On the lattice of [pi]°1 classes

73 Pages·2001·3 MB·English

by  Riazati, Farzan.

ONTHELATTICEOFn°CLASSES
By
FARZANRIAZATI
ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL
OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT
OFTHEREQUIREMENTSFORTHEDEGREEOF
DOCTOROFPHILOSOPHY
UNIVERSITYOFFLORIDA
2001
Copyright2001
by
FarzanRiazati
ACKNOWLEDGMENTS
Iwishtoexpressmygratitudetomyadvisor,ProfessorDouglasCenzer. Ialso
wishtothankAndreNies.
m
TABLEOFCONTENTS
Page
ACKNOWLEDGMENTS
iii
LISTOFTABLES v
ABSTRACT
vi
CHAPTERS
1 PRELIMINARIES 1
1.1 TerminologyandNotation 1
1.2 ExistenceandExamples 5
1.3 BasicProperties 8
1.4 AppearancesandApplications 11
1.5 StoneRepresentationandComputableBooleanAlgebras . . 15
2 LOCALPROPERTIESOFH$CLASSES 17
2.1 PostProgramandtheLatticeCu 17
2.2 PrincipalIdealsofn°Classes 17
2.3 TheWorldofSimpleU°Classes 21
2.3.1 MinimalExtensionsof11°Classes 22
2.3.2 Quasiminimal11°Classes 30
2.3.3 FullyComplementedPrincipalIdeals 33
2.3.4 WhereThereAreNoComplements;Nerode'sTheorem 38
2.4 PrincipalCoveringFilters 41
2.5 TheSplittingProperty 44
3 GLOBALPROPERTIESOFn°CLASSES 48
3.1 Definability,andAutomorphisms 48
3.2 HomogeneityandEmbeddings 51
REFERENCES
58
BIOGRAPHICALSKETCH 62
IV
LISTOFTABLES
Table page
2.1 InitialprincipalidealsofC^ 18
2.2 PrincipalfiltersofE* 22
2.3 StructureofprincipalidealsofC^ 32
AbstractofDissertationPresentedtotheGraduateSchool
oftheUniversityofFloridainPartialFulfillmentofthe
RequirementsfortheDegreeofDoctorofPhilosophy
ONTHELATTICEOFIT?CLASSES
By
FarzanRiazati
August2001
Chairman: ProfessorDouglasCenzer
MajorDepartment: Mathematics
Effectivelyclosedsets,asmodeledby11°classes,haveplayedanimportantrole
incomputabilitytheorygoingbacktotheKleenebasistheorem[1955]. Manyofthe
fundamentalresultsabout11°classesandtheirmemberswereestablishedbyJockusch
andSoarein[1972]. Cenzeret. al[1999]isashortcourseon11°classes.The11°classes
occurnaturallyintheapplicationofcomputabilitytomanyareasofmathematics.
Cenzerand Remmel[1998] isarecentsurveywithmanyexamples. Minimaland
thin 11° classes wereinvestigated by Cenzer et al. [1993]. This dissertation is a
comparativestudyofthelattice£nof11°classeswiththelatticeSofcomputably
enumerable (c.e.) sets. Thework in this thesis concerns thelatticeof11° classes
(modulofinitedifference),comparedandcontrastedwiththelatticeofc.e.sets. The
notion ofaminimalextension Q ofaclass P is definedto meanthat thereisno
classstrictlybetweenPand Q. Previouslyonlytrivialexampleswereknown, but
herewegivegeneralconditionsunderwhichPhasaminimalextension. Recently
initialsegmentsofthelattice(thatis,subsetsofagivenset)havebeenstudied. It
vi
wasshown,incontrasttothelatticeofc.e.sets,thatafinitelatticecanberealized
whichisnotaBooleanalgebra;inparticular,anyfiniteordinalcanberealized. This
thesisannouncesanimprovmentoftheseresultsbyconstructinga11°classPsuch
thatthefamilyofsubclassesofPisisomorphictothesmallestinfiniteordinal(u).
Alsostudiedaredefinabilityofvariousproperties(suchasfiniteness)andinvariance
underautomorphism.
vn
CHAPTER
1
PRELIMINARIES
1.1 TerminologyandNotation
Webeginwithsomebasicdefinitions. Letu={0,1,2,...}denotethesetofnatural
numbers. Foranyset S, £<u; denotesthesetoffinitestrings (cr(0),...,a(n—1))
ofelementsfromS. And S^ denotesthesetofcountablyinfinitesequencesfrom
E. Forastringa=(c(0),0"(l)i •>a{n~1))? ||°"|| denotesthelengthnof<r. The
emptystringhaslength andisdenotedby0. Astringofn manyA;'sisdenoted
bykn. Form< |J<r|j,cr[misthestring(cr(0),...,cr(m—1)). Wesayaisaninitial
segmentofr(writtena-<r)ifcr=rfraforsomem. Giventwostrings<randr,
theconcatenationof<7and r, denotedbycr~r (orsometimesjust o-t), isdefined
by a""t = (<t(0),<t(1),...,<r(m—l),r(0),r(l),...,r(n—1)), where ||cr|| = mand
||r(| = n. Wewritea~aforcr~(a)and a^afor (a)~<r. Foranyx G £"andany
finiten,theinitialsegmentx\nofxis(x(0),...,x(n—1)). ForastringaG £<u/
andanya;€ S1^,wewritecr-<xifcr=a;\nforsomen. ForanyaG Sraandany
I 6 £*",wehavea~x= (er(0),...,a(n—1),x(0),x(l),...). Givenstringsaandr
oflengthn,welet<x(g)r=(cr(0),r(0),...,a{n—1),r(n—1));if||<r||=n+1 and
||t|| =n,thena(g)r= (a\n®r)"cr(n). Giventwoelementsx,yof£•*,x®y=z
wherez(2m)=x(m)andz(2m+1)=y(m).
Weneedtocodeastringa£u><a;asaninteger. Let [m,n]denotethestandard
pairingfunction,thatmapsthepairofnaturalnumbersm,n6wto l(m+")+3m+"l_
ForeachaGuk,welet<a>=[<a\(k-l)>,a(k)},where<>=0,and<m>=m
foreachmGu>.
A tree ToverS<w isasetoffinitestringsfrom£<w whichcontainstheempty
string andwhichisclosedunderinitialsegments.Wesaythatr£Tisanimmediate
successorofastringa £ Tifr = <r^aforsomea £ £. Sinceouralphabetswill
always becountableandeffective,wemayassumethat T C lu<uj. Suchatreeis
saidtobeu-branchingsinceeachnodehaspotentiallyacountablyinfinitenumberof
immediatesuccessors. WeshallidentifyTwiththeset{<a>:a£T}. Thuswesay
thatTiscomputable,computablyenumerable,etc.if{<a>:a£T}iscomputable,
computablyenumerable,etc. Foragivenfunctiong:u<UJ4w,atreeTCw<c"is
saidtobeg-boundedifforeverya£w"^andeveryi£o>,if<r~z£T,thent<g{o).
Thus,forexample,ifg(&)=2forallcr,thenag-boundedtreeissimplyabinarytree.
TissaidtobefinitebranchingifTisg-boundedforsomegthatis,ifeachnodeof
Thasfinitelymanyimmediatesuccessors.
Observethatthisisequivalenttotheexistenceofaboundingfunctionhsuchthat
<r(t)<h(i)foralla£Tandalli< \\a\\. Tissaidtobecomputablybounded(c.b.)
ifitisg-boundedforsomecomputablefunctiong. Asabove,thisisequivalenttothe
existenceofacomputableboundingfunctionhsuchthata(i) <h(i)foralla£T
and alli < \\a\\. IfTis computable,thenthisis alsoequivalenttotheexistence
ofapartialcomputablefunction/suchthat, foranya £ T, ahasat most f(a)
immediatesuccessorsinT. AcomputabletreeTissaidtobehighlycomputableifit
isalsocomputablebounded. ForanytreeT,aninfinitepaththroughTisasequence
(.t(0),x(1),...)suchthatx\n6Tforalln. Welet[T]denotethesetofinfinitepaths
throughT.
ItisimportanttonoteherethatweconsideraIT?settosignifyasubsetofwand
ingenerala11°,S°orA°setisasubsetofuwiththeappropriateformofdefinability
inthearithmeticalhierarchy[23]. Wedenotebycard(A)thecardinalityofthesetor
classA. GiventwotreesSandTcontainedinlo<uj,weletS®T={a®r:a£5,r£
Tsuchthat ||r||<\\a\\<\\r\\+1}. FortwoIT classesP=[S]andQ=[T],define
theamalgamationofPandQ,P<g>Q,byP<g>Q={x$y:xGPAyGQ}- Thenit
isclearthatP®Q=[S®T]. Moregenerally,definetheinfiniteamalgamation0,-S;
tobethosestringsasuchthatforeachi,(er([i,0]),cr([i,1]),...,cr([i,j]))G£#,where
j isthemaximumsuchthat [i,j] < \\a\\. Then [®j5,-] isisomorphictothedirect
productII;[5,-].
Wealsowishtoconsiderthefollowingnotionofdisjointunion. Giventwotrees
SandTcontainedinu<UJ,S T={0}U{<Ta:<rGS}U{l~r:rGP}. Fortwo
n°classesP=[S]andQ=[T],P9Q={O^x:xGPU{l"y:yGQ}. Itiseasy
toseethat[S®T]=[S]@[T]. ClearlyS©TisboundedifandonlyifbothSandP
areboundedandsimilarlyfortheothernotionsofboundedness. Moregenerally,the
infinitedisjointunionQ)iQimaybedefinedtobe{(i)*y:y(zQi]forunbounded
classes Q,. AnodeaofthetreePCco<UJissaidtobe extendible ifthereissome
x G [P]suchthata-<x. ThesetofextendiblenodesofTisdenotedbyExt(T).
Ext(T)maybeviewedastheminimaltreeSsuchthat [S]=[P]. AnodeaGPis
saidtobeadeadendifa£Ext(T),thatis,ifahasnoinfiniteextensionin[P].
Weareinterestedin11°classesinthespaces {0,l}w(theCantorspace) anduiw
(theBairespace). Thetopologyonlouisdeterminedbyabasisofintervals1(a)=
{x€ w*":o-<a?}. Noticethateachintervalisalsoaclosedsetandisthereforesaid
tobeaclopenset. MoreovertheclopensubsetsoftheCantorspacearejustthefinite
unionsofintervals.
TheCantor-BendixsonderivativeD(P) ofacompactsubset Pof{0,l}"isthe
setofnonisolatedpointsofP. ThusapointxGPisnotinD(P)ifandonlyifthere
issomeopensetUcontainingxwhichcontainsnootherpointofP. Equivalently,
x^D(P)ifandonlyifthereissomeclosedsetUsuchthatUPIP={x}. Another
usefulobservationisthat,foranycompactset P, D(P) isemptyifandonlyifP
isfinite. TheiteratedCantor-BendixsonderivativeDa(P) ofaclosedset P C X
is defined for all ordinals a by thefollowing transfinite induction. D°(P) = P;