  In the present paper we examine the relationship between several type $A$
cluster theories and structures. We define a 2D geometric model of a cluster
theory, which generalizes cluster algebras from surfaces, and encode several
existing type $A$ cluster theories into a 2D geometric model. We review two
other cluster theories of type $A$. Then we introduce an abstraction of cluster
structures. We prove two results: the first relates several existing type $A$
cluster theories and the second relates some of these cluster structures using
the new abstraction.
