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ON ISOMORPHISM CLASSES OF COMPUTABLY ENUMERABLE EQUIVALENCE RELATIONS
The Journal of Symbolic Logic ( IF 0.6 ) Pub Date : 2019-06-13 , DOI: 10.1017/jsl.2019.39
URI ANDREWS , SERIKZHAN A. BADAEV

We examine how degrees of computably enumerable equivalence relations (ceers) under computable reduction break down into isomorphism classes. Two ceers are isomorphic if there is a computable permutation of ω which reduces one to the other. As a method of focusing on nontrivial differences in isomorphism classes, we give special attention to weakly precomplete ceers. For any degree, we consider the number of isomorphism types contained in the degree and the number of isomorphism types of weakly precomplete ceers contained in the degree. We show that the number of isomorphism types must be 1 or ω, and it is 1 if and only if the ceer is self-full and has no computable classes. On the other hand, we show that the number of isomorphism types of weakly precomplete ceers contained in the degree can be any member of $[0,\omega ]$. In fact, for any $n \in [0,\omega ]$, there is a degree d and weakly precomplete ceers ${E_1}, \ldots ,{E_n}$ in d so that any ceer R in d is isomorphic to ${E_i} \oplus D$ for some $i \le n$ and D a ceer with domain either finite or ω comprised of finitely many computable classes. Thus, up to a trivial equivalence, the degree d splits into exactly n classes.We conclude by answering some lingering open questions from the literature: Gao and Gerdes [11] define the collection of essentially FC ceers to be those which are reducible to a ceer all of whose classes are finite. They show that the index set of essentially FC ceers is ${\rm{\Pi }}_3^0$-hard, though the definition is ${\rm{\Sigma }}_4^0$. We close the gap by showing that the index set is ${\rm{\Sigma }}_4^0$-complete. They also use index sets to show that there is a ceer all of whose classes are computable, but which is not essentially FC, and they ask for an explicit construction, which we provide.Andrews and Sorbi [4] examined strong minimal covers of downwards-closed sets of degrees of ceers. We show that if $\left( {{E_i}} \right)$ is a uniform c.e. sequence of non universal ceers, then $\left\{ {{ \oplus _{i \le j}}{E_i}|j \in \omega } \right\}$ has infinitely many incomparable strong minimal covers, which we use to answer some open questions from [4].Lastly, we show that there exists an infinite antichain of weakly precomplete ceers.

中文翻译:

关于可计算等价关系的同构类

我们研究了可计算归约下的可计算可枚举等价关系(ceers)的程度如何分解为同构类。如果存在可计算的排列,则两个 ceer 是同构的ω这减少了另一个。作为关注同构类中非平凡差异的一种方法,我们特别关注弱预完备 ceers。对于任何程度,我们考虑该程度中包含的同构类型的数量以及该程度中包含的弱预完备 ceer 的同构类型的数量。我们证明同构类型的数量必须是 1 或ω,当且仅当 ceer 是自满的并且没有可计算的类时,它才为 1。另一方面,我们证明了度中包含的弱预完备 ceers 的同构类型的数量可以是$[0,\omega]$. 事实上,对于任何$n \in [0,\omega ]$, 有度d和弱预完成 ceers${E_1}, \ldots ,{E_n}$d这样任何 ceerRd同构于${E_i} \oplus D$对于一些$i \le n$D具有有限域或域的 ceerω由有限多个可计算类组成。因此,直到一个微不足道的等价,度d精确地分成n我们通过回答文献中一些挥之不去的开放性问题得出结论:Gao 和 Gerdes [11] 将本质上 FC ceer 的集合定义为可简化为所有类都是有限的 ceer 的集合。他们表明,本质上 FC ceers 的索引集是${\rm{\Pi }}_3^0$-hard,虽然定义是${\rm{\Sigma }}_4^0$. 我们通过显示索引集是${\rm{\Sigma }}_4^0$-完全的。他们还使用索引集来表明存在所有类都是可计算的 ceer,但本质上不是 FC,并且他们要求我们提供的显式构造。Andrews 和 Sorbi [4] 研究了向下的强最小覆盖-封闭的度数集。我们证明如果$\left({{E_i}}\right)$是非通用 ceers 的统一 ce 序列,则$\left\{ {{ \oplus _{i \le j}}{E_i}|j \in \omega } \right\}$有无限多个不可比的强最小覆盖,我们用它来回答[4]中的一些开放问题。最后,我们证明存在一个无限的弱预完备 ceers 反链。
更新日期:2019-06-13
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