  In the paper, we define the notion of a state BCK-algebra and a
state-morphism BCK-algebra extending the language of BCK-algebras by adding a
unary operator which models probabilistic reasoning. We present a relation
between state operators and state-morphism operators and measures and states on
BCK-algebras, respectively. We study subdirectly irreducible state (morphism)
BCK-algebras. We introduce the concept of an adjoint pair in BCK-algebras and
show that there is a one-to-one correspondence between adjoint pairs and
state-morphism operators. In addition, we show the generators of quasivarieties
of state-morphism BCK-algebras.
