Computable categoricity and the Ershov hierarchy

In this paper, the notions of F α -categorical and G α -categorical structures are introduced by choosing the isomorphism such that the function itself or its graph sits on the α -th level of the Ershov hierarchy, respectively. Separations obtained by natural graphs which are the disjoint unions of countably many finite graphs. Furthermore, for size-bounded graphs, an easy criterion is given to say when it is computable-categorical and when it is only G 2 -categorical; in the latter case it is not F α -categorical for any recursive ordinal α .