On a structure of the closure semigroup.

In this paper we have studied a certain structure of the closure semigroup. R. E. Johnson has studied a structure of the faithful rings with the aid of a closure operation. While, the transition from the elements of a ring to its ideals can also be regarded as a certain kind of completion process. This method has been studied extensively by H. Prufer in domains of integrity and generalized by Lorenzen (1) to partially ordered groups. The ideal systems are of great importance in partially ordered groups. Now, applying the conception of a ideal system to semigroups, we define a closure semi group and its ideal as usual. A semigroup S is called a closure semigroup which satisties the following conditions