e. mohammadzadeh

In this paper,  first  we introduce the notions of k-nilpotent (solvable) ideals  and k-nilpotent BCK-algebras. Specially, we show  that every commutative ideal is 1-nilpotent (solvable). Second, we state  an equivalent condition to k-nilpotency (solvablity) ideals and  BCK-algebra. Finally,  we study n-fold 2-nilpotent (solvable) ideals and BCK-algebras as a generalization of n-fold commutative ideals and BCK-algebras, and we study the relation between these two concepts. Basically, we compare 2-nilpotent and solvable ideals (BCK-algebras).

In this paper, we introduce nilpotent soft (sub)polygroups. In addition, nilpotency of intersection, extended intersection, restricted union of two nilpotent soft polygroups are studied. Espesialy, a necessary and suficient condition between nilpotency of a polygroup and soft polygroups is obtained. Finally, we define two new soft polygroups (Sα)A∪{c} and (Qα)_A derived from a soft polygroup α_A and study on nilpotency of these structures. Also, we extend a soft homomorphism of groups to polygroups. This helps us to extend some properties of groups to polygroups.