  A C*-algebra is determined to a great extent by the partial order of its
commutative C*-algebras. We study order-theoretic properties of this dcpo. Many
properties coincide: the dcpo is, equivalently, algebraic, continuous,
meet-continuous, atomistic, quasi-algebraic, or quasi-continuous, if and only
if the C*-algebra is scattered. For C*-algebras with enough projections, these
properties are equivalent to finite-dimensionality. Approximately
finite-dimensional elements of the dcpo correspond to Boolean subalgebras of
the projections of the C*-algebra, which determine the projections up to
isomorphism. Scattered C*-algebras are finite-dimensional if and only if their
dcpo is Lawson-scattered. General C*-algebras are finite-dimensional if and
only if their dcpo is order-scattered.
