On some constructions of new triangular norms
Abstract
We discuss the properties of two types of construction of a new t-norm from a given t-norm proposed recently by B. Demant, namely the dilatation and the contraction. In general, the dilatation of a t-norm is an ordinal sum t-norm and the continuity of the outgoing t-norm is preserved. On the other hand, the contraction may violate the continuity as well as the non-continuity of the outgoing t-norm. Several examples are given.
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Math w are&SoftC om puting 2( 19 95)39-45 Onsom econstr uctionso fnew triangula rn orm s Radk oMesiar Slo v akT ec hnicalUniv e rs it yBra ti sl av a Radlins k eho1 1,81368Bratis la v a Slo v aki a Abs tr act W ediscussthepro pertiesoft w ot ypesofconstructionof anewt-normfr omag iv ent-no rmpr opo sedrecen t lyb yB. Demant ,namelythedila tat iona ndth eco n tra ct ion.Ingener al, thed ilat ationo fat-normisa nord inalsumt-normandthe con tin uity of theoutg oingt-normispreserv ed.Ontheo ther hand,the con t ractionma yvio lat ethec on tin uit yasw ellasthe non-co n tin ui ty of theoutg oingt -no rm.Sev eral examplesa re giv en. Keyw ords: co ntract ion,dilat ation,or di nalsum,triangular norm. 1In tro duction Am ong sev eral constructions o f the new t-no rm sform gi ven o ne s [2,3,4], were ca l lt wo bas i c constructi ons ari sen fro m the s e mi g roup in te rpreta t ion of a tri a ngular norm ,se eSc hweiz er a nd Skl ar [3]. Ordinal sum :Let h ] a k ;b k [; k 2K i be a dis jointsystemof open su bintervals of the unit interva l [0,1] and let [ T k ; k 2K ] b e a system 39
40 R.Mesi ar of gi v e nt-nor m s.F or x;y 2 [0 ; 1],p ut T ( x;y )= 8 > > > > > > < > > > > > > : a k +( b k 0 a k ) T k (( x 0 a k ) = ( b k 0 a k ) ; ( y 0 a k ) = ( b k 0 a k )) if x;y 2 [ a k ;b k ] forsom e k 2K mi n( x ;y )othe rwi se Then T isat-normandi ti sc al ledanordinalsumwith sum m ands <a k ;b k ;T k > , k 2K ,briey T [ <a k ;b k ;T k > ; k 2K ] . Semigr oupdefo rmation :l et :[ 0 ; 1] ! [ a; 1], a 2 [ 0 ; 1[,bean i nc reasi ngbi jectio n. Let T beagiv ent-norm .F or x;y 2 [0 ; 1]put T ( x;y )= ( 0 1) ( T ( ( x ) ; ( y )) ; where ( 0 1 ) :[0 ; 1] ! [ 0 ; 1]isthepseudo- i n v erseof , ( 0 1) ( x )= ( 0 1) (m ax( a;x )).N otethatif a =0then T iscal leda -transform ati on of T and T and T areisom or phi c(andhencethepropertiessuc has con ti n uit y ,stric tness,etc. ,arepre serv e d).If a> 0thenthedef orm ati on T prese rv e sthec on tin ui t yandt heA rc him e deanpropert y(and thenil potency )butthestrictne ssm a yb ev iolated.T ak e,e.g.,the productt-norm T P andl et ( x )=2 x 0 1 ,i .e. , a =1 = 2and 0 1 ( x )= 1+log 2 x .The n ( T P ) ( x;y )=1+l og 2 m ax (1 = 2 ; 2 x 0 1 1 2 y 0 1 )=m ax (0 ;x + y; 0 1) : He ncet he -deform ationofthestri ctproduc tt -norm T P i sthe nil poten tLuk as iewicz tnorm T L ,( T P ) = T L . Rec en tl y,Dem an t [1] h as sug gested t wo new t ype s o f t-normc onstructi on s. L e t :[0 ; 1] ! [0 ;a ], a 2 ]0 ; 1], b e a g i veninc reas i ng bije ction. F or x 2 [0 ; 1] w e dene the pseudo -i nverse of by ( 0 1) ( x )= 0 1 (m in( a; x )). Let T be a gi ven t-no rm .F or x; y 2 [0 ; 1], w ede ne: Contr ac tion : T ( ) ( x;y )= ( ( 0 1) ( T ( ( x ) ; ( y ))) ifmax ( x;y ) < 1 T ( x; y )otherwise ;
Onsom ec onstructi on so fn ew triangul arnorms 41 Dila tation : T ( ) ( x ;y )= ( T ( ( 0 1 ) ( x ) ; ( 0 1 ) ( y )) if T ( x; y ) <a T ( x;y )otherwise : Boththecon tractio na ndthedil at ati onofat-n orm T areagai n t-nor ms ,s e e[ 1] .Notethatint hecase a =1boththedil atati onand thecon tracti onarethe us ualsem igrouptransform at iono fSc h w e ize r andSkl ar[ 3] , T ( ) = T and T ( ) = T 0 1 .F urthernotethat the onl yt-norms pre se rv e db yarbi trarydeform ation,transform ation,and con tracti onarethel im itt-nor m s T M an d T W .H o w e v e r,theonl ytnormpre se rv ed b yanarbi trarydilatation is T M ,whil e( T W ) ( ) 6 = T W whenev er (1) 6 =1. 2C on tractionsoft-norm s F oragi v enb i jec tion :[ 0 ; 1] ! [ 0 ;a ]with a< 1andagi v e nt-norm T , thev al uesoftheco n trac tion T ( ) onthehal f-opensq uare[ 0 ; 1[ 2 depend onthev aluesof T onthe ha lf -ope nsquar e[ 0 ;a [ 2 onl y(therem ainde r ofthedomain isc on tai nedintheborders of theuni tsq uarewhereall t-norm scoi ncide).H encethenon-con tin ui ty (a ndtheabse nc eofthe Arc him edeanproperty )o f T ne ednotbet ruefori ts contracti on T ( ) . Ont heotherhand,thecon tin uity of T m a ybevi ol ate db y T ( ) ,to o, whilet he Arc him e deanprope rt yrema insp reser v ed. Examp le 1 i )L et ( x )= a 1 x , a 2 ]0 ; 1[ ,andl et T = T P .Then the c ontr ac tion T ( ) is de ne dby T ( ) ( x;y )= ( a 1 x 1 y if max ( x; y ) < 1 x 1 y ot h er wise : Note that T ( ) is n ot c ontin uous althoug h T is c on tinuous. F urthe r, the strictness T ( ) ( x; y ) <T ( ) ( x; z ) for each x> 0 , y<z , holds true.
42 R.Mesi ar ii)Let ( x ) = a 1 x , a 2 ] 0 ; 1[ ,andlet T h < 0 ;a; T P >; <a ; 1 ;T W > i be anor d inalsumt-no rm. Then T isnon-c ontinuo us(and non- A r chim e de an)but T ( ) = T P isc ontinuousa nd A rchime de an. 4 F oracom posit ion la wo ft w oc on tract ionsw ehav et he fol lo wing resul t. Proposi tion 1 L et :[ 0 ; 1] ! [0 ;a ] and :[ 0 ; 1] ! [0 ;b ] b etwo bije ctionswith a 1 and b 1 .L et T b eagivent-nor m.Then h T ( ) i ( ) = T ( ) ; i.e.,the -c o ntr actionofa -c ontr actionof T isth e -contr action of T . 4 Thep roble moft-norm si n v ari an tu ndergiv en -c on trac tionwillbe partiall ys olv edinthenex tsec tion. 3Dilatationsof tn orm s N on-trivial di latati ons(i .e. ,when a< 1) ar ea l w a ysordi nalsum swith t w osum m ands. Proposi tion2 L et :[ 0 ; 1] ! [0 ;a ] b eagiveni ncr e asingb ijec tion wher e a 2 ] 0 ; 1[ andlet T b eagiv en tnorm.Thenthe -dilatationo f T isanor dinalsumwi thtwos ummands, T ( ) h < 0 ;a;T =a >; < a ; 1 ;T a > i ; whe r e T =a is the tr ansfor mation T of with r esp ec t to t he mapping =a : [0 ; 1] ! [0 ; 1] , whil e T a is the deformation of T with r esp ect to theline ar tr ansformation a :[0 ; 1] ! [ a; 1] , a ( x )= a +( 1 0 a ) 1 x ,dep endin g onl y on T and a (indep en dent of up to t he val ue a = (1) ), T a = T a , T a ( x; y )= max h 0 ;T ( a +(1 0 a ) 1 x; a +(1 0 a ) 1 y ) 0 a i = (1 0 a ) : 4
Onsom ec onstructi on so fn ew triangul arnorms 43 Remark1 F orArc him edeanco ntin uoust-norm sw eha v et hef ollo wing result:le t f be an ad di tiv egeneratorofagiv en t-norm T [5]andle t thel eft derivati v eof f inthepoin t1b en on -tri vial, f 0 0 (1) 2 ] 01 ; 0[. Thenl im a ! 1 0 T a = T L ,wh e re T L i st heLuk asi ewiczt-norm .T he proof follo wsfromthefactthat if T has anaddi tiv egene rator f then T a has anadditiv egenerator f ( a +(1 0 a ) 1 x ). 4 Iti se asytos ee that forarbitr arydi latat iont he t-norm T M re m ains stabl e. F urthe r,foreac h a 2 ]0 ; 1[i tis [ T W ] a = T W andhence for arbi trary i tis h T W i ( ) h < 0 ;a;T W >;<a; 1 ;T W > i .Apply ing the -dilatationt o T W innitel ym an yt im es w ege tanewt-norm T 3 a dependingonlyon a and in va ri an tunder -dil at at io n, T 3 a h <a n ;a n 0 1 ;T W > ; n 2 N i : Anaturalque sti onarises:foragiventransform ation ,arethere som eother -dil atationi n v arian tt-norms upto T M (acon ti n uous tnorm )and T 3 a (adi sc on tin uoust-norm )?Iti sob vi ousthate ach - dil atationin v ari an tt-n orm T 3 diere n tfrom T M shouldbeanordi nal sumoft y pe T 3 h <a n ;a n 0 1 ;T> ; n 2 N i ; where T isat-normsuc hthat T = T a = T = a .Requi ringthecon tin uit y of T ,we ha ve thefollo wi ngresult. Propo siti on3 L et T b eac ontinu oust -norma ndlet a 2 ] 0 ; 1[ .Then T a equal s T if and on ly if T is the memb er of the ex ten de dY ag er 's family h T y p ;p 2 ]0 ; 1 ] i [6], i.e., T y 1 = T M an d for p 2 ] 0 ; 1 [ ,thetno rm T y p is ge ner ate dbyana dditiv e ge ner ator f p , f p ( x )= (1 0 x ) p , x 2 [0 ; 1] . 4 Note tha t the pro o f i sbasedonam o died Ca uc hyfun cti o nal equation. Further, let f b e an additive generator of a given t-no rm T . Th en T =a has a n a dditive generator f ( = a ), see[5],andthus T equa ls
44 R.Mesi ar T =a if an donlyi f f di er sfrom f ( =a )onlyb yam ulti pli cative constan t.F orn il po ten tt-norm T withthenor medg enerator f (this i sthecase of theY age r'st-norm s)thi sm ea ns that =a isthei den tit y, i .e. , ( x )= a 1 x .W eha v ejust shownthene xtresult. Proposi tion4 L et :[0 ; 1] ! [ 0 ;a ] ,wher e a 2 ]0 ; 1[ ,be an incr e a sing bije ction.I f isnotline arthentheo nl yc onti nuoust-nor minvariant under -dila tationisthe strongestt-norm T M .If isline ar ,thenthe onlyc ontinuous tnormsinvariantunder -dila tationar ethememb er s ofthef amily h T 3 a ;p ; p 2 ] 0 ; 1 ] i ,where T 3 a;p h <a n ;a n 0 1 ;T y p > ; n 2 N i . 4 Notethat T 3 1 = T M .F urthe r,itisusualt op ut T y 0 = T W (thele ft l im it m em beroftheY agerfam il y).T hene ac hm em beroft hef am il y h T 3 a;p ; p 2 [0 ; 1 ] i ,where T 3 a; 0 = T 3 a ,isi nv ari an tunde rthe -dil atation for ( x )= a 1 x . Re m ark2 Notethatt he -c on tracti onactsasanin v e rseofthe - dil atati on,theopposite be ingnottrue,i .e .,forarbitr aryt-norm T itis h T ( ) i ( ) = T .No w,i tisob viousthati fagiv en t-norm T i si n v arian t withre spec tt oagiv en -dil atationitha st obein v arian talsowith respecttothecor responding -c on tracti on. Refere nces [1]D em an t, B .,De form atio ne nv on t-Norm e n,i hreSy m m etrien und Sym metri ebrec hungen, pr eprint . [2] F odor, J.C. ,Ar em a rk on cons t ru c ting tnorm s, Fuzzy Se ts and Systems 41 (19 91), 19 5-1 99 . [3] Sc hwei ze r, B . a nd Sklar, A. ,A sso c iativ e functi o ns and statisti ca l tri angle i nequali ties, Publ . Math. Deb r ec en 8 (19 61), 1 69- 186 . [4] Schweizer, B. and Sklar, A., Ass o ciative functio ns a nd a bstract semigro ups, Pub l. Math. De brecen 10 (1 963 ), 69- 81.
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