Engel, Nilpotent and Solvable BCI-algebras

In this paper, we define the concepts of Engel, nilpotent and solvable BCI-algebras and investigate some of their properties. Specially, we prove that any BCK-algebra is a 2-Engel. Then we define the center of a BCI-algebra and prove that in a nilpotent BCI-algebra X, each minimal closed ideal of X is contained in the center of X. In addition, with some conditions, we show that every finite BCI-algebra is solvable. Finally, we investigate the relations among Engel, nilpotent and solvable BCI(BCK)-algebras.