t-Product and t-STP of Cubic Matrices With Application to Hyper-Networked Systems

Motivated by the study of dynamic control systems, this paper proposes novel algebraic operations on cubic matrices to construct both linear and nonlinear controlled dynamics. The standard t-product of cubic matrices imposes strict dimensional constraints; to resolve this, we first introduce the dimension-keeping semi-tensor product (DK-STP), which generalizes the matrix product for arbitrary dimensions. However, the DK-STP yields decoupled subsystem dynamics because it fails to capture interactions across subsystems corresponding to frontal slices. To overcome this limitation, we propose the t-semi-tensor product (t-STP), an integration of the t-product and the DK-STP that allows for coupled subsystems and greater modeling flexibility. We systematically study the algebraic structures derived from the t-STP over cubic matrices, including groups, rings, modules, and Lie groups. Finally, we obtain t-STP-based dynamic control systems over cubic matrices and demonstrate the utility of this framework by applying it to a hyper-networked evolutionary game modeling supply chain interactions.