monotone function in nLab

A function between preordered sets is called monotone if it respects the (pre)ordering. When preordered sets are regarded as categories (namely: ... nLab monotonefunction SkiptheNavigationLinks| HomePage| AllPages| LatestRevisions| Discussthispage| Monotonefunctions Context (0,1)(0,1)-Categorytheory (0,1)-categorytheory:logic,ordertheory (0,1)-category proset,partiallyorderedset(directedset,totalorder,linearorder) top,true, bottom,false monotonefunction implication filter,interval lattice,semilattice meet,logicalconjunction,and join,logicaldisjunction,or compactelement latticeofsubobjects completelattice,algebraiclattice distributivelattice,completelydistributivelattice,canonicalextension hyperdoctrine first-order,Boolean,coherent,tripos (0,1)-topos Heytingalgebra regularelement Booleanalgebra frame,locale Theorems Stoneduality Editthissidebar Monotonefunctions Idea Definition Incomponents Category-theoretic Relatedconcepts Idea Afunctionbetweenpreorderedsetsiscalledmonotoneifitrespectsthe(pre)ordering. Whenpreorderedsetsareregardedascategories(namely:(0,1)-categories)thenmonotonefunctionsareequivalentlythefunctorsbetweenthese. Definition Incomponents LetSSandTTbepreorderedsets,thatissetsequippedwithareflexiveandtransitivebinaryrelation≤\leq.(Byconvention,thesamesymbolisusedforbothsets,eventhoughtechnicallyitisnotthesamerelation.) ThenafunctionfffromSStoTTismonotone(increasing),isotone,weaklyincreasing,ororder-preservingifitpreserves≤\leq: x≤y⇒f(x)≤f(y)x\leqy\;\Rightarrow\;f(x)\leqf(y) forallx,yx,yinSS. Astrictlyincreasingfunctionisaweaklyincreasingfunctionthatisalsoinjective,atleastifSSandTTarepartiallyordered.Betweenarbitrarypreorderedsets,however,itisprobablybettertoacceptasstrictlyincreasinganyweaklyincreasingfunctionthatisweaklyinjectiveinthatx≤yx\leqywheneverf(x)=f(y)f(x)=f(y);suchafunctionmustbeinjectiveifSSisapartialorder(sincey≤xy\leqxwillalsofollow)butnotnecessarilyingeneral. Afunctionffismonotonedecreasing,antitone,weaklydecreasing,ororder-reversingifitreverses≤\leq: x≤y⇒f(y)≤f(x)x\leqy\;\Rightarrow\;f(y)\leqf(x) forallx,yx,yinSS. Astrictlydecreasingfunctionisaweaklydecreasingfunctionthatisalso(weakly)injective. Category-theoretic Asapreorderedsetisthesamethingasacategoryinwhichanytwoparallelmorphismsareequal,soamonotonefunctionissimplyafunctorbetweensuchcategories.Anantitonefunctionisacontravariantfunctor.That‘monotone’maybeusedforbothmatchesthat‘functor’maybeusedforbothcovariantandcontravariantfunctors. Strictlyincreasing(andstrictlydecreasing)functionsareparticularlyimportantbetweenlinearlyorderedsets,wherearethemostnaturalkindofmorphism.Betweenpartiallyorderedsetsingeneral(andbetweenpreorderedsetsusingthestricterdefinition),thestrictlyincreasingfunctionsaresimplythemonomorphisms(ifweaklyincreasingfunctionsaretakenasthemorphisms).Ifweusetheweakerdefinitionbetweenpreorderedsets,thenthestrictlyincreasingfunctionscorrespondtopseudomonicfunctors,whichisanappropriatesortofhighermonomorphism;thisisonereasonforpreferringthatdefinition. ThealternativesortofmonotonefunctiononasingleprosetSSisratherdifferent;wementionitherelargelybecauseofthepotentialterminologicalconfusion,butitmightaswellhaveitsownarticleifwefindanicenameforit.Asafunctor,itisafunctorforwhicheveryobjectisanalgebra;theconditionispartoftherequirementsofaMooreclosure(amonadonSS). Relatedconcepts enrichedmonotone injectivefunction surjectivefunction equivariantfunction oppositeposet LastrevisedonJune9,2022at19:35:10. Seethehistoryofthispageforalistofallcontributionstoit. EditDiscussPreviousrevisionChangesfrompreviousrevisionHistory(15revisions) Cite Print Source