As a generalization of positive implicative $BCK$-algebra,
the notion of generalized $BCK$-algebras is introduced.
A method to make a $gBCK$-algebra from a quasi-ordered set
is provided. The notion of generalized $BCK$-ideals of
generalized $BCK$-algebras is introduced, and then the
connections between such ideals and congruences are considered.
Characterizations of generalized $BCK$-ideals are given.
A generalized $BCK$-ideal generated by a set is established.