  We characterize the canonical diagonal subalgebra of the C*-algebra
associated with a generalized Boolean dynamical system. We also introduce a
particular commutative subalgebra, which we call the abelian core, in our
C*-algebra. We then establish a uniqueness theorem under the assumptions that B
and L are countable, which says that a *-homomorphism of our C*-algebra is
injective if and only if its restriction to the abelian core is injective.
