Octonion valued b -metric spaces and ideal convergence

This paper explores the extension of fundamental concepts such as convergence to \(b\)-metric spaces with octonion valued \(b\)-metric spaces constructed in [Q. B. Cai, S. Cetin, O. Kisi, M. Gurdal, preprint, (2025)]. Octonion-valued metric spaces are based on modifying the triangle inequality of a semi-metric space by multiplying one side of the inequality by a scalar \(b\). This new generalization of metric spaces is very interesting since octonions are not even a ring since they do not have the associative property of multiplication, and the spaces do not satisfy the standard triangle inequality. Through these concepts, statistical convergence and related concepts are generalized. The study emphasizes ideal convergence by defining \(\mathcal{I}\)-convergence and \(\mathcal{I}^{\ast}\)-convergence, and proving their equivalence via property (AP). \(\mathcal{I}\)-Cauchy and \(\mathcal{I}^{\ast}\)-Cauchy sequences are also introduced, along with adaptations of classical theorems.