In this paper, we use the definition of the action on the set of semi-group of the structure of this research .We introduce the concepts of -system which is a triple , , such that is a Hausdorff compact space called phase space, is a semi-group of transformations with a continuous action of on . We study and proof some theoretical properties related with that system. We also introduce the concept of Enfolding semi-group ( , ,and we prove that it is a compact right topological semi-group. In addition, we study the left and right ideals in the Enfolding semi-group. By using the dynamical system, we reflect various properties concerning with its structure for the Enfolding semi-group. Furthermore, we describe the connections between the algebraic and topological properties of the Enfolding semi-group space.
