In this paper we introduce the concept of $n$-fold commutative

$BCK$-algebras which is a kind of generalization of one of commutative

$BCK$-algebras, and investigate the relations among a number of special

types of $BCK$-algebras, and prove the existence of $n$-fold commutative

$BCK$-algebras. Finally we show that if $n\ge 2$, the $n$-fold commutative

$BCK$-algebraic class does not form a variety.