sp-Homogeneous Linear Orderings

Abstract

We study linear orderings expanded by functions for successor and predecessor. The successor and predecessor on linear orderings capture the relatively intrinsically computably enumerable information about orderings in much the same way that dependence captures that for vector spaces. In particular, the sp-homogeneous and weakly sp-homogeneous linear orderings are those which are (ultra-)homogeneous or weakly homogeneous with this additional structure. We demonstrate that these orderings are always relatively $\Delta_4$ categorical and determine exactly which ones are (uniformly) relatively $\Delta_3$ categorical. We also provide a classification for sp-homogeneity and weak sp-homogeneity. We establish that this is the best possible classification by showing that the set of sp-homogeneous linear orderings is $\Pi_5^0$ complete, and that the set of weakly sp-homogeneous linear orderings is $\Sigma_6^0$ complete. These results are obtained in two different ways, one using a hands-on computability theoretic approach and another using more abstract descriptive set theory.