math. sci. appl. e-notes Mathematical Sciences and Applications E-Notes 2147-6268 Murat TOSUN 10.36753/mathenot.1803361 Approximation Theory and Asymptotic Methods Yaklaşım Teorisi ve Asimptotik Yöntemler On Ideal Convergence in a Bi-Complex Valued Metric Space https://orcid.org/0009-0004-2329-829X Hossain Juwel TRIPURA UNIVERSITY https://orcid.org/0000-0003-2804-6564 Debnath Shyamal TRIPURA UNIVERSITY https://orcid.org/0009-0007-8767-8293 Paul Sanat Kumar DHALAI DISTRICT POLYTECHNIC AMBASSA DHALAI TRIPURA, 03 29 2026 14 1 57 66 10 14 2025 03 03 2026 Copyright © 2013, Mathematical Sciences and Applications E-Notes 2013 Mathematical Sciences and Applications E-Notes

In this paper, we investigate some fundamental aspects of bi-complex numbers. Propose two different types of partial order relations and analyze bi-complex valued metric spaces under these orders, highlighting their differences. Moreover, we introduce the concept of a hyperbolic-valued metric space, examine the density of the natural numbers, and explore ideal convergence and the ideal Cauchy property for sequences of bi-complex-valued metric spaces. Finally, we discuss several properties within a bi-complex-valued metric space.

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