FuzzySetsandSystems145(2004)439454www.elsevier.com/locate/fssTriangularnorms. PositionpaperIII: continuoust-normsErichPeterKlementa,, RadkoMesiarb, c, EndrePapdaFuzzyLogicLaboratorium, DepartmentofAlgebra, StochasticsandKnowledge-BasedMathematical Systems,JohannesKeplerUniversity, Linz-Hagenberg, 4040Linz, AustriabDepartmentofMathematicsandDescriptiveGeometry, FacultyofCivil Engineering, SlovakUniversityofTechnology, 81368Bratislava, SlovakiacInstituteofInformationTheoryandAutomation, CzechAcademyofSciences, Prague, CzechRepublicdDepartmentofMathematicsandInformatics, UniversityofNovi Sad, 21000Novi Sad, YugoslaviaReceived17December2002; receivedinrevisedform5June2003; accepted7July2003AbstractThis thirdandlast part of aseries of positionpapers ontriangular norms (for Parts I andII see(E.P.Klement, R. Mesiar, E. Pap, Triangular norms, Positionpaper I: basic analytical andalgebraic properties,FuzzySets andSystems, inpress; E.P. Klement, R. Mesiar, E. Pap, Triangular norms. Positionpaper II:general constructions andparameterizedfamilies, submittedfor publication) presents the representationofcontinuous Archimedean t-norms by means of additive generators, and the representation of continuous t-normsbymeansofordinalsumswithArchimedeansummands,bothwithfullproofs.Finallysomeconsequencesoftheserepresentationtheoremsinthecontext of comparisonandconvergenceof continuoust-norms, andofthedeterminationofcontinuoust-normsbytheirdiagonal sectionsarementioned.c 2003ElsevierB.V. All rightsreserved.Keywords:Continuoustriangularnorm; Additivegenerator; Ordinal sum1. IntroductionThis is the thirdandnal part of a series of positionpapers onthe state of the art of someparticularly important aspects of triangular norms in a condensed form. The monograph [23] providesarathercompleteandself-containedoverviewabout triangularnormsandtheirapplications.Part I [24] consideredsomebasicanalytical properties of t-norms, suchas continuity, andim-portant classes such as Archimedean, strict and nilpotent t-norms. Also the dual operations, the 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/S0165-0114(03)00304-X440 E.P. Klementetal. / FuzzySetsandSystems145(2004)439454triangularconorms, andDeMorgantripleswerementioned. Finally, ashort historical overviewonthedevelopment oft-normsandtheirwayintofuzzysetsandfuzzylogicswasgiven.PartII[25]isdevotedtogeneralconstructionmethodsbasedmainlyonpseudo-inverses,additiveandmultiplicativegenerators, andordinal sums, includingalsosomeconstructionsleadingtonon-continuoust-norms, andtoapresentationofsomedistinguishedfamiliesoft-norms.InthisthirdpartwerstpresenttherepresentationofcontinuousArchimedeant-normsbymeansofadditivegenerators, andthentherepresentationofcontinuoust-normsbymeansofordinalsumswith Archimedean summands. These theorems were rst proved in the framework of triangular normsin [28]. However, they can also be derived from results in [30] in the framework of semigroups. Weincludefull proofsof therepresentationtheoremsmentionedabove, sincetheoriginal sourcesarenot soeasilyaccessibleand/or theyheavilyusethespecial languageof semigrouptheory. Finallyweincludesomeresultsandexampleswhichfollowfromtheserepresentationtheorems.Several notions andresults fromParts I andII will beneededinthis paper, andtheycanbefound there in full detail [24,25]. For the convenience of the reader, we briey recall some of them.Recall that atriangular norm(brieyt-norm)isabinaryoperation1 ontheunit interval [0, 1]whichis commutative, associative, monotone andhas 1as neutral element, i.e., it is a function1 : [0, 1]2[0, 1]suchthat forall x, ,, : [0, 1]:(T1) 1(x, ,) =1(,, x),(T2) 1(x, 1(,, :)) =1(1(x, ,), :),(T3) 1(x, ,)61(x, :)whenever ,6:,(T4) 1(x, 1) =x.Observethatforacontinuoust-norm1theArchimedeanpropertyisequivalentto1(x, x)xforall x ]0, 1[, and that each continuous Archimedean t-norm is either strict or nilpotent [24, Theorem6.15].Givenat-norm1,anelementx [0, 1]issaidtobeidempotentif1(x, x) =x(clearly,0and1areidempotent elementsofeacht-norm, theso-calledtrivial idempotent elements).Observethat thepseudo-inverseisdenedfor arbitrarymonotonefunctions[25, Denition2.1].Inour special settingwemostlydeal withcontinuous, decreasingfunctiont : [0, 1] [0, ] witht(1) =0, inwhichcasethepseudo-inverset(1)reducestot(1)(x) = t1(min(x, t(0))).2. Representation of continuous Archimedean t-normsFortheclassofall t-norms(whichincludesnon-continuoust-normsandevent-normswhicharenot Borel measurable) the only existing characterization is by the axioms (T1)(T4). The importantsubclass of continuous t-norms, however, has nicerepresentations interms of one-placefunctionsandordinal sums.Theorem 2.1. Forafunction1 : [0, 1]2[0, 1]thefollowingareequivalent:(i) 1isacontinuousArchimedeant-norm.E.P. Klementetal. / FuzzySetsandSystems145(2004)439454 441(ii) 1has a continuous additive generator, i.e., there exists a continuous, strictly decreasing func-tion t : [0, 1] [0, ] with t(1) =0, which is uniquely determined up to a positive multiplicativeconstant, suchthatforall (x, ,) [0, 1]21(x, ,) = t(1)(t(x) + t(,)). (1)Proof. Assume rst that t : [0, 1] [0, +] is a continuous, strictly decreasing function with t(1) =0and that 1is constructed by (1), i.e., tis an additive generator of 1. The commutativity (T1) and themonotonicity(T3)of1areobvious.Also,theboundarycondition(T4)holdssinceforallx [0, 1]1(x, 1) = t(1)(t(x) + t(1)) = t(1)(t(x)) = x.Concerningtheassociativity(T2), forall x, ,, : [0, 1]weobtain1(1(x, ,), :) =t(1)(t(1(x, ,)) + t(:))=t(1)(t(t(1)(t(x) + t(,))) + t(:))=t(1)(t(x) + t(,) + t(:))=t(1)(t(x) + t(t(1)(t(,) + t(:))))=t(1)(t(x) + t(1(,, :)))=1(x, 1(,, :)),wherethethirdequalityisaconsequenceoft(t(1)(t(x) + t(,))) = min(t(x) + t(,), t(0)).Toprovetheconverse, let 1beacontinuousArchimedeant-norm. Concerningthenotionx(n)1wewill use, recall that, foreachx [0, 1], wehavex(0)1=1and, forn N, byrecursionx(n)1= 1(x, x(n1)1).Denenowforx [0, 1]andm, n Nx(1}n)1= sup{, [0, 1]|,(n)1 x},x(m}n)1= (x(1}n)1)(m)1.Since1isArchimedean, wehaveforall x ]0, 1]limnx(1}n)1= 1. (2)Notethat theexpressionx(m}n)1iswell-denedbecauseofx(m}n)1=x(km}kn)1forall k N. If, forsomex [0, 1] andsomen N{0}, wehavex(n)1=x(n+1)1then, inthestandardwaybyinduction, weobtainx(n)1= x(2n)1= (x(n)1)(2)1and, since 1is continuous Archimedean, x(n)1{0, 1}. This means that we have x(n)1x(n+1)1wheneverx(n)1]0, 1[.442 E.P. Klementetal. / FuzzySetsandSystems145(2004)439454Nowchooseandxanarbitraryelement a ]0, 1[, anddenethefunctionh : Q[0, [ [0, 1]byh(r) =a(r)1. Since1iscontinuousandsince(2)holds, hisacontinuousfunction. Moreover, wehaveforall x [0, 1]andm, n, , q Nx(m}n)+(}q)1=x((mq+n)}nq)1=(x(1}nq)1)(mq+n)1=1((x(1}nq)1)(mq)1, (x(1}nq)1)(n)1)=1(x(m}n)1, x(}q)1)and, asaconsequence, forall r, s Q[0, [h(r + s) = a(r+s)1= 1(a(r)1, a(s)1) 6a(r)1= h(r),i.e., his alsonon-increasing. Thefunctionhis evenstrictlydecreasingonthepreimageof ]0, 1]sinceforall m}n, }q Q[0, [withh(m}n)0wegeth_mn+ q_6h_mq + 1nq_= (a(1}nq)1)(mq+1)1 (a(1}nq)1)(mq)1= h_mn_.Themonotonicityandcontinuityof honQ[0, [ allowsustoextendit uniquelytoafunction

h : [0, ] [0, 1]via

h(x) = inf {h(r) | r Q [0, x]}.Then

hiscontinuousandnon-increasing, andwehaveforall x, , [0, ]

h(x + ,) = 1(

h(x), h(,)).Moreover,

h is strictly decreasing on the preimage of ]0, 1]. Dene the function t : [0, 1] [0, ] byt(x) = sup{, [0, ] | h(,)x}withtheusual conventionsup =0(observethat t isjust thepseudo-inverseof

handviceversa).Thent iscontinuous, strictlydecreasing, andsatisest(1) =0[23, Remark3.4]. Acombinationofall theargumentssofaryieldsthat t isindeedacontinuousadditivegeneratorof1sinceforeach(x, ,) [0, 1]21(x, ,) = 1(

h(t(x)), h(t(,))) =

h(t(x) + t(,)) = t(1)(t(x) + t(,)).Toshowthat thecontinuous additivegenerator t of 1 constructedaboveis uniqueuptoaposi-tivemultiplicativeconstant, assumethat thetwofunctionst1, t2 : [0, 1] [0, ]arebothcontinuousadditivegeneratorsof1, i.e., wehaveforeach(x, ,) [0, 1]2theequalityt(1)1(t1(x) + t1(,)) = t(1)2(t2(x) + t2(,)).Substituting u =t2(x) and t =t2(,), we obtain that, for all u, t [0, t2(0)] satisfying u+t [0, t2(0)[,t1 t(1)2(u) + t1 t(1)2(t) = t1 t(1)2(u + t). (3)Thenfromthecontinuityof t1andt(1)2it followsthat (3)holdsfor all u, t [0, t2(0)] withu +t [0, t2(0)].E.P. Klementetal. / FuzzySetsandSystems145(2004)439454 443Eq. (3)isaCauchyfunctional equation(see[2]), whosecontinuous, strictlyincreasingsolutionst1t(1)2: [0, t2(0)] [0, ] must satisfy t1t(1)2=bid[0, t2(0)]for some b ]0, [. As a consequence,weget t1 =bt2forsomeb ]0, [, thuscompletingtheproof.Becauseof thespecial formof thepseudo-inverset(1), representation(1)inTheorem2.1canalsobewrittenas1(x, ,) = t1(min(t(x) + t(,), t(0))).We already have seen in [24, Proposition 6.13 and Theorem 6.17] that a continuous Archimedeant-norm is either strict or nilpotent, a distinction which can be made also with the help of their additivegenerators. Indeed, generators t of strict t-norms satisfyt(0) =whilegenerators of nilpotent t-normssatisfyt(0)[25, Corollary2.8].Recall that fortheproduct 1PandfortheLukasiewiczt-norm1Ladditivegeneratorst : [0, 1] [0, ]aregivenby, respectively,t(x) = log x,t(x) = 1 x.BasedontheproofofTheorem2.1, it ispossibletogivesomeconstructivewaytoobtainadditivegenerators of continuous Archimedean t-norms. As an illustrating example, we include the followingresult of [11] (compare also[1,4,33]) for the case of strict t-norms whichcanbe derivedinastraightforwardmannerfromtheproofofTheorem2.1.Corollary 2.2. Let 1be a strict t-norm. Fix an arbitrary element x0]0, 1[, and dene the functiont : [0, 1] [0, ]byt(x) = inf_m nkm, n, k Nand(x0)(m)1 1((x0)(n)1, x(k)1)_.Thent isanadditivegeneratorof1.Example 2.3. IfweconsidertheHamacherproduct 1[15]denedby1(x, ,) =x,x + , x,,whenever (x, ,) =(0, 0), observethat weget (takingintoaccount 1}=0and1}0 =)for all(x, ,) [0, 1]21(x, ,) =1(1}x) + (1},) 1and, foreachx [0, 1]andeachn Nx(n)1=1(n}x) n + 1.444 E.P. Klementetal. / FuzzySetsandSystems145(2004)439454Forx0 =0.5theinequality(x0)(m)1 1((x0)(n)1, x(k)1)is easily seen to be equivalent to mnk((1}x)1), yielding the additive generator t : [0, 1] [0, ]of1speciedbyt(x) = inf_m nkm, n, k Nandm nk

1x 1_=1 xx.The representationof continuous Archimedeant-norms giveninTheorem2.1is basedontheadditionontheinterval [0, +]. Thereisacompletelyanalogousrepresentationthereof basedonthe multiplicationon[0, 1], thus leadingtoa representationof continuous Archimedeant-normsbymeansofmultiplicativegenerators[25, Section2]. Byduality, therearealsorepresentationsofcontinuous Archimedeant-conorms bymeans of additivegenerators andmultiplicativegenerators,respectively.Remark 2.4. (i)If1isacontinuousArchimedeant-normwithadditivegeneratort : [0, 1] [0, ],thenthefunction0 : [0, 1] [0, 1]denedby0(x) =et(x)isamultiplicativegeneratorof1.(ii) If Sis a continuous Archimedean t-conorm then the dual t-norm 1is continuous Archimedeanand, therefore, hasanadditivegeneratort : [0, 1] [0, ]. Thens : [0, 1] [0, ]denedbys(x) =t(1 x)isanadditivegenerator of S, and : [0, 1] [0, 1] denedby (x) = et(1x)isamulti-plicativegeneratorofS.(iii) Given a continuous Archimedean t-norm 1and a strictly increasing bijection : [0, 1] [0, 1],it isclearthat thefunction1 : [0, 1]2[0, 1]givenby1(x, ,) = 1(1((x), (,)))is a continuous Archimedean t-norm too. By Theorem 2.1, there are additive generators t, t : [0, 1] [0, ]of1and1, respectively. Takingintoaccount [25, Proposition2.9], tequalst uptoamultiplicativeconstant.It isstraightforwardthat eachisomorphism: [0, 1] [0, 1]preserves(amongmanyotherprop-erties) the continuity, the strictness and the existence of zero divisors. Therefore, each t-norm whichisisomorphictoastrict ortoanilpotent t-norm, itselfisstrict ornilpotent, respectively.Conversely, if 11and12are twostrict t-norms withadditive generators t1andt2(whicharebijectivefunctionsfrom[0, 1]into[0, ]inthiscase), respectively, then: [0, 1] [0, 1]givenby=t11 t2isastrictlyincreasingbijectionand12 =(11). If11and12aretwonilpotent t-normswith additive generators t1and t2, respectively, then we have 12 =(11), where the strictly increasingbijection: [0, 1] [0, 1] isgivenby=t11 ((t1(0)}t2(0))t2)(observethat inthiscasethetwofunctionst1and(t1(0)}t2(0))t2canbeviewedasbijectionsfrom[0, 1]into[0, t1(0)]).Wethereforehaveshownthefollowingresult:Lemma 2.5. TwocontinuousArchimedeant-normsareisomorphicif andonlyif theyareeitherbothstrictorbothnilpotent.E.P. Klementetal. / FuzzySetsandSystems145(2004)439454 445Animmediateconsequenceof Remark2.4(iii) andLemma2.5is that theproduct 1PandtheLukasiewiczt-norm1Larenot onlyprototypical examplesof strict andnilpotent t-norms, respec-tively, but that eachcontinuousArchimedeant-normisisomorphiceitherto1Porto1L:Theorem 2.6. (i)Afunction1 : [0, 1]2[0, 1]isastrictt-normifandonlyifitisisomorphictotheproduct1P.(ii) Afunction1 : [0, 1]2[0, 1] is anilpotent t-normif andonlyif it is isomorphic tothe Lukasiewiczt-norm1L.Eachmultiplicativegenerator0 : [0, 1] [0, 1]ofastrict t-norm1canbeviewedasanisomor-phismbetween1Pand1, i.e., 1 =(1P)0. Inparticular, this means that thereareinnitelymanyisomorphismsbetween1Pand1. Ontheotherhand, if1isanilpotent t-normwithadditivegen-erator t : [0, 1] [0, ], thenthere is a unique isomorphism: [0, 1] [0, 1] between1Land1,namely, =1 (1}t(0))t.Recall that eachcontinuoust-norm1satisfying1(x, x)xforall x ]0, 1[isArchimedean[23,Proposition2.15].Corollary 2.7. If 1isacontinuoust-normwithtrivial idempotent elementsonly, i.e., 1(x, x) =xonlyifx {0, 1}, then1isArchimedeanand, therefore, hasacontinuousadditivegenerator.Remark 2.8. Note that the representation in Theorem 2.1 holds for continuous Archimedean t-normsonly. However, there are several possibilities to show the existence of continuous additive generatorsforafunction1 : [0, 1]2[0, 1]underweakerhypothesesthaninTheorem2.1.For example, it is possible to drop the commutativity (T1) [30] (see also [23, Theorem 2.43]) or toweakentheassociativity(T2)[6,27].Inthecaseofleft-continuoust-norms,eithertheArchimedeanproperty[26]ortheexistenceofa(not necessarilycontinuous)additivegenerator[38]impliestheexistence of a continuous additive generator. In the case of a strictly monotone Archimedean t-norm1, the continuityof 1 at the point (1, 1) is sucient for the existence of a continuous additivegenerator[14].3. Representation of continuous t-normsThe construction of a new semigroup from a family of given semigroups using ordinal sums goesbacktoA. H. Cliord[8](seealso[9,17,34]), anditisbasedonideaspresentedin[10,21]. Ithasbeensuccessfullyappliedtot-normsin[13,24,28,36].Denition 3.1. Let (1:):Abe a family of t-norms and (]a:, e:[):Abe a family of non-empty,pairwisedisjoint opensubintervalsof[0, 1]. Thet-norm1denedby1(x, ,) =___a: + (e: a:)1:_x a:e: a:,, a:e: a:_if (x, ,) [a:, e:]2,min(x, ,) otherwise,446 E.P. Klementetal. / FuzzySetsandSystems145(2004)439454iscalledtheordinal sumofthesummands a:, e:, 1:, : A, andweshall write1= (a:, e:, 1:):A.Observethat theindexset Aisnecessarilyniteorcountablyinnite. It alsomaybeempty, inwhichcasetheordinal sumequalstheidempotent t-norm1M.Notethat therepresentationofcontinuousArchimedeant-normsbymeansofmultiplicativegen-erators can be derived directly frommore general results for I -semigroups (see [23,28,30,37]).Similarly, the following representation of continuous t-norms by means of ordinal sums follows alsofromresultsof[30]inthecontext ofI -semigroups.Theorem 3.2. Afunction1 : [0, 1]2[0, 1] isacontinuoust-normif andonlyif 1 isanordinalsumofcontinuousArchimedeant-norms.Proof. Obviously, eachordinal sumofcontinuoust-normsisacontinuoust-norm.Conversely, if1isacontinuoust-norm, werstshowthatthesetI1ofallidempotentelementsof 1is a closed subset of [0, 1]. Indeed, if (xn)nNis a sequence of idempotent elements of 1whichconvergestosomex [0, 1], thenthecontinuityof1impliesx =limnxn =limn1(xn, xn) = 1(x, x),soxisalsoanidempotent element of1, andI1isclosed.InthecaseI1 =[0, 1]wehave1 =1M,i.e.,anemptyordinalsum.IfI1 = [0, 1]itcanbewrittenas the (non-trivial) union of a nite or countably innite family of pairwise disjoint open subintervals(]a:, e:[):Awhere, of course, eacha:andeache:(but noelement in]a:, e:[) is anidempotentelement of1.For thetimebeing, assumethat A = andxanarbitrary: A. Thenthemonotonicityof 1impliesthat forall (x, ,) [a:, e:]2a: = 1(a:, a:) 61(x, ,) 61(e:, e:) = e:and, forall x [a:, 1]a: = 1(a:, a:) 61(x, a:) 61(1, a:) = a:,showingthat ([a:, e:], 1|[a:,e:]2 )isasemigroupwithannihilator a:andwithtrivial idempotent ele-ments only (actually, a:acts as an annihilator on [a:, 1]). Because of the monotonicity and continuityof1wealsohaveforeach: A{1(:, e:) | : [0, 1]} = [0, e:],whichmeans that eachx [0, e:] canbewrittenas x =1(:, e:)for some: [0, 1]. This, togetherwiththeassociativityof1, impliesthat1(x, e:) = 1(1(:, e:), e:) = 1(:, 1(e:, e:)) = 1(:, e:) = x,showing that e:acts as a neutral element on [0, e:] and, subsequently, in the I -semigroup([a:, e:], 1|[a:,e:]2 ).E.P. Klementetal. / FuzzySetsandSystems145(2004)439454 447Let : : [0, 1] [a:, e:]bethestrictlyincreasingbijectiongivenby:(x) = a: + (e: a:)x,thenforeach: Athefunction1: : [0, 1]2[0, 1]denedby1:(x, ,) = 1:(1(:(x), :(,))isacontinuoust-normwhichhasonlytrivial idempotent elements, andwhichisalsoArchimedeanbecause of Corollary 2.7. A simple computation veries that for all : A and for all (x, ,) [a:, e:]2wehave1(x, ,) = a: + (e: a:)1:_x a:e: a:,, a:e: a:_.If (x, ,) [0, 1]2(without loss of generalitywe mayassume x6,) is containedinnone of thesquares [a:, e:]2then there exists some idempotent element b [x, ,] which acts as a neutral elementon[0, b]andasanannihilatoron[b, 1], andwehave1(x, ,) = 1(1(x, b), ,) = 1(x, 1(b, ,)) = 1(x, b) = x = min(x, ,),completingtheproofthat 1 =(a:, e:, 1:):A.The uniqueness of the representation of 1 is an immediate consequence of the one-to-onecorrespondence between the set of idempotent elements of 1 and the family of intervals(]a:, e:[):A.The combination of Theorem 3.2 and of the results of Section 2 yields the following representationsofcontinuoust-norms:Corollary 3.3. Forafunction1 : [0, 1]2[0, 1]thefollowingareequivalent:(i) 1isacontinuoust-norm.(ii) 1isisomorphictoanordinal sumwhosesummandscontainonlythet-norms1Pand1L.(iii) Thereisafamily(]a:, e:[):Aofnon-empty, pairwisedisjointopensubintervalsof[0, 1]andafamilyh: : [a:, e:] [0, ] of continuous, strictlydecreasingfunctionswithh:(e:) =0foreach: Asuchthatforall (x, ,) [0, 1]21(x, ,) =_h(1):(h:(x) + h:(,)) if (x, ,) [a:, e:]2,min(x, ,) otherwise.(4)Example 3.4. Considerthecontinuoust-norm1(seeFig. 1)givenby1(x, ,) =___max_3x + 3, + 9x, 16, 0_if (x, ,) [0,13]2,4x + 4, 3x, 49x + 9, 9x, 8if (x, ,) [23, 1]2,min(x, ,) otherwise.448 E.P. Klementetal. / FuzzySetsandSystems145(2004)439454Fig. 1. Contourplotsoftheisomorphict-norms1(left)and10.7,0.8fromExample3.4.This t-norm1 canbe writtenas the ordinal sum(0, 1}3, 11, 2}3, 1, 12) with11and12beinggivenby11(x, ,) = max_x + , + x, 12, 0_,12(x, ,) =x,x + , x,.Observethat thenilpotent t-norm11was introducedin[39], andthat thestrict t-norm12is theHamacherproduct 1H0[25], andthat thefunctionst1, t2givenbyt1(x) = log1 + x2,t2(x) =1 xx.arecontinuousadditivegeneratorsof 11and12, respectively. Deningthefunctionsh1 : [0, 1}3] [0, ]andh2 : [2}3, 1] [0, ]byh1(x) = log1 + 3x2,h2(x) =3 3x3x 2,wecanrepresent our t-norm1 inform(4). For anynumbersa, b ]0, 1[ withabconsider thet-norm 1ab =(0, a, 1L, b, 1, 1P) (see Fig. 1). Then 1is isomorphic to 1ab, i.e., we have 1 =(1ab)wherethestrictlyincreasingbijection: [0, 1] [0, 1]isgivenby(x) =___a log (1 + 3x)log 2if x [0,13],a + (b a)(3x 1) if x ]13,23],b + (1 b)e(3x3)}(3x2)otherwise.E.P. Klementetal. / FuzzySetsandSystems145(2004)439454 449Analogousrepresentationsforcontinuoust-conormscanbeobtainedbyduality(makingthenec-essarychanges, e.g., replacingminbymax).4. Consequences of the representation theoremsTheorems2.1and3.2simplifytheworkwithcontinuoust-normsinthesensethat it sucestoconsider (a family of) continuous Archimedean t-norms and, subsequently, their additive generators.Inparticular, theadditivegenerator (whichisaone-placefunction)of acontinuousArchimedeant-norm1carriesall theinformationofthewholet-norm1.Knowingthestructureof continuous t-norms allows us alsotodeducegeneral properties frompartial information. Forinstance, ifforacontinuoust-norm1andforsomex0]0, 1[theverticalsection[: [0, 1] [0, 1]givenby[(,) =1(x0, ,)isstrictlymonotoneandsatises[(,),forall, ]0, 1], then1isastrict t-norm.In this section, we demonstrate the impact of Theorems 2.1 and 3.2 on the problems of (pointwise)comparison and convergence of continuous t-norms, and on the determination of continuous t-normsbytheirdiagonal sections.Thefollowingnecessaryandsucient conditionfor thecomparisonof continuousArchimedeant-norms canbefoundin[37, Lemma5.5.8] (seealso[23, Theorem6.2], for thespecial caseofstrict t-normsit wasrst provedin[35](seealso[5]).Theorem 4.1. Let 11and 12be two continuous Archimedean t-norms with additive generatorst1, t2 : [0, 1] [0, ], respectively. Thefollowingareequivalent:(i) 11612.(ii) The functiont1 t12: [0, t2(0)] [0, ] is subadditive, i.e., for all u, t [0, t2(0)] withu +t [0, t2(0)]wehavet1 t12(u + t) 6t1 t12(u) + t1 t12(t).There exist criteria (some of which are only sucient) for the comparability of continuousArchimedeant-normswhichsometimesareeasier tocheckthanthesubadditivityinTheorem4.1.Thefollowingsucient conditionscanbederivedeasilyfromTheorem4.1andfrom[16, (103)](recall that afunction[: [a, b] [0, ]iscalledconcaveif[(zx + (1 z),) z[(x) + (1 z)[(,)forall x, , [a, b]andforall z [0, 1]).Corollary 4.2. Let 11and12be twocontinuous Archimedeant-norms withadditive generatorst1, t2 : [0, 1] [0, ], respectively.Thenwehave11612ifoneofthefollowingconditionsissatis-ed:(i) Thefunctiont1 t12: [0, t2(0)] [0, ]isconcave.450 E.P. Klementetal. / FuzzySetsandSystems145(2004)439454(ii) Thefunction[: ]0, t2(0)] [0, ]denedby[(x) =(t1 t12)(x)xisnon-increasing.(iii) Thefunctiont

1t

2: ]0, 1[ [0, [isnon-decreasing.Example 4.3. In[25, Example2.10(i)]wehaveseenthatforeachcontinuousArchimedeant-norm1withadditivegeneratort : [0, 1] [0, ], andforeachz ]0, [, thefunctiontz: [0, 1] [0, ]isanadditivegenerator of acontinuousArchimedeant-normwhichwasdenotedthere1(z). Nowweareabletoshowthatthefamily(1(z))z]0,[isstrictlyincreasingwithrespecttotheparameterz. Indeed, forz, j ]0, [thecompositefunctiontz (tj)1: [0, t(0)j] [0, ]isgivenbytz (tj)1(x) = xz}j,and it is concave whenever z6j, showing that (1(z))z ]0,[ is a strictly increasing family of t-norms.Consequently, the families of Yager t-norms [40], of Acz elAlsina t-norms [3], and of Dombi t-norms[12]arestrictlyincreasingfamiliesoft-norms.Anontrivial problemwasthemonotonicityof thefamilyof Frankt-norms(1Fz )z[0, ][13]. Arst proofthereofappearedin[7, Proposition1.12]. Inthefollowingwegiveasimplerproof[22]basedonCorollary4.2(iii)(seealso[23, Proposition6.8]).Proposition 4.4. Thefamily(1Fz )z[0, ]ofFrankt-normsisstrictlydecreasing.Proof. Recallthat1F0=1M, 1F1=1P,whoseadditivegeneratortF1isgivenbytF1 (x) = log x,and1F=1LwhoseadditivegeneratortFisgivenbytF(x) =1 x. Foreachz ]0, 1[ [1, ], 1Fzisastrict t-norm, anditsadditivegeneratortFzisgivenbytFz (x) = log (z 1)}(zx1).Triviallywehave1F0=1M1Fzforall z ]0, ]. From(tF)

(tFz)

(x) =___x if z = 1,zx1zxlog zif z ]0, 1[]1, [,it follows that for each z ]0, [ the function (tF)

}(tFz )

is non-decreasing, implying 1F61Fzand,since1Fisnilpotent and1Fzisstrict, even1F1Fz .Nowlet usshowthat 1Fj61Fzwhenever1zj. Observethat forall x ]0, 1[weget(tFj)

(tFz)

(x) =jx(zx1) log jzx(jx1) log z=log jlog z1 (1}z)x1 (1}j)x.E.P. Klementetal. / FuzzySetsandSystems145(2004)439454 451Then(tFj)

}(tFz )

isnon-decreasingon]0, 1[ifandonlyif_1 _1j_x__1z_xlog1z6_1 _1z_x__1j_xlog1j,i.e., ifandonlyifwehavetheinequality(1}z)xlog1z(1}j)xlog1j1 (1}z)x1 (1}j)x. (5)Considernowthefunctions[, q : ]0, 1[ [0, [denedby[(x) =1 (1}z)xandq(x) =1 (1}j)x.Then, bytheCauchymeanvaluetheorem, foreachx ]0, 1[thereexistsa, ]0, x[suchthat1 (1}z)x1 (1}j)x=[(x) [(0)q(x) q(0)=[

(,)q

(,)=(1}z),log (1}z)(1}j),log (1}j)(1}z)xlog (1}z)(1}j)xlog (1}j).This proves inequality (5) and, consequently, the function (tFj)

}(tFz )

is non-decreasing, i.e., 1Fj61Fzand, because of 1Fj=1Fz , even 1Fj 1Fzin this case. Similarly we can show1F1 1Fzfor allz ]1, [.Thecase0zj61canbetransformedinto161}j1}z, andthecase0z1jisprovedcombiningthetwolattercases.The comparison of arbitrary continuous t-norms is much more complicated, and it is fully describedin[22](seealso[23, Theorem6.12]).Whencomparingt-normsit isevident that theincomparabilityoftheirdiagonal sectionsimpliestheincomparabilityof thet-normsthemselves. Theconverse, however, isnot trueingeneral, noteveninthecaseofcontinuousArchimedeant-norms.Example 4.5. Consider thefunctiont : [0, 1] [0, ] denedby(theindexnmaybeanynumberin Z)t(x) =___ if x = 0,2n(2 (4x1}2n1)2) if x _122n+1 ,122n_,0 if x = 1,then t is an additive generator of some strict t-norm1. Asimple computation shows that thediagonal sections of 1 and1Pcoincide, but theoppositediagonal sections d1, d1P : [0, 1] [0, 1]givenbyd1(x) =1(x, 1 x)andd1P(x) =1P(x, 1 x)areincomparable(seeFig. 2), implyingtheincomparabilityof1and1P.This shows that dierent continuous t-norms may have identical diagonal sections. Note that therearemethods todescribeall continuous t-norms havingagivendiagonal section[20,23,29]. Hereweonlymentiononeof thesemethodsappliedtostrict t-norms[20,29] (seealso[23, Proposition7.11]):452 E.P. Klementetal. / FuzzySetsandSystems145(2004)439454Fig. 2. Contour plot of the strict t-norm1 (left) consideredinExample 4.5together withthe incomparable oppositediagonal sectionsof1and1P(right).Proposition 4.6. Leto : [0, 1] [0, 1]beastrictlyincreasingbijectionsuchthato(x, x)xforallx ]0, 1[. Thenacontinuous t-norm1 hasdiagonal sectionoif andonlyif 1 isstrict andthefunctiont : [0, 1] [0, ]givenbyt(x) =___ if x = 0,2n[(o(n)(x)) if x ]o(n+1)(0.5), o(n)(0.5)],0 if x = 1,is an additive generator of 1, where [: [o(0.5), 0.5] [1, 2] is a strictly decreasing bijection,o(0)=id[0,1], o(n)=o o(n1)whenevern N, ando(n)=(o(n))1whenever n N.As a consequence of Proposition 4.6, two dierent strict t-norms with the same diagonal section arenecessarilyincomparable, comparealsoExample4.5(thesameresultholdsforarbitrarycontinuoust-norms).Additive generators characterize also analytical properties of the continuous Archimedean t-norms.For instance, a continuous Archimedean t-norm is 1-Lipschitz if and only if it has a convex additivegenerator[31,37].Also, convergence properties can be expressed by means of additive generators [18] (see also [23,Corollary8.21]).Proposition 4.7. Let (1n)nNbeasequenceof continuous Archimedeant-norms andlet 1 beacontinuousArchimedeant-norm. Thenthefollowingareequivalent:(i) limn 1n =1.(ii) There exists a sequence of additive generators (tn : [0, 1] [0, ])nNof (1n)nNsuch that therestriction_limntn_]0,1]coincideswiththerestrictionofsomeadditivegeneratorof1to]0, 1].E.P. Klementetal. / FuzzySetsandSystems145(2004)439454 453Notethat, wheneverinProposition4.7thelimitt-norm1isstrict, thenlimntnisanadditivegeneratorof1.Forexample, foreachn1thefunctiontn : [0, 1] [0, ]givenbytn(x) =12 log(1 +n)logn 1nx1is an additive generator of the (strict) Frank t-norm1Fn[13]. Then for all x]0, 1] we havelimn tn(x)=1x, i.e., the sequence (tn|]0,1])n1converges to the restriction of an additive genera-tor of the Lukasiewicz t-norm 1Lto ]0, 1]. Therefore, the sequence (1Fn )n1converges to 1L(=1F).Finallywementionthat thecontinuousArchimedeant-normsformadensesubclassoftheclassof all continuoust-norms(withrespect totheuniformtopology), i.e., eachcontinuoust-normcanbeapproximatedbysomecontinuousArchimedeant-normwitharbitraryprecision[19,23,32].Morepreciselywehave:Theorem 4.8. Let1beacontinuoust-norm. Thenforeachc0thereisastrictt-norm11andanilpotentt-norm12suchthatforall (x, ,) [0, 1]2| 1(x, ,) 11(x, ,) | c,| 1(x, ,) 12(x, ,) | c.AcknowledgementsThisworkwassupportedbytwoEuropeanactions(CEEPUSnetworkSK-42andCOSTaction274)aswell asbythegrantsVEGA1/0273/03, APVT-20-023402andMNTRS-1866. Theauthorsalsowouldliketothanktherefereesfortheirvaluablecomments.References[1] J. Acz el, Surlesop erationsdeniespourdesnombresr eels, Bull. Soc. Math. France76(1949)5964.[2] J. Acz el, LecturesonFunctional EquationsandtheirApplications, AcademicPress, NewYork, 1966.[3] J. Acz el, C. 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