Linearization of the BCK-logic

In the paper we obtain a new characterization of the BCK-algebras which are subdirect product of BCK-chains. We give an axiomatic algebraizable extension of the BCK-calculus, by means of a recursively enumerable set of axioms, such that its equivalent algebraic semantics is definitionally equivalent to the quasivariety of BCK-algebras generated by the BCK-chains. We propose the concept of "linearization of a system" and we give some examples.