Given a structure ℳ over ω and a syntactic complexity class $$ \mathfrak{E} $$, we say that a subset is $$ \mathfrak{E} $$-definable in ℳ if there exists a C-formula Θ(x) in the language of ℳ such that for all x ∈ ω, we have x ∈ A iff Θ(x) is true in the structure. S. S. Goncharov and N. T. Kogabaev [Vestnik NGU, Mat., Mekh., Inf., 8, No. 4, 23-32 (2008)] generalized an idea proposed by Friedberg [J. Symb. Log., 23, No. 3, 309-316 (1958)], introducing the notion of a $$ \mathfrak{E} $$-classification of M: a computable list of $$ \mathfrak{E} $$-formulas such that every $$ \mathfrak{E} $$-definable subset is defined by a unique formula in the list. We study the connections among$$ {\varSigma}_1^0- $$, $$ d-{\varSigma}_1^0- $$, and $$ {\varSigma}_2^0 $$-classifications in the context of two families of structures, unbounded computable equivalence structures and unbounded computable injection structures. It is stated that every such injection structure has a $$ {\varSigma}_1^0- $$classification, a $$ {\varSigma}_1^0- $$classification, and a $$ {\varSigma}_2^0 $$-classification. In equivalence structures, on the other hand, we find a richer variety of possibilities.

中文翻译：

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可定义子集的分类

给定 ω 上的结构 ℳ 和句法复杂性类 $$ \mathfrak{E} $$，如果存在 C-公式 Θ(x ) 在 ℳ 的语言中，对于所有 x ∈ ω，我们有 x ∈ A 且仅当 Θ(x) 在结构中为真。SS Goncharov 和 NT Kogabaev [Vestnik NGU, Mat., Mekh., Inf., 8, No. 4, 23-32 (2008)] 概括了弗里德伯格 [J. 符号。Log., 23, No. 3, 309-316 (1958)]，介绍了 $$ \mathfrak{E} $$-分类 M 的概念：$$ \mathfrak{E} $$- 的可计算列表公式使得每个 $$ \mathfrak{E} $$ 可定义子集由列表中的唯一公式定义。我们研究了 $$ {\varSigma}_1^0- $$、$$ d-{\varSigma}_1^0- $$ 和 $$ {\varSigma}_2^0 $$-classifications 在上下文中的联系两个结构族，无界可计算等价结构和无界可计算注入结构。据说每个这样的注入结构都有一个 $$ {\varSigma}_1^0- $$classification，一个 $$ {\varSigma}_1^0- $$classification，和一个 $$ {\varSigma}_2^0 $$-分类。另一方面，在等价结构中，我们发现了更丰富的可能性。