Radical approach in BCH-algebras

We define the notion of radical in BCH-algebra and investigate the structure of [X; k], a viewpoint of radical in BCH-algebras. 1. Introduction. In 1966, Imai and Iséki [8] and Iséki [9] introduced two classes of abstract algebras: BCK-algebras and BCI-algebras. It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. In 1983, Hu and Li [5, 6] introduced a wide class of abstract algebras: BCH-algebras. They have shown that the class of BCI-algebras is a proper subclass of the class of BCH-algebras. They have studied some properties of these algebras. As we know, the primary aim of the theory of BCH-algebras is to determine the structure of all BCH-algebras. The main task of a structure theorem is to find a complete system of invariants describing the BCH-algebra up to isomorphism, or to establish some connection with other mathematics branches. In addition, the ideal theory plays an important role in studying BCI-algebras, and some interesting results have been ob-introduced nil ideals in BCH-algebras. They introduced the concept of nil subsets by using nilpotent elements, and investigated some related properties. In this note, we define the notion of radical in BCH-algebra, and some fundamental results concerning this notion are proved.