The study of automorphisms of computable and other structures connects computability theory with classical group theory. Among the noncomputable countable structures, computably enumerable structures are one of the most important objects of investigation in computable model theory. Here we focus on the lattice structure of computably enumerable substructures of a given canonical computable structure. In particular, for a Turing degree d , we investigate the groups of d -computable automorphisms of the lattice of d -computably enumerable vector spaces, of the interval Boolean algebra B η of the ordered set of rationals, and of the lattice of d -computably enumerable subalgebras of B η . For these groups, we show that Turing reducibility can be used to substitute the group-theoretic embedding. We also prove that the Turing degree of the isomorphism types for these groups is the second Turing jump d ′′ of d .

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子结构格的图灵度和自同构群

可计算结构和其他结构的自同构研究将可计算性理论与经典群论联系起来。在不可计算的可数结构中，可计算的可枚举结构是可计算模型理论中最重要的研究对象之一。在这里，我们关注给定规范可计算结构的可计算可枚举子结构的晶格结构。特别地，对于图灵度 d ，我们研究了 d 可计算可枚举向量空间的格、有理数有序集的区间布尔代数 B η 和 d 的格的 d 可计算自同构群B η 的可计算可枚举子代数。对于这些群，我们展示了图灵可约性可用于替代群论嵌入。