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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:[email protected] (E.P.
Klement), [email protected](R. Mesiar),
[email protected],[email protected](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. Abel,
Untersuchungen der Funktionen zweier unabh angigen ver anderlichen
Gr oen x und y wie f(x; y), welchedieEigenschaft haben, dassf(z;
f(x; y))cinesymmetrischeFunktionvonx; yundzist, J. ReineAngew.
Math. 1(1826)1115.[2] J. Acz el, Surlesop
erationsdeniespourdesnombresr eels, Bull. Soc. Math.
France76(1949)5964.[3] J. Acz el, Vorlesungen
uberFunktionalgleichungenundihreAnwendungen, Birkh auser, Basel,
1961.[4] J. Acz el, LecturesonFunctional
EquationsandtheirApplications, AcademicPress, NewYork, 1966.[5] C.
Alsina, G. Mayor, M.S. Tom as, J. Torrens,
Acharacterizationofclassofaggregationfunctions,
FuzzySetsandSystems53(1993)3338.[6] C. Alsina, E. Trillas, L.
Valverde, On non-distributive logical connectives for fuzzy sets
theory, BUSEFAL 3 (1980)1829.[7] J.M. Antony, H. Sherwood,
Fuzzygroupsredened, J. Math. Anal. Appl. 69(1979)124130.[8] R.
Bellman, M. Giertz, Ontheanalyticformalismofthetheoryoffuzzysets,
Inform. Sci. 5(1973)149156.[9] L.E.J. Brouwer,
DieTheoriederendlichenkontinuierlichenGruppenunabh
angigvondenAxiomenvonLie, Math.Ann. 67(1909)246267.[10] M. Budin
cevi c, M.S. Kurili c, Afamilyof strict anddiscontinuous triangular
norms, FuzzySets andSystems 95(1998)381384.[11] D. Butnariu, E.P.
Klement, TriangularNorm-BasedMeasuresandGameswithFuzzyCoalitions,
KluwerAcademicPublishers, Dordrecht, 1993.[12] J.H. Carruth, J.A.
Hildebrant, R.J. Koch, TheTheoryof Topological Semigroups,
LectureNotes inMathematics,Marcel Dekker, NewYork, 1983.[13]
E. Cartan, La th eorie des groupes nis et continus et lAnalysis
Situs, in: M em. Sci. Math., vol. 42, Gauthier-Villars,Paris,
1930.[14] A.H. Cliord,
Naturallytotallyorderedcommutativesemigroups, Amer. J. Math.
76(1954)631646.[15] A.C. Climescu, Surl
equationfonctionelledelassociativit e, Bull.
EcolePolytechn. Iassy1(1946)116.[16] J. Dombi, Basicconceptsfor
atheoryof evaluation: theaggregativeoperator, EuropeanJ. Oper. Res.
10(1982)282293.[17] D. Dubois, Triangular norms for fuzzysets, in:
E.P. Klement (Ed.), Proc. 2ndInternat. Seminar onFuzzySetTheory,
Linz, 1980, pp. 3968.[18] D. Dubois, H. Prade, FuzzySetsandSystems:
TheoryandApplications, AcademicPress, NewYork, 1980.[19] W. M.
Faucett, Compact semigroups
irreduciblyconnectedbetweentwoidempotents, Proc. Amer. Math. Soc.
6(1955)741747.[20] J.C. Fodor,
Contrapositivesymmetryoffuzzyimplications,
FuzzySetsandSystems69(1995)141156.[21] J.C. Fodor, M. Roubens,
Fuzzy Preference Modelling and Multicriteria Decision Support,
Kluwer AcademicPublishers, Dordrecht, 1994.[22] M.J. Frank, On the
simultaneous associativity of F(x; y) and x+y F(x; y), Aequationes
Math. 19 (1979)194226.[23] S. Gottwald,
UntersuchungenzurmehrwertigenMengenlehre. I, Math. Nachr.
72(1976)297303.[24] S. Gottwald,
UntersuchungenzurmehrwertigenMengenlehre. II, Math. Nachr.
74(1976)329336.[25] S. Gottwald,
UntersuchungenzurmehrwertigenMengenlehre. III, Math. Nachr.
79(1977)207217.[26] M. Grabisch, H.T. Nguyen, E.A. Walker,
Fundamentals of Uncertainty Calculi with Applications to Fuzzy
Inference,KluwerAcademicPublishers, Dordrecht, 1995.[27] P. H ajek,
MetamathematicsofFuzzyLogic, KluwerAcademicPublishers, Dordrecht,
1998.E.P. Klementetal. / FuzzySetsandSystems143(2004)526 25[28]
K.H. Hofmann, J.D. Lawson, Linearlyorderedsemigroups:
historicoriginsandA.H. Cliordsinuence, in: K.H.Hofmann, M.W.
Mislove (Eds.), Semigroup Theory and its Applications, London
Mathematics Society Lecture Notes,vol. 231,
CambridgeUniversityPress, Cambridge, 1996, pp. 1539.[29] M. Hossz
u, Somefunctional equationsrelatedwiththeassociativitylaw, Publ.
Math. Debrecen3(1954)205214.[30] S. Jenei, Structureof
left-continuoustriangular normswithstronginducednegations,
(I)Rotationconstruction, J.Appl. Non-Classical
Logics10(2000)8392.[31]
S.Jenei,StructureofGirardmonoidson[0,1],in:S.E.Rodabaugh,E.P.Klement(Eds.),TopologicalandAlgebraicStructures
in Fuzzy Sets. A Handbook of Recent Developments in the Mathematics
of Fuzzy Sets, Kluwer AcademicPublishers, Dordrecht, 2003. pp.
277308.[32] D. Klaua,
Uber einen Ansatz zur mehrwertigen Mengenlehre, Monatsb.
Deutsch. Akad. Wiss. Berlin 7 (1965)859867.[33] D. Klaua,
Einbettungder klassischenMengenlehre indie mehrwertige, Monatsb.
Deutsch. Akad. Wiss. Berlin9(1967)258272.[34] F. Klein-Barmen,
Uber gewisse Halbverb ande und kommutative Semigruppen II, Math.
Z 48 (19421943) 715734.[35]
E.P.Klement,Someremarksont-norms,fuzzy-algebrasandfuzzymeasures,in:E.P.Klement(Ed.),ProceedingsSecondInternational
SeminaronFuzzySet Theory, Linz, 1980, pp. 125142.[36] E.P. Klement,
Operations on fuzzy sets and fuzzy numbers related to triangular
norms, in: Proc. 11th Internat. Symp.onMultiple-ValuedLogic,
Norman, IEEEPress, NewYork, 1981, pp. 218225.[37] E.P. Klement, R.
Mesiar, E. Pap,
Ontherelationshipofassociativecompensatoryoperatorstotriangularnormsandconorms,
Internat. J. Uncertain.
FuzzinessKnowledge-BasedSystems4(1996)129144.[38] E.P. Klement, R.
Mesiar, E. Pap, TriangularNorms, KluwerAcademicPublishers,
Dordrecht, 2000.[39] E.P. Klement, S. Weber, Generalizedmeasures,
FuzzySetsandSystems40(1991)375394.[40] A. Koles arov a,
AnoteonArchimedeantriangularnorms, BUSEFAL80(1999)5760.[41] C.M.
Ling, Representationofassociativefunctions, Publ. Math.
Debrecen12(1965)189212.[42] M.K. Luhandjula, Compensatoryoperators
infuzzylinear programmingwithmultipleobjectives, FuzzySets
andSystems8(1982)245252.[43] K. Menger, Statistical metrics, Proc.
Nat. Acad. Sci. USA8(1942)535537.[44] R. Mesiar, H. Thiele,
OnT-quantiersandS-quantiers, in: V. Nov ak, I. Perlieva(Eds.),
DiscoveringtheWorldwithFuzzyLogic, Physica-Verlag, Heidelberg,
2000, pp. 310326.[45] P.S. Mostert, A.L. Shields,
Onthestructureofsemi-groupsonacompact manifoldwithboundary, Ann.
ofMath.Ser. II65(1957)117143.[46] C.V. Negoita, D.A. Ralescu,
ApplicationsofFuzzySetstoSystemsAnalysis, Wiley, NewYork, 1975.[47]
R.B. Nelsen, AnIntroductiontoCopulas, in: LectureNotesinStatistics,
vol. 139, Springer, NewYork, 1999.[48] H.T. Nguyen, E. Walker,
AFirst CourseinFuzzyLogic, CRCPress, BocaRaton, FL, 1997.[49] A.B.
Paalman-deMiranda, Topological Semigroups, MatematischCentrum,
Amsterdam, 1964.[50] E. Pap, Null-AdditiveSet Functions,
KluwerAcademicPublishers, Dordrecht, 1995.[51] P. Perny, Mod
elisation, agr egation et exploitation des pr ef erences oues dans
une probl ematique de rangement, Ph.D.Thesis, Universit
eParis-Dauphine, Paris, 1992.[52] P. Perny, B. Roy, The use of
fuzzy outranking relations in preference modelling, Fuzzy Sets and
Systems 49 (1992)3353.[53] B. Schweizer, A. Sklar, Espacesm
etriquesal eatoires, C. R. Acad. Sci. ParisS er.
A247(1958)20922094.[54] B. Schweizer, A. Sklar, Statistical
metricspaces, PacicJ. Math. 10(1960)313334.[55] B. Schweizer, A.
Sklar, Associativefunctions andstatistical triangleinequalities,
Publ. Math. Debrecen8(1961)169186.[56] B. Schweizer, A. Sklar,
Associativefunctionsandabstract semigroups, Publ. Math.
Debrecen10(1963)6981.[57] B. Schweizer, A. Sklar,
ProbabilisticMetricSpaces, North-Holland, NewYork, 1983.[58]
A.N.
Serstnev,Randomnormedspaces:problemsofcompleteness,Kazan.Gos.Univ.U
cen.Zap.122(1962)320.[59] D. Smutn a, Anoteonnon-continuoust-norms,
BUSEFAL76(1998)1924.[60] D. Smutn a, Onapeculiart-norm,
BUSEFAL75(1998)6067.[61] M. Sugeno,
Theoryoffuzzyintegralsanditsapplications, Ph.D. Thesis,
TokyoInstituteofTechnology, 1974.[62] E. Trillas,
Sobrefuncionesdenegacionenlateoradeconjuntasdifusos,
Stochastica3(1979)4760.[63] I.B. T urksen,
Inter-valuedfuzzysetsandcompensatoryAND,
FuzzySetsandSystems51(1992)295307.26 E.P. Klementetal. /
FuzzySetsandSystems143(2004)526[64] S. Weber,
-decomposablemeasuresandintegralsforArchimedeant-conorms , J. Math.
Anal. Appl. 101(1984)114138.[65] R.R. Yager, D.P. Filev,
EssentialsofFuzzyModellingandControl, Wiley, NewYork, 1994.[66]
R.R. Yager, A. Rybalov, Uninormaggregationoperators,
FuzzySetsandSystems80(1996)111120.[67] L.A. Zadeh, Fuzzysets,
Inform. Control 8(1965)338353.[68] H.-J. Zimmermann, P. Zysno,
Latent connectives inhumandecisionmaking, FuzzySets andSystems
4(1980)3751.