A Study of Tricomplex Numbers: Representation, Subalgebras, Idempotent Forms

Tricomplex numbers, as natural extensions of complex and bicomplex systems [cf. 1–10], provide a broad algebraic framework enriched with diverse subalgebras and distinctive computational features. This study focuses on three main aspects: the representation of tricomplex numbers, the structure and classification of their subalgebras, and the idempotent decomposition which simplifies calculations and clarifies algebraic behavior. By employing these representations, intricate operations become more tractable, providing insights into the interplay between subalgebras and their idempotent components. The results establish a coherent framework for tricomplex analysis, paving the way for both theoretical exploration and practical applications in advanced mathematical contexts.