (0, 1]. Then (x, t) ∈ ∨qmµ and (y, t) ∈ ∨qmµ, that is, µ(x) > 1 or µ(x) + t > 1 −m, and µ(y) > 1 or µ(y) + t > 1 − m. By Theorem 1, we have µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1−m 2 }. T. Oner et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6500 9 of 13 Case 1: If µ(x) ≥ t and µ(y) ≥ t, then µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1−m 2 } = 1−m 2 . Hence, µ((x|(y|y))|(x|(y|y))) + t > 1−m 2 + 1−m 2 = 1−m, and so ((x|(y|y))|(x|(y|y)), t)qmµ. If t ≤ 1−m 2 , then µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1−m 2 } = 1−m 2 ≥ t, and thus ((x|(y|y))|(x|(y|y)), t) ∈ µ. Hence, ((x|(y|y))|(x|(y|y)), t) ∈ ∨qmµ. Therefore, (x|(y|y))|(x|(y|y)) ∈ [µ]t. Case 2: If µ(x) ≥ t and µ(y) + t > 1 −m. If t > 1−m 2 , then µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1−m 2 } = µ(y) ∧ 1−m 2 > (1−m− t) ∧ 1−m 2 = 1−m− t, and so ((x|(y|y))|(x|(y|y)), t)qmµ. If t ≤ 1−m 2 , then µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1−m 2 } ≥ min{t, 1−m−t, 1−m 2 } = t. Hence, ((x|(y|y))|(x|(y|y)), t) ∈ µ, and hence ((x|(y|y))|(x|(y|y)), t) ∈ ∨qmµ. Therefore, (x|(y|y))|(x|(y|y)) ∈ [µ]t. Case 3: If µ(x) + t > 1 −m and µ(y) ≥ t. If t > 1−m 2 , then µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1−m 2 } = µ(x) ∧ 1−m 2 > (1−m− t) ∧ 1−m 2 = 1−m− t, and so ((x|(y|y))|(x|(y|y)), t)qmµ. If t ≤ 1−m 2 , then µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1−m 2 } ≥ min{1−m−t, t, 1−m 2 } = t. Hence, ((x|(y|y))|(x|(y|y)), t) ∈ µ, and hence ((x|(y|y))|(x|(y|y)), t) ∈ ∨qmµ. Therefore, (x|(y|y))|(x|(y|y)) ∈ [µ]t. Case 4: If µ(x)+t > 1−m and µ(y)+t > 1−m. If t > 1−m 2 , then µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1−m 2 } > (1−m− t) ∧ 1−m 2 = 1−m− t, and so ((x|(y|y))|(x|(y|y)), t)qmµ. If t ≤ 1−m 2 , then µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1−m 2 } ≥ min{1−m−t, t, 1−m 2 } ≥ (1−m− t) ∧ 1−m 2 = 1−m 2 ≥ t. Hence, ((x|(y|y))|(x|(y|y)), t) ∈ µ, and hence ((x|(y|y))|(x|(y|y)), t) ∈ ∨qmµ. Therefore, (x|(y|y))|(x|(y|y)) ∈ [µ]t. Consequently, [µ]t is a subalgebra of H. Conversely, let µ be a fuzzy set in H and t ∈ (0, 1] be such that [µ]t is a subalgebra of H. If it is possible, let µ((x|(y|y))|(x|(y|y))) < t ≤ min{µ(x), µ(y), 1−m 2 } for some t ∈ (0, 1). Then x, y ∈ U(µ, t) ⊆ [µ]t, which implies that (x|(y|y))|(x|(y|y)) ∈ [µ]t. Hence, µ((x|(y|y))|(x|(y|y))) ∈ [µ]t or µ((x|(y|y))|(x|(y|y))) + t+m > 1, a contradiction. Therefore, µ((x|(y|y))|(x|(y|y))) ≥ min{µ(x), µ(y), 1−m 2 } for all x, y ∈ H. By Theorem 1, we conclude that µ is an (∈,∈ ∨qm)-fuzzy subalgebra of H. T. Oner et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6500 10 of 13 Theorem 11. Let µ be a proper (∈,∈ ∨qm)-fuzzy subalgebra of H having at least two values t1, t2 < 1−m 2 . If all [µ]t, t ∈ (0, 1−m 2 ], are subalgebras, then µ can be decomposed into the union of two proper nonequivalent (∈,∈ ∨qm)-fuzzy subalgebras of H. Proof. Let µ be a proper (∈,∈ ∨qm)-fuzzy subalgebra of H with values 1−m 2 > t1 > t2 > . . . > tn, where n > 2. Let H0 = [µ] 1−m 2 and Hk = [µ]tk for k = 1, 2, . . . , n. Then µ 1−m 2 = H0 ⊂ H1 ⊂ . . . ⊂ Hn = H is the chain of (∈,∈ ∨qm)-subalgebras of H. Consider the fuzzy sets λ1, λ2 ≤ µ defined by λ1(x) = { t1 if x ∈ H1 tk if x ∈ Hk\Hk−1, k = 2, . . . , n, λ2(x) =  µ(x) if x ∈ H0 t2 if x ∈ H2\H0 tk if x ∈ Hk\Hk−1, k = 3, . . . , n. Then λ1 and λ2 are (∈,∈ ∨qm)-fuzzy subalgebras of H with H0 ⊂ H1 ⊂ . . . ⊂ Hn and H0 ⊂ H1 ⊂ . . . ⊂ Hn being respectively chains of (∈,∈ ∨qm)-fuzzy subalgebras of H. Obviously, µ = λ1 ∨ λ2. Moreover, λ1 and λ2 are non-equivalent since H0 ̸= H1. Definition 8. Let ⟨A, |A, 0A⟩ and ⟨B, |B, 0B⟩ be Sheffer stroke Hilbert algebras. Then a mapping f : A → B is called a homomorphism if f(x|Ay) = f(x)|Bf(y) for all x, y ∈ A and f(0A) = 0B. Theorem 12. Let A = ⟨A, |A, 0A⟩ and B = ⟨B, |B, 0B⟩ be Sheffer stroke Hilbert algebras, f : A → B be a surjective homomorphism. If µ is an (∈,∈ ∨qm)-fuzzy subalgebra of B for m ∈ (0, 1), then f−1(B) is an (∈,∈ ∨qm)-fuzzy subalgebra of A. Proof. Let µ be an (∈,∈ ∨qm)-fuzzy subalgebra of B for m ∈ (0, 1) and x, y ∈ A. Then f−1(µ)((x|A(y|Ay))|A(x|A(y|Ay))) = µ(f((x|A(y|Ay))|A(x|A(y|Ay)))) = µ((f(x)|B(f(y)|Bf(y)))|B(f(x)|B(f(y)|Bf(y)))) ≥ min{µ(f(x)), µ(f(y)), 1−m 2 } = min{f−1(µ(x)), f−1(µ(y)), 1−m 2 }. Hence, f−1(µ) is an (∈,∈ ∨qm)-fuzzy subalgebra of A. Definition 9. Let f : X → Y be a function. An (∈,∈ ∨qm)-fuzzy subalgebra µ is said to be f -invariant if f(x) = f(y) implies that µ(x) = µ(y) for all x, y ∈ X. Theorem 13. Let A = ⟨A, |A, 0A⟩ and B = ⟨B, |B, 0B⟩ be Sheffer stroke Hilbert algebras, f : A → B be a homomorphism and µ an (∈,∈ ∨qm)-fuzzy subalgebra of A. If µ is f -invariant, then f(µ) is an (∈,∈ ∨qm)-fuzzy subalgebra of B, where f(µ)(x) = { sup x∈f−1(y) µ(x) if f−1(y) ̸= ∅ 0 otherwise. T. Oner et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6500 11 of 13 Proof. Let y1, y2 ∈ B. If f−1(y1) = ∅ or f−1(y2) = ∅, then the proof is obvious. Otherwise, let f−1(y1) ̸= ∅ or f−1(y2) ̸= ∅. Then there exist x1, x2 ∈ A such that and f(x1) = y1 and f(x2) = y2. Thus, f(µ)((y1|B(y2|By2))|B(y1|B(y2|By2))) = sup x∈f−1((y1|B(y2|By2))|B(y1|B(y2|By2))) µ(x) = sup x∈f−1((f(x1)|B(f(x2)|Bf(x2)))|B(f(x1)|B(f(x2)|Bf(x2)))) µ(x) = sup x∈f−1(f((x1|A(x2|Ax2))|A(x1|A(x2|Ax2)))) µ(x) = µ((x1|A(x2|Ax2))|A(x1|A(x2|Ax2))) ≥ min{µ(x1), µ(x2), 1−m 2 } = min { sup x∈f−1(f(x1)) µ(x), sup x∈f−1(f(x2)) µ(x), 1−m 2 } = min { sup x∈f−1(y1) µ(x), sup x∈f−1(y2) µ(x), 1−m 2 } = min { f(µ)(y1), f(µ)(y2), 1−m 2 } . Hence, f(µ) is an (∈,∈ ∨qm)-fuzzy subalgebra of B. Remark 2. Let A = ⟨A, |A, 0A⟩ and B = ⟨B, |B, 0B⟩ be Sheffer stroke Hilbert algebras. Then A×B = ⟨A×B, |A×B, 0A×B⟩ is a Sheffer stroke Hilbert algebra, where the set A×B is the Cartesian product of A and B and the operation |A×B on this set is defined by (a1, b1)|A×B(a2, b2) = (a1|Aa2, b1|Bb2), and the fixed element is 0A×B = (0A, 0B). Let µA and µB be (∈,∈ ∨qm)-fuzzy subalgebras of Sheffer stroke Hilbert algebras A = ⟨A, |A, 0A⟩ and B = ⟨B, |B, 0B⟩, respectively, for m ∈ (0, 1). The cartesian product of µA and µB is defined by µ = µA × µB, where µ(x, y) = min{µA(x), µB(y)}. Theorem 14. If µA and µB are (∈,∈ ∨qm)-fuzzy subalgebras of Sheffer stroke Hilbert algebras A = ⟨A, |A, 0A⟩ and B = ⟨B, |B, 0B⟩, respectively, then µ is an (∈,∈ ∨qm)-fuzzy subalgebra of A×B = ⟨A×B, |A×B, 0A×B⟩. Proof. For any (x1, y1), (x2, y2) ∈ A×B, we have µ(((x1, y1)|A×B((x2, y2)|A×B(x2, y2)))|A×B((x1, y1)|A×B((x2, y2)|A×B(x2, y2)))) = µ((x1|A(x2|Ax2))|A(x1|A(x2|Ax2)), (y1|B(y2|By2))|B(y1|B(y2|By2))) = min{µA((x1|A(x2|Ax2))|A(x1|A(x2|Ax2))), µB((y1|B(y2|By2))|B(y1|B(y2|By2)))} ≥ min { min { µA(x1), µA(x2), 1−m 2 } ,min { µB(y1), µB(y2), 1−m 2 }} = min { min { µA(x1), µB(y1), 1−m 2 } ,min { µA(x2), µB(y2), 1−m 2 }} = min { µ(x1, y1), µ(x2, y2), 1−m 2 } . Hence, ⟨A×B, |A×B⟩ is an (∈,∈ ∨qm)-fuzzy subalgebra of A×B. T. Oner et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6500 12 of 13 4. Conclusion In this paper, we introduce and study generalized fuzzy subalgebras within the frame- work of Sheffer stroke Hilbert algebras. By defining (∈,∈ ∨qm)-fuzzy subalgebras, we have extended classical subalgebra concepts to the fuzzy setting, providing a robust the- oretical foundation for further research. Our key results include the characterization of these subalgebras through their level subsets, as well as necessary and sufficient conditions for their existence. Additionally, we have demonstrated the algebraic properties of these subalgebras, including their behavior under intersection, union, and homomorphism. The implications of this work are twofold. First, it enriches the theoretical understand- ing of Sheffer stroke Hilbert algebras by incorporating fuzzy set theory. Second, it opens new avenues for applications in logical systems and algebraic structures where uncertainty and vagueness are inherent. 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