Table Of Content

I CENTRE DEMATHEMATIGUE SOCIALE \l‘ FECOLE DES HAUTES E‘TuDEs EN SCIENCES SOCIALES‘ E'OMBINATORICS ”E‘GRAPHS AND-ALGEBRA . fi / w / fi / fi / fi / é / M / wf‘? MOUTON-THEHAGUE-_PARIS combinatorics graphsandalgebra methods and models in the social sciences MOUTON-THEHAGUE-PARIS combinatorics graphs and algebra byCENTREDEMATHEMATIQUESOCIALE ECOLEDESHAUTESETUDESENSCIENCESSOCIALES MOUTON-THEHAGUE-PARIS Thisvolumeha:beenpublilhedwiththerapport ‘ oftheDepartmentofSocialScience:ofUnma,Pan: ISBN:9027975116 0 OriginalFrenchedition:UNESCO Entitled:Cambimtalre,gmphexetalgébre PublishedbyMoutonandGauthier-Villas,Paris—TheHague,incolhhorationwith thefinaledosHaukesElude:enSciencesSociales. 0 Englishedition:UNESCO.1976 TnnslatedfromtheFrenchby:VictoriaVOI'IHagen Cove!designbyJum'unSchrofer PrintedIntheNethzrlandr preface The purpose ofthis manual, is tointroducestudentsandresearchersinthe humanandsocialsciencestosomemathematicaldomainsofparticularimpor- tanceL oftheir, ’' _ " inthese Thisfirstvolume isdevotedtofundamentalchaptersinalgebraandcombiv natorics.Ithasbeenconceivedasanintroductiontothemeaningandproper usageofcertainkeywordswhichareencounteredmoreandmorefrequentlyin theconstructionofmodelsorinthepresentationofmathematizedtheoriesin psychology, sociology, anthropology, linguistics ormusicalcomposition.The indexoftermsfoundattheendofthismanualisthusofmajorimportance:for eachterm,thisindexrefersthereadertoadefinitionandtothechaptersinthis textwherethetermhasbeenusedinvariouscontexts. ‘ These contexts... what are they? Those found in the most elementary ‘structures‘ which can beassignedafinitesetorasetwhichcanbe-finitely constructed, structures of ordering, classifications, trees, Boolean algebras, groups, ' .' and scales. “ , these topics are nowpartofwhatiscommonlytaughtinmanyuniversities,especiallyinFrench highereducation,tostudentsinthesocialsciences. Thevariousarticlesinthismanualhavebeenwrittensothattheycanberead independently: thatis,theorderinwhichtheycanbereadisoflittleimpor- tanceandislefttothediscretionofthereader.Nevertheless,wehavemade everyefforttoshowtheinterrelationshipsthatunitethedomainsintroducedin thisvolumebyreferringthereadertovariousotherarticles.Inaddition,the bibliographyofeachchapterfurnishesreferenceswhichwillpermitthereaderto improveonthisbriefintroductiontothesemathematicaldomains,aswellas acquaintinghim with the detailedapplicationsofthesetopicsin thehuman sciences. Thereaderisnot ,, ’tohave, ' ‘L ' " ’ 1,,other 6 Preface than that normally acquiredinhighschool.However,thosereaders whoen- countertoomuchdifficultycanrefertointroductorytextssuchasthosealready publishedbycertainauthorsofthismanual‘.Intheseworksthereaderwillfind theelementarybasesofcombinatorics(subsetsofafinitesetandtheirsimplicial organization)towhichthearticlesheresometimesreferr The readercanalso consultthesymbolindexplacedatthebeginningofthismanual. M.Burbut 1.M.-Barbut,C.d’Adhémnr,B. LeclercandP.Jullien,Malhéman'quuélémenminr. applicanorrsa la statisn'que e!aux sciences miller. Paris, P.U.F.. 1973:M. Barbut, MarhérmquesdessciencesdcI'Itomme.Paris,P.U.F.,I968;G.Th.Guilbuud,Mnthéma- Driquuneord,,P1a9ri6s5,.l’.U.F.,1966;P.RosenstiehlandI.Mothes,Marhlmariq‘uesd:I‘m-11ml'Paris' contents Preface,byM.Barbut ........................... 5 Symbolindex ............................... 9 1.Trees byP.RosenstiehlandB.lgclerc Introduction ......... 15 1.Treewords .............. 16 2(Rootedsemi-lattices 31 3.Modulesontrees ............................ 39 Bibliography .............................. 55 2.Orderingandchasificalion byB.Monjardet Introduction ............................... 59 1.Quasiorderings 60 2.Equivalencerelations .......................... 68 34Orderings ................................ 7S 4.Quasiorderingsrevisited ........................ 84 5.Someapplications ........................... 87 Bibliography .......-......... 90 8 Contents 3.Booleanalgebras,Boole-nrings byG.Th.Guilbaud 1.ThepresentstatusofBooleanalgebras ................. 2.DescriptionofaBooleanalgebra ............ 3.Booleanrings .............................. 4.Simplicialobjects byG.Th.Guilbaud 1.Thefirstmodel:simplecomplexorsimplex _ .............. 107 2.Someothermodels ...................‘........ 112 3.Simplicialcategory(abstractcomplex) ................. 118 4.Simplicialsets(setcomplexes) ....... 5.Productsofsirnplexes ...... Bibliographicnotice ........ 5.Monoidsandgroups byC.d’Adhémar Introduction ............................... l27 l.Monoids ................................ 128 2.Groups ................................. 134 3.Homomorphisrns ............................ 139 4.Permutations .............................. 141 Bibliography ............................... 145 6.Measurescales byM.Barbut Introduction ............................... 147 1.Elementsofthetypologyofordinalscales(typesoforder) ....... 149 2.Theadditivecalculus(abeliangroups) ................. 152 3.Additivenumericalscales(totallyorderedabelians) ........... 159 4.Theadditiveandmultiplicativescales ...... .'.......... 165 S.Intervalscales.Affinescales ..................... '. 171 Bibliography ............................... 176 Subjectindex..........................‘...... 178