基于布尔矩阵的Scott拓扑子基（英文）-豆柴文库

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基于布尔矩阵的Scott拓扑子基（英文）
Introduction
Thetheoryoftopologyisanimportantconceptinmathematicsthatinvestigatesthepropertiesofobjectsthatremainunchangedwhentheyarestretchedordeformed.Oneofthefundamentalconceptsintopologyisthenotionofasubbasis.Inatopology,asubbasisisacollectionofsetsthatgeneratethetopologybytakingfiniteintersectionsandarbitraryunions.OnenotablesubbasisistheScottsubbasis,whichisthecollectionofallsetsoftheform[a,∞),whereaisanelementinapartiallyorderedset.Inthispaper,weexploretheuseofBooleanmatricestogeneratetheScotttopologyonapartiallyorderedset.
BooleanMatrices
ABooleanmatrixisamatrixwhereeachelementiseither0or1.Booleanmatricesarecommonlyusedincomputersciencetorepresentlogicalrelationsandoperations.Intopology,Booleanmatriceshavebeenusedtodescribetherelationsbetweenelementsinapartiallyorderedset.Givenapartiallyorderedset,wecandefineaBooleanmatrixwherethe(i,j)thentryis1ifandonlyifiislessthanorequaltojinthepartialorder.
PartialOrderandScottTopology
Apartialorderisabinaryrelationthatisreflexive,antisymmetric,andtransitive,oftendenotedas≤.Inapartiallyorderedset,twoelementscaneitherberelatedbylessthanorequalto(≤)ornotrelatedbyit.Achaininapartiallyorderedsetisasubsetoftheorderedsetwhereanytwoelementsarecomparablebytherelation≤.Anantichainisasubsetwherenotwoelementsarecomparable.
Atopologyonapartiallyorderedsetisgeneratedbyasubbasisofsetsthatsatisfycertainproperties.TheScotttopologyonapartiallyorderedsetisgeneratedbytheScottsubbasis,whichisacollectionofallsetsoftheform[a,∞),whereaisanelementinthepartiallyorderedset.AsubsetXofapartiallyorderedsetissaidtobeScott-closedif,foranychainSinX,thesupremumofSexistsinX.AsubsetXissaidtobeScott-openifitscomplementisScott-closed.TheScotttopologyisthetopologygeneratedbythecollectionofallScott-opensets.
UsingBooleanMatricestoGenerateScottTopology
Givenapartiallyorderedset,wecandefineitscorrespondingBooleanmatrixanduseittogeneratetheScotttopology.LetPbeapartiallyorderedsetandletBbeitscorrespondingBooleanmatrix.TheScottsubbasisisgivenb