Computability and Categoricity of Weakly Ultrahomogeneous Structures

This paper investigates the effective categoricity of ultrahomogeneous structures. It is shown that any computable ultrahomogeneous structure is $\Delta^0_2$ categorical. A structure A is said to be weakly ultrahomogeneous if there is a finite (exceptional) set of elements $a_1,\dots,a_n$ such that A becomes ultrahomogeneous when constants representing these elements are added to the language. Characterizations are obtained for weakly ultrahomogeneous linear orderings, equivalence structures, injection structures and trees, and these are compared with characterizations of the computably categorical and $\Delta^0_2$ categorical structures. Index sets are used to determine the complexity of the notions of ultrahomegenous and weakly ultrahomogeneous for various families of structures.