Cluster algebras generated by projective cluster variables

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.