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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.25005 |
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| _version_ | 1866918444592529408 |
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| author | Calvert, Wesley Cenzer, Douglas Gonzalez, David Harizanov, Valentina Ng, Keng Meng |
| author_facet | Calvert, Wesley Cenzer, Douglas Gonzalez, David Harizanov, Valentina Ng, Keng Meng |
| contents | 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 $Δ_4$ categorical and determine exactly which ones are (uniformly) relatively $Δ_3$ 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 $Π_5^0$ complete, and that the set of weakly sp-homogeneous linear orderings is $Σ_6^0$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_25005 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | sp-Homogeneous Linear Orderings Calvert, Wesley Cenzer, Douglas Gonzalez, David Harizanov, Valentina Ng, Keng Meng Logic 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 $Δ_4$ categorical and determine exactly which ones are (uniformly) relatively $Δ_3$ 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 $Π_5^0$ complete, and that the set of weakly sp-homogeneous linear orderings is $Σ_6^0$ 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. |
| title | sp-Homogeneous Linear Orderings |
| topic | Logic |
| url | https://arxiv.org/abs/2509.25005 |