n-fold sub-implicative ideals and n-fold sub-commutative ideals of BCI-algebras

Abstract

The notions of n-fold sub-implicative ideals and n-fold sub-commutative ideals of BCI-algebras are introduced. We show that a nonempty subset of a BCI-algebra is an n-fold sub-implicative ideal if and only if it is both an n-fold sub-commutative ideal and an n-fold positive implicative ideal. We prove that any p-ideal is always an n-fold sub- implicative ideal and an n-fold sub-commutative ideal. We give a new characterization of n-fold positive implicative ideals of BCI-algebras. Moreover some other properties about n-fold sub-commutative ideals of BCI-algebras are given.