Generated new classes of BCK-algebras using permutation quasi-ordered sets

This work aims to develop new techniques for introducing new BCK-algebra classes via permutation in symmetric groups. permutation generalized BCK-algebras are introduced as a generalization of positive implicative permutation BCK-algebras. It shows how to create a permutation generalized BCK-algebra from a permutation quasi-ordered set. The concept of permutation generalized ideals of permutation generalized BCK-algebras is introduced, followed by examining the relationships between such ideals and congruence. Permutation generalized BCK-ideals are described in detail. It is established that a set generates a permutation generalized BCK-ideal.