Mathematical Foundations, Structural Embeddings, and Complement Operations in Hyperbolic Fuzzy Set Theory

Mathematical Foundations, Structural Embeddings, and Complement Operations in Hyperbolic Fuzzy Set Theory | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Mathematical Foundations, Structural Embeddings, and Complement Operations in Hyperbolic Fuzzy Set Theory Palash Dutta This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8602843/v1 This work is licensed under a CC BY 4.0 License Status: Under Revision Version 1 posted 14 You are reading this latest preprint version Abstract This paper presents a comprehensive study of Hyperbolic Fuzzy Sets (HyFS), a generalized fuzzy framework characterized by independent optimistic and pessimistic membership degrees constrained by a hyperbolic relation. We explore the embedding of q-Rung Orthopair Fuzzy Sets, (n,m)-Rung Orthopair Fuzzy Sets, and (n,m)-Power Root Fuzzy Sets into HyFS, thereby establishing its flexibility and generality. A comparative analysis is carried out based on computational complexity, runtime, and scalability, where HyFS demonstrates superior performance due to its simple structure. The study further investigates complement operations in HyFS, including classical and mixed-type complements using different fuzzy negation functions. Theoretical properties such as De Morgan’s laws, distributivity, and difference operations are examined. Our results highlight the robustness, efficiency, and logical consistency of HyFS, making it a promising tool for uncertainty modeling and decision-making applications. q-ROFSs (n m)-ROFSs (n m)-PRFSs HyFSs Fuzzy Complement Mixed- Type Complement Computational Complexity Scalability Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Revision Version 1 posted Editorial decision: Revision requested 07 May, 2026 Reviews received at journal 18 Mar, 2026 Reviews received at journal 11 Mar, 2026 Reviews received at journal 03 Mar, 2026 Reviewers agreed at journal 26 Feb, 2026 Reviews received at journal 23 Feb, 2026 Reviewers agreed at journal 19 Feb, 2026 Reviewers agreed at journal 19 Feb, 2026 Reviewers agreed at journal 19 Feb, 2026 Reviewers agreed at journal 18 Feb, 2026 Reviewers invited by journal 17 Feb, 2026 Editor assigned by journal 01 Feb, 2026 Submission checks completed at journal 16 Jan, 2026 First submitted to journal 14 Jan, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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