The foldness of
$PI(\ll,$ $\ll,$ $\subseteq)_{BCK}$-ideals and
$PI(\subseteq,$ $\subseteq,$ $\subseteq)_{BCK}$-ideals is 
considered, and their properties are investigated. 
Hyper homomorphic inverse image of
$PI(\ll,$ $\ll,$ $\subseteq)_{BCK}$-ideals,
$PI(\ll,$ $\subseteq,$ $\subseteq)_{BCK}$-ideals,
$PI(\subseteq,$ $\subseteq,$ $\subseteq)_{BCK}$-ideals, and
$PI(\ll,$ $\ll,$ $\ll)_{BCK}$-ideals
are discussed.