We study the following open question in computable model theory: does there exist a structure of computable dimension two which is the prime model of its first-order theory? We construct an example of such a structure by coding a certain family of c.e. sets with exactly two one-to-one computable enumerations into a directed graph. We also show that there are examples of such structures in the classes of undirected graphs, partial orders, lattices, and integral domains.
"Prime models of finite computable dimension." J. Symbolic Logic 74 (1) 336 - 348, March 2009. https://doi.org/10.2178/jsl/1231082315