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1、ACategoricalViewonAlgebraicLatticesinFormalConceptAnalysisPascalHitzler1MarkusKr?tzsch2Guo-QiangZhang31InstitutAIFB,Universit?tKarlsruhe,Germany.2Fakult?tfürInformatik,TechnischeUniversit?tDresden,Germany.3DepartmentofElectricalEngineeringandComputerScience,CaseWesternReserveUniversi
2、ty,Cleveland,Ohio,U.S.A.AbstractFormalconceptanalysishasgrownfromanewbranchofthemathemat-ical?eldoflatticetheorytoawidelyrecognizedtoolinComputerScienceandelsewhere.Inordertofullybene?tfromthistheory,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—calledcompactor?nite—standforcompu-t
6、ationallyrepresentableinformation.Everyelementorinformationitemnotdi-rectlyrepresentablecanbeapproximatedbyrepresentable,i.e.compact,items.1Soalgebraiclatticescanbeidenti?edascomputationallyrelevantstructures,andassuchhavefoundapplicationsinComputerScience,mostprominentlyinthetheoryo
7、fdenotationalsemantics,domaintheory(see,e.g.[AJ94]),butre-centlyalsoinaspectsregardingknowledgerepresentationandreasoning(seee.g.[RZ01,ZR04,Hit04]).Ascanbeexpectedfromrichmathematicalstructuressuchasalgebraiclat-tices,amultitudeofpossiblecharacterizationshavebeenestablished,rangingfr
8、omtheclassic
