Computability and Categoricity of Ultrahomogeneous Structures

Abstract

This paper investigates the effective categoricity of ultrahomogeneous structures. It is shown that any computable ultrahomogeneous structure is $\varDelta^0_2$ categorical. A structure ${\mathcal A}$ is said to be weakly ultrahomogeneous if there is a finite (exceptional) set of elements a ₁,…,a _n such that ${\mathcal A}$ becomes ultrahomogeneous when constants representing these elements are added to the language. Characterizations are obtained for the weakly ultrahomogeneous linear orderings, equivalence structures, and injection structures, and compared with characterizations of the computably categorical and $\varDelta^0_2$ categorical structures.