A Theoretical Exploration of Paraletrix Calculus as an Extension of Rhotrix Mathematics - Isa Yahaya; Salisu Lukunti; Umar Mujahid Aliyu; Imafidor Hassan Ibrahim; Mohammed Abubakar Kolo; Sulaiman Ahmad; Nura Hashim; Mohammed Yusuf Marafa (African Multidisciplinary Journal of Sciences and Artificial Intelligence, 2026)
This paper, titled A Theoretical Exploration of Paraletrix Calculus as an Extension of Rhotrix Mathematics, builds upon earlier studies in generalized matrix theory by extending the structural and operational framework of non-standard matrix-like objects. Atanassov and Shannon [1] first introduced matrix-tertions and matrix-ngittrets as entities that interpolate between 2-dimensional vectors and 2×2 matrices, thereby enriching the conceptual landscape of generalized matrices. Ajibade [2] subsequently advanced the field by proposing thotrices as intermediates between 2×2 and 3×3 matrices, while further developments in rhotrix theory have established various multiplication techniques, such as heart-oriented and row–column multiplications—and yielded several important results. Recognizing the diversity of both rectangular and square matrices, the paraletrix structure was formulated as a generalization of the thotrix, allowing unequal numbers of rows and columns and thus providing a more flexible algebraic setting. This study extends the mathematical framework by introducing differentiation and integration within paraletrix calculus, defining these operations for paraletrix-valued functions with respect to an independent variable. In doing so, it lays the groundwork for a coherent calculus on paraletrices as a theoretical extension of rhotrix mathematics and generalized matrix theory.