Towards Finding a Lattice that Characterizes the ${>}\ \omega ^2$ -Fickle Recursively Enumerable Turing Degrees

Liling Ko
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Abstract

Abstract Given a finite lattice L that can be embedded in the recursively enumerable (r.e.) Turing degrees $\langle \mathcal {R}_{\mathrm {T}},\leq _{\mathrm {T}}\rangle $ , we do not in general know how to characterize the degrees $\mathbf {d}\in \mathcal {R}_{\mathrm {T}}$ below which L can be bounded. The important characterizations known are of the $L_7$ and $M_3$ lattices, where the lattices are bounded below $\mathbf {d}$ if and only if $\mathbf {d}$ contains sets of “fickleness” $>\omega $ and $\geq \omega ^\omega $ respectively. We work towards finding a lattice that characterizes the levels above $\omega ^2$ , the first non-trivial level after $\omega $ . We introduced a lattice-theoretic property called “ $3$ -directness” to describe lattices that are no “wider” or “taller” than $L_7$ and $M_3$ . We exhaust the 3-direct lattices L, but they turn out to also characterize the $>\omega $ or $\geq \omega ^\omega $ levels, if L is not already embeddable below all non-zero r.e. degrees. We also considered upper semilattices (USLs) by removing the bottom meet(s) of some 3-direct lattices, but the removals did not change the levels characterized. This leads us to conjecture that a USL characterizes the same r.e. degrees as the lattice on which the USL is based. We discovered three 3-direct lattices besides $M_3$ that also characterize the $\geq \omega ^\omega $ -levels. Our search for a $>\omega ^2$ -candidate therefore involves the lattice-theoretic problem of finding lattices that do not contain any of the four $\geq \omega ^\omega $ -lattices as sublattices. Abstract prepared by Liling Ko. E-mail: ko.390@osu.edu URL: http://sites.nd.edu/liling-ko/
关于寻找表征${>}\ \ ω ^2$ -易变递归可枚举图灵度的格
给定一个有限格L,它可以嵌入到递归可枚举(r。图灵度$\langle \mathcal {R}_{\mathrm {T}},\leq _{\mathrm {T}}\rangle $,我们通常不知道如何描述度$\mathbf {d}\in \mathcal {R}_{\mathrm {T}}$以下L可以有界。已知的重要特征是$L_7$和$M_3$晶格,当且仅当$\mathbf {d}$分别包含“可变”集$>\omega $和$\geq \omega ^\omega $时,晶格被限定在$\mathbf {d}$以下。我们努力寻找一个格来表征$\omega ^2$以上的水平,这是$\omega $之后的第一个非平凡水平。我们引入了一种称为“$3$ -直接性”的晶格理论性质来描述不比$L_7$和$M_3$“宽”或“高”的晶格。我们耗尽了3-direct格L,但如果L还没有嵌入到所有非零r.e.度以下,它们也可以表征$>\omega $或$\geq \omega ^\omega $级别。我们还考虑了上半格(USLs),通过去除一些3-直格的底部会合,但去除并没有改变表征的水平。这使我们推测,USL的特征与USL所基于的晶格具有相同的r.e.度。除了$M_3$之外,我们还发现了三个3-直格,它们也表征了$\geq \omega ^\omega $ -水平。因此,我们对$>\omega ^2$ -候选者的搜索涉及到寻找不包含四个$\geq \omega ^\omega $ -格中的任何一个作为子格的格的晶格理论问题。[摘要]柯丽玲。电子邮件:ko.390@osu.edu URL: http://sites.nd.edu/liling-ko/
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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