Ultrahomogeneity and $ω$-categoricity of monounary algebras
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911548443721728 |
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| author | Quinn-Gregson, Thomas |
| author_facet | Quinn-Gregson, Thomas |
| contents | Ultrahomogeneity and $ω$-categoricity are two central concepts arising from model theory, with strong connections with oligomorphic permutation groups and quantifier elimination. In particular, both are conditions on the automorphism group of a structure.
The aim of this paper is to describe both the $ω$-categorical monounary algebras and the ultrahomogeneous monounary algebras of arbitrary cardinalities. We show that a monounary algebra is $ω$-categorical [ultrahomogeneous] if and only if every element has finite height and Aut$(\mathcal{A})$ has only finitely many 1-orbits [$\mathcal{A}$ is 1-ultrahomogeneous]. Our classification of ultrahomogeneous monounary algebras is then viewed in the context of previously studied variants of ultrahomogeneity, including (partial)-homogeneity and transitivity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_26616 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Ultrahomogeneity and $ω$-categoricity of monounary algebras Quinn-Gregson, Thomas Logic Rings and Algebras 08A60, 03C35, 03C10 Ultrahomogeneity and $ω$-categoricity are two central concepts arising from model theory, with strong connections with oligomorphic permutation groups and quantifier elimination. In particular, both are conditions on the automorphism group of a structure. The aim of this paper is to describe both the $ω$-categorical monounary algebras and the ultrahomogeneous monounary algebras of arbitrary cardinalities. We show that a monounary algebra is $ω$-categorical [ultrahomogeneous] if and only if every element has finite height and Aut$(\mathcal{A})$ has only finitely many 1-orbits [$\mathcal{A}$ is 1-ultrahomogeneous]. Our classification of ultrahomogeneous monounary algebras is then viewed in the context of previously studied variants of ultrahomogeneity, including (partial)-homogeneity and transitivity. |
| title | Ultrahomogeneity and $ω$-categoricity of monounary algebras |
| topic | Logic Rings and Algebras 08A60, 03C35, 03C10 |
| url | https://arxiv.org/abs/2603.26616 |