Non-Commutative Neutrosophic MR-Metric Spaces: A Unified Framework for Quantum Structures and Fixed Point Theory

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DOI: https://doi.org/10.28924/2291-8639-24-2026-171
Published: May 29, 2026
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Abed Al-Rahman M. Malkawi, Ayat M. Rabaiah, Ala Amourah, Abdullah Alsoboh, Ahmed Al Kasbi, Tala Sasa, Non-Commutative Neutrosophic MR-Metric Spaces: A Unified Framework for Quantum Structures and Fixed Point Theory, Int. J. Anal. Appl., 24 (2026), 171.

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Abed Al-Rahman M. Malkawi, Ayat M. Rabaiah, Ala Amourah, Abdullah Alsoboh, Ahmed Al Kasbi, Tala Sasa

Abstract

This paper introduces a novel mathematical structure called Non-Commutative Neutrosophic MR-Metric Spaces (NC-NMR-S). This framework provides a unified approach combining concepts from non-commutative geometry, neutrosophic logic, and fixed point theory. Building upon the foundations of generalized metric spaces and neutrosophic fuzzy metrics, this work extends these ideas to the realm of operator algebras. The proposed framework employs a ternary commutator metric and defines quantum neutrosophic membership functions via the Gelfand-Naimark-Segal (GNS) construction. We establish the well-definedness of this structure and demonstrate its reduction to classical neutrosophic MR-metric spaces in the commutative limit. The theory is illustrated with concrete quantum examples, including canonical commutation relations, Pauli algebras, and quantum harmonic oscillators. Furthermore, we present significant applications in quantum entanglement detection, measurement incompatibility quantification, quantum error correction, and quantum machine learning. This work provides a comprehensive bridge between algebraic quantum structures and neutrosophic analysis, extending previous research in fixed point theory and fractional calculus.

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