Tahsin Oner

In this paper, we introduce and formalize the theory of Sheffer stroke hoop algebras, providing a minimal axiomatization for bounded hoop algebras utilizing a single binary operation. We systematically establish the fundamental algebraic properties of this novel structure, beginning with the logical independence of its core axioms. We equip the algebra with an induced partial order, proving it constitutes a well-defined ∧-semilattice, and demonstrate a bidirectional structural translation: bounded hoop algebras satisfying the Double Negation Property (DNP) can be equivalently expressed as Sheffer stroke hoop algebras, and vice versa. Furthermore, we investigate the internal algebraic architecture by introducing sub-algebras, filters, positive implicative filters, and ideals, rigorously establishing the inherent structural duality between filters and ideals. By proving that proper filters naturally induce full algebraic congruences, we successfully construct quotient Sheffer stroke hoop algebras. We characterize prime filters by demonstrating that a quotient algebra forms a chain if and only if its generating filter is prime. Finally, we complete the theoretical framework by establishing the Correspondence Theorem and the First Isomorphism Theorem, seamlessly embedding classical universal algebraic principles into the Sheffer stroke setting.

In recent years, the study of generalized fuzzy structures in algebraic systems has attracted considerable attention due to their ability to represent uncertainty and bipolar information. In this paper, we introduce the notion of cubic bipolar ideals in the framework of Sheffer stroke Hilbert algebras. This concept integrates the descriptive capability of cubic sets with the dual representation of bipolar information, providing a broader perspective for investigating algebraic structures associated with the Sheffer stroke operation. We establish the definition of cubic bipolar ideals and investigate several of their fundamental properties. In particular, the structural behavior of these ideals is examined within Sheffer stroke Hilbert algebras. Furthermore, the preservation of cubic bipolar ideals under algebraic homomorphisms is analyzed through the study of images and preimages. The Cartesian product of cubic bipolar ideals is also discussed, and conditions ensuring the stability of the resulting structures are obtained. The results presented here contribute to the development of fuzzy algebraic theory and extend existing approaches to Sheffer stroke-based algebraic systems.

This paper introduces the concept of interval-valued intuitionistic fuzzy n-fold implicative filters in hoop algebras and explores their fundamental properties. We establish several new and equivalent characterizations, investigate their closure properties, and provide explicit algorithms for their construction and verification. Furthermore, we examine the relationships between interval-valued intuitionistic fuzzy n-fold implicative filters, interval-valued intuitionistic fuzzy filters, and classical n-fold implicative filters. The results presented here extend beyond straightforward generalizations, offering both practical tools and theoretical insights that are not previously available in the literature. The results presented here build upon earlier studies by systematically characterizing interval-valued intuitionistic fuzzy n-fold implicative filters in hoop algebras. Several new equivalent conditions and computational methods are introduced, and relationships with existing filter concepts are clarified.

This study investigates linear Diophantine fuzzy structures within the framework of Sheffer stroke BCK-algebras (SBCK-algebras). We introduce and characterize linear Diophantine fuzzy SBCK-subalgebras and linear Diophantine fuzzy SBCK-ideals, establishing fundamental connections between these fuzzy structures and their corresponding crisp subalgebras and ideals. In particular, we prove that the level sets of linear Diophantine fuzzy SBCK-subalgebras form SBCK-subalgebras, and, conversely, every SBCK-subalgebra gives rise to such a fuzzy structure. Additionally, we show that every linear Diophantine fuzzy SBCK-ideal induces a linear Diophantine fuzzy SBCK-subalgebra; however, the converse does not necessarily hold. Several structural properties, homomorphic images, and intersections of such fuzzy ideals are also examined. These results demonstrate how linear Diophantine logic naturally integrates with Sheffer stroke BCK-algebras and enriches their algebraic behavior.

The theory of order‐based mappings plays a significant role in analyzing the structural and logical properties of nonclassical algebraic systems. This paper introduces and thoroughly investigates the concepts of inferior subalgebras and inferior ideals within the framework of Sheffer stroke Hilbert algebras. The fundamental properties of these mappings are examined, and characterization theorems are established by employing upper α ‐inferior sets. It is demonstrated that these upper sets naturally induce classical subalgebras and ideals under specific conditions, providing a structural link between crisp algebraic substructures and inferior mappings. Furthermore, closure properties under inferior intersections are discussed, and the operational behavior of these systems is analyzed. A key result of this study shows that every inferior ideal inherently determines an inferior subalgebra, delineating the exact relationship between these two structures. The findings presented herein not only extend the theory of inferior mappings to a broader class of logical algebras but also contribute to the ongoing classification of order‐theoretic mappings in nonclassical logic.

This paper explores the structure of interval-valued intuitionistic fuzzy (IVIF) subsets 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 subsets 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 modelfor 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.