Let 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}} be the class of all structures 𝔄{\mathfrak{A}} such that the automorphism group of 𝔄{\mathfrak{A}} has at most c⁢nd⁢n{cn^{dn}} orbits in its componentwise action on the set of n -tuples with pairwise distinct entries, for some constants c,d{c,d} with d<1{d<1}. We show that 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}} is precisely the class of finite covers of first-order reducts of unary structures, and also that 𝒦exp+{\mathcal{K}_{{\operatorname{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 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}}. We also show that Thomas’ conjecture holds for 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}}: all structures in 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}} have finitely many first-order reducts up to first-order interdefinability.

中文翻译：

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小轨道增长的置换群

令 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}} 是所有结构 𝔄{\mathfrak{A}} 的类，使得 𝔄{\mathfrak{A}} 的自同构群对于具有成对不同条目的 n 元组集合，对于某些常数 c,d{c,d} 和 d<1{d<，在其分量操作中最多具有 c⁢nd⁢n{cn^{dn}} 轨道1}。我们证明 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}} 正是一元结构的一阶约简的有限覆盖类，而且 𝒦exp+{\mathcal{K} _{{\operatorname{exp}}{+}}} 正是一元结构有限覆盖的一阶约简类。因此，一元结构的有限覆盖的一阶约化类在采用模型同伴和模型完备核的情况下是封闭的，这是研究来自 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}} 的结构的约束满足问题时的一个重要性质。我们还证明了托马斯猜想对 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}} 成立：𝒦exp+{\mathcal{K}_{{\operatorname{exp}} 中的所有结构{+}}} 具有有限多个一阶约简，直到一阶可互定义性。