MP and MT-implications on a finite scale

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1 MP ad MT-implicatios o a fiite scale M Mas Dpt de Matemàtiques i If Uiversitat de les Illes Balears Palma de Mallorca Spai dmimmg0@uibes M Moserrat Dpt de Matemàtiques i If Uiversitat de les Illes Balears Palma de Mallorca Spai dmimma0@uibes J Torres Dpt de Matemàtiques i If Uiversitat de les Illes Balears Palma de Mallorca Spai dmijts0@uibes Abstract This paper is devoted to the study of discrete implicatios that satisfy modus poes (MP), modus tolles (MT) or both (MPT) The mai goal is to characterize all R, S, QL ad D-implicatios o a fiite chai L satisfyig these properties for a give smooth t-orm T 1 The o-smooth case is also discussed for a special family of t-orms Keywords: Discrete implicatios, modus poes, modus tolles, smooth t-orms, fiite scale 1 Itroductio It is well kow that fuzzy implicatio fuctios (see the survey [12]) are used i approximate reasoig, ot oly to represet fuzzy coditioal statemets of the form If p the q (with p, q fuzzy statemets), but also to perform ifereces i ay fuzzy rule based system I this iferece process, the two mai classical rules are modus poes (MP) ad modus tolles (MT) that allow to perform, respectively, forward ad backward ifereces I terms of fuzzy logic, these implicatios are operators I : [0, 1] 2 [0, 1] extedig the classical material implicatio, that is, satisfyig I(0, 0) = I(0, 1) = I(1, 1) = 1 ad I(1, 0) = 0 Sice cojuctios, disjuctios ad egatios are usually performed by t-orms (T ), t-coorms (S) ad strog egatios (N), i fuzzy set theory as much as i fuzzy logic ad approximate reasoig, the majority of the kow implicatio fuctios are directly derived from these operators 1 The four most usual ways to defie these implicatio fuctios are: 1 Although some authors have derived also implicatios from other aggregatio fuctios, specially uiorms (see [1], [11],[14]) i) R-implicatios defied by I(x, y) = sup{z [0, 1] T (x, z) y} (1) for all x, y [0, 1] ii) S-implicatios defied by I(x, y) = S(N(x), y), x, y [0, 1] (2) iii) QL-implicatios defied by I(x, y) = S(N(x), T (x, y)), x, y [0, 1] (3) iv) D-implicatios, that are the cotrapositio with respect to N of QL-implicatios ad are give by I(x, y) = S(T (N(x), N(y)), y), x, y [0, 1] (4) Moreover, i this cotext, give ay t-orm T 1 ad ay strog egatio N 1, modus poes ad modus tolles for a implicatio I ca be writte as ad T 1 (x, I(x, y)) y, x, y [0, 1] (5) T 1 (N 1 (y), I(x, y)) N 1 (x), x, y [0, 1] (6) respectively These two equatios have bee recetly solved i [15] for the first three metioed classes of implicatios O the other had, the study of operators defied o fiite scales is a area of icreasig iterest (see [2], [4], [5], [8], [9], [10], ad [13]) Maily, because it allows to deal with fiite families of liguistic labels avoidig umerical iterpretatios (ecessaries i the fuzzy logic approach) I this cotext, the book chapter [13] brigs a survey o (smooth) discrete t- orms ad t-coorms o a fiite chai L, ad the four classes of implicatios, R, S, QL ad D-implicatios derived from discrete t-orms, are studied i [9] ad

2 [10] From these discrete implicatios, ew possibilities for approximate reasoig with fiite families of liguistic labels appear with their cosequet applicatios i computig with words However, i this lie, the study of MP ad MT rules for discrete implicatios, equivalet to the oe give i [15] for fuzzy implicatios, is essetial This is precisely the mai goal of this paper After some prelimiaries give i sectio 2, we devote sectio 3 to characterize those R, S, QL ad D-implicatios o a fiite chai L, derived from smooth t-orms, that satisfy equatio (5), equatio (6) or both Fially, sectio 4 is devoted to the o-smooth case for a special family of t-orms 2 Prelimiaries We recall here the smooth t-orms ad the smooth t- coorms o a fiite chai L ad their characterizatio, that will be used alog the paper It is well kow that for our purposes (see [13]) all fiite chais with the same umber of elemets are equivalet ad the, from ow o, we will deal with the simplest fiite chai of + 1 elemets: L = {0, 1, 2, 1, } where 1 Such a L ca be uderstood as a set of liguistic terms or labels The followig two defiitios are adapted from [5] (see also [13]) Defiitio 1 A fuctio f : L L is said to be smooth if it satisfies oe of the followig coditios: f is odecreasig ad f(x) f(x 1) 1 for all x L with x 1 f is oicreasig ad f(x 1) f(x) 1 for all x L with x 1 Defiitio 2 A biary operator F o L is said to be smooth if it is smooth i each variable The importace of the smoothess coditio lies i the fact that it is geerally used as a discrete couterpart of cotiuity o [0,1] Although t-orms, t-coorms ad strog egatios are usually biary operators o [0,1], they ca be defied as i [2] o ay bouded partially ordered set ad, i particular, o L I this last case, they are usually kow as discrete t-orms ad discrete t-coorms I this way, recall that smoothess for discrete t-orms (ad also for t-coorms) is equivalet to the divisibility coditio, that is, x y if ad oly if there exists z L such that T (y, z) = x, (see [13]) Propositio 1 There is oe ad oly oe strog egatio o L that is give by N(x) = x for all x L (7) From ow o, N will always deote the egatio o L give by (7) Smooth t-orms have bee characterized as ordial sums of Archimedea oes as follows Propositio 2 (See [13]) There is oe ad oly oe Archimedea smooth t-orm o L, deoted by T L, give by T L (x, y) = max{0, x + y } for all x, y L (8) which is kow as the Lukasiewicz t-orm Moreover, give ay subset J of L cotaiig 0,, there is oe ad oly oe smooth t-orm o L that has J as the set of idempotet elemets, that will be deoted by T J I fact, if J is the set J = {0 = i 0 < i 1 < < i m 1 < i m = } the T J is give by: max{i k, x + y i k+1 } if x, y [i k, i k+1 ] T J (x, y) = for some i k J mi{x, y} otherwise (9) Smooth t-coorms have a classificatio theorem like the above oe for t-orms which ca be easily deduced by N-duality The expressio of the oly Archimedea smooth t-coorm o L is give by S L (x, y) = mi{, x + y} for all x, y L (10) which is also kow as the Lukasiewicz t-coorm I geeral, we have Propositio 3 (See [13]) Give ay subset J of L cotaiig 0,, there is oe ad oly oe smooth t- coorm o L, S J, that has J as the set of idempotet elemets I fact, if J is the set J = {0 = i 0 < i 1 < < i m 1 < i m = } the S J is give by: mi{i k+1, x + y i k } if x, y [i k, i k+1 ] S J (x, y) = for some i k J max{x, y} otherwise (11) Note that with these otatios, the Lukasiewicz t- orm ad t-coorm ca be writte, respectively as T L = T {0,} ad S L = S {0,} The followig result follows from the previous propositios

3 Propositio 4 (See [13]) There are exactly 2 1 differet smooth t-orms (t-coorms) o L Defiitio 3 A biary operator I : L L L is said to be a (discrete) implicatio if it satisfies: (I1) I is oicreasig i the first variable ad odecreasig i the secod oe (I2) I(0, 0) = I(, ) = ad I(, 0) = 0 Note that, from the defiitio, it follows that I(0, x) = ad I(x, ) = for all x L The four ways to defie fuzzy implicatios apply here to defie discrete implicatios However, sice the oly strog egatio o L is the oe give by (7) ad we deal with a fiite scale, i our case they ca be rewritte as follows: I(x, y) = max{z L T (x, z) y}, x, y L (12) I(x, y) = S( x, y), x, y L (13) I(x, y) = S( x, T (x, y)), x, y L (14) I(x, y) = S(T ( x, y), y), x, y L (15) All these classes of discrete implicatios have bee already studied: R ad S-implicatios i [9], ad QL ad D-implicatios i [10] Thus we refer to these cited papers for details o these kids of discrete implicatios that we will use i the paper Although the o-smooth case is cosidered i these refereces, i the preset work we will deal oly with R, S, QL ad D-operators derived from smooth t-orms ad smooth t-coorms 3 Mai results I this sectio we deal with (implicatio) operators o the fiite chai L that satisfy the modus poes, the modus tolles or both, with respect to a smooth t-orm T 1 Agai, sice we have oly oe strog egatio o L, i our case equatio (6) ca be rewritte depedig oly o the t-orm T 1 ad thus, we ca adopt the followig defiitios: Defiitio 4 Let T 1 be a t-orm o L A fuctio I : L 2 L will be called: - a MP-operator for T 1 wheever it satisfies T 1 (x, I(x, y)) y for all x, y L (16) - a MT-operator for T 1 wheever it satisfies T 1 ( y, I(x, y)) x for all x, y L (17) - a MPT-operator for T 1 wheever it is both, a MP ad a MT-operator Moreover, we will say that I is a MP-implicatio (or MT, or MPT-implicatio) if it is a implicatio ad also a MP-operator (or MT or MPT-operator, respectively) Note that whereas all R ad S-operators (give by equatios (12) ad (13), respectively) are always implicatios, this is ot the case for QL ad D-operators (give by equatios (14) ad (15), respectively), see for istace [10] Thus, we will divide our study of properties MP, MT ad MPT i three subsectios, the first oe devoted to R-implicatios, the secod oe devoted to S-implicatios ad, fially, the third oe devoted to QL ad D-operators i geeral, icludig of course QL ad D-implicatios Before this, let us begi with two easy but importat propositios i the discussio of the metioed properties The first oe deals with MP-operators Propositio 5 Let T 1 be a t-orm o L ad I : L 2 L a MP-operator for T 1 The, T 1 (x, I(x, 0)) = 0 for all x L The secod oe deals with MT-operators Propositio 6 Let T 1 be a t-orm o L ad I : L 2 L a MT-operator for T 1 The, T 1 ( y, I(, y)) = 0 for all y L 31 R-implicatios Give ay t-orm T we will deote by I T its residual implicatio, that is, the operator give by equatio (12): I T (x, y) = max{z L T (x, z) y} We deal i this subsectio with implicatios I T : L 2 L, where T is a smooth t-orm Let us recall here the structure of these implicatios For a easier uderstadig we give their graphical structure i Figure 1 istead of their formulas, that ca be foud i [9] Thus, R ad S-implicatios ca be viewed i Figures 1 ad 2, respectively We begi with the property of MP-implicatios Give ay t-orm T, let us deote also by Idemp T the set of all idempotet elemets of the t-orm T, that is: Idemp T = {x L T (x, x) = x} The we have the followig propositio

4 i m 1 i m 2 i 2 i 1 I i2 I im 1 I i1 0 i 1 i 2 i m 2 i m 1 y I im Figure 1 The structure of the R-implicatio derived from T J where J = {0 = i 0 < i 1 < < i m 1 < i m = } ad I ik+1 (x, y) = i k+1 + y x for k = 0,, m 1 Propositio 7 Let T, T 1 be smooth t-orms o L ad I T : L 2 L the R-implicatio associated to T The followig statemets are equivalet: i) I T is a MP-implicatio for T 1 ii) I T (x, y) I T1 (x, y) for all x, y L iii) Idemp T1 Idemp T Thus, it is clear that we ca deduce the followig particular cases Corollary 1 Let T, T 1 be smooth t-orms o L ad I T : L 2 L the R-implicatio associated to T The, i) If T = mi, I mi is a MP-implicatio for ay smooth t-orm T 1 ii) If T = T L, I TL is a MP-implicatio for T 1 if ad oly if T 1 = T L iii) If T 1 = T L, I T is a MP-implicatio for ay smooth t-orm T iv) If T 1 = mi, I T is a MP-implicatio for mi if ad oly if T = mi With respect to MT-implicatios we obtai solutios oly whe T 1 is the Lukasiewicz t-orm ad the I T works for ay smooth t-orm T This ca be proved through the followig two propositios Propositio 8 Let T, T 1 be t-orms o L with T smooth ad let I T : L 2 L be the R-implicatio associated to T If I T is a MT-implicatio for T 1 the ecessarily T 1 (x, x) = 0 for all x L Remark 1 It is proved i Lemma 1 of [9] that, for smooth t-orms, the previous coditio is equivalet to be T 1 the Lukasiewicz t-orm That is, for smooth t-orms, T 1 (x, x) = 0 for all x L T 1 = T L Propositio 9 Let T, T 1 be smooth t-orms o L ad I T : L 2 L the R-implicatio associated to T The I T is a MT-implicatio for T 1 if ad oly if T 1 = T L Now, joitly the obtaied results for MP ad for MTimplicatios we obtai the followig corollary Corollary 2 Let T, T 1 be smooth t-orms o L ad I T : L 2 L the R-implicatio associated to T The,I T is a MPT-implicatio for T 1 if ad oly if T 1 = T L 32 S-implicatios Let us ow deal with S-implicatios Give ay t- coorm S we will deote by I S the correspodig S- implicatio give by equatio (13) That is: I S (x, y) = S( x, y) for all x, y L The structure of S-implicatios ca be viewed i Figure 2 (their expressio ca be foud i [9]) i 1 i 2 i m 2 i m 1 I i1 I i2 I max 0 i 1 i 2 i m 2 i m 1 I max I im 1 I im Figure 2 The structure of the S-implicatio derived from the dual t-coorm of T J, where J = {0 = i 0 < i 1 < < i m 1 < i m = }, I max (x, y) = max{ x, y} ad I ik+1 (x, y) = mi{ i k, i k+1 + y x} for k = 0,, m 1 For S-implicatios, the study of the modus poes is solved by the followig two propositios Propositio 10 Let T 1 be a t-orm, S a smooth t- coorm o L ad I S : L 2 L the correspodig S-

5 implicatio If I S is a MP-implicatio for T 1 the ecessarily T 1 (x, x) = 0 for all x L Propositio 11 Let T 1 be a smooth t-orm ad S a smooth t-coorm o L ad I S : L 2 L the correspodig S-implicatio The I S is a MP-implicatio for T 1 if ad oly if T 1 = T L I this case, the study of modus tolles gives exactly the same solutios Specifically, Propositio 12 Let T 1 be a t-orm ad S a smooth t-coorm o L ad I S : L 2 L the correspodig S-implicatio If I S is a MT-implicatio for T 1 the ecessarily T 1 (x, x) = 0 for all x L Propositio 13 Let T 1 be a smooth t-orm ad S a smooth t-coorm o L ad I S : L 2 L the correspodig S-implicatio The I S is a MT-implicatio for T 1 if ad oly if T 1 = T L Ad cosequetly we have: Corollary 3 Let T 1 be a smooth t-orm ad S a smooth t-coorm o L ad I S : L 2 L the correspodig S-implicatio The I S is a MPTimplicatio for T 1 if ad oly if T 1 = T L 33 QL ad D-operators I this subsectio we deal with QL ad D-operators give by equatios (14) ad (15), respectively As we have commeted, ot all of them are implicatios i the sese of Defiitio 3 I fact, it is proved i [10] that this occurs i the smooth case (for both QL ad D) if ad oly if S = S L However we will study the properties MP, MT ad MPT i geeral for all QL ad D-operators, regardless of they are or they are ot implicatios Agai we ca begi with this first propositio Propositio 14 Let T 1 be a t-orm, I QL a QLoperator ad I D a D-operator The: i) If I QL (I D ) is a MP-operator for T 1 the ecessarily T 1 (x, x) = 0 for all x L ii) If I QL (I D ) is a MT-operator for T 1 the ecessarily T 1 (x, x) = 0 for all x L Whe we deal with QL ad D-implicatios, sice the the t-coorm S i equatios (14) ad (15) must be S = S L, they are derived simply from a smooth t- orm as: I QL (x, y) = S L ( x, T (x, y)) = x + T (x, y) (18) for all x, y L, ad I D (x, y) = S L (T ( x, y), y) = y + T ( x, y) (19) for all x, y L, respectively Moreover, it is proved i [10] that the set of QL-implicatios ad the set of D-implicatios coicide whe we derive them from smooth t-orms ad t-coorms, ad cosequetly we ca study both kid of implicatios at the same time Let us recall the structure of QL ad D-implicatios i Figure 3 (agai their formulas ca be foud i [10]) i m 1 i m 2 i 2 i 1 I i0 I i1 I im 2 x + y I im 1 0 i 1 i 2 i m 2 i m 1 Figure 3 The structure of the QL-implicatio derived from T J where J = {0 = i 0 < i 1 < < i m 1 < i m = } For j = 0,, m 1, each I ij is give by I ij (x, y) = max{ x + i j, + y i j+1 } for all x, y [i j, i j+1 ] The same structure correspods to the D-implicatio derived from T N(J) I the study of QL ad D-implicatios that are MP, MT ad MPT for T 1 we obtai always the same result Ay of these coditios is satisfied if ad oly if T 1 = T L For QL-operators ad D-operators i geeral, the result is the same, but ow we eed to study both kids of operators separately I ay case, we have: Propositio 15 Let T 1 be a smooth t-orm, I QL a QL-operator ad I D a D-operator The, the followig statemets are equivalet: i) I QL (I D ) is a MP-operator for T 1 ii) I QL (I D ) is a MT-operator for T 1 iii) I QL (I D ) is a MPT-operator for T 1 iv) T 1 = T L A table summarizig all results i this sectio ca be viewed i Table 1

6 I T MP for T 1 MT for T 1 MPT for T 1 Idemp T1 Idemp T T 1 = T L T 1 = T L I S T 1 = T L T 1 = T L T 1 = T L I QL T 1 = T L T 1 = T L T 1 = T L I D T 1 = T L T 1 = T L T 1 = T L Moreover, i this case both T k ad T k+1 coicide with the ilpotet miimum Thus, from ow o, we will cosider oly the cases whe k < k without ay loose of geerality We will cosider also 3 sice for the cases = 1, 2 it is always T k = T L The idexed family of t-orms T k ca be viewed i Figure 4 Table 1 Characterizatio of R, S, QL ad D- operators that are MP, MT ad MPT-operators for a smooth t-orm T 1 4 The o-smooth case I our previous study we have see that, give ay t- orm T 1 ad ay biary operator I : L 2 L, to be I a MP, MT, or MPT-operator, the coditio k k mi x + y k 0 T 1 (x, x) = 0 for all x L (20) is ecessary i almost all cases From Remark 1 we kow that i the smooth case this is equivalet to be T 1 the Lukasiewicz t-orm However, i the o-smooth case we have may others t-orms o L satisfyig such coditio Namely, for ay k L such that k k we have the followig idexed family of t-orms T k give by 0 if x + y T k x + y k if x + y > ad (x, y) = (21) k x, y k mi{x, y} otherwise Each t-orm of this family satisfies equatio (20), see for istace [9] I fact, these t-orms T k are the discrete couterpart of the family of t-orms itroduced by J C Fodor i [3] (see also [6]) whe he studies cotrapositive symmetry (geuie property of S-implicatios) for R-implicatios For each t-orm of this family the correspodig R ad S-implicatios coicide (see also [7], Jeei family of t-orms i pages 97 98) This fact is also true for our family T k i the discrete case (see [9]) Note that T k is o-smooth except for the case k = ad i this extreme case T agrees with the Lukasiewicz t-orm T L Note also that the ilpotet miimum (see agai [9]) is obtaied i the other extreme case give by k = /2 where /2 meas the floor of /2, that is, the greatest iteger which is smaller tha or equal to /2 Remark 2 Note that, i the particular case whe k = k, must be a eve umber ad k = /2 0 k k Figure 4 The structure of the t-orm T k Thus, sice each T k satisfies the ecessary coditio (20), we ca study which R, S, QL ad D-implicatios derived from smooth t-orms are MP, MT or MPTimplicatios for each T k I this lie we have the followig results with respect to the modus poes We begi with R-implicatios Propositio 16 Let T be a smooth t-orm ad I T its correspodig R-implicatio The, I T is a MPimplicatio for T k if ad oly if Idemp T cotais the set [k, ] = {x L k x} The case of S-implicatios is solved i the followig propositio Propositio 17 Let S be a smooth t-coorm ad I S its correspodig S-implicatio The, I S is a MPimplicatio for T k if ad oly if Idemp S cotais the set [0, k] The case of QL ad D-implicatios is also easy Sice they are derived from S L ad a smooth t-orm T, they are give by equatios (18) ad (19) respectively We will oly deal with T k whe k < k < sice the case k = correspods to the Lukasiewicz t-orm that has bee studied i the previous sectio Propositio 18 Suppose k < k < Let T be a smooth t-orm ad I QL ad I D the correspodig QL

7 ad D-implicatios derived from T The, the followig statemets are equivalet: i) I QL is a MP-implicatio for T k ii) I D is a MP-implicatio for T k iii) T is the Lukasiewicz t-orm T L iv) I QL = I D = I is the Kleee-Diees implicatio give by I(x, y) = max( x, y) I table 2 we ca view summarized all results i this sectio cocerig MP-implicatios for the t-orm T k R-implicatio from T S-implicatios from S MP-implicatio for T k [k, ] Idemp T [0, k] Idemp S tolles, to the oes obtaied for the modus poes, just by cotrapositio The oly exceptio of this is for R-implicatios I this case the results ca ot be derived from cotrapositio ad we eed to study MT idepedetly of MP However, we also obtai a idetical result to the oe obtaied for modus poes Propositio 19 Let T be a smooth t-orm ad I T its correspodig R-implicatio The, I T is a MTimplicatio for T k if ad oly if Idemp T cotais the set [k, ] I all remaiig cases all results ca be derived from cotrapositio First of all, ote that S-implicatios always satisfy cotrapositio with respect to the uique egatio N(x) = x O the other had, D-operators are the cotrapositio (with respect to N(x) = x) of QL-operators ad vice versa Usig these facts ad the duality betwee MP ad MT we ca easily prove the results cocerig MT QL-implicatios from T D-implicatios from T T = T L T = T L Propositio 20 Let S be a smooth t-coorm ad I S its correspodig S-implicatio The, the followig statemets are equivalet: Table 2 Characterizatio of MP-implicatios for the t-orm T k with k < k < O the other had, the geeral case of QL ad D- operators is ot so easy That is, whe these operators are derived from a smooth t-coorm S differet from S L I this case, we ca give oly some partial results, as follows: - Whe S = max, for ay smooth t-orm T, the QLoperator give by I QL (x, y) = max( x, T (x, y)) ad the D-operator give by I D (x, y) = max(t ( x, y), y) are MP-operators for T k ( k k ) - Whe S(x, x) = x for all x k, both the QL ad the D-operator derived from S ad ay smooth t- orm T are MP-operators for T k Fially, we wat to deal with the MT-property for T k From the duality betwee MP ad MT, we will be able to derive idetical results for the case of modus i) I S is a MP-implicatio for T k ii) I S is a MT-implicatio for T k iii) I S is a MPT-implicatio for T k iv) Idemp S cotais the set [0, k] Propositio 21 Suppose k < k < Let T be a smooth t-orm ad I QL ad I D the correspodig QL ad D-implicatios derived from T The, the followig statemets are equivalet: i) I QL (ad I D ) is a MP-implicatio for T k ii) I QL (ad I D ) is a MT-implicatio for T k iii) I QL (ad I D ) is a MPT-implicatio for T k iv) T is the Lukasiewicz t-orm T L v) I QL = I D = I is the Kleee-Diees implicatio The results for MT are summarized ow i table 3

8 R-implicatio from T S-implicatios from S QL-implicatios from T D-implicatios from T MT-implicatio for T k [k, ] Idemp T [0, k] Idemp S T = T L T = T L Table 3 Characterizatio of MT-implicatios for the t-orm T k with k < k < To fiish, ote that the geeral case of modus poes for QL ad D-operators ca be traslated also for modus tolles via duality, obtaiig agai exactly the same results I view of the characterizatios of MP ad MT coditios, sice both coicide i all four cases, it is clear that i each case the correspodig characterizatio also works i fact for the MPT-coditio 5 Coclusio The two mai iferece rules, modus poes (MP) ad modus tolles (MT), are studied for the four most usual classes of discrete implicatios: R, S, QL ad D- implicatios A characterizatio of MP ad a characterizatio of MT is give for all these kids of implicatios, obtaiig i the majority of cases the coditio T 1 (x, x) = 0, which directly derives (i the smooth case) ito the Lukasiewicz t-orm For this reaso, the o-smooth case is also studied for a geeral class of discrete t-orms T 1 that satisfy the coditio above I this study, a lot of ew solutios amog R, S, QL ad D-implicatios, derived from smooth t-orms, is obtaied for both properties MP ad MT Ackowledgemet This paper has bee partially supported by the Spaish grat MTM ad the Govermet of the Balearic Islads grat PCTIB-2005GC1-07 Refereces [1] B De Baets ad J C Fodor, Residual operators of uiorms, Soft Computig 3 (1999) [3] JC Fodor, Cotrapositive symmetry o fuzzy implicatios, Fuzzy Sets ad Systems 69 (1995) [4] J C Fodor, Smooth associative operatios o fiite ordial scales, IEEE Tras o Fuzzy Systems 8 (2000) [5] Ll Godo ad C Sierra, A ew approach to coective geeratio i the framework of expert systems usig fuzzy logic, i: Proceedigs of the XVI- IIth ISMVL, Palma de Mallorca, 1988, pp [6] S Jeei (2000), New family of triagular orms via cotrapositive symmetrizatio of residuated implicatios, Fuzzy Sets ad Systems, Vol 110, [7] EP Klemet, R Mesiar ad E Pap, Triagular orms, Kluwer Academic Publishers, Dordrecht (2000) [8] M Mas, G Mayor ad J Torres, t Operators ad uiorms o a fiite totally ordered set, It J of Itelliget Systems 14 (1999) [9] M Mas, M Moserrat ad J Torres, S- implicatios ad R-implicatios o a fiite chai, Kyberetika 40 (2004) 3 20 [10] M Mas, M Moserrat ad J Torres, O two types of discrete implicatios, It J of Approximate Reasoig 40 (2005) [11] M Mas, M Moserrat ad J Torres, QL ad D-Implicatios from uiorms, i : Proceedigs of IPMU-2006, Paris, Frace, 2006, pp [12] M Mas, M Moserrat, J Torres ad E Trillas, A survey o fuzzy implicatio fuctios, IEEE Trasactios o Fuzzy Systems, accepted [13] G Mayor ad J Torres, Triagular orms i discrete settigs, i: EP Klemet ad R Mesiar (Eds), Logical, Algebraic, Aalytic, ad Probabilistic Aspects of Triagular Norms Elsevier, Amsterdam, 2005, pp [14] D Ruiz ad J Torres, Residual implicatios ad co-implicatios from idempotet uiorms, Kyberetika 40 (2004) [15] E Trillas, C Alsia ad A Pradera, O MPTimplicatio fuctios for fuzzy logic, Rev R Acad Cie Serie A Mat 98 (2004) [2] B De Baets ad R Mesiar, Triagular orms o product lattices, Fuzzy Sets ad Systems 104 (1999) 61 75