In paper[1], the author had shown that commuta- tive BCK--algebras with condition $(S)$ under natural partial order are distributive lattices. In this paper, we shall extend above mentioned conclusion, and obtain seven important properties on commutative BCK--algebra $(X,*,\leq;0)$ if it is a lattice under natural partial order, and answer problem 1 in paper [1] if X is a finite set.etc.