We define measures and states on BCK-algebras. We show that
the set of elements of measure 0 is an ideal, and the
corresponding quotient BCK-algebra is commutative with a
lifted original measure. We present different examples of
commutative as well as positively implicative BCK-algebras
on subsets of $[0,\infty)$. Finally, we characterize
commutative BCK-algebras as well as positively implicative
BCK-algebras via order-determining systems of
state-morphisms and states, respectively.
