Characterization of P-Semi Homogeneous System of Difference Equations of Three dimension

     The aim of this paper is to define new concepts, namely a homogenous system of difference equations x(n+1)=Bx(n)  where  B is a matrix of real numbers, which is called P-semi homogenous of order m if there exists a non-zero matrix A and integer number m such that the following equation holds: F (A(c)x(n))= 〖P(A(c))〗^m F(x(n)), Where F is a function,  m and  P are integer numbers and c is a real number. This definition is a generalization to the (3×3)-semi-homogeneous system of difference equations of order m. Special cases are studied of this definition and illustrative examples are given and some characterizations of this definition are also given. The necessary and sufficient conditions for a homogenous system of difference Equations to be P-semi homogenous of order one or greater than one as well as some examples and theorems about there are given.