Residual implications derived from uninorms satisfying Modus Ponens

Transcription

1 6th World Congrss of th Intrnational Fuzzy Systms Association (IFSA) 9th Confrnc of th Europan Socity for Fuzzy Logic and Tchnology (EUSFLAT) Rsidual implications drivd from uninorms satisfying Modus Ponns M. Mas, M. Monsrrat, D. Ruiz-Aguilra, J. Torrns Dpartmnt of Mathmatics and Computr Scinc Univrsity of th Balaric Islands Cra. d Valldmosa, km 7,5 7 Palma, Ills Balars, Spain -mails: {mmg448, mma, danil.ruiz, jts4}uib.s Abstract Modus Ponns is a ky proprty for fuzzy implication functions that ar going to b usd in fuzzy infrnc procsss. In this papr it is invstigatd whn fuzzy implication functions drivd from uninorms via rsiduation, usually calld RUimplications, satisfy th modus ponns with rspct to a continuous t-norm T, or quivalntly, whn thy ar T -conditionals. For RU-implications it is provd that T -conditionality only dpnds on th undrlying t-norm T U of th uninorm U usd to driv th rsidual implication and this fact lads to a lot of nw solutions of th Modus Ponns proprty. Along th papr th particular cass whn th uninorm lis in any of th most usual classs of uninorms ar considrd. Kywords: Fuzzy implication function, rsidual implication, Modus Ponns, uninorm. Introduction Fuzzy implication functions play a fundamntal rol in fuzzy logic and approximat rasoning. This kind of logical oprations ar ssntial in modlling all fuzzy conditionals and also in th infrnc procss. Morovr, thy ar also usful in many application filds not only drivd from th propr approximat rasoning, but also in othr aspcts as fuzzy subst-hood masurs, fuzzy rlational quations, fuzzy mathmatical morphology, and computing with words among othrs. For this rason, invstigations on fuzzy implication functions hav bn xtnsivly dvlopd in last dcads vn from th pur thortical point of viw, as it can b sn in th survy [8] and in th books [3, 4], ntirly dvotd to this kind of logical oprations. On of th main topics in this thortical study consists on th invstigation of additional proprtis of implication functions, proprtis that usually com from th concrt applications whr implications functions ar going to b applid. Th study of ach on of ths additional proprtis usually lads to solv a functional quation (or inquality) involving fuzzy implication functions (s for instanc Chaptr 7 in [4] and th rfrncs thrin). On of ths additional proprtis, that in this cas coms from approximat rasoning, is known as th (gnralizd) Modus Ponns. In fact, forward infrnc schms in approximat rasoning ar usually basd on th Modus Ponns that is carrid out through th wll known Compositional Rul of Infrnc (CRI) of Zadh, basd on th sup T composition, whr T is a t-norm (s for instanc, Sction 8.3 in [4]). Thus, if I is a fuzzy implication function and T is a t-norm, th Modus Ponns proprty for I with rspct to T bcoms th functional inquality: T (x,i(x,y)) y for all x,y [,], proprty that is also known as T -conditionality. Th Modus Ponns has bn xtnsivly studid in th litratur by som authors (namly [, 4, 6, 4, 5, 6, 7]). Howvr, all ths studis involv only th main classs of implication functions:. R-implications drivd from (lft-continuous) t-norms, I T (x,y) = sup{z [,] T (x,z) y, x,y [,]},. (S, N)-implications drivd from a t-conorm S and a fuzzy ngation N, I S,N (x,y) = S(N(x),y), x,y [,], 3. QL-implications drivd from a t-norm T, a t- conorm S and a fuzzy ngation N, I S,N,T (x,y) = S(N(x),T (x,y)), x,y [,]. On th othr hand, not that thr xist othr kinds of implication functions lik D-implications and Yagr s implications. Morovr, som gnralizations of R,(S, N), and QL-implications hav bn introducd, by substituting th t-norm and th t- conorm by mor gnral aggrgation functions (for mor dtails s [4] and also [9] with th rfrncs thrin). On of ths gnralizations is basd on uninorms obtaining th so-calld RU-implications ([7]), (U,N)-implications ([5]), and vn QL and D- implications drivd from conjunctiv and disjunctiv uninorms ([5]). 5. Th authors - Publishd by Atlantis Prss 33

2 For ths kinds of implications th Modus Ponns has not bn studid yt and this is th ida of th prsnt papr. In particular w want to dal with th cas of RU-implications, laving th othr cass for a futur work. Spcifically, w want to study th T -conditionality with rspct to any continuous t-norm T for th cas of RU-implications. W will prov that thr ar a lot of thm that satisfy th Modus Ponns with rspct to any t-norm T, and w will charactriz th spcial cas whn th t-norm T and th undrlying oprations of th uninorm ar continuous.. Prliaris W will suppos th radr to b familiar with th thory of t-norms, t-conorms and fuzzy ngations (all ncssary rsults and notations can b found in []). W also suppos that som basic facts on uninorms ar known (s for instanc [9]) as wll as thir most usual classs, that is, uninorms in U and U max ([9]), rprsntabl uninorms ([9]), idmpotnt uninorms ([6, 4, 3]) and uninorms continuous in th opn unit squar ([]). W rcall hr only som facts on implications and uninorms in ordr to stablish th ncssary notation that w will us along th papr. Dfinition A binary oprator I : [,] [,] [,] is said to b a fuzzy implication function, or an implication, if it satisfis: (I) I(x,z) I(y,z) whn x y, for all z [,]. (I) I(x,y) I(x,z) whn y z, for all x [,]. (I3) I(,) = I(,) = and I(,) =. Not that, from th dfinition, it follows that I(,x) = and I(x,) = for all x [,] whras th symmtrical valus I(x,) and I(,x) ar not drivd from th dfinition. Dfinition A uninorm is a two-plac function U : [,] [,] which is associativ, commutativ, incrasing in ach plac and such that thr xists som lmnt [,], calld nutral lmnt, such that U(,x) = x for all x [,]. Evidntly, a uninorm with nutral lmnt = is a t-norm and a uninorm with nutral lmnt = is a t-conorm. For any othr valu ],[ th opration works as a t-norm in th [,] squar, as a t-conorm in [,] and its valus ar btwn imum and maximum in th st of points A() givn by A() = [,[ ],] ],] [,[. W will usually dnot a uninorm with nutral lmnt and undrlying t-norm and t-conorm, T and S, by U T,, S. For any uninorm it is satisfid that U(,) {,} and a uninorm U is calld conjunctiv if U(,) = and disjunctiv whn U(,) =. On th othr hand, lt us rcall th most studid classs of uninorms in th litratur. Thorm ([9]) Lt U : [,] [,] b a uninorm with nutral lmnt ],[. (a) If U(,) =, thn th sction x U(x,) is continuous xcpt in x = if and only if U is givn by U(x,y) = T ( x, y ) ( ) if (x,y) [,], + ( )S x, y if (x,y) [,], (x,y) if (x,y) A(), whr T is a t-norm, and S is a t-conorm. (b) If U(,) =, thn th sction x U(x,) is continuous xcpt in x = if and only if U is givn by th sam structur as abov, changing imum by maximum in A(). Th st of uninorms as in cas (a) will b dnotd by U and th st of uninorms as in cas (b) by U max. W will dnot a uninorm in U with undrlying t-norm T, undrlying t-conorm S and nutral lmnt as U T,,S and in a similar way, a uninorm in U max as U T,,S max. Idmpotnt uninorms wr charactrizd first in [6] for thos with a latral continuity and in [4] for th gnral cas. An improvmnt of this last rsult was don in [3] as follows. Thorm ([3]) U is an idmpotnt uninorm with nutral lmnt [,] if and only if thr xists a non incrasing function g : [, ] [, ], symmtric with rspct to th idntity function, with g() =, such that U(x,y) = (x,y) if y < g(x) or (y = g(x) and x < g (x)), max(x,y) if y > g(x) or (y = g(x) and x > g (x)), x or y if y = g(x) and x = g (x), bing commutativ in th points (x, y) such that y = g(x) with x = g (x). Any idmpotnt uninorm U with nutral lmnt and associatd function g, will b dnotd by U g, id and th class of idmpotnt uninorms will b dnotd by U id. Obviously, for any of ths uninorms th undrlying t-norm T is th imum and th undrlying t-conorm S is th maximum. Dfinition 3 ([9]) Lt b in ],[. A binary opration U : [,] [,] is a rprsntabl uninorm if and only if thr xists a strictly incrasing function h : [,] [,+ ] with h() =, h() = and h() = + such that U(x,y) = h (h(x) + h(y)) 34

3 for all (x,y) [,] \ {(,),(,)} and U(,) = U(,) {,}. Th function h is usually calld an additiv gnrator of U. Rmark Rcall that th ar no continuous uninorms with nutral lmnt ],[. In fact, rprsntabl uninorms wr charactrizd as thos uninorms that ar continuous in [,] \ {(,),(,)} (s []) as wll as thos that ar strictly incrasing in th opn unit squar (s [8]). Any rprsntabl uninorm U with nutral lmnt and additiv gnrator h, will b dnotd by U h, rp and th class of rprsntabl uninorms will b dnotd by U rp. For any of ths uninorms th undrlying t-norm T is always strict and th undrlying t-conorm S is strict as wll. A mor gnral class containing rprsntabl uninorms is givn by thos uninorms that ar continuous in th opn unit squar ],[. This class was charactrizd in [] as follows. Thorm 3 ([] and [] for th currnt vrsion) Suppos U is a uninorm continuous in ],[ with nutral lmnt ],[. Thn ithr on of th following cass is satisfid: (a) Thr xist u [, [, λ [, u], two continuous t-norms T and T and a rprsntabl uninorm R such that U can b rprsntd as U(x,y) = ( λt xλ, y ) λ ( ) if x,y [,λ], λ + (u λ)t x λ u λ, y λ u λ if x,y [λ,u], ) u + ( u)r( x u u, y u u if x,y ]u,[, if (x,y) ]λ,] and max(x,y) =, λ or if (x,y) = (λ,) or (x,y) = (,λ), (x, y) lswhr. () (b) Thr xist v ],], ω [v,], two continuous t-conorms S and S and a rprsntabl uninorm R such that U can b rprsntd as U(x,y) = ( ) v + (ω v)s x v ω v, y v ω v if x,y [v,ω], ( ) ω + ( ω)s x ω ω, y ω ω if x,y [ω,], vr ( x v, y ) v x,y ],v[, if max(x,y) [,ω[ and (x,y) =, ω or if (x,y) = (,ω) or (x,y) = (ω,), max(x, y) lswhr. () Th class of all uninorms continuous in ],[ will b dnotd by U cos. A uninorm as in () will b dnotd by U T,λ,T,u,(R,) cos, and th class of all uninorms continuous in th opn unit squar of this form will b dnotd by U cos,. Analogously, a uninorm as in () will b dnotd by U (R,),v,S,ω,S cos,max and th class of all uninorms continuous in th opn unit squar of this form will b dnotd by U cos,max. For any uninorm U T,λ,T,u,(R,) cos,, th undrlying t-norm of U is givn by an ordinal sum of thr t-norms, T,T and a strict t-norm, whras th undrlying t-conorm is always strict. Similarly, for any uninorm U (R,),v,S,ω,S cos,max, th undrlying t-norm of T is always strict, whras th undrlying t-conorm is givn by an ordinal sum of thr t-conorms, a strict t-conorm, S, and S. On th othr hand, diffrnt classs of implications drivd from uninorms hav bn studid. W rcall hr RU-implications. Dfinition 4 Lt U b a uninorm. Th rsidual opration drivd from U is th binary opration givn by I U (x,y) = sup{z [,] U(x,z) y} for all x,y [,]. Proposition 4 ([7]) Lt U b a uninorm and I U its rsidual opration. Thn I U is an implication if and only if th following condition holds U(x,) = for all x <. (3) In this cas I U is calld an RU-implication. This includs all conjunctiv uninorms but also many disjunctiv ons, for instanc in th classs of rprsntabl uninorms (s [7]), idmpotnt uninorms (s []), and uninorms continuous in th unit opn squar (s []). Howvr, whn w dal with lft-continuous uninorms U w clarly hav that U satisfis condition (3) if and only if it is conjunctiv. Som proprtis of RU-implications hav bn studid involving th main classs of uninorms, thos prviously statd: uninorms in U, rprsntabl uninorms, idmpotnt uninorms and uninorms continuous in th opn unit squar (for mor dtails s [, 4, 7, 7,, ]). Howvr, although th strong intrst of th Modus Ponns proprty, its study is not among th proprtis invstigatd for implications drivd from uninorms. Lt us rcall th dfinition of th Modus Ponns in th framwork of fuzzy logic. Dfinition 5 Lt I b an implication function and T a t-norm. It is said that I satisfis th Modus Ponns proprty with rspct to T, or that I is a T -conditional if T (x,i(x,y)) y for all x,y [,]. (4) A wll known gnral rsult on T -conditionality was provd in [4]. Proposition 5 Lt I b an implication function and T a lft-continuous t-norm. Thn I is a T - conditional if and only if I I T, whr I T dnots th rsidual implication drivd from T. 35

4 3. RU-implications that ar T -conditionals In this sction w want to dal with th cas of RUimplications. Thus, th main goal of this sction is to charactriz whn an RU-implication drivd from a uninorm U is a T -conditional for a t-norm T, spcially whn T is continuous. All along this sction it will b undrstood that any considrd uninorm U satisfis U(x, ) = for all x <, in ordr to nsur that th corrsponding rsidual I U is an RU-implication, according to Proposition 4. Proposition 6 Lt U b a uninorm with nutral lmnt ],[ and undrlying t-norm T U, and lt I U b th corrsponding RU-implication. Thn th following itms ar quivalnt: i) I U is a T -conditional. ii) I U satisfis Equation (4) for all y < x <. iii) Th inquality ( ( x T x,i TU, y )) y holds for all x,y such that y < x <, whr I TU dnots th rsidual implication drivd from th t-norm T U. This rsult provs that th undrlying t-conorm S U of th uninorm U and th valus of U in th rgion A() ar not rlvant in ordr I U to b a T -conditional. Only th undrlying t-norm T U is rlvant and th inquality corrsponding to T - conditionality only nds to b chckd in th rgion R = {(x,y) [,] y < x < }. Th rgion R is picturd in Figur 3. Figur : Rgion R. R Thus, w continu our study dpnding on how th undrlying t-norm T U is. Not that if T (,) =, thn condition iii) in Proposition 6 is always satisfid. Thn, any t-norm T U (continuous or not) will work in this cas. From now on, w will rstrict ourslvs to th cas whn T U is continuous. Taking into account th classification of continuous t-norms (s for instanc []), w will divid our study in thr stps rspctivly dvotd to th cass whn T U =, T U is Archimdan or T U is givn by an ordinal sum. Lt us bgin with th cas whn T U =. Proposition 7 Lt U b a uninorm with nutral lmnt ],[ and undrlying t-norm T U givn by th imum. Thn th RU-implication I U is a T -conditional for any t-norm T. In particular, this is th cas for any idmpotnt uninorm U. Exampl Lt N b a strong ngation. Among th class of idmpotnt uninorms, an important xampl is givn by uninorms whnvr g = N (s []), that is, thy hav th form (x,y) U(x,y) = max(x,y) (x, y) or max(x, y) if y < N(x), if y > N(x), othrwis, bing commutativ in th points (x, y) such that y = N(x). In ths cass th corrsponding RUimplication is givn by { (N(x),y) if y < x, I U (x,y) = max(n(x),y) if y x. From th proposition abov, all ths implications ar T -conditionals for any t-norm T. Figur 3 shows th idmpotnt uninorm U N, id and its RU-implication givn in this xampl whn th considrd ngation is th classical on N c (x) = x. Lt us now dal with th cas whn th undrlying t-norm T U is Archimdan and th t-norm T is continuous. It is wll known that whn a t-norm is Archimdan it is rprsntd by a dcrasing additiv gnrator ϕ : [,] [,+ ], with ϕ() = + if th t-norm is strict, and with ϕ() = whn th t-norm is nilpotnt. Morovr, in this last cas th function N(x) = ϕ ( ϕ(x)) for all x [,] is a strong ngation usually calld th associatd ngation of th nilpotnt t-norm T. Lt us considr both cass sparatly. Proposition 8 Lt U b a uninorm with nutral lmnt ],[ and undrlying t-norm T U strict. Lt I U b th RU-implication drivd from U and T a continuous t-norm. i) If I U is a T -conditional, thn thr xists a such that T is an ordinal sum of th form T = (,a,t a, a,,t ), whr T a is Archimdan and T continuous. ii) Lt ϕ and ϕ a b th additiv gnrators of T U and T a, rspctivly. Thn I U is a T - conditional if and only if th function g dfind from [,+ ] to [ϕ a ( a ),ϕ a()] givn by th xprssion g(u) = ϕ a a ϕ (u) ) is sub-additiv. 36

5 max y x x y Figur : Structur of U N, (bottom) with Nc (x) = x. id (top) and IU Rmark i) Not that whn a =, th rsult abov can b rlatd to th rsults about T conditionality for rsidual implications drivd from continuous t-norms givn in [4]. Spcifically, w can driv that IU is a T -conditional if and only T = (h,, T i, h,, T i), whr T is Archimdan, T continuous and ITU is a T -conditional. ii) Of cours that on can tak a = in th proposition abov obtaining RU -implications that ar T -conditional for Archimdan t-norms T. iii) Not that uninorms with TU strict includ, but ar not limitd to, all rprsntabl uninorms as wll as thos uninorms continuous in ], [ lying in Ucos,max. In addition, th subst of all uninorms with TU and SU strict hav bn rcntly charactrizd in [3]. Figur 3: Rprsntabl U with additiv uninorm x gnrator h(x) = log x (top) and its rsidual implication IU (bottom). Namly, it is wll known that U is a rprsntabl uninorm with nutral lmnt = and additiv x gnrator h(x) = log x. In this cas, th undrlying t-norm TU is strict with additiv gnrator givn by ϕ(x) = log x (s [9] or Sction. in x []). Morovr, this situation corrsponds to tak a = and ϕ (x) = log(x) in Proposition 8 and consquntly th corrsponding function g is givn by g(x) = ϕ ϕ (x) = log + x = log( + x ), Exampl Lt us tak, for instanc, th conjunctiv uninorm givn by U (x, y) = ( if (x, y) {(, ), (, )}, xy othrwis, xy+( x)( y) which is clarly sub-additiv. In Figur 3 uninorm U and its rsidual implication IU hav bn dpictd. Proposition 9 Lt U b a uninorm with nutral lmnt ], [ and undrlying t-norm TU nilpotnt with associatd ngation NU. Lt IU b th RU -implication drivd from U and T a continuous t-norm. whos rsidual implication IU is givn by ( if (x, y) {(, ), (, )}, IU (x, y) = ( x)y othrwis, x+y xy and tak also T = TP th product t-norm. It is asy to s that in this cas IU is a T -conditional. i) If IU is a T -conditional with rspct to T, thn thr xists a such that T is an ordinal sum 37

6 of th form T = (,a,t a, a,,t ), whr T a is nilpotnt with associatd ngation N a such that N a covrs N U, i.., N U ( x ) an a( x a ) for all x [,], and T is continuous. ii) Lt ϕ and ϕ a b th additiv gnrators of T U and T a, rspctivly. Thn I U is a T - conditional with rspct to T if and only if th function ( g : [,] [ϕ a ( a ),] givn by g(u) = ϕ a a ϕ (u) ) is sub-additiv. Rmark 3 i) Not that in th prvious proposition, th fact that g is subadditiv alrady nsurs that th corrsponding ngations N a and N U ar such that N a covrs N U. Namly, w hav that N a covrs N U if and only if ( ( x )) ( ( ϕ x )) ϕ aϕ a ϕ a, a but taking th chang x = ϕ (z), this is quivalnt to g( z) g(z), which is clarly satisfid whn g is sub-additiv. ii) Again not that whn a =, th rsult abov can b statd as follows: I U is a T -conditional if and only T = (,,T,,,T ), whr T is Archimdan, T continuous and I TU is a T -conditional. Of cours a = can b takn in th proposition abov obtaining RU-implications that ar T -conditional for nilpotnt t- norms T. iii) Rcall that uninorms U with T U nilpotnt and S U Archimdan wr charactrizd in [] and [3]. Exampl 3 Lt U b a uninorm in U with nutral lmnt = and undrlying t-norm T U = T L th Łukasiwicz t-norm, that is, U is givn by th xprssion max(,x + y ) if (x,y) [, ], +S U(x,y) = U (x,y ) if (x,y) [,], (x, y) othrwis, whr S U can b any t-conorm. In this cas I U is givn by (s [7]): if x < and x y, x + y if x < and x > y, I U (x,y) = y if y x, if y < x, +R SU (x,y ) if x y, whr R SU (x,y) = sup{z [,] S U (x,z) y}. Thn I U is always a T L -conditional bcaus, using Proposition 9, w hav a = and ϕ(x) = ϕ (x) = x, obtaining g(x) = x+ which is clarly subadditiv. In Figur 4 w can s th structur of this gnral uninorm in U as in this xampl and th corrsponding RU-implication. x + y x + y S U R SU y Figur 4: Structur of U (top) and I U (bottom) whn U U, T U = T L and =. Proposition Lt U b a uninorm with nutral lmnt ],[ and undrlying t-norm givn by th ordinal sum T U = ( a i, b i,t i ) i I with a i < b i and T i Archimdan for all i I. Lt I U b th RU-implication drivd from U and T a continuous t-norm. Thn I U is a T -conditional if and only if th following itms hold: i) Th st of idmpotnt lmnts of T, that ar lss than or qual to, ar containd in [,] \ { i I (a i,b i )}. ii) Th t-norm T is an ordinal sum of th form T = ( c j,d i,t j ) j J in such a way that for all i I thr xists a j J such that (a i,b i ) (c j,d j ), and if T i is nilpotnt with associatd ngation N i for som i I and a i = c j thn T j must b also nilpotnt with associatd ngation N j such that N j covrs N i. If ϕ i and ϕ j ar th rspctiv additiv gnrators of T i and T j, thn th function [ g ij that is dfind ] from [,ϕ i ()] to ϕ j ( b i c j d j c j ),ϕ j ( a i c j d j c j ) givn by ( ) a i + (b i a i )ϕ i (u) c j g ij (u) = ϕ j d j c j is sub-additiv. From th prvious rsult w can asily driv th following rsult. 38

7 Proposition Lt U b a uninorm with nutral lmnt ],[ and undrlying t-norm givn by th ordinal sum T U = ( a i, b i,t i ) i I with a i < b i and T i Archimdan for all i I. Lt T b th ordinal sum T = ( a i,b i,t i ) i I with T j Archimdan for all i I and suppos that I T is T i -conditional i for all i I. Thn I U is a T -conditional. Exampl 4 Lt U b a uninorm in U with nutral lmnt = and undrlying t-norm T U givn by th ordinal sum T U = (,,T P,,,T L ) that is, U is givn by th xprssion 4xy if (x,y) [, 4 ], max( U(x,y) = 4,x + y ) if (x,y) [ 4, ], +S U (x,y ) if (x,y) [,], (x, y) othrwis, whr S U can b any t-conorm. Lt us considr th t-norm T givn by th xprssion T P if (x,y) [, ], T (x,y) = T L if (x,y) [,], (x, y) othrwis. Using Proposition, I U is a T -conditional bcaus, I TP is a T P -conditional and I TL is a T L - conditional. In Figur 5 w can s th structur of this gnral uninorm in U as in this xampl and th corrsponding RU-implication. 4. Conclusions and futur work Forward infrnc schms in approximat rasoning ar basd on th Modus Ponns proprty, also calld T -conditionality. Thus, fuzzy implication functions usd in th infrnc procss of any fuzzy rul basd systm ar rquird to satisfy this proprty, which bcoms ssntial in approximat rasoning and fuzzy control. Fixd a continuous t- norm T modlling th conjunction, w studid in this papr which fuzzy implication functions satisfy T -conditionality among a spcial kind of implications drivd from uninorms: RU-implications. In this cas w hav charactrizd all th solutions of th Modus Ponns proprty with rspct to a continuous t-norm T and from ths charactrizations w obtain a lot of nw fuzzy implication functions satisfying th T -conditionality. Morovr, w want to xtnd this study to th cass of othr classs of fuzzy implication functions drivd from uninorms, lik (U,N)-implications and QL and D-implications. It is worth to point out that w hav alrady startd with th cas of (U,N)- implications obtaining again a lot of nw solutions. Morovr, contrariously to what happns with RU-implications, T -conditionality for (U, N)- implications only dpnds on th undrlying t- conorm S U and only in som cass, dpnding on 4 4 T P 4 T P 4 T L S U T L Figur 5: Structur of U (top) and T (bottom) such that U U, and I U is a T -conditional. th valu α N = inf{z [,] N(z) = }, whr N is th ngation usd to driv th corrsponding (U, N)-implication, and is th nutral lmnt of th uninorm U. Anothr possibl xtnsion that dsrvs to b invstigatd is rlatd to th Modus Ponns with rspct to a conjunctiv uninorm U instad of a continuous t-norm T. Of cours that, as futur work, it should b also includd th study of th Modus Tollns for all ths classs of implications. Acknowldgmnts This papr has bn partially supportd by th Spanish grant TIN P. Rfrncs [] I. Aguiló, J. Suñr, J. Torrns, A charactrization of rsidual implications drivd from lftcontinuous uninorms, Information Scincs, Vol. 8() () [] C. Alsina, E. Trillas, Whn (S, N)-implications ar (T,T )-conditional functions?, Fuzzy Sts and Systms, Vol. 34, 35 3 (3). [3] M. Baczyński, G. Bliakov, H. Bustinc-Sola, A. Pradra (Eds.), Advancs in Fuzzy Implication Functions, Studis in Fuzzinss and Soft 39

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