2025/09/29 by Wesley Calvert, Douglas Cenzer, Calvert, Wesley +7
Computer Science · #Advanced Algebra and Logic #FOS: Mathematics #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2509.25005
openalex publication_date 2025/09/29 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28
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 Δ4 categorical and determine exactly which ones are (uniformly) relatively Δ3 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 Π50 complete, and that the set of weakly sp-homogeneous linear orderings is Σ60 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.