Nilpotent BCK-algebras

We use the notion of pseudocommutators to define the derived ideal of a BCK-algebra. Using the derived ideal we define the commutativization of a BCK-algebra; this construction is functorial and the commutativization functor is left adjoint to the inclusion functor from the category of commutative BCK-algebras to the category of BCK-algebras. This yields a new proof that commutative BCK-algebras are a reflective subcategory of BCK-algebras. After this, we introduce central series and define a notion of nilpotence for BCK-algebras and prove some properties of nilpotence. In particular, for any variety of BCK-algebras, the subclass of nilpotent algebras is a subpseudovariety, though in general not a subvariety. We also show that the class of BCK-algebras of nilpotence class at most c is a subquasivariety of all BCK-algebras, and is a variety if and only if c = 1. We close by showing that every BCK-algebra of finite height is nilpotent.