We study the computable reducibility ≤c for equivalence relations in the Ershov hierarchy. For an arbitrary notation a for a nonzero computable ordinal, it is stated that there exist a $$ {\varPi}_a^{-1} $$ -universal equivalence relation and a weakly precomplete $$ {\varSigma}_a^{-1} $$ - universal equivalence relation. We prove that for any $$ {\varSigma}_a^{-1} $$ equivalence relation E, there is a weakly precomplete $$ {\varSigma}_a^{-1} $$ equivalence relation F such that E ≤cF. For finite levels $$ {\varSigma}_m^{-1} $$ in the Ershov hierarchy at which m = 4k +1 or m = 4k +2, it is shown that there exist infinitely many ≤c-degrees containing weakly precomplete, proper $$ {\varSigma}_m^{-1} $$ equivalence relations.

中文翻译：

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Ershov 层次结构中的弱预完备等价关系

我们研究 Ershov 层次结构中等价关系的可计算归约性≤c。对于非零可计算序数的任意记号 a，说明存在 $$ {\varPi}_a^{-1} $$ -普遍等价关系和弱预完备 $$ {\varSigma}_a^{- 1} $$ - 通用等价关系。我们证明对于任何 $$ {\varSigma}_a^{-1} $$ 等价关系 E，存在弱预完备 $$ {\varSigma}_a^{-1} $$ 等价关系 F 使得 E ≤cF . 对于 Ershov 层次结构中 m = 4k +1 或 m = 4k +2 的有限级 $$ {\varSigma}_m^{-1} $$，表明存在无限多个 ≤c-degrees 包含弱预完备, 正确的 $$ {\varSigma}_m^{-1} $$ 等价关系。