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Publié par | universitat_bremen |
Publié le | 01 janvier 2007 |
Nombre de lectures | 37 |
Langue | English |
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ONA
CATEGORICALGENERALIZATIONOFTHE
CONCEPTOFFUZZYSET
SergejsSolovjovs
ionatDissert
zurErlangungdesGradeseinesDoktors
derNaturwissenschaften
–Dr.rer.nat.–
VorgelegtimFachbereich3(Mathematik&Informatik)
derUniversit¨atBremen
imJuni2007
Datum
esd
Promotionskolloquiums:
ErsterGutachter:
ZweiterGutachter:
22.
inuJ
2007
Prof.Dr.Hans-EberhardPorst
Prof.Dr.AlexanderˇSostak
(Universit¨at
(Universit¨at
)enemBr
Lettlands)
tsentCon
Abstract
tsledgemenwknoAc
ntioductroIn
1Categoriesoflattice-valuedsetsascategoriesofarrows
1.1ThecategoryX(A)ofA-valuedobjects....................
1.1.1Someexamples..............................
1.1.2Basicsubcategories............................
1.2TopologicalpropertiesofX(A).........................
1.3Arelationbetweenthefunctors(−)∗and(−)◦................
2OnageneralizationofGoguen’scategorySet(L)
2.1ThecategoryX(A)anditstopologicalproperties...............
2.2Anexampleofanon-topologicalcategoryX(A)................
2.3OnconcretecartesianclosednessofX(A)...................
2.4X(A)isnotatopos...............................
2.5OnrepresentabilityofpartialmorphismsinX(A)...............
2.6X(A)isaquasitopos...............................
2.6.1TheinnerstructureofX(A).......................
2.6.2ArelationbetweenthestructuresgeneratedbyΩandΔ.......
2.7SomeremarksonrepresentabilityofpartialmorphismsinX(A).......
3Aspectsofcommacategories
3.1DefinitionofthecommacategoryX∗(A)anditsalgebraicproperties....
3.2AfactorizationstructureforsourcesonX∗(A)................
3.3CoalgebraicpropertiesofX∗(A)........................
3.4Afactorizationstructure∗forsinksonX∗(A).................
3.5MonadicpropertiesofX(A)..........................
3.6SomeremarksonthemonadT.........................
iii
v
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557901812112326282922323367314417405456585
4Onfuzzificationofalgebraicandtopologicalstructures
4.1Fuzzificationmachineryforalgebraicstructures.......
4.2Fuzzificationmachineryfortopologicalstructures......
5Quantalemodules
5.1DefinitionofthecategoryQ-ModofQ-modules..
5.2Fromquantalemodulestotopologicalspaces....
5.3Q-Modisamonadicconstruct............
5.4OnsomespecialmorphismsinQ-Mod........
5.5Q-Modisnotanabeliancategory..........
5.6Quantalemodulesdonotformatopos........
5.7Tensorproductofquantalemodules.........
5.8Quantale-valuedpower-setfunctors..........
5.9FactorizationstructuresonQ-Mod..........
5.10Completionofpartiallyorderedsets.........
yBibliograph
iv
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92
Abstract
Thetheoryoffuzzystructureshasbeendevelopingratherfastrecently.Inparticular,
[22,23]presentdifferentapproachestothefoundationsoffuzzysets.Byanalogywiththe
aforesaidconceptwepresentthenotionofafuzzyobjectinacategory.
AfuzzysetintroducedbyZadehin[62]isasetequippedwithafunctiontotheunit
finterval[0,1],i.e.,justamapX✲[0,1].In[18]Goguenreplacedtheunitintervalby
fanarbitrary(butfixed)partiallyorderedsetLandconsideredmapsoftypeX✲L
calledL-fuzzysets.LateronsomeauthorsputastructureonX,e.g.,Rosenfeldin[45]
usedgroups.Someobviousgeneralizationsarise,namely:
•considerso-calledlattice-valuedsets,i.e.,allowchangeofbasisL;
•usedifferentlattice-theoreticalstructuresinsteadofaposet,e.g.,quantalesoreven
quantaloids(see,e.g.,[46,47]);
•usedifferentmathematicalstructuresinsteadofaset.
Bearingtheaforesaidideasinmindweproceedasfollows:givenaconcretecategory(A,U)
foverX,consideranA-valuedobjecttobeanU-structuredarrowX✲UA.IfAis
equippedwiththestructureof2-categoryandthefunctorUisadjointonegetsaconcrete
categoryX(A)ofA-valuedobjects.Ouraimistostudypropertiesofthiscategory.
InthefirstchapterweshowthenecessaryandsufficientconditionsforX(A)tobe
topological.ThenexttwochaptersaredevotedtothestudyoftwosubcategoriesofX(A).
OneofthemgeneralizestheGoguen’scategorySet(L)ofL-fuzzysetswithafixedbasisL
(see,e.g.,[18]),theotheristhecommacategory(idX↓U).Theformersubcategorygives
risetoafuzzificationprocedureofalgebraicandtopologicalstructuresconsideredinthe
subsequentchapter.Thelastchapterisdevotedtothecategoryofquantalemodules(see,
v
e.g.,[46,47])
constructions
asaparticularrealizationof
nda
stsulre
mrof
het
category
thecategoryA.Thechapter
ofmodules
vi
revo
a
gnir.
si
atedotivm
yb
tsledgemenwknoAc
Iwouldliketothankallwhohelpedmeinpreparationofthethesis.Specialthanksaredue
tomysupervisorprof.A.ˇSostakswhowasthefirstmantoguidemethroughmyresearch.
I’malsogratefultotheteamoftheUniversityofBremen,Germany.Inthefirstplaceto
professorsH.-E.PorstandH.Herrlichaswellasallparticipantsoftheseminar”KatMAT”
formanyusefulsuggestionsandremarks.SpecialthanksaretoC.Schubert,C.Dzierzonand
K.Freundfortheirpatienceinansweringmyquestions.Aninterestingexampleconcerning
T0-spaceswassuggestedbyprof.R.-E.HoffmannwhenIwasparticipatingathisseminar
”OrderedSetsandLattices”.
Duringthe”SummerSchoolonGeneralAlgebraandOrderedSets2005”prof.J.Paseka
ofMasarykUniversityinBrno,ChechRepublicgavemesomegoodadvicesonquantale
modulesaswellassentmehishabilitationthesisonthistopic.
ThefinancialsideofmyresearchwassuppliedbytheEuropeanSocialFund(ESF).
Riga,Latvia
200727,yMa
vii
SergejsSolovjovs
ductiontroIn
Itisawell-acceptedobservationthattherealworldproblemsinvolvedifferentkindsof
vagueenvironments.Tofacethechallengeweneednewmathematicalconceptsor,quoting
Zadeh[61],
”...aradicallydifferentkindofmathematics,themathematicsoffuzzyorcloudyquan-
titieswhicharenotdescribedintermsofprobabilitydistributions.”
ThiswasZadeh’smainmotivationwhenhestartedfuzzysetsin[62]asfollows:
”Afuzzyset(class)Ain[agivenset]Xischaracterizedbyamembership(characteristic)
functionfA(x)whichassociateswitheachpointxinXarealnumberintheinterval
[0,1],withthevalueoffA(x)atxrepresentingthe”gradeofmembership”ofxinA.”
ThenextstepwasdonebyGoguenin[18]wherehewrote:
”Afuzzysetisasettogetherwithafunctiontoatransitivepartiallyorderedset
(hereaftercalledaposet);afuzzysetisthereforeasortofgeneralizedcharacteristic
function.WehabituallydenotetheposetbyLandcallthefuzzysetanL-fuzzysetor
.”set-Lan
ByanalogywiththecategorySetofsetsGoguenconsidersthecategorySet(L)ofL-sets
withafixedbasisL.
Anotherimportantnotionof[18]isthePrincipleofFuzzificationwhichsaysthat”a
fuzzy(orL-fuzzyorL-)somethingisanL-setofsomethings(i.e.,anL-fuzzysetontheset
ofsomethings)”.AsaresultRosenfeldin[45]fuzzifiedthenotionsofgroupoidandgroup,
butChang[11]andLowen[32]consideredfuzzytopologicalspaces.
StartingwithHutton[27],Rodabaugh[40],Eklund[15],andothers,thefollowingideas
eisar
•considerso-calledlattice-valuedsets,i.e.,allowthechangeofbasisLalongwiththe
changeofset;
1
2
•usedifferentlattice-theoreticalstructuresinsteadofposets,e.g.,quantalesoreven
quantaloids(see,e.g.,[46,47]);
•fuzzifydifferentmathematicalstructures,e.g.,modules,fields(incaseoftherealline
Rtheconceptoffuzzynumberappears[42])orcategories[59].
Thisleadstoanaturalgeneralizationofthenotionoffuzzyset,namely,tothenotionof
fuzzyobjectincategoryintroducedinthisthesisasfollows.Startwithaconcretecategory
(A,U)overX.
Definition.Afuzzy(A-fuzzyorA-valued)objectinthecategoryXisanU-structured
arrowXf✲UA.
Followingthestandardterminologyofthefuzzycommunity(see,e.g.,[25])wepreferthe
termA-valued(cf.lattice-valued)object.IfAisequippedwiththestructureof2-category
andthefunctorUisadjointonegetstheconcretecategoryX(A)ofA-valuedobjects
(Definition1.1.3).Thedefinitionallowsonetoconsiderdifferentrealizationsofthecategory
A,forexample,suchunexpectedcategoriesasthecategoryTopoftopologicalspacesor
thecategoryGrpofgroups(Examples1.1.9and1.1.12).Ouraimistostudypropertiesof
thecategoryX(A).
InthefirstchapterweshowthenecessaryandsufficientconditionsforX(A)tobe
topological(Proposition1.2.24).Inaword,theexistenceofcertainfunctorsisrequired.
SincethepropertyofbeingtopologicalprovidesthecategoryX(A)withmanyfeaturesof
itsbasecategory,i.e.,thecategoryX×A,onecanputtherequirementsasaxiomson
X(A).Thechapterendsbyrelationsbetweentheaforesaidfunctors.
ThenexttwochaptersaredevotedtothestudyoftwosubcategoriesofX(A).Thefirst
onegeneralizesGoguen’scategorySet(L)(andthereforeisdenotedbyX(A)),thesecond
oneisthecommacategory(idX↓U)(denotedbyX∗(A)).Weshowthenecessaryand
sufficientconditionsforX(A)tobeaquasitopos(Proposition2.6.1).Itfollowsthatas
suchthecategoryhasanadditionalrichinnerstructure(Definitions2.6.8–2.6.16).The
mainresultsforthecategoryX∗(A)includethenecessaryandsufficientconditionstobe
algebraic(coalgebraic)andmonadic(Propositions3.1.14,3.3.9and3.5.3).Wealsoconsider
factorizationstructuresonX∗(A)(Propositions3.2.6and3.4.5).
3
WecontinuebyconsideringageneralizationoftheaforesaidPrincipleofFuzzification.
Theissuehasarichhistorysincethenotionoffuzzysetinducedfuzzificationofdiffer-
entmathematicalstructures.WehavealreadymentionedL-topologicalspaces[11,24,32]
andL-groups[45].Bothapproachesarefixed-basedanduseimplicitlyGoguen’scategory
Set(L).Avariable-basisapproachoverthecategoryofsemi-quantales(whichisgood
enoughsincethecategoriesoffuzzifiedstructuresaretopologicalo