On a categorical generalization of the concept of fuzzy set [Elektronische Ressource] / Sergejs Solovjovs
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On a categorical generalization of the concept of fuzzy set [Elektronische Ressource] / Sergejs Solovjovs

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ONACATEGORICAL GENERALIZATION OF THECONCEPT OF FUZZY SETSergejs SolovjovsDissertationzur Erlangung des Grades eines Doktorsder Naturwissenschaften– Dr. rer. nat. –Vorgelegt im Fachbereich 3 (Mathematik & Informatik)der Universitat¨ Bremenim Juni 2007Datum des Promotionskolloquiums: 22. Juni 2007Erster Gutachter: Prof. Dr. Hans-Eberhard Porst (Universitat¨ Bremen)ˇZweiter Gutachter: Prof. Dr. Alexander Sostak (Universitat¨ Lettlands)ContentsAbstract vAcknowledgements viiIntroduction 11 Categories of lattice-valued sets as categories of arrows 51.1 The category X(A)ofA-valuedobjects.................... 51.1.1 Someexamples.................. 71.1.2 Basicsubcategories............................ 91.2 Topological properties of X(A)............. 10∗ ◦1.3 A relation between the functors (−) and (−) ................ 182 On a generalization of Goguen’s category Set(L) 212.1 The category X(A)anditstopologicalproperties............... 212.2 An example of a non-topological category X(A)........ 232.3 On concrete cartesian closedness of X(A)................... 262.4 X(A)isnotatopos ....................... 282.5 On representability of partial morphisms in X(A)............... 292.6 X(A)isaquasitopos....................... 322.6.1 The inner structure of X(A)................... 322.6.2 ArelationbetwenthestructuresgeneratedbyΩandΔ... 362.7 Some remarks on representability of partial morphisms in X(A)....... 373 Aspects of comma categories 41∗3.

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Publié par
Publié le 01 janvier 2007
Nombre de lectures 37
Langue English

Extrait

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

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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

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stsulre

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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

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