Engel BCI-algebras: an application of left and right commutators

Abstract

. We introduce Engel elements in a BCI-algebra by using left and right normed commutators, and some properties of these elements are studied. The notion of n -Engel BCI-algebra as a natural generalization of commutative BCI-algebras is introduced, and we discuss Engel BCI-algebra, which is defined by left and right normed commutators. In particular, we prove that any nilpotent BCI-algebra of type 2 is an Engel BCI-algebra, but solvable BCI-algebras are not Engel, generally. Also, it is proved that 1-Engel BCI-algebras are exactly the commutative BCI-algebras.