On the Conjugation of Tricomplex Numbers: Algebraic Operations, Moduli, and Decompositions

We investigate the algebraic structure and symmetry properties of tricomplex numbers, emphasizing the seven distinct involutive conjugations arising from its independent imaginary units. After defining these conjugations, we derive their fundamental algebraic properties, including relations under addition, subtraction, and moduli behavior. By systematically comparing pairs of conjugations, we establish canonical identities that characterize elements of C_3 in terms of invariant subalgebras and twisted modules. A total of 141 identities is obtained through analysis of equality, negation, addition, subtraction, and multiplicative relations among conjugation operators; among these are 112 canonical conjugation identities. These identities correspond to idempotent projection operators that decompose any tricomplex number into invariant and anti-invariant components, yielding a full canonical decomposition into symmetry-adapted subspaces. We also examine the action of conjugations on primitive idempotent elements. The results extend classical conjugation theory from complex analysis to higher-order multicomplex systems, providing a unified framework for coordinate-wise analysis, subalgebra classification, and functional formulations.