  The class of weak BCK-algebras is obtained by weakening one of standard BCK
axioms. It is known that every weak BCK-algebra is completely determined by the
structure of its initial segments. We review several natural classes of
commutative weak BCK-algebras, prove that they are equationally definable, and
show that the order duals of a multitude of algebras with implication known in
the literature in connection with various quantum logics are, in fact,
commutative weak BCK-algebras belonging to that or other of these classes. We
also characterize initial segments of algebras in each of the classes as
lattices equipped with a suitable kind of complementation. In particular,
commutative weak BCK-algebras are just those meet semilattices with the least
element in which all initial segments are non-distributive de Morgan lattices.
