  We introduce the notion of a lower bound cluster algebra generated by
projective cluster variables as a polynomial ring over the initial cluster
variables and the so-called projective cluster variables. We show that under an
acyclicity assumption, the cluster algebra and the lower bound cluster algebra
generated by projective cluster variables coincide. In this case we use our
results to construct a basis for the cluster algebra. We also show that any
coefficient-free cluster algebra of types $A_n$ or $\widetilde{A}_n$ is equal
to the corresponding lower bound cluster algebra generated by projective
cluster variables.
