Available online at www.sciencedirect.comFuzzySetsandSystems143(2004)526www.elsevier.com/locate/fssTriangularnorms. PositionpaperI: basicanalyticalandalgebraicpropertiesErichPeterKlementa;, RadkoMesiarb, EndrePapcaDepartmentofAlgebra, StochasticsandKnowledge-BasedMathematical Systems,JohannesKeplerUniversity, 4040Linz, AustriabDepartmentofMathematicsandDescriptiveGeometry, FacultyofCivil Engineering,SlovakTechnical University, 81368Bratislava, SlovakiacDepartmentofmathematicsandInformatics, UniversityofNovi Sad, 21000Novi Sad, SerbiaandMontenegroReceived14May2002; receivedinrevisedform30October2002; accepted2June2003AbstractWepresentthebasicanalyticalandalgebraicpropertiesoftriangularnorms. Wediscusscontinuityaswellas theimportant classes of Archimedean, strict andnilpotent t-norms. Triangular conorms andDeMorgantriplesarealsomentioned. Finally, abriefhistorical surveyontriangularnormsisgiven.c 2003ElsevierB.V. All rightsreserved.Keywords:Triangularnorms1. IntroductionTriangular norms (brieyt-norms) are anindispensable tool for the interpretationof the con-junctioninfuzzylogics [27] and, subsequently, for theintersectionof fuzzysets [67]. Theyare,however, interestingmathematical objectsforthemselves.Triangular norms, as we use themtoday, were rst introducedinthe context of probabilisticmetricspaces[54,57,58], basedonsomeideaspresentedin[43] (seeSection7for details). Theyalsoplayanimportant roleindecisionmaking[21,26], instatistics[47]aswell asinthetheoriesofnon-additivemeasures[39,50,61,64]andcooperativegames[11].Someparameterizedfamiliesoft-norms(see, e.g. [22])turnout tobesolutionsofwell-knownfunctional equations.Algebraicallyspeaking, t-normsarebinaryoperationsontheclosedunit interval [0,1] suchthat([0; 1]; T; 6)isanabelian, totallyorderedsemigroupwithneutral element 1[28]. Correspondingauthor. Tel.: +43-732-2468-9151; fax: +43-732-2468-1351.E-mail addresses:ep.klement@jku.at (E.P. Klement), mesiar@math.sk(R. Mesiar), pap@im.ns.ac.yu,pape@eunet.yu(E. Pap).0165-0114/$ - seefront matter c 2003ElsevierB.V. All rightsreserved.doi:10.1016/j.fss.2003.06.0076 E.P. Klementetal. / FuzzySetsandSystems143(2004)526For the closelyrelatedconcept of uninorms (whichturn[0,1] intoanabelian, totallyorderedsemigroupwithneutral element e ]0; 1[)see[38,66].Arecent monograph[38] providesarather completeoverviewabout triangular normsandtheirapplications.Inaseriesofthreepaperswewant tosummarizeinacondensedformthemost important factsabout t-norms. ThisPart I dealswiththebasicanalytical properties, suchascontinuity, andwithimportant classes such as Archimedean, strict and nilpotent t-norms. We also mention the dualoperations,thetriangularnorms,andDeMorgantriples.Finallywegiveashorthistoricaloverviewonthedevelopment oft-normsandtheirwayintofuzzysetsandfuzzylogics.Tokeepthepaper readable, wehaveomittedall proofs(usuallygivingasourcefor thereaderinterestedinthem)andratherincludedanumberof(counter-)examples,inordertomotivateandtoillustratetheabstract notionsused.Part II will be devoted to general construction methods based mainly on pseudo-inverses, additiveandmultiplicative generators, andordinal sums, addingalsosome constructions leadingtonon-continuoust-norms, andtoapresentationofsomedistinguishedfamiliesoft-norms.Finally, Part III will concentrateoncontinuoust-norms, inparticular, ontheir representationbyadditiveandmultiplicativegeneratorsandordinal sums.2. Triangular normsThetermtriangularnormappearedforthersttime(withslightlydierentaxioms)in[43]. Thefollowing set of independent axioms for triangular norms goes back to Schweizer and Sklar [5361].Denition 2.1. Atriangular norm(briey t-norm) is a binary operation T on the unit interval[0, 1] which is commutative, associative, monotone and has 1 as neutral element, i.e., it is a functionT : [0; 1]2 [0; 1]suchthat forall x; y; z [0; 1]:(T1) T(x; y) =T(y; x),(T2) T(x; T(y; z)) =T(T(x; y); z),(T3) T(x; y)6T(x; z) whenevery6z,(T4) T(x; 1) =x.Sinceat-normisanalgebraicoperationontheunit interval [0,1], someauthors(e.g., in[48])prefertouseaninxnotationlikex yinsteadoftheprexnotationT(x; y). Infact, someoftheaxioms(T1)(T4)thenlookmorefamiliar: forall x; y; z [0; 1](T1) x y =y x,(T2) x (y z) =(x y) z,(T3) x y6x z whenevery6z,(T4) (x 1) =x.Becauseoftheimportanceofsomefunctionalaspects(e.g.,continuity)andsinceweprefertokeepauniednotationthroughout this paper, weshall consistentlyusetheprexnotationfor t-norms(andt-conorms).E.P. Klementetal. / FuzzySetsandSystems143(2004)526 700.250.5 0.50.50.50.250.50.7510.250.50.7510.250.50.7510.250.50.7510.50.75100.250.50.75100.250.50.75100.250.50.751TMTPTLTD0 0.25 0.5 0.75 100.250.50.7510 0.25 0.5 0.75 100.250.50.7510 0.25 0.5 0.75 100.250.50.75100.250.50.7510 0.25 0.5 0.75 1Fig. 1. 3Dplots(top)andcontourplots(bottom)ofthefourbasict-normsTM; TP; TL, andTD(observethattherearenocontourlinesforTD).Since t-norms are obviously extensions of the Boolean conjunction, they are usually used asinterpretationsoftheconjunctionin[0, l]-valuedandfuzzylogics.There exist uncountably many t-norms. In [38, Section 4] some parameterized families of t-normsarepresentedwhichareinterestingfromdierent pointsofview.The following are the four basic t-norms, namely, the minimumTM, the product TP, theLukasiewiczt-normTL, andthedrasticproduct TD(seeFig. 1for 3Dandcontour plots), whicharegivenby, respectively:TM(x; y) = min(x; y); (1)TP(x; y) = x y; (2)TL(x; y) = max(x + y 1; 0); (3)TD(x; y) =_0 if (x; y) [0; 1[2;min(x; y) otherwise:(4)These four basic t-norms are remarkable for several reasons. The drastic product TD and the minimumTMarethesmallest andthelargest t-normrespectively(withrespect tothepointwiseorder). TheminimumTMistheonlyt-normwhereeachx [0; 1]isanidempotentelement(compareDenition6.1), whereas the product TPand the Lukasiewicz t-normTLare prototypical examples of twoimportantsubclassesoft-norms, namely, oftheclassesofstrictandnilpotentt-norms, respectively.It should be mentioned that the t-norms TM; TP; TL, and TDwere denoted M; ; W, and Z, respec-tively, in[57].Sometimes weshall visualizet-norms (andfunctions F : [0; 1]2[0; 1] ingeneral)indierentforms: as 3Dplots, i.e., as surfaces in the unit cube, as contour plots showing the curves (or,8 E.P. Klementetal. / FuzzySetsandSystems143(2004)526more generally, the sets) where the function in question has constant (equidistant) values, and,occasionally, asdiagonal sections, i.e., asgraphsofthefunctionx F(x; x).Theboundarycondition(T4)andthemonotonicity(T3)weregivenintheir minimal form. To-getherwith(T1)it followsthat, forall x [0; 1], eacht-normTsatisesT(0; x) = T(x; 0) = 0; (5)T(1; x) = x: (6)Therefore, all t-normscoincideontheboundaryoftheunit square[0; 1]2.The monotonicity of a t-norm Tin its second component (T3) is, together with the commutativity(T1), equivalent tothe(joint)monotonicityinbothcomponents, i.e., toT(x1; y1) 6T(x2; y2) whenever x16x2and y16y2: (7)Sincet-norms arejust functions fromtheunit squareintotheunit interval, thecomparisonoft-normsisdoneintheusual way, i.e., pointwise.Denition 2.2. If,fortwot-normsT1andT2,wehaveT1(x; y)6T2(x; y)forall(x; y) [0; 1]2,thenwesaythatT1isweakerthanT2or, equivalently, thatT2isstrongerthanT1, andwewriteinthiscaseT16T2.Weshall writeT1T2if T16T2andT1 =T2, i.e., if T16T2andif T1(x0; y0)T2(x0; y0) forsome(x0; y0) [0; 1]2.Asanimmediateconsequenceof(T1),(T3)and(T4),thedrasticproductTDistheweakest,andtheminimumTMisthestrongest t-norm, i.e., foreacht-normTwehaveTD6T6TM: (8)Betweenthefourbasict-normswehavethesestrict inequalitiesTD TL TP TM: (9)Aslight modicationofaxiom(T4)leadstothefollowingnotion, introducedin[30,31].Denition 2.3. AfunctionF : [0; 1]2[0; 1] whichsatises, for all x; y; z [0; 1], the properties(T1)(T3)andF(x; y) 6min(x; y) (10)iscalledat-subnorm.Clearly, each t-normis a t-subnorm, but not vice versa: for example, the zero function is at-subnormbut not at-norm.Eacht-subnormcanbetransformedintoat-normbyredening(if necessary)itsvaluesontheupperright boundaryoftheunit square[38, Corollary1.8].E.P. Klementetal. / FuzzySetsandSystems143(2004)526 9Proposition 2.4. If F : [0; 1]2[0; 1] is a t-subnormthen the function T : [0; 1]2[0; 1]denedbyT(x; y) =_F(x; y) if(x; y) [0; 1[2;min(x; y) otherwise;isatriangularnorm.Aninterestingquestioniswhetherat-normisdetermineduniquelybyitsvaluesonthediagonalof theunit square. Ingeneral, this is not thecase, but thetwoextremal t-norms TDandTMarecompletelydeterminedbytheir diagonal sections, i.e., bytheir valuesonthediagonal of theunitsquare.The associativity(T2) allows us toextendeacht-normT (whichwas introducedas a binaryoperation)inauniquewaytoannaryoperationforarbitraryn N {0}byinduction:nTi=1xi =_1 if n = 0;T_xn;Tn1i=1xi_otherwise:(11)Wealsoshall usethenotationT(x1; x2; : : : ; xn) =nTi=1xi:If, inparticular, x1 = x2 = = xn = x, weshall brieywritex(n)T= T(x; x; : : : ; x): (12)Then-aryextensions of theminimumTMandtheproduct TPareobvious. For theLukasiewiczt-normTLandthedrasticproduct TDwegetTL(x1; x2; : : : ; xn) = max_n

i=1xi (n 1); 0_;TD(x1; x2; : : : ; xn) =_xiif xj = 1for all j = i;0 otherwise:The fact that each t-norm Tis weaker than TMimplies that, for each sequence (xi)iNof elementsof[0,1], thesequence_nTi=1xi_nNisnon-increasingandboundedfrombelowand, subsequently, convergent. WethereforecanextendTtoa(countably)innitaryoperationputtingTi=1xi =limnnTi=1xi: (13)However, similarlyas for innite series of numbers, thensome desirable properties suchas thegeneralizedassociativitymaybeviolated(formoredetailssee[44]).10 E.P. Klementetal. / FuzzySetsandSystems143(2004)5263. Triangular conormsIn[55]triangularconormswereintroducedasdualoperationsoft-norms. Wegivehereaninde-pendent axiomaticdenition.Denition 3.1. A triangular conorm (t-conorm for short) is a binary operation Son the unit interval[0,1] which is commutative, associative, monotone and has 0 as neutral element, i.e., it is a functionS : [0; 1]2 [0; 1]whichsatises, forall x; y; z [0; 1], (T1)(T3)and(S4) S(x; 0) = x:Thefollowingarethefourbasict-conorms, namely, themaximumSM, theprobabilisticsumSP,theLukasiewiczt-conormor (boundedsum)SL, andthedrasticsumSD(seeFig. 2for 3Dandcontourplots), whicharegivenby, respectively:SM(x; y) = max(x; y); (14)SP(x; y) = x + y x y; (15)SL(x; y) = min(x + y; 1); (16)SD(x; y) =_1 if (x; y) ]0; 1]2;max(x; y) otherwise:(17)Thet-conormsSM; SP; SL, andSDweredenotedM; ; WandZ, respectively, in[57].Theoriginal denitionoft-conormsgivenin[55]iscompletelyequivalent totheaxiomaticde-nitiongivenabove: afunctionS : [0; 1]2 [0; 1]isat-conormifandonlyifthereexistsat-norm00.251 0.5 0.5 0.50.50.250.50.7510.250.50.7510.250.50.7510.250.50.7510.50.7500.2510.50.7500.2510.50.7500.2510.50.75TMTPTLTD0 0.25 0.5 0.75 100.250.50.7510 0.25 0.5 0.75 100.250.50.7510 0.25 0.5 0.75 100.250.50.7510 0.25 0.5 0.75 100.250.50.751Fig. 2. 3Dplots(top)andcontourplots(bottom)ofthefourbasict-conormsSM; SP; SL, andSD.E.P. Klementetal. / FuzzySetsandSystems143(2004)526 11Tsuchthat forall (x; y) [0; 1]2eitheroneofthetwoequivalent equalitiesholds:S(x; y) = 1 T(1 x; 1 y); (18)T(x; y) = 1 S(1 x; 1 y): (19)Thet-conormgivenby(18)iscalledthedualt-conormofTand, analogously, thet-normgivenby(19)issaidtobethedual t-normof S. Obviously, (TM; SM); (TP; SP); (TL; SL), and(TD; SD)arepairsoft-normsandt-conormswhicharemutuallydual toeachother.ConsideringthestandardnegationNs(x) =1 x(compare(20))ascomplement ofxintheunitinterval, Eq. (18) explains the name t-conorm. We shall keep this original notion and avoid the terms-normwhichsometimesisusedsynonymouslyintheliterature.Thedualityexpressedin(18)allowsustotranslatemanypropertiesof t-normsintothecorre-spondingpropertiesoft-conorms, includingthenaryandinnitaryextensionsofat-conorm.Thedualitychangestheorder:if, forsomet-normsT1andT2wehaveT16T2, andifS1andS2arethedual t-conormsofT1andT2, respectively, thenweget S1S2.If (T; S)isapair of mutuallydual t-normsandt-conorms, thendualities(18)and(19)canbegeneralizedasfollows(hereI canbeanarbitraryniteorcountablyinniteindexset):SiIxi = 1 TiI(1 xi);TiIxi = 1 SiI(1 xi):Infuzzylogics, t-conormsareusuallyusedasaninterpretationofthedisjunction .4. Negations and De Morgan triplesFinally, let ushaveabrieflookat negations.Denition 4.1.(i) Anon-increasingfunctionN : [0; 1] [0; 1]iscalledanegationif(N1) N(0) = 1 and N(1) = 0:(ii) AnegationN : [0; 1] [0; 1]iscalledastrictnegationif, additionally,(N2) Niscontinuous:(N3) Nisstrictlydecreasing:(iii) Astrict negationN : [0; 1] [0; 1]iscalledastrongnegationifit isaninvolution, i.e., if(N4) N N= id[0;1]:It isobviousthat N : [0; 1] [0; 1] isastrict negationif andonlyif it isastrictlydecreasingbijection.12 E.P. Klementetal. / FuzzySetsandSystems143(2004)526Themost important andmost widelyusedstrongnegationisthestandardnegationNs : [0; 1] [0; 1]givenbyNs(x) = 1 x: (20)Note that N : [0; 1] [0; 1] is a strong negation if and only if there is a monotone bijection g : [0; 1] [0; 1]suchthat forall x [0; 1](x) = g1(Ns(g(x))); (21)i.e., eachstrongnegationisamonotonetransformationofthestandardnegation[62].ThenegationN : [0; 1] [0; 1]givenbyN(x) =1 x2isstrict, but not strong.Anexampleofanegationwhichisnotstrictand,subsequently,notstrong,isthe G odel negationNG[0; 1] [0; 1]givenbyNG(x) =_1 if x = 0;0 if x ]0; 1]:(22)The standard negation Nswas used, e.g., in [53,54] when introducing t-conorms as duals oft-norms, orin[67]whenmodelingthecomplement ofafuzzyset.Givenat-normT andastrict negationN, oneobtainsat-conormS : [0; 1]2[0; 1], whichisN-dual toTinthesenseofS(x; y) = N1(T(N(x); N(y))): (23)Note, however, that ifNisanon-strict negation, formula(23)cannot beapplied.IfNisastrongnegation, then, applyingtheconstructionin(23)tothet-conormS, wegetbackthet-normTwestartedwith.Atriple (T; S; N), where T is a t-norm, S is a t-conormandN is a negationis calleda DeMorgantripleifforall (x; y) [0; 1]2wehaveT(x; y) = N(S(N(x); N(y)));S(x; y) = N(T(N(x); N(y))):This means that, givena t-normT; (T; S; N) is a De Morgantriple if andonlyif Nis a strongnegationandSistheN-dual ofT.Let s : [0; 1] [0; 1]beastrictlyincreasingbijection. ThenS : [0; 1]2 [0; 1]denedbyS(x; y) = s1(min(s(x) + s(y); 1))is at-conorm(infact, S is anilpotent t-conormwithadditivegenerator s [38, Denition3.39]).Moreover, N : [0; 1] [0; 1]givenbyN(x) = inf {y [0; 1] | S(x; y) = 1}isastrongnegation. IfTist-normwhichisN-dual toSthenwehaveT(x; y) = s1(TL(s(x); s(y)));S(x; y) = s1(SL(s(x); s(y)));N(x) = s1(Ns(s(x)));E.P. Klementetal. / FuzzySetsandSystems143(2004)526 13which means that the De Morgan triple (T; S; N) is isomorphic to the Lukasiewicz De Morgan triple(TL; SL; Ns).Even if (T; S; N) is a De Morgan triple, we do not necessarily have T(x; N(x)) =0 and S(x; N(x))=1for all x [0; 1], i.e., the lawof the excludedmiddle (whichis one of the crucial featuresof the classical, two-valued Boolean logic) may be violated. For instance, if the t-normT intheDeMorgantriple(T; S; Ns)has nozerodivisors, i.e., if T(x; y)0whenever x0andy0(seeDenition6.1(iii)),thenthelawoftheexcludedmiddleneverholds.Ontheotherhand,intheDeMorgantriple(TL; SL; Ns)and, afortiori, ineachDeMorgantriple(T; S; Ns)withT6TL, wehaveamany-valuedanalogueoftheclassical lawoftheexcludedmiddle.It isnoteworthythat, givenaDeMorgantriple(T; S; N), thetuple([0; 1]; T; S; N; 0; 1)canneverbe a Boolean algebra: in order to satisfy distributivity we must have T =TMand S =SM(seeProposition6.18), inwhichcaseitisimpossibletohavebothT(x; N(x)) =0andS(x; N(x)) =1forall x [0; 1].5. ContinuityAscanbeseenfromthedrasticproduct TDanditsdual SD, t-normsandt-conorms(viewedasfunctions in two variables) need not be continuous (in fact, they need not, even be Borel measurablefunctions [38, Example 3.75]). Nevertheless, for a number of reasons continuous t-norms andt-conorms playanimportant role. Therefore, we shall discuss here continuityas well as left- andright-continuity.Recall that a t-normT : [0; 1]2[0; 1] is continuous if for all convergent sequences (xn)nN;(yn)nN[0; 1]NwehaveT_limnxn; limnyn_=limnT(xn; yn):Obviously, the continuityof a t-conormS is equivalent tothe continuityof the dual t-normT.Sincetheunit square[0; 1]2isacompact subset of thereal planeR2, thecontinuityof at-normT : [0; 1]2 [0; 1]isequivalent toitsuniformcontinuity.Obviously, thebasict-normsTM; TPandTLaswell astheir dual t-conormsSM; SPandSLarecontinuous, andthedrasticproduct TDandthedrasticsumSDarenot continuous.Ingeneral, areal functionoftwovariables, e.g, withdomain[0; 1]2, maybecontinuousineachvariablewithout beingcontinuouson[0; 1]2. Becauseof their monotonicity, triangular norms(andconorms)areexceptionsfromthis:Proposition 5.1. At-normT : [0; 1]2 [0; 1] iscontinuousif andonlyif it iscontinuousineachcomponent, i.e., if for all x0; y0[0; 1] both the vertical section T(x0;.) : [0; 1] [0; 1] and thehorizontal sectionT(.; y0) : [0; 1] [0; 1]arecontinuousfunctionsinonevariable.Obviously,becauseofthecommutativity(T1),forat-normorat-conormitscontinuityisequiv-alent toitscontinuityintherst component.For applications, e.g., inprobabilisticmetricspaces, many-valuedlogics or decomposablemea-sures, quite often weaker forms of continuity are sucient. Since we have a similar result as14 E.P. Klementetal. / FuzzySetsandSystems143(2004)52600.250.50.7510.500.250.50.7510 20.50.75100 0.25 0.5 0.75 100.250.50.751Fig. 3. 3Dplot (left)andcontourplot ofthenilpotent minimumTnMdenedby(24).Proposition5.1forleft-andright-continuoust-norms, thesedenitionsaregiveninonecomponentonly.Denition 5.2. At-normT : [0; 1]2[0; 1] is saidtobeleft-continuous (right-continuous)if foreachy [0; 1]andforall non-decreasing(non-increasing)sequences(xn)nNwehavelimnT(xn; y) = T_limnxn; y_:Clearly, at-normiscontinuousifandonlyifit isbothleft-andright-continuous.ThenilpotentminimumTnM(mentionedin[20,51,52], foravisualizationseeFig. 3)denedbyTnM(x; y) =_0 if x + y 61;min(x; y) otherwise(24)isat-normwhichisleft-continuousbut not right-continuous. Thedrasticproduct TD, ontheotherhand, isright-continuousbutnotleft-continuous. Anexampleofat-normwhichisneitherleft-norright-continuouscanbefoundinExample6.14(iv).Clearly, at-normT is left-continuous if andonlyif its dual t-conormgivenby(18) is right-continuous, andviceversa.6. Algebraic propertiesIn the language of algebra, Tis a t-norm if and only if ([0; 1]; T; 6) is a fully ordered commutativesemigroup with neutral element 1 and annihilator (zero element) 0. Therefore, it is natural to consideradditional algebraicpropertiesat-normmayhave.Our rst focus areidempotent andnilpotent elements, andzerodivisors. Sincefor eachn Nwetriviallyhave0(n)T=0and1(n)T=1, onlyelementsof ]0,1[ will beconsideredascandidatesfornilpotent elementsandzerodivisorsinthefollowingdenition.E.P. Klementetal. / FuzzySetsandSystems143(2004)526 15Denition 6.1. Let Tbeat-norm.(i) Anelement a [0; 1]iscalledanidempotentelementofTifT(a; a) =a. Thenumbers0and1(whichareidempotentelementsforeacht-normT)arecalled trivialidempotentelementsofT, eachidempotent element in]0, 1[will becalledanon-trivial idempotent element ofT.(ii) Anelement a ]0; 1[ iscalledanilpotent element of T if thereexistssomen Nsuchthata(n)T=0.(iii) Anelement a ]0; 1[ is calleda zerodivisor of T if there exists some b ]0; 1[ suchthatT(a; b) =0.The set of idempotent elements of the minimum TMequals [0, 1] (actually, TMis the only t-normwith this property). For the Lukasiewicz t-norm TLas well as for the drastic product TD, both the setofnilpotentelementsandthesetofzerodivisorsequal]0,1[.TheminimumTMandtheproductTPhave neither nilpotent elements nor zero divisors, and TP; TL, and TDpossess only trivial idempotentelements.The set of idempotent elements of the nilpotent minimum TnMdened in (24) equals {0}]0:5; 1],itsset ofnilpotent elementsis]0,0.5], anditsset ofzerodivisorsequals]0, 1[.Theidempotent elementsof t-normscanbecharacterizedinthefollowingway, whichinvolvestheoperationminimum[38, Proposition2.3].Proposition 6.2. (i)Anelementa[0; 1]isanidempotentelementofat-normTifandonlyifforall x [a; 1]wehaveT(a; x) = min(a; x).(ii)IfTisacontinuoust-norm,thena [0; 1]isanidempotentelementofTifandonlyifforall x [0; 1]wehaveT(a; x) = min(a; x).Remark 6.3. For arbitraryt-normssomegeneral observationsconcerningidempotent andnilpotentelementsandzerodivisorscanbeformulated.(i) Noelement of]0,1[canbebothidempotent andnilpotent.(ii) Eachnilpotent element aofat-normTisalsoazerodivisorofT, but not conversely(TnMisacounterexample).(iii) If at-normT has anilpotent element athenthereis always anelement b ]0; 1[ suchthatb(2)T=0.(iv) Ifa ]0; 1[isanilpotent element ofat-normTtheneachnumberb ]0, a[isalsoanilpo-tent element of T, i.e., theset of nilpotent elements of at-normT caneither betheemptyset (as for TMor TP) or aninterval of theform]0; c[ or ]0; c]. Thesameis truefor zerodivisors.Example 6.4. Forthet-normT[57, Example5.3.13]givenbyT(x; y) =___0 if (x; y) [0; 0:5]2;2(x 0:5)(y 0:5) + 0:5 if (x; y) ]0:5; 1]2;min(x; y) otherwise;(25)16 E.P. Klementetal. / FuzzySetsandSystems143(2004)526itsset ofnilpotent elementsanditsset ofzerodivisorsbothequal ]0,0.5], andforeachelement ofthefamily(Tc)c]0;1]oft-normsdenedbyTc(x; y) =_max(0; x + y c) if (x; y) [0; c]2min(x; y) otherwise;theset ofnilpotent elementsandtheset ofzerodivisorsofTcequal ]0; c[.Althoughtheset ofnilpotent elementsisingeneral asubset oftheset ofzerodivisors, foreacht-normthe existence of zerodivisors is equivalent tothe existence of nilpotent elements, i.e., at-normhaszerodivisorsifandonlyifit hasnilpotent elements[38, Proposition2.5].Forright-continuoust-norms(infact, theright-continuityofTonthediagonaloftheunitsquareis sucient) it is possible to obtain each idempotent element as the limit of the powers of a suitablex [0; 1][38, Proposition2.6].Proposition 6.5. LetTbeat-normwhichisright-continuousonthediagonal {(x; x) | x [0; 1]}oftheunitsquare[0; 1]2, andleta [0; 1]. Thefollowingareequivalent:(i) aisanidempotentelementofT.(ii) Thereexistsanx [0; 1]suchthata = limnx(n)T.Itiswell-knownthat,forcontinuoust-norms,itssetofidempotentelementsisaclosedsubsetoftheunit interval [0,1]. Asaconsequenceof[38, Corollary2.8], thisisalsotruefort-normswhichareright-continuous insomespecicpoints of thediagonal of theunit squareand, consequently,fort-normswhichareright-continuous:Corollary 6.6. LetTbeat-normsuchthatforeacha [0; 1[T(a; a) = a whenever limxaT(x; x) = a:ThenthesetofidempotentelementsofTisaclosedsubsetof[0,1].The t-normT givenin(25) shows that the converse implicationdoes not necessarilyholdinCorollary6.6(just considerthecasea = 0:5).Some t-norms have additional algebraic properties. The rst group of such properties centersaroundthenotionsof strict monotonicityandtheArchimedeanproperty, whichplayanimportantroleinmanyalgebraicconcepts, e.g., insemigroups.Denition 6.7. Foranarbitraryt-normTweconsiderthefollowingproperties:(i) Thet-normTissaidtobestrictlymonotoneif(SM) T(x; y) T(x; z) whenever x 0 and y z:(ii) Thet-normTsatisesthecancellationlawif(CL) T(x; y) = T(x; z) implies x = 0 or y = z:E.P. Klementetal. / FuzzySetsandSystems143(2004)526 17(iii) Thet-normTsatisestheconditional cancellationlawif(CCL) T(x; y) = T(x; z) 0 impliesy = z:(iv) Thet-normTiscalledArchimedeanif(AP) for each(x; y) ]0; 1[2thereisann Nwithx(n)T y:(v) Thet-normThasthelimitpropertyif(LP) for all x ]0; 1[: limnx(n)T= 0:Example 6.8. (i) TheminimumTMhasnoneoftheseproperties, andtheproduct TPsatisesallofthem. TheLukasiewiczt-normTLandthedrasticproduct TDareArchimedeanandsatisfythe conditional cancellationlaw(CCL) andthe limit property(LP), but none of the otherproperties.(ii) If a t-normT satises the cancellation law(CL) then it obviously fullls the conditionalcancellationlaw(CCL), but not conversely(see, e.g., TL).(iii) The algebraic properties introducedinDenition6.7are independent of the continuity: thecontinuoust-normTMshowsthat continuityimpliesnoneoftheseproperties. Conversely, TDandthenon-continuoust-normTgivenbyT(x; y) =_xy2if (x; y) [0; 1[2;min(x; y) otherwise;(26)whichisstrictlymonotoneandsatisesthecancellationlaw(CL),areexamplesdemonstratingthat noneofthealgebraicpropertiesimpliesthecontinuityofthet-normunderconsideration.Thestrictmonotonicity(SM)ofat-normisrelatedtotheotherpropertiesasfollows[38,Propo-sition2.11]:Proposition 6.9. LetTbeat-norm. Thenwehave:(i) Tisstrictlymonotoneifandonlyifitsatisesthecancellationlaw(CL).(ii) IfTisstrictlymonotonethenithasonlytrivial idempotentelements.(iii) IfTisstrictlymonotonethenithasnozerodivisors.The Archimedean property (AP) of a t-normcan be characterized in the following way [38,Theorem2.12].Proposition 6.10. Forat-normTthefollowingareequivalent:(i) TisArchimedean.(ii) Tsatisesthelimitproperty(LP).18 E.P. Klementetal. / FuzzySetsandSystems143(2004)526(iii) Thasonlytrivial idempotentelementsand, wheneverlimxx0T(x; x) = x0forsomex0]0; 1[, thereexistsay0]x0; 1[suchthatT(y0; y0) =x0.Combining the continuity with some algebraic properties, we obtain two extremely importantclassesoft-norms.Denition 6.11.(i) At-normTiscalledstrictifit iscontinuousandstrictlymonotone.(ii) At-normT is called nilpotent if it is continuous and if each a ]0; 1[ is a nilpotentelement ofT.Example 6.12. (i)Theproduct TPisastrict t-norm, andtheLukasiewiczt-normTLisanilpotentt-norm. In fact [38, Propositions 5.9, 5.10] each strict t-norm is isomorphic to TPand each nilpotentt-normisisomorphictoTL.(ii)BecauseofProposition6.9(i), at-normTisstrictifandonlyifitiscontinuousandsatisesthecancellationlaw(CL).(iii)Eachstrict andeachnilpotent t-normfulllstheconditional cancellationlaw(CCL).Thefollowingresultgivesanumberofsucientconditionsforat-normtobeArchimedean[38,Proposition2.15].Proposition 6.13. Foranarbitraryt-normTwehave:(i) IfTisright-continuousandhasonlytrivial idempotentelementsthenitisArchimedean.(ii) If T is right-continuous and satises the conditional cancellation law(CCL) then it isArchimedean.(iii) Iflimxx0 T(x; x)x0foreachx0]0; 1[thenTisArchimedean.(iv) IfTisstrictthenitisArchimedean.(v) Ifeachx ]0; 1[isanilpotentelementofTthenTisArchimedean.In[40]it wasshownthat eachleft-continuousArchimedeant-normisnecessarilycontinuous.All the implications between the algebraic properties of t-norms considered so far are summarizedandvisualizedinFig. 4. Thefollowingarecounterexamplesshowingthattherearenootherlogicalrelationsbetweenthesealgebraicproperties.Example 6.14. (i) The Lukasiewicz t-normTLshows that an Archimedean t-normneed not bestrictlymonotone, andthat thelimit property(LP)doesnot implythecancellationlaw(CL). Theproduct TPis anexample of a continuous Archimedeant-normwithout nilpotent elements. ThedrasticproductTDisanexampleofanon-continuousArchimedeant-normforwhicheacha ]0; 1[isanilpotent element.(ii)Thet-normgivenin(26)showsthatastrictlymonotonet-normneednotbecontinuousand,subsequently, not necessarilystrict.E.P. Klementetal. / FuzzySetsandSystems143(2004)526 19Fig. 4. Thelogical relationshipbetweenvariousalgebraicpropertiesoft-norms: adoublearrowindicatesanimplication,adottedarrowmeansthat thecorrespondingimplicationholdsforcontinuoust-norms.(iii)Thenon-continuoust-normgivenin(25)showsthat at-normwithonlytrivial idempotentelementsisnot necessarilystrictlymonotoneorArchimedean.(iv) At-normmay satisfy both the strict, monotonicity (SM) and the Archimedean property(AP)without beingcontinuousand, subsequently, without beingstrict. Oneexampleforthisisthet-normintroducedin(26), anothert-normwiththesefeaturesisthefollowing[10]:recallthateach(x; y) ]0; 1]2isinaone-to-onecorrespondencewithapair((xn)nN; (yn)nN)ofstrictlyincreasingsequencesofnatural numbersgivenbytheuniqueinnitedyadicrepresentationsx =

n=112xnand y =

n=112ynofthenumbersxandy, respectively. Usingthisnotion, thenthefunctionT : [0; 1]2 [0; 1]givenbyT(x; y) =___

n=112xn+ynif (x; y) ]0; 1[2;min(x; y) otherwiseisat-normwhichisstrictlymonotone, Archimedean, andleft-continuouson]0; 1[2. However, Tisdiscontinuousineachpoint(x; y) ]0; 1]2whereatleastonecoordinateisadyadicrationalnumber(i.e., of theformm=2nfor somem; n Nwithm62n; observethat theset of discontinuitypointsofTisdensein[0; 1]2). Consequently, Tisnot strict.(v)Amodicationof thet-normin(iv)yieldsat-normwhichisstrictlymonotonebut neitherArchimedeannorcontinuous(compare[67]):keepingthenotationof(iv),thefunctionT : [0; 1]2[0; 1], whichisdenedbyT(x; y) =___

n=112xn+ynnif (x; y) ]0; 1]2;0 otherwise;20 E.P. Klementetal. / FuzzySetsandSystems143(2004)526isat-normwhichisstrictlymonotone, left-continuouson[0; 1]2, but discontinuousineachpoint(x; y)]0; 1[2where at least one coordinate is a dyadic rational number. However, T is not Archimedean.(vi)ThefunctionT : [0; 1]2 [0; 1]denedbyT(x; y) =___xy if (x; y) [0; 0:5]2;2(x 0:5)(y 0:5) + 0:5 if (x; y) ]0:5; 1]2;min(x; y) otherwise;isat-normwhichhasonlytrivial idempotent elements, nozerodivisors, isnot Archimedeanandnot strictlymonotone.(vii)Recall that eachx ]0; 1]hasauniqueinnitedyadicrepresentationx = n=1 1=2xn, where(xn)n Nis a strictlyincreasingsequence of natural numbers, andconsider the functionf: [0; 1] [0; 1]denedbyf(x) =___

n=123xnif x =

n=112xn;0 if x = 0:ThenthefunctionT : [0; 1]2 [0; 1](introducedin[59], compare[38, Example3.21])givenbyT(x; y) =_f(f(1)(x) f(1)(y)) if (x; y) [0; 1[2;min(x; y) otherwise;wheref(1): [0; 1] [0; 1]isthepseudoinverseoff(observethat f(1)isalsoknownasCantorfunction)givenbyf(1)(x) = sup{z [0; 1] | f(z) x};is an Archimedean t-norm which is continuous in the point (1,1), but which has no zero divisors andwhichisnotstrictlymonotone. AmorecomplicatedexampleofthistypeistheKrauset-norm[38,AppendixB.1], whichisalsoanon-continuoust-normwithacontinuousdiagonal, thusprovidingacounterexampletoanopenproblemstatedin[57].It turns out that among the continuous Archimedean t-norms there are only two classes: thenilpotent andthestrict t-norms. Theexistenceof nilpotent elements (or zerodivisors)provides asimplecheckforthat [38, Theorem2.18], seealso(Fig. 5).Theorem 6.15. LetTbeacontinuousArchimedeant-norm. Thenthefollowingareequivalent:(i) Tisnilpotent.(ii) ThereexistssomenilpotentelementofT.(iii) ThereexistssomezerodivisorofT.(iv) Tisnotstrict.Remark 6.16. (i)AconsequenceofProposition6.10isthatat-normTisArchimedeanifandonlyif it fullls thelimit property(LP). Notethat, e.g., for topological semigroups, theArchimedeanpropertyisusuallydenedbymeansofthelimit property(LP)(see[12,45]).E.P. Klementetal. / FuzzySetsandSystems143(2004)526 21Fig. 5. Dierent classes of t-norms, eachof themwithatypical representative: withinthecentral circleonends thecontinuoust-norms, andtheclassesof strict andnilpotent t-normsaremarkedingray(for thedenitionof theordinalsums(0; 0:5; TL)and(0:5; 1; TD)see[38, Denition3.44]).(ii) An immediate consequence of Theorem 6.15 and Example 6.12(iii) is that a continuous t-normisArchimedeanifandonlyifit satisestheconditional cancellationlaw(CCL).(iii)FromTheorem6.15it followsthat acontinuoust-normT isstrict if andonlyif for eachx ]0; 1[ thesequence(x(n)T)nNisstrictlydecreasingandconvergesto0. Again, thisistheusualwaytodenethestrictnessoftopological semigroups.The strict monotonicity of t-conorms as well as strict, Archimedean and nilpotent t-conorms can beintroduced using dualities (18) and (19). Without presenting all the technical details, we only mentionthatitsucestointerchangethewordst-normandt-conormandtherolesof0and1, respectively,andsometimes toreverse the inequalities involved, inorder toobtainthe proper denitions andresultsfort-conorms. Forinstance, at-conormSisstrictlymonotoneif(SM) S(x; y) S(x; z) whenever x 1 and y z:The Archimedean property is an example where it is necessary to reverse the inequality, so at-conormSisArchimedeanif(AP) for each(x; y) ]0; 1[2thereisann Nsuchthat x(n)S y:Ofcourse, at-conormfulllsanyofthesepropertiesifandonlyifthedual t-normfulllsit.Finallylet ushaveabrieflookat thedistributivityoft-normsandt-conorms.Denition 6.17. LetTbeat-normandSbeat-conorm. ThenwesaythatTisdistributiveoverSifforall x; y; z [0; 1]T(x; S(y; z)) = S(T(x; y); T(x; z));22 E.P. Klementetal. / FuzzySetsandSystems143(2004)526andthat SisdistributiveoverTifforall x; y; z [0; 1]S(x; T(y; z)) = T(S(x; y); S(x; z)):IfTisdistributiveoverSandSisdistributiveoverT, then(T; S)iscalledadistributivepair(oft-normsandt-conorms).Inthecontext ofdistributivitytheminimumTMandthemaximumSMplayadistinguishedrole(comparealso[8]).Proposition 6.18. LetTbeat-normandSat-conorm. Thenwehave:(i) SisdistributiveoverTifandonlyifT =TM.(ii) TisdistributiveoverSifandonlyifS =SM.(iii) (T; S)isadistributivepairifandonlyifT =TMandS =SM.7. Historical remarksThehistoryoftriangularnormsstartedwithMengerspaperStatistical metrics[43]. Themainidea was to study metric spaces where probability distributions rather than numbers are used to modelthedistancebetweentheelementsof thespaceinquestion. Triangular normsnaturallycameintothe picture in the course of the generalization of the classical triangle inequality to this more generalsetting. Theoriginal set ofaxiomsfort-normswassomewhat weaker, includingamongothersalsotriangularconorms.Consequently, therst eldwheret-norms playedamajor rolewas thetheoryof probabilisticmetric spaces (as statistical metric spaces were calledafter 1964). Schweizer andSklar [5361]provided the axioms of t-norms, as they are used today, and a redenition of statistical metricspacesgivenin[58]ledtoarapiddevelopmentoftheeld. Manyresultsconcerningt-normswereobtainedinthecourseofthisdevelopment, most ofwhicharesummarizedinthemonograph[57]ofSchweizerandSklar.Mathematicallyspeaking, thetheoryof (continuous) t-norms has tworather independent roots,namely, the eld of (specic) functional equations and the theory of (special topological) semigroups.Concerning functional equations, t-norms are closely related to the equation of associativity (whichisstillunsolvedinitsmostgeneralform). TheearliestsourceinthiscontextseemstobeAbel[1],further results inthis directionwereobtainedin[9,13,2,29]. EspeciallyAcz els monograph[3,4].had(andstill has) a bigimpact onthe development of t-norms. The mainresult basedonthisbackgroundwasthefull characterizationofcontinuousArchimedeant-normsbymeansofadditivegeneratorsin[41](forthecaseofstrict t-normssee[55]).Another direction of research was the identication of several parameterized families of t-norms assolutions of some (more or less) natural functional equations. The perhaps most famous result in thiscontext has been proven in [22], showing that the family of Frank t-norms andt-conorms (together with ordinal sums thereof) are the only solutions of the so-called Frank functionalequation.E.P. Klementetal. / FuzzySetsandSystems143(2004)526 23Thestudyof aclass of compact, irreduciblyconnectedtopological semigroups was initiatedin[19], includingacharacterizationofsuchsemigroups, wheretheboundarypoints(at thesametimeannihilatorandneutralelement,respectively)aretheonlyidempotentelementsandwherenonilpo-tent elements exist. In the language of t-norms, this provided a full representation of strict t-norms. In[45] all such semigroups, where the boundary points play the role of annihilator and neutral element,werecharacterized(seealso[49]). Againinthelanguageoft-norms, thisprovidedarepresentationofall continuoust-norms[41].Several construction methods from the theory of semigroups, such as (isomorphic) transformations(whicharecloselyrelatedtogenerators mentionedabove) andordinal sums (basedontheworkof Cliord[14], andforeshadowedin[34,15]), havebeensuccessfullyappliedtoconstruct wholefamiliesof t-normsfromafewgivenprototypical examples[56]. Summarizing, startingwithonlythree t-norms, namely, the minimumTM, the product TPand the Lukasiewicz t-normTL, it ispossible toconstruct all continuous t-norms bymeans of isomorphic transformations andordinalsums[41].Non-continuous t-norms, suchas the drastic product TD, have beenconsideredfromthe verybeginning[54]. In[41] evenanadditivegenerator for thist-normwasgiven. However, ageneralclassicationofnon-continuoust-normsisstill not known.Inhis seminal paper Fuzzysets, Zadeh[67] introducedthetheoryof fuzzysets as agener-alizationof theclassical Cantorianset theorywhoselogical basisisthetwo-valuedBooleanlogic(compare alsoKlaua [32,33]). It was suggestedin[67] touse the minimumTM, the maximumSM, andthestandardnegationNstomodel theintersection, union, andcomplement of fuzzysets,respectively. However, alsotheproductTP, theprobabilisticsumSPandtheLukasiewiczt-conormSL(thelatter inarestrictedform)werealreadymentionedaspossiblecandidatesfor intersectionandunionoffuzzysets, respectively, inthisveryrst paper.Theuseof general t-normsandt-conormsfor modelingtheintersectionandtheunionof fuzzysets seems tohaveat least twoindependent roots. Ontheonehand, therewas aseries of semi-narsdevotedtothistopic, heldintheseventiesbyTrillasat theDepartament deMatem atiquesiEstadsticadelEscolaT ecnicaSuperiordArquitecturaoftheUniversitat PolitecnicadeBarcelona.Ontheother hand, thereweresuggestionsbyH ohleduringtheFirst International SymposiumonPolicy Analysis and Information Systems (Durham, NC, 1979) and the First International Sem-inar on Fuzzy Set Theory (Linz, Austria, 1979). The canonical reason for this was that theaxiomsofcommutativity, associativity, monotonicityaswell astheboundaryconditionswere(andstill are) generally considered as reasonable, even indispensable properties of meaningfulextensions of the Cantorian intersection and union (a notable exception fromthis are the com-pensatoryoperatorswhichmaybenon-associative, compareZimmermannandZysno[68], Dombi[16], Luhandjula [42], T urksen[63], Alsina et al. [5], Yager andFilev[65], andKlement et al.[37]).Very early traces of (some slight variations of) t-norms and t-conorms in the context of integrationoffuzzysetswithrespecttonon-additivemeasurescanbefoundinthePh.D.ThesisofM.Sugeno[61], rstconceptsforauniedtheoryoffuzzysets(basedonTMandSM)werepresentedin[46]andS. Gottwald[2326]. Therst papers usinggeneral t-norms andt-conorms for operations onfuzzy sets were Anthony and Sherwood [7], Alsina et al. [6], Dubois [17], and Klement [35,36] (seealso Dubois and Prade [18]). A full characterization of strong negations as models of the complementoffuzzysetscanbefoundin[62].24 E.P. Klementetal. / FuzzySetsandSystems143(2004)526AcknowledgementsThisworkwassupportedbytwoEuropeanactions(CEEPUSnetworkSK-42andCOSTaction274)aswell asbygrantsVEGA1/8331/01andMNTRS-1866.References[1] N.H. 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