Reducible solution to a quaternion tensor equation

We establish necessary and sufficient conditions for the existence of the reducible solution to the quaternion tensor equation A*N C*NB=C via Einstein product using Moore-Penrose inverse,and present an expression of the reducible solution to the equation when it is solvable.Moreover,to have a general solution,we give the solvability conditions for the quaternion tensor equation A1*N C1*MB1+a1*C2*MB2+A2*NC3*MB2=e,which plays a key role in investigating the reducible solution to A*NC*NB=e.The expression of such a solution is also presented when the consistency conditions are met.In addition,we show a numerical example to illustrate this result.