Abstract
We study the cubic finite switchboard state machines (cubic fssms) intensively. The notions of (resp., good, strong) subsystem of cubic fssm are introduced. We prove that the Cartesian composition of two (resp., good, strong) subsystems of cubic fssms is (resp., good, strong) subsystem of cubic fssm. Similarly, the product of two (resp., good, strong) subsystems of cubic fssms is (resp. good, strong) subsystem of cubic fssm. We define the homomorphism between two of cubic fssms and proved some related results. Many examples on each case are provided.