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, £a{n~1))? ||°"|| denotesthelengthnof=[,a(k)},where<>=0,and=m foreachmGu>. A tree ToverS:a£T}. Thuswesay thatTiscomputable,computablyenumerable,etc.if{:a£T}iscomputable, computablyenumerable,etc. Foragivenfunctiong:u,ifQ,byPQ={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

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