Structures with Small Orbit Growth

2018/10/12 by Manuel Bodirsky, Bertalan Bodor, Bodirsky, Manuel +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #Advanced Topology and Set Theory #Connective tissue disorders research #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO) #Microtubule and mitosis dynamics

paper · pdf · doi:10.48550/arxiv.1810.05657

openalex publication_date 2018/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K_exp+ be the class of all structures A such that the automorphism group of A has at most c n^{d n} orbits in its componentwise action on the set of n-tuples with pairwise distinct entries, for some constants c,d with d < 1. We show that K_exp+ is precisely the class of finite covers of first-order reducts of unary structures, and also that K_exp+ is precisely the class of first-order reducts of finite covers of unary structures. It follows that the class of first-order reducts of finite covers of unary structures is closed under taking model companions and model-complete cores, which is an important property when studying the constraint satisfaction problem for structures from K_exp+. We also show that Thomas' conjecture holds for K_exp+: all structures in K_exp+ have finitely many first-order reducts up to first-order interdefinability.

Citations

• 2^ℵ₀ pairwise non-isomorphic maximal-closed subgroups of Sym(ℕ) via the classification of the reducts of the Henson digraphs

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