In his algebraic theory of differential equations, J. F. Rittt has developed a decomposition theory for systems of algebraic differential equations by introducing the idea of irreducible systems and proving that every system is equivalent to one and essentially only one finite set of irreducible systems. The analogous theorem for algebraic difference equations was given by Ritt and Doob.t The purpose of the present paper is to derive a decomposition theorem for algebraic mixed difference equations; i.e., equations which contain algebraically one or more unknown functions y(x), their "transforms" y(x), y(x+l), y(x+2), * , and the derivatives of those transforms.