Isomorphism types and theories of Rogers semilattices of arithmetical numberings

We investigate differences in isomorphism types and elementary theories of Rogers semilattices of arithmetical numberings, depending on different levels of the arithmetical hierarchy. It is proved that new types of isomorphism appear as the arithmetical level increases. It is also proved the incompleteness of the theory of the class of all Rogers semilattices of any fixed level. Finally, no Rogers semilattice of any infinite family at arithmetical level $ngeq 2$ is weakly distributive, whereas Rogers semilattices of finite families are always distributive.