EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7074 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Interval-Valued Intuitionistic Fuzzy Subalgebras of Sheffer Stroke Hilbert Algebras Aiyared Iampan1,∗, Neelamegarajan Rajesh2, Tahsin Oner3, Firat Ates4, Kannan Geetha2, Arsham Borumand Saeid5 1 Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand 2 Department of Mathematics, Rajah Serfoji Government College, Thanjavur-613005, Tamil Nadu, India 3 Department of Mathematics, Faculty of Science, Ege University, 35100 Izmir, Turkey 4 Department of Mathematics, Balıkesir University, Balıkesir, Turkey 5 Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran Abstract. This paper explores the structure of interval-valued intuitionistic fuzzy (IVIF) sub- sets within the framework of Sheffer stroke Hilbert algebras (SSHAs). After establishing the foundational definitions of interval-valued intuitionistic fuzzy Sheffer stroke subalgebras (IVIFSS- subalgebras), we investigate their algebraic properties, closure under Sheffer stroke operations, and stability under set-theoretic intersections and unions. A key result characterizes IVIFSS- subalgebras through a pair of membership conditions, and it is further shown that the level sub- sets corresponding to IVIF-degrees form classical subalgebras in the crisp setting. These findings demonstrate that IVIF extensions preserve core algebraic behaviors while offering a robust model for uncertainty. The results contribute to the ongoing generalization of fuzzy algebraic systems and lay the groundwork for further developments involving fuzzy ideals and logical applications. 2020 Mathematics Subject Classifications: 20N05, 94D05, 03E72 Key Words and Phrases: Sheffer stroke Hilbert algebra, interval-valued intuitionistic fuzzy set, interval-valued intuitionistic fuzzy Sheffer stroke subalgebra 1. Introduction Sheffer stroke Hilbert algebras (SSHAs) are essential algebraic structures that play a significant role in logical systems and Boolean algebras. They serve as a foundation ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7074 Email addresses: aiyared.ia@up.ac.th (A. Iampan), nrajesh topology@yahoo.co.in (N. Rajesh), tahsin.oner@ege.edu.tr (T. Oner), firat@balikesir.edu.tr (F. Ates), geeethaak@gmail.com (K. Geetha), arsham@uk.ac.ir (A. Borumand Saeid) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Iampan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7074 2 of 14 for representing logical operations and have broad applications across fields like formal logic, artificial intelligence, and quantum computing. The Sheffer stroke operation, typi- cally defined by the NAND operation, is central to these algebras and is fundamental in constructing more advanced logical systems. The application of the Sheffer stroke in various algebraic structures has been thoroughly explored, making significant contributions to multiple fields of study. Research has focused on Sheffer stroke reducts within basic algebras [1, 2], Sheffer stroke MTL-algebras [3], and ortholattices [4]. Additionally, the role of fuzzy sets in Sheffer stroke BE-algebras [5] has been investigated, underscoring the far-reaching effects of the Sheffer stroke operation across diverse algebraic systems. The concept of fuzzy sets, proposed by Zadeh [6], revolutionized the study of un- certainty and vagueness, providing a powerful tool for modeling imprecision. Fuzzy set theory has since been applied in a variety of real-world contexts, sparking considerable research into its expansion. Following the introduction of fuzzy sets, numerous studies focused on generalizing the theory. One key direction has been the integration of fuzzy sets with other uncertainty models, such as soft sets and rough sets, as explored in vari- ous works [7–9]. Another significant extension of fuzzy sets is the notion of intuitionistic fuzzy sets, introduced by Atanassov [10]. These sets enhance the applicability of fuzzy sets by incorporating both membership and non-membership degrees, making them suit- able for situations involving incomplete information or uncertainty. Intuitionistic fuzzy sets have been widely applied in domains such as medical diagnostics, optimization, and multi-criteria decision-making [11–13]. Hilbert algebras were initially introduced in the 1950s by Henkin [14] as a means to investigate implications within intuitionistic and other non-classical logics. In the 1960s, scholars such as Horn and Diego further developed these algebras from an alge- braic standpoint. Diego demonstrated that Hilbert algebras form a locally finite variety [15]. Researchers such as Busneag [16, 17] and Jun [18] also contributed to the study of Hilbert algebras, focusing on the role of their filters in forming deductive systems. In the context of fuzzy logic, Dudek [19] explored the fuzzification of subalgebras and deductive systems in Hilbert algebras, adding a new layer of complexity to these structures. The algebraic framework of SSHAs has evolved considerably over the past few years, establishing itself as a fertile ground for generalizations in fuzzy and soft logic. The foun- dational connection between Sheffer stroke operations and Hilbert algebras was rigorously explored by Oner et al. [20], leading to a formal algebraic structure that integrates logical minimalism with algebraic expressiveness. This foundation has since been extended in several directions. For instance, stabilization properties via ideals have been analyzed in [21] to explore internal algebraic regularities. The incorporation of fuzzy logic began with fuzzy filters [22], followed by the introduction of fuzzy ideals [23], and then generalized fuzzy subalgebras [24]. Parallel developments include investigations into length-based and mean-fuzzy structures [25, 26], as well as soft set extensions using N-structures [27]. Col- lectively, these contributions not only demonstrate the flexibility of the SSHA model but also set the stage for more refined fuzzy generalizations, such as those based on intuition- istic and interval-valued constructs. A. Iampan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7074 3 of 14 Despite the broad utilization of SSHAs, their connection with fuzzy logic and intu- itionistic fuzzy set theory remains largely unexplored. The introduction of interval-valued intuitionistic fuzzy sets (IVIFSs) offers a novel approach to representing uncertainty and partial membership in algebraic structures. By replacing fixed values with intervals to denote membership and non-membership degrees, IVIFS provide a more flexible approach to dealing with vagueness and uncertainty. The paper introduces the concept of fuzzy and intuitionistic fuzzy sets in algebraic structures, particularly within the context of Sheffer-styled Hilbert algebras. It empha- sizes the importance of fuzzy subsets in capturing uncertainty and partial membership in algebraic systems. The study then focuses explicitly on interval-valued intuitionistic fuzzy (IVIF) subsets, providing fundamental definitions and properties. Section 3 elaborates on these IVIF subsets, examining how they can be characterized as subalgebras and their behavior under various set operations. The section aims to establish a solid theoretical foundation for integrating fuzzy logic principles into algebraic structures, thereby paving the way for future research and applications. The list of acronyms is given in Table 1. Table 1: List of acronyms Acronyms Representation SSHA Sheffer stroke Hilbert algebra IFS intuitionistic fuzzy set IVIFS interval-valued intuitionistic fuzzy set IVIFSS-subalgebra interval-valued intuitionistic fuzzy Sheffer stroke subalgebra 2. Preliminaries In this section, to deepen the understanding of the foundational structures introduced earlier, we will elaborate on the key properties and relationships of the relevant algebraic and fuzzy constructs. Specifically, we will define certain operations and structures, and examine their characteristics and interconnections. This groundwork will ensure clarity and provide a solid theoretical basis for the subsequent sections. Our aim is to present these fundamental concepts and structures in a clear and precise manner, establishing the essential building blocks for the developments that follow. Definition 1. [28] Let A := (A, |) be a groupoid. Then the operation | is said to be Sheffer stroke or Sheffer operation if it satisfies: (s1) (∀d, g ∈ A) (d | g = g | d), (s2) (∀d, g ∈ A) ((d | d) | (d | g) = d), (s3) (∀d, g,w ∈ A) (d | ((g | w) | (g | w)) = ((d | g) | (d | g)) | w), A. Iampan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7074 4 of 14 (s4) (∀d, g,w ∈ A) ((d | ((d | d) | (g | g))) | (d | ((d | d) | (g | g))) = d). To improve the clarity of this manuscript, we introduce the following notation, which will be used consistently throughout the text: w | (g | g) := wg. Definition 2. [20] A Sheffer stroke Hilbert algebra (SSHA) is a groupoid SH := (SH , |, 0) equipped with a Sheffer stroke operation that satisfies specific conditions: (sH1) (d | (gw | gw)) | ((dg | (dw | dw)) | (dg | (dw | dw))) = dd, (sH2) dg = gd = dd ⇒ d = g for all d, g,w ∈ SH . Proposition 1. [20] Let SH := (SH , |, 0) be an SSHA. Then the binary relation d ≤ g ⇔ dg = 0 is a partial order on SH . Definition 3. [20] Let SH := (SH , |, 0) be an SSHA. A nonempty subset A of SH is called a subalgebra of SH if dg | dg ∈ A for all d, g ∈ A. Definition 4. [10] Let X be a nonempty set. The intuitionistic fuzzy set (IFS) on X is defined as a structure A := {⟨d, µA(d), γA(d)⟩ | d ∈ X}, (1) where µA : X → [0, 1] represents the degree of membership of d in A, and γA : X → [0, 1] represents the degree of non-membership of d in A such that 0 ≤ µA(d) + γA(d) ≤ 1. The IFS in (1) is simply denoted by A = (µA, γA). Let D [0, 1] be the set of all closed subintervals of the interval [0, 1]. Consider I1, I2 ∈ D [0, 1]. If I1 = [m1, n1] and I2 = [m2, n2], then rmin{I1, I2} = [min{m1,m2},min{n1, n2}] and rmax{I1, I2} = [max{m1,m2},max{n1, n2}]. Thus, if Ii = [mi, ni] ∈ D [0, 1] for i = 1, 2, . . ., then we define rsupi{Ii} = [sup i {mi}, sup i {ni}] and rinf i{Ii} = [inf i {mi}, inf i {ni}]. Now, we call I1 ≥ I2 if and only if m1 ≥ m2 and n1 ≤ n2. Similarly, the relations I1 ≤ I2 and I1 = I2 are defined. A. Iampan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7074 5 of 14 Definition 5. An interval-valued intuitionistic fuzzy set (IVIFS) A over a universe X is defined as an object of the form A = {⟨d, µA(d), γA(d)⟩ | d ∈ X}, where µA(d) : X → D [0, 1] and γA(d) : X → D [0, 1]. The functions µA(d) and γA(d) represent the intervals of the degree of membership and non-membership of the element d in A, respectively, where µA(d) = [µl A(d), µ u A(d)] and γA(d) = [γlA(d), γ u A(d)] for all d ∈ X, subject to condition 0 ≤ µl A(d) + γuA(d) ≤ 1. For simplicity, we denote the IVIFS A as A = (µA, γA), where A = {⟨d, µA(d), γA(d)⟩ | d ∈ X}. Additionally, the complements of µA and γA are given by µA(d) = [1−µu A(d), 1−µl A(d)] and γA(d) = [1− γuA(d), 1− γlA(d)], where [µA(d), γA(d)], represents the complement of d in A. 3. Interval-valued intuitionistic fuzzy Sheffer stroke subalgebras The integration of IVIFSs with SSHAs provides a fertile ground for generalizing clas- sical algebraic structures under uncertainty. While intuitionistic fuzzy frameworks have been extensively applied to various algebraic systems, their interval-valued extensions remain relatively underexplored within the context of SSHA. This motivates a deeper investigation into the algebraic behavior of such fuzzy structures. In this section, we introduce the concept of interval-valued intuitionistic fuzzy Sheffer stroke subalgebras (IVIFSS-subalgebras) and examine their fundamental properties. The aim is to establish criteria under which IVIF subsets of an SSHA preserve subalgebraic structure with respect to the Sheffer operation. Special attention is given to their closure properties, interaction under set-theoretic operations, and logical coherence. These results not only enrich the theory of fuzzy algebraic systems but also offer tools for modeling imprecise reasoning in logic and computational intelligence. Definition 6. Let SH := (SH , |, 0) be an SSHA. An IVIFS A = (µA, γA) in SH is called an interval-valued intuitionistic fuzzy Sheffer stroke subalgebra (IVIFSS-subalgebra) of SH if (∀d, g ∈ SH) ( µA(d g | dg) ≥ rmin{µA(d), µA(g)} γA(d g | dg) ≤ rmax{γA(d), γA(g)} ) . (2) Proposition 2. Let SH := (SH , |, 0) be an SSHA. Every IVIFSS-subalgebra A = (µA, γA) of SH satisfies (∀d ∈ SH) ( µA(0) ≥ µA(d) γA(0) ≤ γA(d) ) . (3) Proof. For any d ∈ SH , we have µA(0) = µA(d d | dd) ≥ rmin{µA(d), µA(d)} = rmin{[µl A(d), µ u A(d)], [µ l A(d), µ u A(d)]} = [µl A(d), µ u A(d)] = µA(d) A. Iampan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7074 6 of 14 and γA(0) = γA(d d | dd) ≤ rmax{γA(d), γA(d)} = rmax{[γlA(d), γuA(d)], [γlA(d), γuA(d)]} = [γlA(d), γ u A(d)] = γA(d). Proposition 3. Let SH := (SH , |, 0) be an SSHA. Every IVIFSS-subalgebra A = (µA, γA) of SH satisfies the following: (∀d, g ∈ SH) ( µA(d g | dg) ≥ µA(g) γA(d g | dg) ≤ γA(g) ) (4) if and only if µA(0) = µA(d) and γA(0) = γA(d) for all d ∈ SH . Proof. Let d ∈ SH . Then µA(d) = µA((d|0)|(d|0)) = µA(d 0 | d0) ≥ µA(0) and γA(d) = γA((d|0)|(d|0)) = γA(d 0 | d0) ≤ γA(0). Then by Proposition 2, µA(0) = µA(d) and γA(0) = γA(d). The converse is clear. Theorem 1. Let SH := (SH , |, 0) be an SSHA. An IVIFS A = ([µl A, µ u A], [γ l A, γ u A]) in SH is an IVIFSS-subalgebra of SH if and only if µl A, µ u A, γ l A and γuA are fuzzy subalgebras of SH . Proof. Let µl A and µu A be fuzzy subalgebras of SH and d, g ∈ SH . Then µl A(d g | dg) ≥ min{µl A(d), µ l A(g)} and µu A(d g | dg) ≤ min{µu A(d), µ u A(g)}. Now, µA(d g | dg) = [µl A(d g | dg), µu A(d g | dg)] ≥ [min{µl A(d), µ l A(g)},min{µu A(d), µ u A(g)}] = rmin{[µl A(d), µ u A(d)], [µ l A(g), µ u A(g)]} = rmin{µA(d), µA(g)}. A. Iampan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7074 7 of 14 Let γlA and γuA be fuzzy subalgebras of SH and d, g ∈ SH . Then γlA(d g | dg) ≤ max{γlA(d), γlA(g)} and γuA(d g | dg) ≤ max{γuA(d), γuA(g)}. Now, γA(d g | dg) = [γlA(d g | dg), γuA(dg | dg)] ≤ [max{γlA(d), γlA(g)},max{γuA(d), γuA(g)}] = rmax{[γlA(d), γuA(d)], [γlA(g), γuA(g)]} = rmax{γA(d), γA(g)}. Hence, A = ([µl A, µ u A], [γ l A, γ u A]) is an IVIFSS-subalgebra of SH . Conversely, assume that A = ([µl A, µ u A], [γ l A, γ u A]) is an IVIFSS-subalgebra of SH . For any d, g ∈ SH , [µl A(d g | dg), µu A(d g | dg)] = µA(d g | dg) ≥ rmin{µA(d), µA(g)} = rmin{[µl A(d), µ u A(d)], [µ l A(g), µ u A(g)]} = [min{µl A(d), µ l A(g)},min{µu A(d), µ u A(g)}], and [γlA(d g | dg), γuA(dg | dg)] = γA(d g | dg) ≤ rmax{γA(d), γA(g)} = rmax{[γlA(d), γuA(d)], [γlA(g), γuA(g)]} = [max{γlA(d), γlA(g)},max{γuA(d), γuA(g)}]. Thus, µl A(d g | dg) ≥ min{µl A(d), µ l A(g)}, µu A(d g | dg) ≥ min{µu A(d), µ u A(g)}, γlA(dg | dg) ≤ max{γlA(d), γlA(g)} and γuA(d g | dg) ≤ max{γuA(d), γuA(g)}. Therefore, µl A, µ u A, γ l A and γuA are fuzzy subalgebras of SH . Theorem 2. Let SH := (SH , |, 0) be an SSHA. If A = (µA, γA) and B = (µB, γB) are IVIFSS-subalgebras of SH , then A ∩B = (µA∩B, γA∪B) is an IVIFSS-subalgebra of SH . Proof. Let d, g ∈ A∩B. Since A = (µA, γA) and B = (µB, γB) are IVIFSS-subalgebras of SH , µA∩B(d g | dg) = [µl A∩B(d g | dg), µu A∩B(d g | dg)] = [min{µl A(d g | dg), µl B(d g | dg)},min{µu A(d g | dg), µu B(d g | dg)}] ≥ [min{µl A∩B(d), µ l A∩B(g))},min{µu A∩B(d), µ u A∩B(g))}] = rmin{µA∩B(d), µA∩B(g)} and γA∪B(d g | dg) = [γlA∪B(d g | dg), γuA∪B(d g | dg)] = [max{γlA(dg | dg), γlB(dg | dg)},max{γuA(dg | dg), γuB(dg | dg)}] ≤ [max{γlA∪B(d), γ l A∪B(g)},max{γuA∪B(d), γ u A∪B(g)}] = rmax{γA∪B(d), γA∪B(g)}. Hence, A ∩B = (µA∩B, γA∪B) is an IVIFSS-subalgebra of SH . A. Iampan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7074 8 of 14 Definition 7. Let SH := (SH , |, 0) be an SSHA. Let A = (µA, γA) be an IVIFS defined on SH . The operators ⊕A and ⊗A are defined as ⊕A = {⟨d, µA(d), µA(d)⟩ | d ∈ SH} and ⊗A = {⟨d, γA(d), γA(d)⟩ | d ∈ SH}. Theorem 3. Let SH := (SH , |, 0) be an SSHA. If A = (µA, γA) is an IVIFSS-subalgebra of SH , then ⊕A and ⊗A are both IVIFSS-subalgebras. Proof. Let d, g ∈ SH . Then µA(d g | dg) = [1, 1]− µA(d g | dg) ≤ [1, 1]− rmin{µA(d), µA(g)} = rmax{1− µA(d), 1− µA(g)} = rmax{µA(d), µA(g)}. Hence, ⊕A is an IVIFSS-subalgebra of SH . Let d, g ∈ SH . Then γA(d g | dg) = [1, 1]− γA(d g | dg) ≥ [1, 1]− rmax{γA(d), γA(g)} = rmin{1− γA(d), 1− γA(g)} = rmin{γA(d), γA(g)}. Hence, ⊗A is an IVIFSS-subalgebra of SH . The sets {d ∈ SH | µA(d) = µA(0)} and {d ∈ SH | γA(d) = γA(0)} are denoted by µ∗ A and γ∗A, respectively. Theorem 4. Let SH := (SH , |, 0) be an SSHA. Let A = (µA, γA) be an IVIFSS-subalgebra of SH , then the sets µ∗ A and γ∗A are subalgebras of SH . Proof. Let d, g ∈ µ∗ A. Then µA(d) = µA(0) = µA(g) and so µA(d g | dg) ≤ rmin{µA(d), µA(g)} = µA(0). By using Proposition 2, we have µA(d g | dg) = µA(0) and hence dg | dg ∈ µ∗ A. Again, let d, g ∈ γ∗A. Then γA(d) = γA(0) = γA(g) and so γA(d g | dg) ≤ rmax{γA(d), γA(g)} = γA(0). Again, by Proposition 2, we hece γA(x g | dg) = γA(0); hence dg | dg ∈ γ∗A. Therefore, the sets µ∗ A and γ∗A are subalgebras of SH . Theorem 5. Let SH := (SH , |, 0) be an SSHA. Let B be a nonempty subset of SH and A = (µA, γA) be an IVIFS in SH defined by µA(d) = { [ϑ1, ϑ2] if d ∈ B [η1, η2] otherwise and γA(d) = { [ϖ1, ϖ2] if d ∈ B [ς1, ς2] otherwise for all [ϑ1, ϑ2], [η1, η2], [ϖ1, ϖ2], [δ1, δ2] ∈ D [0, 1] with [ϑ1, ϑ2] ≥ [η1, η2] and [ϖ1, ϖ2] ≤ [ς1, ς2] and ϑ2 +ϖ2 ≤ 1 and η2 + δ2 ≤ 1. Then A = (µA, γA) is an IVIFSS-subalgebra of SH if and only if B is a subalgebra of SH . Moreover, µ∗ A = B = γ∗A. A. Iampan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7074 9 of 14 Proof. Let A = (µA, γA) be an IVIFSS-subalgebra of SH . Let d, g ∈ SH be such that d, g ∈ B. Then µA(d g | dg) ≥ rmin{µA(d), µA(g)} = rmin{[ϑ1, ϑ2], [ϑ1, ϑ2]} = [ϑ1, ϑ2] and γA(d g | dg) ≤ rmax{γA(d), γA(g)} = rmax{[ϑ1, ϑ2], [ϑ1, ϑ2]} = [ϑ1, ϑ2]. So, dg | dg ∈ B. Hence, B is a subalgebra of SH . Conversely, suppose that B is a subalgebra of SH . Let d, g ∈ SH . Consider two cases: Case (i): If d, g ∈ B, then dg | dg ∈ B. Thus, µA(d g | dg) = [ϖ1, ϖ2] = rmin{µA(d), µA(g)} and γA(d g | dg) = [θ1, θ2] = rmax{γA(d), γA(g)}. Case (ii): If d /∈ B or g /∈ B, then µA(d g | dg) ≥ [η1, η2] = rmin{µA(d), µA(g)} and γA(d g | dg) ≤ [θ1, θ2] = rmax{γA(d), γA(g)}. Hence, A = (µA, γA) is an IVIFSS-subalgebra of SH . Now, µ∗ A = {d ∈ SH | µA(d) = µA(0)} = {d ∈ SH | µA(d) = [ϑ1, ϑ2]} = B and γ∗A = {d ∈ SH | γA(d) = γA(0)} = {d ∈ SH | γA(d) = [ϖ1, ϖ2]} = B. To confirm that an IVIFSS-subalgebra aligns with the underlying crisp structure, we examine its level subsets. Specifically, we define α- and β-level sets and verify that these subsets form classical subalgebras of the SSHA. This ensures that the fuzzy extension preserves the essential algebraic properties. Definition 8. Let SH := (SH , |, 0) be an SSHA. Let A = (µA, γA) is an IVIFSS-subalgebra of SH . For [p1, p2], [q1, q2] ∈ D [0, 1], the set U (µA : [p1, p2]) = {d ∈ SH | µA(d) ≥ [p1, p2]} is called an upper [p1, p2]-level of A and L (γA : [q1, q2]) = {d ∈ SH | γA(d) ≤ [q1, q2]} is called a lower [q1, q2]-level of A. Theorem 6. Let SH := (SH , |, 0) be an SSHA. If A = (µA, γA) is an IVIFSS-subalgebra of SH , then the upper [p1, p2]-level and lower [q1, q2]-level of A are subalgebras of SH . Proof. Let d, g ∈ U (µA : [s1, s2]). Then µA(d) ≤ [p1, p2] and µA(g) ≤ [p1, p2]. It follows that µA(d g | dg) ≤ rmin{µA(d), µA(g)} ≤ [p1, p2] so that dg | dg ∈ U (µA : [p1, p2]). Hence, U (µA : [p1, p2]) is a subalgebra of SH . Let d, g ∈ L (γA : [q1, q2]). Then γA(d) ≤ [q1, q2] and γA(g) ≤ [q1, q2]. It follows that γA(d g | dg) ≤ rmax{γA(d), γA(g)} ≤ [q1, q2] so that dg | dg ∈ L (γA : [q1, q2]). Hence, L (γA : [q1, q2]) is a subalgebra of SH . Theorem 7. Let SH := (SH , |, 0) be an SSHA. Let A = (µA, γA) be an IVIFS in SH such that the sets U (µA : [p1, p2]) and L (γA : [q1, q2]) are subalgebras of SH for every [p1, p2], [q1, q2] ∈ D [0, 1]. Then A = (µA, γA) is an IVIFSS-subalgebra of SH . A. Iampan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7074 10 of 14 Proof. Let for every [p1, p2], [q1, q2] ∈ D [0, 1], U (µA : [p1, p2]) and L (γA : [q1, q2]) are subalgebras of SH . In contrary, let d0, g0 ∈ SH be such that µA(d g0 0 | dg00 ) < rmin{µA(d0), µA(g0)}. Let µA(d0) = [θ1, θ2], µA(g0) = [θ3, θ4] and µA(d g0 0 | dg00 ) = [p1, p2]. Then [p1, p2] < rmin{[θ1, θ2], [θ3, θ4]} = [min{θ1, θ3},min{θ2, θ4}]. So, p1 < min{θ1, θ3} and p2 < min{θ2, θ4}. Consider, [ρ1, ρ2] = 1 2 [µA(d g0 0 | dg00 ) + rmin{µA(d0), µA(g0)}] = 1 2 [[p1, p2] + [min{θ1, θ3},min{θ2, θ4}]] = [12(p1 +min{θ1, θ3}), 12(p2 +min{θ2, θ4})]. Therefore, min{θ1, θ3} > ρ1 = 1 2(p1 + min{θ1, θ3}) > p1 and min{θ2, θ4} > ρ2 = 1 2(p2 + min{θ2, θ4}) > p2. Hence, [min{θ1, θ3},min{θ2, θ4}] > [ρ1, ρ2] > [p1, p2], so that dg00 | dg00 /∈ U (µA : [p1, p2]), a contradiction, since µA(d0) = [θ1, θ2] ≥ [min{θ1, θ3},min{θ2, θ4}] > [ρ1, ρ2] and µA(g0) = [θ3, θ4] ≥ [min{θ1, θ3},min{θ2, θ4}] > [ρ1, ρ2]. This implies dg00 | dg00 ∈ U (µA : [p1, p2]). Thus µA(d g | dg) ≤ rmin{µA(d), µA(g)} for all d, g ∈ SH . Again, in contrary, let d0, g0 ∈ SH be such that γA(d g0 0 | dg00 ) > rmax{γA(d0), γA(g0)}. Let γA(d0) = [η1, η2], γA(g0) = [η3, η4] and γA(d g0 0 | dg00 ) = [q1, q2]. Then [q1, q2] > rmax{[η1, η2], [η3, η4]} = [max{η1, η3},max{η2, η4}]. So, q1 > max{η1, η3} and q2 > max{η2, η4}. Let us consider, [λ1, λ2] = 1 2 [γA(d g0 0 | dg00 ) + rmax{γA(d0), γA(g0)}] = 1 2 [[q1, q2] + [max{η1, η3},max{η2, η4}] = 1 2(q1 +max{η1, η3}), 12(q2 +max{η2, η4}). Therefore, max{η1, η3} < λ1 = 1 2 [(q1+max{η1, η3})] < q1 and max{η2, η4} < λ2 = 1 2 [(q2+ max{η2, η4})] < q2. Hence, [max{η1, η3},max{η2, η4}] < [λ1, λ2] < [q1, q2] so that dg00 | dg00 /∈ L (γA : [q1, q2]), a contradiction, since γA(x0) = [η1, η2] ≤ [max{η1, η3},max{η2, η4}] > [β1, β2] and γA(g0) = [η3, η4] ≥ [max{η1, η3},max{η2, η4}] > [β1, β2]. Hence, dg00 | dg00 ∈ L (γA : [q1, q2]). Thus, γA(d g | dg) ≥ rmax{γA(d), γA(g)} for all d, g ∈ SH . Theorem 8. Let SH := (SH , |, 0) be an SSHA. Any subalgebra of SH can be realized as both the upper [p1, p2]-level and lower [q1, q2]-level of some IVIFSS-subalgebra of SH . Proof. Let B be a subalgebra of SH , and A = (µA, γA) be an IVIFS on SH defined by µA(d) = { [ϱ1, ϱ2] if d ∈ B [0, 0] otherwise and γA(d) = { [ϑ1, ϑ2] if d ∈ B [1, 1] otherwise for all [ϱ1, ϱ2], [ϑ1, ϑ2] ∈ D [0, 1] and ϱ2 + ϑ2 ≤ 1. We consider the following cases: A. Iampan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7074 11 of 14 Case (i): If d, g ∈ B, then µA(d) = [ϱ1, ϱ2], γA(d) = [ϑ1, ϑ2] and µA(g) = [ϱ1, ϱ2], γA(g) = [β1, β2]. Thus, µA(d g | dg) = [ϱ1, ϱ2] = rmin{[ϱ1, ϱ2], [α1, α2]} = rmin{µA(d), µA(g)} and γA(d g | dg) = [ϑ1, ϑ2] = rmax{[ϑ1, ϑ2], [ϑ1, ϑ2]} = rmax{γA(d), γA(g)}. Case (ii): If d ∈ B and g /∈ B, then µA(d) = [ϱ1, ϱ2], γA(d) = [ϑ1, ϑ2] and µA(g) = [0, 0], γA(g) = [1, 1]. Thus, µA(d g | dg) ≥ [0, 0] = rmin{[ϱ1, ϱ2], [0, 0]} = rmin{µA(d), µA(g)} and γA(d g | dg) ≥ [1, 1] = rmax{[ϑ1, ϑ2], [1, 1]} = rmax{γA(d), γA(g)}. Case (iii): If d /∈ B and g ∈ B, then µA(d) = [0, 0], γA(d) = [1, 1], µA(g) = [α1, α2], γA(g) = [ϑ1, ϑ2]. Thus, µA(d g | dg) ≥ [0, 0] = rmin{[0, 0], [ϱ1, ϱ2]} = rmin{µA(d), µA(g)} and γA(d g | dg) ≤ [1, 1] = rmax{[1, 1], [β1, β2]} = rmax{γA(d), γA(g)}. Case (iv): If d /∈ B and g /∈ B, then µA(d) = [0, 0], γA(d) = [1, 1] and µA(g) = [0, 0], γA(g) = [1, 1]. Now, µA(d g | dg) ≤ [0, 0] = rmin{[0, 0], [0, 0]} = rmin{µA(d), µA(g)} and γA(d g | dg) ≥ [1, 1] = rmax{[1, 1], [1, 1]} = rmax{γA(d), γA(g)}. Therefore, A = (µA, γA) is an IVIFSS-subalgebra of SH . Theorem 9. Let SH := (SH , |, 0) be an SSHA. Let B be a subset of SH and A = (µA, γA) be an IVIFS on SH defined by µA(d) = { [ϱ1, ϱ2] if d ∈ B [0, 0] otherwise and γA(d) = { [ϑ1, ϑ2] if d ∈ B [1, 1] otherwise for all [ϱ1, ϱ2], [ϑ1, ϑ2] ∈ D [0, 1] and ϱ2+ϑ2 ≤ 1. If A = (µA, γA) is realized as a lower-level subalgebra and an upper-level subalgebra of some IVIFSS-subalgebra of SH , then B is a subalgebra of SH . A. Iampan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7074 12 of 14 Proof. Let A = (µA, γA) be an IVIFSS-subalgebra of SH , and d, g ∈ B. Then µA(d) = [ϱ1, ϱ2] = µA(g) and γA(d) = [β1, β2] = γA(g). Thus, µA(d g | dg) ≤ rmin{µA(d), µA(g)} = rmin{[ϱ1, ϱ2], [ϱ1, ϱ2]} = [ϱ1, ϱ2] and γA(d g | dg) ≥ rmax{γA(d), γA(g)} = rmax{[ϑ1, ϑ2], [ϑ1, ϑ2]} = [ϑ1, β2], which imply that dg | dg ∈ B. 4. Conclusion This study explored the fundamental properties of interval-valued intuitionistic fuzzy (IVIF) subsets and subalgebras within Sheffer stroke Hilbert algebras, highlighting their al- gebraic behavior and relationships under various set operations. The results demonstrated that these fuzzy structures could serve as effective extensions of classical algebraic systems, allowing for the flexible capture of uncertainty and partial membership. The theoretical insights provided by this work lay a foundation for integrating fuzzy logic principles into algebraic frameworks, with potential applications in logical reasoning, decision-making, and information processing. Future research could focus on extending the concept of IVIF subsets and ideals to other algebraic systems such as Boolean algebras, orthomodular lattices, and residuated structures. Developing computational algorithms for the automatic identification and manipulation of IVIF subalgebras will facilitate practical applications in areas such as ar- tificial intelligence, fuzzy decision-making, and data analysis. Additionally, investigating the application of these fuzzy algebraic structures in real-world scenarios—such as med- ical diagnosis, control systems, and information security—may provide valuable insights and enhance their practical utility. Further exploration of their interactions with proba- bilistic and other uncertainty-based frameworks can also open new avenues for theoretical advancement. Acknowledgements This research was supported by University of Phayao and Thailand Science Research and Innovation Fund (Fundamental Fund 2026, Grant No. 2252/2568). References [1] T. Oner and I. Senturk. The Sheffer stroke operation reducts of basic algebra. Open Mathematics, 15(1):926–935, 2017. [2] I. Senturk, T. Oner, and A. Borumand Saeid. Congruences of Sheffer stroke basic algebras. Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica, 28(2):209–228, 2020. A. Iampan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7074 13 of 14 [3] I. Senturk and T. Oner. A construction of very true operator on Sheffer stroke MTL- algebras. International Journal of Maps in Mathematics, 4(2):93–106, 2021. [4] I. Chajda. Sheffer operation in ortholattices. Acta Universitatis Palackianae Olomu- censis. Facultas Rerum Naturalium. Mathematica, 44(1):19–23, 2005. [5] N. Chunsee, P. Julatha, and A. Iampan. Fuzzy set approach to ideal theory on Sheffer stroke BE-algebras. Journal of Mathematics and Computer Science, 34(3):283–294, 2024. [6] L. A. Zadeh. Fuzzy sets. Information and Control, 8(3):338–353, 1965. [7] B. Ahmad and A. Kharal. On fuzzy soft sets. Advances in Fuzzy Systems, 2009:Article ID 586507, 6 pages, 2009. [8] M. Atef, M. I. Ali, and T. M. Al-shami. Fuzzy soft covering-based multi-granulation fuzzy rough sets and their applications. Computational and Applied Mathematics, 40:Article number 115, 2021. [9] N. Cǎgman, S. Enginǒglu, and F. Citak. Fuzzy soft set theory and its application. Iranian Journal of Fuzzy Systems, 8(3):137–147, 2011. [10] K. T. Atanassov. Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1):87–96, 1986. [11] H. Garg and S. Singh. A novel triangular interval type-2 intuitionistic fuzzy sets and their aggregation operators. Iranian Journal of Fuzzy Systems, 15(5):69–93, 2018. [12] H. Garg and K. Kumar. An advanced study on the similarity measures of intuitionistic fuzzy sets based on the set pair analysis theory and their application in decision making. Soft Computing, 22:4959–4970, 2018. [13] H. Garg and K. Kumar. Distance measures for connection number sets based on set pair analysis and its applications to decision-making process. Applied Intelligence, 48:3346–3359, 2018. [14] L. Henkin. An algebraic characterization of quantifiers. Fundamenta Mathematicae, 37(1):63–74, 1950. [15] A. Diego. Sur les algèbres de Hilbert, volume 21 of Collection de Logique Mathema- tique, Serie A. Gauthier-Villars, Paris, 1966. [16] D. Busneag. A note on deductive systems of a Hilbert algebra. Kobe Journal of Mathematics, 2:29–35, 1985. [17] D. Busneag. Hilbert algebras of fractions and maximal Hilbert algebras of quotients. Kobe Journal of Mathematics, 5:161–172, 1988. [18] Y. B. Jun. Deductive systems of Hilbert algebras. Mathematica Japonica, 43:51–54, 1996. [19] W. A. Dudek. On fuzzification in Hilbert algebras. In Proceedings of the Olomouc Workshop ’98 and Summer School ’98, volume 11, pages 77–83. Contributions to General Algebra, 1999. [20] T. Oner, T. Katican, and A. Borumand Saeid. Relation between Sheffer stroke and Hilbert algebras. Categories and General Algebraic Structures with Applications, 14(1):245–268, 2021. [21] T. Katican and H. Bordbar. Sheffer stroke Hilbert algebras stabilizing by ideals. Axioms, 13(2):97, 2024. [22] T. Oner, T. Katican, and A. Borumand Saeid. Fuzzy filters of Sheffer stroke Hilbert A. Iampan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7074 14 of 14 algebras. Journal of Intelligent and Fuzzy Systems, 40(1):759–772, 2021. [23] T. Oner, T. Katican, and A. Borumand Saeid. Fuzzy ideals of Sheffer stroke Hilbert algebras. Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 93:85–94, 2023. [24] N. Rajesh, A. Iampan, T. Oner, and A. Borumand Saeid. Generalized fuzzy sub- algebras of Sheffer stroke Hilbert algebras. European Journal of Pure and Applied Mathematics, 18(3):6500, 2025. [25] N. Rajesh, T. Oner, A. Iampan, and A. Rezaei. Investigating length and mean-fuzzy subalgebras in Sheffer stroke Hilbert algebras. European Journal of Pure and Applied Mathematics, 18(2):5914, 2025. [26] N. Rajesh, T. Oner, A. Iampan, and I. Senturk. On length and mean fuzzy ideals of Sheffer stroke Hilbert algebras. European Journal of Pure and Applied Mathematics, 18(1):5779, 2025. [27] T. Oner, N. Rajesh, A. Iampan, and A. Borumand Saeid. Soft subalgebras and ideals of Sheffer stroke Hilbert algebras based on N -structures. European Journal of Pure and Applied Mathematics, 18(2):6018, 2025. [28] H. M. Sheffer. A set of five independent postulates for Boolean algebras, with ap- plication to logical constants. Transactions of the American Mathematical Society, 14(4):481–488, 1913.