EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6018 ISSN 1307-5543 – ejpam.com Published by New York Business Global Soft Subalgebras and Ideals of Sheffer Stroke Hilbert Algebras based on N -Structures Tahsin Oner1, Neelamegarajan Rajesh2, Aiyared Iampan3,∗, Arsham Borumand Saeid4 1 Department of Mathematics, Faculty of Science, Ege University, 35100 Izmir, Turkey 2 Department of Mathematics, Rajah Serfoji Government College, Thanjavur-613005, Tamil Nadu, India 3 Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand 4 Department of Pure Mathematics, Faculty of Mathematics and Computer, Shadid Bahonar University of Kerman, Kerman, Iran Abstract. In this study, we introduce the concepts ofN -ideals of types (∈,∈) and (∈,∈ ∨q), along with soft N∈-sets, soft Nq-sets, and soft N∈∨q-sets. Additionally, we define soft N -subalgebras and N -ideals within the context of Sheffer stroke Hilbert algebras and explore various properties of these structures. The paper also provides characterizations of N -subalgebras of types (∈,∈) and (∈,∈ ∨q), as well as N -ideals for both of these types. Furthermore, we further examine their corresponding soft versions, extending the algebraic framework in the setting of Sheffer stroke Hilbert algebras. 2020 Mathematics Subject Classifications: 06F35, 03G25, 03E72. Key Words and Phrases: Sheffer stroke Hilbert algebra, N -subalgebra of types (∈,∈) and (∈,∈ ∨q), N -ideal of types (∈,∈) and (∈,∈ ∨q). 1. Introduction The Sheffer operation, also known as the Sheffer stroke or NAND operator, was first introduced by Henry Maurice Sheffer [1]. Its significance arises from its ability to function as a fundamental logical operator capable of constructing an entire logical system on its own. This feature allows any axiom within a logical framework to be reformulated solely using the Sheffer operation, simplifying the manipulation and control of the system’s intrinsic properties. Moreover, the axioms of Boolean algebra, which serve as the algebraic foundation for classical propositional logic, can also be completely expressed through the ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6018 Email addresses: tahsin.oner@ege.edu.tr (T. Oner), nrajesh topology@yahoo.co.in (N. Rajesh), aiyared.ia@up.ac.th (A. Iampan), arsham@uk.ac.ir (A. Borumand 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 (2) (2025), 6018 2 of 11 Sheffer operation. This highlights the Sheffer operation’s pivotal role and versatility in both logical and algebraic settings. The Sheffer stroke has been utilized in various algebraic structures, including Boolean algebra, MV-algebra, BL-algebra, BCK-algebra, BE-algebra, ortholattices, and Hilbert algebra, among others (refer to [2–8]). Sheffer stroke Hilbert algebras are crucial structures in algebraic logic, providing an essential framework for modeling logical systems. Defined through the Sheffer stroke, these algebras offer a means to represent negation and other key logical operations. As such, they are indispensable in the study of algebraic logic, particularly in areas like Boolean algebra and lattice theory. Recent studies have further demonstrated their versatility—for instance, Rajesh et al. [9] investigated the notions of length and mean fuzzy ideals within this algebraic framework, revealing new insights into their structural depth and fuzzy generalizations. This line of research continues to highlight the richness of Sheffer stroke Hilbert algebras in both crisp and fuzzy logical environments. In 1999, Molodtsov [10] introduced the soft set theory, offering a novel mathematical tool designed to address uncertainties, free from the limitations of traditional theoretical models. Soft sets have recently emerged as a promising approach to modeling imprecision and flexibility in mathematical frameworks. They enable the handling of uncertainty within algebraic structures, allowing for a more adaptable study of these systems under condi- tions that are not precisely defined. Building on soft set theory, soft subalgebras extend these concepts into classical algebraic systems, providing a broader understanding of their properties and behaviors. This paper explores new ideas within Sheffer stroke Hilbert algebras, such as soft N - sets, soft N -subalgebras, and soft N -ideals, with a focus on types (∈,∈) and (∈,∈ ∨q). These concepts are analyzed through the lens of N -structures, a generalized framework that investigates algebraic systems in more flexible conditions. The study examines the features of N -subalgebras and N -ideals, along with their soft variants, to provide a deeper understanding of their behavior in various contexts. The main objective of this research is to study soft N -subalgebras and N -ideals within Sheffer stroke Hilbert algebras. Additionally, the paper aims to investigate the interconnec- tions between soft N -sets and N -structures, contributing to the broader field of algebraic logic. Through this analysis, we seek to further our comprehension of the role that Sheffer stroke Hilbert algebras play in both algebraic and logical systems. 2. Preliminaries In this section, we revisit essential concepts and results concerning Sheffer stroke Hilbert algebras, which will be referenced in the subsequent sections. Definition 1. [1] Let A := (A, |) be a groupoid. Then the operation | is said to be a Sheffer stroke or a Sheffer operation if it satisfies: (s1) (∀ξ, ζ ∈ A) (ξ | ζ = ζ | ξ), T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6018 3 of 11 (s2) (∀ξ, ζ ∈ A) ((ξ | ξ) | (ξ | ζ) = ξ), (s3) (∀ξ, ζ, τ ∈ A) (ξ | ((ζ | τ) | (ζ | τ)) = ((ξ | ζ) | (ξ | ζ)) | τ), (s4) (∀ξ, ζ, τ ∈ A) ((ξ | ((ξ | ξ) | (ζ | ζ))) | (ξ | ((ξ | ξ) | (ζ | ζ))) = ξ). To improve the clarity of this manuscript, we introduce the following notation, which will be used consistently throughout the text: ξ | (ζ | ζ) := ξζ . for all elements ξ, ζ ∈ A. Definition 2. [11] A Sheffer stroke Hilbert algebra (SSH-algebra) is a structure HS := (H, |, 0) of type (2, 0), in which H is a nonempty set, | is a Sheffer stroke on H, and 0 is the fixed element in H satisfying specific conditions: (sH1) (∀ξ, ζ, τ ∈ H) ((ξ | (ζτ | ζτ )) | ((ξζ | (ξτ | ξτ )) | (ξζ | (ξτ | ξτ ))) = ξξ), (sH2) (∀ξ, ζ ∈ H) (ξζ = ζξ = ξξ ⇒ ξ = ζ). Proposition 1. [11] Let HS := (H, |, 0) be an SSH-algebra. Then the binary relation (∀ξ, ζ ∈ H) (ξ ≤ ζ ⇔ ξζ = 0) is a partial order on H. Definition 3. [11] Let HS := (H, |, 0) be an SSH-algebra. A nonempty subset S of H is said to be a subalgebra of H if ξζ | ξζ ∈ S for all ξ, ζ ∈ S. Definition 4. [11] Let HS := (H, |, 0) be an SSH-algebra. A nonempty subset I of H is called an ideal of H if (1) 0 ∈ I, (2) (∀ξ, ζ ∈ H) (ξζ | ξζ ∈ I and ζ ∈ I ⇒ ξ ∈ I). 3. Soft subalgebras/ideals of Sheffer stroke Hilbert algebras Let HS := (H, |, 0) represent an SSH-algebra. For a subset Υ of [−1, 0], the pair (S,Υ) is referred to as a soft N -set over H, where S is a mapping from Υ to P (H), i.e., S : Υ → P (H). Given an N -structure (H, f) and Υ = [−1, 0], we introduce two mappings: S∈ : Υ → P (H); η 7→ {ξ ∈ H : (H, ξη) is an N∈-subset of (H, f)} and Sq : ∆ → P (H); η 7→ {ξ ∈ H : (H, ξη) is an Nq-subset of (H, f)}. Then, the pairs (S∈,Υ) and (Sq,Υ) are soft N -sets over H. If S∈(η) ̸= ∅ (or Sq(η) ̸= ∅) for some η ∈ Υ, we say that (S∈,Υ) (or (Sq,Υ)) forms a soft N∈-set (or soft N q-set) over H. A soft N∈ ∨q-set over H is defined as the union of a soft N∈-set and a soft N q-set over H, and is denoted by (S∈∨q,Υ), where S∈∨q(η) = S∈(η) ∨ Sq(η) for all η ∈ Υ. T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6018 4 of 11 Definition 5. Let HS := (H, |, 0) be an SSH-algebra. A soft N -set (S,Υ) over H is called a soft N -subalgebra over H if (∀η ∈ Υ) (S(η) ̸= ∅ ⇒ S(η) is a subalgebra of H). (1) Theorem 1. Let HS := (H, |, 0) be an SSH-algebra. Given an N -structure (H, f) and Υ = [−1, 0), the soft N∈-set (S∈,Υ) is a soft N -subalgebra over H if and only if (H, f) is an N -subalgebra of type (∈,∈). Proof. Assume that (S∈,Υ) is a soft N -subalgebra over H. If (H, f) is not an N - subalgebra of type (∈,∈), then there exist ϱ,ϖ ∈ H and t ∈ Υ such that (H, ϱt) and (H,ϖt) are N∈-subsets of (H, f), but (H, (ϱϖ | ϱϖ)t) is not an N∈-subset of (H, f). Therefore, ϱ,ϖ ∈ S∈(t) and ϱϖ | ϱϖ /∈ S∈(t), implying that S∈(t) is not a subalgebra of H. This leads to a contradiction, which means that (H, f) must be an N -subalgebra of type (∈,∈). Conversely, suppose (H, f) is an N -subalgebra of type (∈,∈). Let η ∈ Υ be such that S∈(η) ̸= ∅. If ξ, ζ ∈ S∈(η), then both (H, ξη) and (H, ζη) are N∈-subsets of (H, f). Consequently, (H, (ξζ | ξζ)η) = (H, (ξζ | ξζ))∨{η,η}) is an N∈-subset of (H, f). This implies that (ξζ | ξζ) ∈ S∈(η), showing that S∈(η) is a subalgebra of H for all η ∈ Υ. Hence, (S∈,Υ) is a soft N -subalgebra over H. Lemma 1. Let HS := (H, |, 0) be an SSH-algebra. An N -structure (H, f) is an N - subalgebra of type (∈,∈) if and only if (∀ξ, ζ ∈ H) (f(ξζ | ξζ) ≤ ∨ {f(ξ), f(ζ)}). (2) Proof. Suppose that (H, f) is an N -subalgebra of type (∈,∈). Then for all ξ, ζ ∈ H and for all η ∈ Υ, if f(ξ) ≤ η and f(ζ) ≤ η, then f(ξζ | ξζ) ≤ η. This implies that f(ξζ | ξζ) ≤ ∨ {f(ξ), f(ζ)}. Conversely, suppose that the inequality holds for all ξ, ζ ∈ H. Let η ∈ Υ, and suppose that f(ξ) ≤ η and f(ζ) ≤ η. Then,∨ {f(ξ), f(ζ)} ≤ η ⇒ f(ξζ | ξζ) ≤ η. Thus, (H, f) is closed under the Sheffer operation for type (∈,∈), and hence is an N - subalgebra of type (∈,∈). Theorem 2. Let HS := (H, |, 0) be an SSH-algebra. Given an N -structure (H, f) and Υ = [−1, 0), the soft Nq-set (Sq,Υ) is a soft N -subalgebra over H if and only if (H, f) is an N -subalgebra of type (∈,∈). Proof. Assume that (H, f) is an N -subalgebra of type (∈,∈) and let η ∈ Υ be such that Sq(η) ̸= ∅. If ξ, ζ ∈ Sq(η), then (H, ξη) and (H, ζη) are Nq-subsets of (H, f), and so f(ξ) + η + 1 < 0 and f(ζ) + η + 1 < 0. It follows from (2) that f(ξζ | ξζ) + η + 1 ≤ ∨ {f(ξ), f(ζ)}+ η + 1 < 0, T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6018 5 of 11 and so (H, (ξζ | ξζ)η) = (H, (ξζ | ξζ)∨{η,η}) is an Nq-subset of (H, f). Hence, ξζ | ξζ ∈ Sq(η), and thus Sq(η) is a subalgebra of H for all η ∈ Υ with Sq(η) ̸= ∅. Therefore, (Sq,Υ) is a soft N -subalgebra over H. Conversely, suppose that the soft Nq-set (Sq,Υ) is a soft N -subalgebra over H, and assume that f(ϱϖ | ϱϖ) > ∨ {f(ϱ), f(ϖ)} for some ϱ,ϖ ∈ H. Then there exists t ∈ Υ such that f(ϱϖ | ϱϖ) + t+ 1 < 0 and ∨ {f(ϱ), f(ϖ)}+ t+ 1 < 0. It follows that (H, ϱt) and (H,ϖt) are Nq-subsets of (H, f) but (H, (ϱϖ | ϱϖ)t) is not an Nq-subset of (H, f). This is a contradiction, and hence (∀ξ, ζ ∈ H) (f(ξζ | ξζ) ≤ ∨ {f(ξ), f(ζ)}). Therefore, (H, f) is an N -subalgebra of type (∈,∈) by Lemma 1. Theorem 3. Let HS := (H, |, 0) be an SSH-algebra. Given an N -structure (H, f) and the soft N∈-set (S∈,Υ) with Υ = [−1,−0.5), the following are equivalent: (1) (S∈,Υ) is a soft N -subalgebra over H. (2) (∀ξ, ζ ∈ H) ( ∧ {f(ϱϖ | ϱϖ),−0.5} ≤ ∨ {f(ξ), f(ζ)}). Proof. Assume that the soft N∈-set (S∈,Υ) is a soft N -subalgebra over H. Then S∈(η) is a subalgebra of H for all η ∈ Υ with S∈(η) ̸= ∅. If there exist ϱ,ϖ ∈ H such that∧ {f(ϱϖ | ϱϖ),−0.5} > t = ∨ {f(ϱ), f(ϖ)}, then t ∈ Υ and (H, ϱt) and (H,ϖt) are N∈-subsets of (H, f), that is, ϱ,ϖ ∈ S∈(t), but (H, (ϱϖ | ϱϖ)t) is not an N∈-subset of (H, f), that is, ϱϖ | ϱϖ /∈ S∈(t), a contradiction. Thus, ∧ {f(ξζ | ξζ),−0.5} ≤ ∨ {f(ξ), f(ζ)} for all ξ, ζ ∈ h. Conversely, suppose that (2) is valid. Let ξ, ζ ∈ S∈(η) for every η ∈ Υ. Then (H, ξη) and (H, ζη) are N∈-subsets of (H, f), and so∧ {f(ξζ | ξζ),−0.5} ≤ ∨ {f(ξ), f(ζ)} ≤ η < −0.5. It follows that (H, (ξζ | ξζ)η) is an N∈-subset of (H, f), that is, ξζ | ξζ ∈ S∈(η). Thus, S∈(η) is a subalgebra of H, and therefore, (S∈,Υ) is a soft N -subalgebra over H. Lemma 2. Let HS := (H, |, 0) be an SSH-algebra. An N -structure (H, f) is an N - subalgebra of type (∈,∈ ∨q) if and only if (∀ξ, ζ ∈ H) (f(ξζ | ξζ) ≤ ∨ {f(ξ), f(ζ),−0.5}). (3) T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6018 6 of 11 Proof. Suppose that (H, f) is an N -subalgebra of type (∈,∈ ∨q). Let ξ, ζ ∈ H. Then, if both f(ξ) ≤ η and f(ζ) ≤ η for some η ∈ Υ, it must follow that f(ξζ | ξζ) ≤ ∨ {f(ξ), f(ζ),−0.5}. Conversely, suppose the above inequality holds for all ξ, ζ ∈ H. Let η ∈ Υ, and assume that f(ξ) ≤ η and f(ζ) ≤ η. Then,∨ {f(ξ), f(ζ),−0.5} ≤ η ⇒ f(ξζ | ξζ) ≤ η. This confirms that (H, f) is an N -subalgebra of type (∈,∈ ∨q). Theorem 4. Let HS := (H, |, 0) be an SSH-algebra. Given an N -structure (H, f) and a soft N∈-set (S∈,Υ), the following assertions are equivalent: (1) (H, f) is an N -subalgebra of type (∈,∈ ∨q). (2) (S∈,Υ) is a soft N -subalgebra over H for Υ = [−0.5, 0). Proof. Assume that (H, f) is an N -subalgebra of type (∈,∈ ∨q). Let ξ, ζ ∈ H and η ∈ Υ be such that ξ, ζ ∈ S∈(η). Then (H, ξη) and (H, ζη) are N∈-subsets of (H, f). It follows from (3) that (∀ξ, ζ ∈ H) (f(ξζ | ξζ) ≤ ∨ {f(ξ), f(ζ),−0.5} ≤ ∨ {η,−0.5} = η). Then (H, (ξζ | ξζ)η) is an N∈-subset of (H, f). Thus ξζ | ξζ ∈ S∈(η), and so (S∈,Υ) is a soft N -subalgebra over H. Conversely, suppose that the soft N∈-set (S∈,Υ) with Υ = [−0.5, 0) is a soft N - subalgebra over H. Assume that (3) is not valid. Then f(ϱϖ | ϱϖ) > t ≥ ∨ {f(ϱ), f(ϖ),−0.5} for some t ∈ Υ and ϱ,ϖ ∈ H. It follows that (H, ϱt) and (H,ϖt) are N∈-subsets of (H, f), and so ϱ,ϖ ∈ S∈(t). But f(ϱ ϖ | ϱϖ) > t induces that (H, (ϱϖ | ϱϖ)t) is not an N∈-subset of (H, f). This is a contradiction, and thus (∀ξ, ζ ∈ H) (f(ξζ | ξζ) ≤ ∨ {f(ξ), f(ζ),−0.5}). Using Lemma 2, we know that (H, f) is an N -subalgebra of type (∈,∈ ∨q). Theorem 5. Let HS := (H, |, 0) be an SSH-algebra. Let (S∈,Υ) be a soft N∈-set over H. If Υ = [−0.5, 0), then for any subalgebra L of H, there exists an N -subalgebra (H, f) of type (∈,∈ ∨q) such that S∈(η) = L for all η ∈ Υ. T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6018 7 of 11 Proof. Take an N -subalgebra (H, f) in which f is given as follows: f : H → [−1, 0]; x 7→ { η ∈ Υ if ξ ∈ L 0 otherwise Obviously, S∈(η) = L for all η ∈ Υ. Assume that f(ϱϖ | ϱϖ) > ∨ {f(ϱ), f(ϖ),−0.5} for some ϱ,ϖ ∈ H. Then f(ϱϖ | ϱϖ) = 0 and ∨ {f(ϱ), f(ϖ),−0.5} = η, since |Im(f)| = 2. It follows that f(ϱ) = η = f(ϖ), so that ϱ,ϖ ∈ L. But ϱϖ | ϱϖ /∈ L, since f(ϱϖ | ϱϖ) = 0. This is impossible, and so (∀ξ, ζ ∈ H) (f(ξζ | ξζ) ≤ ∨ {f(ξ), f(ζ),−0.5}). Therefore, (H, f) is an N -subalgebra of type (∈,∈ ∨q) by Lemma 2. Definition 6. Let HS := (H, |, 0) be an SSH-algebra. An N -structure (H, f) is called an N -ideal of type (∈,∈) (resp., type (∈,∈q)) if the following assertions are valid: (1) If a point N -structure (H, ξη) is an N∈-subset of (H, f), then (H, 0η) is an N∈-subset (resp., N∈∨q-subset) of (H, f). (2) If two point N -structures (H, (ξζ | ξζ)η) and (H, ζδ) are N∈-subsets of (H, f), then the point N -structure (H, ξη∨δ) is an N∈-subset (resp., N∈∨q-subset) of (H, f). Lemma 3. Let HS := (H, |, 0) be an SSH-algebra. Let (H, f) be an N -structure. Then f(0) ≤ f(ξ) for all ξ ∈ H if and only if (H, 0η) is an N∈-subset of (H, f) whenever (H, ξη) is an N∈-subset of (H, f) for all ξ ∈ H and η ∈ Υ = [−1, 0). Proof. Assume that f(0) ≤ f(ξ) for all ξ ∈ H and let η ∈ Υ be such that (H, ξη) is an N∈-subset of (H, f). Then f(0) ≤ f(ξ) ≤ η, and so (H, 0η) is an N∈-subset of (H, f). Conversely, suppose that (H, 0η) is an N∈-subset of (H, f) whenever (H, ξη) is an N∈- subset of (H, f) for all ξ ∈ H and η ∈ Υ = [−1, 0). If we take η = f(ξ) for any ξ ∈ H, then (H, ξη) is an N∈-subset of (H, f), and thus (H, 0η) is an N∈-subset of (H, f). Hence, f(0) ≤ η = f(ξ) for all ξ ∈ H. Lemma 4. Let HS := (H, |, 0) be an SSH-algebra. Given an N -structure (H, f), the following are equivalent: (1) f(ξ) ≤ ∨ {f(ξζ | ξζ), f(ζ)} for all ξ, ζ ∈ H. (2) For any ξ, ζ ∈ H and η, δ ∈ Υ = [−1, 0), if (H, (ξζ | ξζ)η) and (H, ζδ) are N∈-subsets of (H, f), then (H, ξη∨δ) is an N∈-subset of (H, f). T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6018 8 of 11 Proof. Assume that f(ξ) ≤ ∨ {f(ξζ | ξζ), f(ζ)} for all ξ, ζ ∈ H. Let ξ, ζ ∈ H and η, δ ∈ Υ be such that (H, (ξζ | ξζ)η) and (H, ζδ) are N∈-subsets of (H, f). Then f(ξζ | ξζ) ≤ η and f(ζ) ≤ δ, which imply that f(ξ) ≤ ∨ {f(ξζ | ξζ), f(ζ)} ≤ η ∨ δ. Hence, (H, ξη∨δ) is an N∈-subset of (H, f). Conversely, suppose (2) is valid. If we take η = f(ξζ | ξζ) and δ = f(ζ), then (H, (ξζ | ξζ)η) and (H, ζδ) are N∈-subsets of (H, f). It follows that (H, ξη∨δ) is an N∈-subset of (H, f) and thus, we have f(ξ) ≤ η ∨ δ = ∨ {f(ξζ | ξζ), f(ζ)}. Combining Lemmas 3 and 4, we have the following theorem. Theorem 6. Let HS := (H, |, 0) be an SSH-algebra. An N -structure (H, f) is an N -ideal of type (∈,∈) if and only if (∀ξ, ζ ∈ H) (f(0) ≤ f(ξ) ≤ ∨ {f(ξζ | ξζ), f(ζ)}). (4) Definition 7. Let HS := (H, |, 0) be an SSH-algebra. A soft N -set (S,Υ) over H is called a soft N -ideal over H if (∀η ∈ Υ) (S(η) ̸= ∅ ⇒ S(η) is an ideal of H). (5) Theorem 7. Let HS := (H, |, 0) be an SSH-algebra. Given an N -structure (H, f) and the soft N∈-set (S∈,Υ), the following are equivalent: (1) (S∈,Υ) is a soft N -ideal over H for Υ = [−1, 0). (2) (H, f) is an N -ideal of type (∈,∈). Proof. Assume that (S∈,Υ) is a soft N -ideal over H for Υ = [−1, 0). If there exists ϱ ∈ H such that f(0) > f(ϱ), then we can take η ∈ Υ such that f(0) > η ≥ f(ϱ). Thus, (H, 0η) is not an N∈-subset of (H, f), and so 0 /∈ S∈(η). This is a contradiction, and so f(0) ≤ f(ξ) for all ξ ∈ H. Suppose that there exist ϱ,ϖ ∈ H such that f(ϱ) > ∨ {f(ϱϖ | ϱϖ), f(ϖ)}. Taking η = ∨ {f(ϱϖ | ϱϖ), f(ϖ)} implies that (H, (ϱϖ | ϱϖ)η) and (H,ϖη) are N∈-subsets of (H, f). That is, ϱϖ | ϱϖ ∈ S∈(η) and ϖ ∈ S∈(η). Since S∈(η) is an ideal of H, ϱ ∈ S∈(η). Hence, (H, aη) is an N∈-subset of (H, f), and so f(ϱ) ≤ η. This is a contradiction, and therefore f(ξ) ≤ ∨ {f(ξζ | ξζ), f(ζ)} for all ξ, ζ ∈ H. Hence, (H, f) is an N -ideal of type (∈,∈). Conversely, assume that (H, f) is an N -ideal of type (∈,∈) and let η ∈ Υ be such that S∈(η) ̸= ∅. Then there exists ξ ∈ S∈(η), and so (H, ξη) is an N∈-subset of (H, f). It follows that f(0) ≤ f(ξ) ≤ η and so that (H, 0η) is an N∈-subset of (H, f), that is, 0 ∈ S∈(η). Let ξζ | ξζ ∈ S∈(η) and ζ ∈ S∈(η). Then (H, (ξζ | ξζ)η) and (H, ζη) are N∈-subsets of (H, f). Thus, f(ξζ | ξζ) ≤ η and f(ζ) ≤ η. It follows from (4) that f(ξ) ≤ ∨ {f(ξζ | ξζ), f(y)} ≤ η. Hence, (H, ξη) = (H, ξη∨η) is an N∈-subset of (H, f), and so ξ ∈ S∈(η). Thus, S∈(η) is an ideal of H for all η ∈ Υ, and therefore (S∈,Υ) is a soft N -ideal over H for Υ = [−1, 0). T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6018 9 of 11 Theorem 8. Let HS := (H, |, 0) be an SSH-algebra. The soft Nq-set (Sq,Υ) is a soft N -ideal over H for Υ = [−1, 0) if and only if the N -structure (H, f) is an N -ideal of type (∈,∈). Proof. Assume that (H, f) is an N -ideal of type (∈,∈), and let η ∈ Υ be such that Sq(η) ̸= ∅. This implies there exists ξ ∈ Sq(η), so (H, ξη) is an Nq-subset of (H, f). If 0 /∈ Sq(η), then (H, 0q) is not an Nq-subset of (H, f), which implies f(0)+ η+1 < 0. Using (4), we obtain f(ξ) + η + 1 ≥ f(0) + η + 1 ≥ 0, leading to a contradiction, thus 0 ∈ Sq(η). Next, consider ξζ | ξζ ∈ Sq(η) and ζ ∈ Sq(η). Then (H, (ξζ | ξζ)η) and (H, ζη) are Nq-subsets of (H, f). If (H, ξη) is not an Nq-subset of (H, f), then f(ξ) + η + 1 ≥ 0. From (4), we have ∨ {f(ξζ | ξζ), f(ζ)}+η+1 ≥ f(ξ)+η+1 ≥ 0. Hence, either f(ξζ | ξζ)+η+1 ≥ 0 or f(ζ) + η + 1 ≥ 0, leading to a contradiction. Thus, ξ ∈ Sq(η). Therefore, (Sq,Υ) is a soft N -ideal over H for Υ = [−1, 0). Conversely, suppose that the softNq-set (Sq,Υ) is a softN -ideal overH for Υ = [−1, 0). If f(0) > f(ϱ) for some ϱ ∈ H, then there exists δ ∈ Υ such that f(0) + δ + 1 ≥ 0 and f(ϱ) + δ + 1 ≥ 0. This implies that (H, ϱδ) is an Nq-subset of (H, f), and (H, 0) is an Nq-subset of (H, f), leading to a contradiction. Thus, f(0) ≤ f(ξ) for all ξ ∈ H. Suppose there exist ϱ,ϖ ∈ H such that f(ϱ) > ∨ {f(ϱϖ | ϱϖ), f(ϖ)}. Then for some δ ∈ Υ, we have f(ϱ)+δ+1 ≥ 0 and ∨ {f(ϱϖ | ϱϖ), f(ϖ)}+δ+1 ≥ 0. This implies that f(ϱϖ | ϱϖ)+δ+1 < 0 and f(ϖ) + δ + 1 < 0, meaning that (H, (ϱϖ | ϱϖ)δ) and (H,ϖδ) are Nq-subsets of (H, f). Therefore, ϱϖ | ϱϖ ∈ Sq(δ) and ϖ ∈ Sq(δ). Since Sq(δ) is an ideal of H, we conclude that ϱ ∈ Sq(δ), and thus (H, ϱδ) is an Nq-subset of (H, f). This is a contradiction, so f(ξ) ≤ ∨ {f(ξζ | ξζ), f(ζ)} for all ξ, ζ ∈ H. Thus, (H, f) is an N -ideal of type (∈,∈). Theorem 9. Let HS := (H, |, 0) be an SSH-algebra. Given an N -structure (H, f) and the soft N∈-set (S∈,Υ), the following are equivalent: (1) (H, f) is an N -ideal of type (∈,∈ ∨q). (2) (S∈,Υ) is a soft N -ideal over H for Υ = [−0.5, 0). Proof. Assume that (H, f) is an N -ideal of type (∈,∈ ∨q). We first show that (∀ξ ∈ H) (f(0) ≤ ∨ {f(ξ),−0.5}). Suppose that f(0) > f(ξ) > −0.5. Then, there exists η ∈ (−0.5, 0) such that f(0) > η ≥ f(ξ), which implies that (H, ξη) is an N∈-subset of (H, f). However, (H, 0η) is not an N∈-subset of (H, f), nor is it an Nq-subset of (H, f) because f(0) + η + 1 ≥ 0. This is a contradiction. Hence, f(0) ≤ f(ξ) for all ξ ∈ H. Now, if f(ξ) ≤ −0.5, then (H, ξ−0.5) is an N∈-subset of (H, f), and thus (H, 0−0.5) is an N∈∨q-subset of (H, f). Therefore, f(0) ≤ −0.5, as if f(0) > −0.5, we would have f(0)− 0.5+1 > 0, which is a contradiction. Let η ∈ Υ = [−0.5, 0). Then for all ξ ∈ S∈(η), f(0) ≤ ∨ {f(ξ),−0.5}, and so f(0) ≤ ∨ {f(ξ),−0.5} ≤ ∨ {η,−0.5} = η. Thus, (H, 0η) is an N∈-subset of (H, f), implying that 0 ∈ S∈(η). Now, we show that (∀ξ, ζ ∈ H) (f(ξ) ≤ ∨ {f(ξζ | ξζ), f(ζ),−0.5}). T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6018 10 of 11 If ∨ {f(ξζ | ξζ), f(ζ)} > −0.5, then f(ξ) ≤ ∨ {f(ξζ | ξζ), f(ζ)}. Otherwise, there exists δ ∈ (−0.5, 0) such that f(ξ) > δ ≥ ∨ {f(ξζ | ξζ), f(ζ)}. Then (H, (ξζ | ξζ)δ) and (H, yδ) are N∈-subsets of (H, f), but (H, ξδ) is not an N∈-subset of (H, f), and neither is it an Nq-subset of (H, f), as f(ξ) + δ+1 > 2δ+1 > 0. This leads to a contradiction. Therefore, we must have f(ξ) ≤ ∨ {f(ξζ | ξζ), f(ζ),−0.5} for all ξ, ζ ∈ H. Let ξ, ζ ∈ H be such that ξζ | ξζ ∈ S∈(η) and ζ ∈ S∈(η). Then (H, (ξζ | ξζ)η) and (H, ζδ) are N∈-subsets of (H, f), and so f(ξζ | ξζ) ≤ η and f(ζ) ≤ η. Hence, f(ξ) ≤ ∨ {f(ξζ | ξζ), f(ζ),−0.5} ≤ ∨ {η,−0.5} = η, which implies that (H, ξδ) is anN∈-subset of (H, f), and therefore ξ ∈ S∈(η). Thus, (S∈,Υ) is a soft N -ideal over H for Υ = [−0.5, 0). Conversely, suppose that (2) is valid. If f(0) > ∨ {f(ϱ),−0.5} for some ϱ ∈ H, then there exists η ∈ Υ such that f(0) > η ≥ ∨ {f(ϱ),−0.5}. Then η ∈ Υ, and (H, ϱη) is an N∈-subset of (H, f). However, (H, 0η) is not an N∈-subset of (H, f), meaning 0 /∈ S∈(η). This is a contradiction, and so f(0) ≤ ∨ {f(ξ),−0.5} for all ξ ∈ H. Let ξ ∈ H and η ∈ Υ be such that (H, ξδ) is an N∈-subset of (H, f). Then f(ξ) ≤ η. Suppose that (H, 0η) is not an N∈-subset of (H, f). Then f(0) > η. If f(ξ) > −0.5, then f(0) ≤ ∨ {f(ξ),−0.5} = f(ξ) ≤ η, which is impossible. Thus, f(ξ) ≤ −0.5, and f(0) + η + 1 < 2f(0) + 1 ≤ 2 ∨ {f(ξ),−0.5}+ 1 = 0, implying that (H, 0η) is an Nq-subset of (H, f). Therefore, (H, 0η) is an N∈∨q-subset of (H, f). Assume that there exist ϱ,ϖ ∈ H such that f(ϱ) > ∨ {f(ϱϖ | ϱϖ), f(ϖ),−0.5}. Let η = ∨ {f(ϱϖ | ϱϖ), f(ϖ),−0.5}. Then η ∈ Υ, and (H, (ϱϖ | ϱϖ)η) and (H,ϖη) are N∈-subsets of (H, f). But (H, ϱη) is not an N∈-subset of (H, f), leading to a contradiction. Therefore, (∀ξ, ζ ∈ H) (f(ξ) ≤ ∨ {f(ξζ | ξζ), f(ζ),−0.5}). Let ξ, ζ ∈ H and η, δ ∈ Υ be such that (H, (ξζ | ξζ)η) and (H, ζδ) are N∈-subsets of (H, f), and suppose that (H, ξη∨δ) is not an N∈-subset of (H, f). Then, f(ξζ | ξζ) ≤ η, f(ζ) ≤ η, and f(ξ) > η ∨ δ. If ∨ {f(ξζ | ξζ), f(ζ)} > −0.5, then f(ξ) ≤ ∨ {f(ξζ | ξζ), f(ζ),−0.5} = ∨ {f(ξζ | ξζ), f(ζ)} ≤ η ∨ δ, which is a contradiction. Thus,∨ {f(ξζ | ξζ), f(ζ)} ≤ −0.5, and so f(ξ) + (η ∨ δ) + 1 < 2f(ξ) + 1 ≤ 2 ∨ {f(ξζ | ξζ), f(ζ),−0.5}+ 1 = 0, which shows that (H, ξη∨δ) is anNq-subset of (H, f). Therefore, (H, ξη∨δ) is anN∈∨q-subset of (H, f). Consequently, (H, f) is an N -ideal of type (∈,∈ ∨q). T. Oner et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6018 11 of 11 4. Conclusion In this study, we have introduced and explored the notions of soft N -subalgebras and soft N -ideals in the context of Sheffer stroke Hilbert algebras, incorporating the framework of N -structures. By defining and characterizing these algebraic constructs, we have established fundamental properties and the interrelationships between different types of soft N -subalgebras and soft N -ideals, particularly under the types (∈,∈) and (∈,∈ ∨q). Our results provide a deeper understanding of the algebraic behavior of Sheffer stroke Hilbert algebras under soft set theory, highlighting their flexibility in modeling uncertainty within logical systems. These findings not only contribute to the ongoing development of algebraic logic but also open pathways for further research in generalizing soft algebraic structures in broader mathematical settings. Future work may focus on extending these concepts to other non-classical algebraic frameworks and exploring their potential applications in computational logic and fuzzy systems. 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 appli- cation to logical constants. Trans. Am. Math. Soc., 14(4):481–488, 1913. [2] I. Chajad. Sheffer operation in ortholattices. Acta Univ. Palacki. 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