Under [Formula: see text]-amalgamation, we obtain the canonical hyperdefinable group from the group configuration.

Makkai [10] produced an arithmetical structure of Scott rank $\omega _{1}^{\mathit{CK}}$. In [9]. Makkai's example is made computable. Here we show that there are computable trees of Scott rank $\omega _{1}^{\mathit{CK}}$. We introduce a notion of "rank homogeneity". In rank homogeneous trees, orbits of tuples can be understood relatively easily. By using these trees, we avoid the need to pass to the more complicated "group trees" of [10] and [9]. Using the same kind of trees, we obtain one of rank (...) $\omega _{1}^{\mathit{CK}}$ that is "strongly computably approximable". We also develop some technology that may yield further results of this kind. (shrink)

There exists a countable structure of Scott rank where and where the -theory of is not ω-categorical. The Scott rank of a model is the least ordinal β where the model is prime in its -theory. Most well-known models with unbounded atoms below also realize a non-principal -type; such a model that preserves the Σ1-admissibility of will have Scott rank . Makkai [M. Makkai, An example concerning Scott heights, J. Symbolic Logic 46 301–318. [4]] produces a hyperarithmetical model of Scott (...) rank whose -theory is ω-categorical. A computable variant of Makkai’s example is produced in [W. Calvert, S.S. Goncharov, J.F. Knight, J. Millar, Categoricity of computable infinitary theories, Arch. Math. Logic . [1]; J. Knight, J. Millar, Computable structures of rank J. Math. Logic . [2]]. (shrink)

Computable structures of Scott rank ${\omega_1^{CK}}$ are an important boundary case for structural complexity. While every countable structure is determined, up to isomorphism, by a sentence of ${\mathcal{L}_{\omega_1 \omega}}$ , this sentence may not be computable. We give examples, in several familiar classes of structures, of computable structures with Scott rank ${\omega_1^{CK}}$ whose computable infinitary theories are each ${\aleph_0}$ -categorical. General conditions are given, covering many known methods for constructing computable structures with Scott rank ${\omega_1^{CK}}$ , which guarantee that the (...) resulting structure is a model of an ${\aleph_0}$ -categorical computable infinitary theory. (shrink)