!!!!!!A categorical view of algebraic lattices in formal concept analysis (2006)

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1、ACategoricalViewonAlgebraicLatticesinFormalConceptAnalysisPascalHitzler1MarkusKrötzsch2Guo-QiangZhang31InstitutAIFB,UniversitätKarlsruhe,Germany.2FakultätfürInformatik,TechnischeUniversitätDresden,Germany.3DepartmentofElectricalEngineeringandComputerScience,CaseWesternReserveUniversi

2、ty,Cleveland,Ohio,U.S.A.AbstractFormalconceptanalysishasgrownfromanewbranchofthemathemat-icalfieldoflatticetheorytoawidelyrecognizedtoolinComputerScienceandelsewhere.Inordertofullybenefitfromthistheory,webelievethatitcanbeenrichedwithnotionssuchasapproximationbycomputationorrep-resenta

3、bility.Thelatterarecommonlystudiedindenotationalsemanticsanddomaintheoryandcapturedmostprominentlybythenotionofalgebraicity,e.g.oflattices.Inthispaper,weexplorethenotionofalgebraicityinfor-malconceptanalysisfromacategory-theoreticalperspective.Tothisend,webuildonthethenotionofapproxi

4、mableconceptwithasuitablecategoryandshowthatthelatterisequivalenttothecategoryofalgebraiclattices.Atthesametime,thepaperprovidesarelativelycomprehensiveaccountoftherepresentationtheoryofalgebraiclatticesintheframeworkofStonedual-ity,relatingwell-knownstructuressuchasScottinformations

5、ystemswithfurtherformalismsfromlogic,topology,domainsandlatticetheory.1IntroductionAlgebraiclatticesconvenientlyrepresentcomputationallyrelevantproperties.Aspartialorderstheyallowfortheexpressionofamountsofinformationcon-tent.Distinguishedelements—calledcompactorfinite—standforcompu-t

6、ationallyrepresentableinformation.Everyelementorinformationitemnotdi-rectlyrepresentablecanbeapproximatedbyrepresentable,i.e.compact,items.1Soalgebraiclatticescanbeidentifiedascomputationallyrelevantstructures,andassuchhavefoundapplicationsinComputerScience,mostprominentlyinthetheoryo

7、fdenotationalsemantics,domaintheory(see,e.g.[AJ94]),butre-centlyalsoinaspectsregardingknowledgerepresentationandreasoning(seee.g.[RZ01,ZR04,Hit04]).Ascanbeexpectedfromrichmathematicalstructuressuchasalgebraiclat-tices,amultitudeofpossiblecharacterizationshavebeenestablished,rangingfr

8、omtheclassic