{"title":"Towards Finding a Lattice that Characterizes the \n${>}\\ \\omega ^2$\n -Fickle Recursively Enumerable Turing Degrees","authors":"Liling Ko","doi":"10.1017/bsl.2021.56","DOIUrl":null,"url":null,"abstract":"Abstract Given a finite lattice L that can be embedded in the recursively enumerable (r.e.) Turing degrees \n$\\langle \\mathcal {R}_{\\mathrm {T}},\\leq _{\\mathrm {T}}\\rangle $\n , we do not in general know how to characterize the degrees \n$\\mathbf {d}\\in \\mathcal {R}_{\\mathrm {T}}$\n below which L can be bounded. The important characterizations known are of the \n$L_7$\n and \n$M_3$\n lattices, where the lattices are bounded below \n$\\mathbf {d}$\n if and only if \n$\\mathbf {d}$\n contains sets of “fickleness” \n$>\\omega $\n and \n$\\geq \\omega ^\\omega $\n respectively. We work towards finding a lattice that characterizes the levels above \n$\\omega ^2$\n , the first non-trivial level after \n$\\omega $\n . We introduced a lattice-theoretic property called “ \n$3$\n -directness” to describe lattices that are no “wider” or “taller” than \n$L_7$\n and \n$M_3$\n . We exhaust the 3-direct lattices L, but they turn out to also characterize the \n$>\\omega $\n or \n$\\geq \\omega ^\\omega $\n levels, if L is not already embeddable below all non-zero r.e. degrees. We also considered upper semilattices (USLs) by removing the bottom meet(s) of some 3-direct lattices, but the removals did not change the levels characterized. This leads us to conjecture that a USL characterizes the same r.e. degrees as the lattice on which the USL is based. We discovered three 3-direct lattices besides \n$M_3$\n that also characterize the \n$\\geq \\omega ^\\omega $\n -levels. Our search for a \n$>\\omega ^2$\n -candidate therefore involves the lattice-theoretic problem of finding lattices that do not contain any of the four \n$\\geq \\omega ^\\omega $\n -lattices as sublattices. Abstract prepared by Liling Ko. E-mail: ko.390@osu.edu URL: http://sites.nd.edu/liling-ko/","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Bulletin of Symbolic Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/bsl.2021.56","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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/