Definition 3.1 (see [8]). A BCK-alg

Definition 3.1 (see [8]). A BCK-algebra X is called positive implicative if (x∗y)∗z
= (x ∗ z) ∗ (y ∗ z). It is called implicative if x ∗ (y ∗ x) = x. It is commutative if
x∗(x∗y) = y ∗(y ∗x).
It is well known that positive implicative BCI-algebras, implicative BCI-algebras
and commutative BCI-algebras are positive implicative BCK-algebras, implicative
BCK-algebras and commutative BCK-algebras, respectively [9].
In [1], Chaudhry defined three classes of proper BCI-algebras, namely, weakly positive
implicative BCI-algebras, weakly implicative BCI-algebras, and weakly commutative
BCI-algebras. He also investigated a few properties of these algebras. We recall
these definitions and the following result.