International Journal of Approximate Reasoning

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1 International Journal of Approximate Reasoning 49 (2008) Contents lists available at ScienceDirect International Journal of Approximate Reasoning journal omepage: Modus ponens and modus tollens in discrete implications M. Mas, M. Monserrat, J. Torrens * Departament de Ciències Matemàtiques i Informàtica, Universitat de les Illes Balears, Palma de Mallorca, Spain article info abstract Article istory: Received 5 October 2007 Received in revised form 11 April 2008 Accepted 15 April 2008 Available online 22 April 2008 Tis paper is devoted to te study of discrete implications tat satisfy modus ponens (MP), modus tollens (MT) or bot (MPT). Te main goal is to caracterize all R, S, QL and D-implications on a finite cain L, derived from smoot t-norms, satisfying tese properties for a given smoot t-norm T 1. Te case wen T 1 is non-smoot is also discussed for a special family of t-norms. Ó 2008 Elsevier Inc. All rigts reserved. Keywords: Discrete implications Modus ponens Modus tollens Smoot t-norms Finite scale 1. Introduction It is well known tat fuzzy implication functions (see te survey [14]) are used in approximate reasoning, not only to represent fuzzy conditional statements of te form If p ten q (wit p; q fuzzy statements), but also to perform inferences in any fuzzy rule based system. In tis inference process, te two main classical rules are modus ponens (MP) and modus tollens (MT) tat allow to perform, respectively, forward and backward inferences. In terms of fuzzy logic, tese implications are operations I : ½0; 1Š 2!½0; 1Š extending te classical material implication, tat is, satisfying Ið0; 0Þ ¼Ið0; 1Þ ¼Ið1; 1Þ ¼1 and Ið1; 0Þ ¼0. Since conjunctions, disjunctions and negations are usually performed by t-norms (T), t-conorms (S) and strong negations (N), in fuzzy set teory as muc as in fuzzy logic and approximate reasoning, te majority of te known implication functions are directly derived from tese operations. 1 Te four most usual ways to define tese implication functions are: (i) R-implications defined by Iðx; yþ ¼supfz 2½0; 1ŠjTðx; zþ 6 yg for all x; y 2½0; 1Š: ð1þ (ii) S-implications defined by Iðx; yþ ¼SðNðxÞ; yþ for all x; y 2½0; 1Š: ð2þ (iii) QL-implications defined by Iðx; yþ ¼SðNðxÞ; Tðx; yþþ for all x; y 2½0; 1Š: ð3þ * Corresponding autor. addresses: dmimmg0@uib.es (M. Mas), dmimma0@uib.es (M. Monserrat), dmijts0@uib.es (J. Torrens). 1 Altoug some autors ave derived also implications from oter aggregation functions, specially uninorms (see [1,12,17]) X/$ - see front matter Ó 2008 Elsevier Inc. All rigts reserved. doi: /j.ijar

2 M. Mas et al. / International Journal of Approximate Reasoning 49 (2008) (iv) D-implications, tat are te N-reciprocal of QL-implications wit respect to te strong negation N, and are given by Iðx; yþ ¼SðTðNðxÞ; NðyÞÞ; yþ for all x; y 2½0; 1Š: ð4þ Moreover, in tis context, given any t-norm T 1 and any strong negation N 1, modus ponens and modus tollens for an implication I can be written as and T 1 ðx; Iðx; yþþ 6 y for all x; y 2½0; 1Š ð5þ T 1 ðn 1 ðyþ; Iðx; yþþ 6 N 1 ðxþ for all x; y 2½0; 1Š; ð6þ respectively. Tese two inequalities ave been recently solved in [18] for te first tree mentioned classes of implications. On te oter and, te study of operations defined on finite scales is an area of increasing interest (see [2,4,6,9 11] and [16]). Mainly, because it allows to deal wit finite families of linguistic labels avoiding numerical interpretations (necessaries in te fuzzy logic approac). In tis context, te book capter [16] brings a survey on (smoot) discrete t-norms and t- conorms on a finite cain L, and te four classes of implications, R, S, QL and D-implications derived from discrete t-norms, are studied in [10,11]. Discrete implications sow new possibilities for approximate reasoning wit finite families of linguistic labels, wit teir consequent applications in computing wit words. However, in tis line, te study of MP and MT rules for discrete implications, equivalent to te one given in [18] for fuzzy implications, is essential. Tis paper is precisely devoted to tis study and it is an extended version of te Eusflat-07 communication [13], were te results were presented but witout proofs. After some preliminaries given in Section 2, we devote Section 3 to caracterizations of tose R, S, QL and D-implications on a finite cain L, derived from smoot t-norms, tat satisfy inequality (5), inequality (6) or bot. Finally, Section 4 is devoted to te non-smoot case for a special family of t-norms. 2. Preliminaries We recall ere te smoot t-norms and te smoot t-conorms on a finite cain L and teir caracterizations, tat will be used along te paper. It is well known tat for our purposes (see [16]) all finite cains wit te same number of elements are equivalent and tus, from now on, we will deal wit te simplest finite cain of n þ 1 elements: L ¼f0; 1; 2...; n 1; ng; were n P 1. Suc an L can be understood as a set of linguistic terms or labels. Te following two definitions are adapted from [6] (see also [16]). Definition 1. ([16], Definition 7.3.1) A function f : L! L is said to be smoot if it satisfies one of te following conditions: f is nondecreasing and f ðxþ f ðx 1Þ 6 1 for all x 2 L wit x P 1. f is nonincreasing and f ðx 1Þ f ðxþ 6 1 for all x 2 L wit x P 1. Definition 2. ([16], Definition 7.3.1) A binary operation F on L is said to be smoot if it is smoot in eac variable. Te importance of te smootness condition lies in te fact tat it is generally used as a discrete counterpart of continuity on [0,1]. Altoug t-norms, t-conorms and strong negations are usually binary operations on [0,1], tey can be defined as in [2] on any bounded partially ordered set and, in particular, on L. In tis last case, tey are usually known as discrete t-norms and discrete t-conorms. In tis way, recall tat smootness for discrete t-norms (and also for t-conorms) is equivalent to te divisibility condition, tat is, x 6 y if and only if tere exists z 2 L suc tat Tðy; zþ ¼x (see Proposition in [16]). Proposition 1. ([16], Example 7.2.6) Tere is one and only one strong negation on L tat is given by NðxÞ ¼n x for all x 2 L: ð7þ From now on, N will always denote te negation on L given by (7). Moreover, given any t-norm T (t-conorm S) onl, we will denote by Idemp T (Idemp S ) te set of all idempotent elements of T (S), tat is Idemp T ¼fx 2 LjTðx; xþ ¼xg: It is known tat a t-norm T on L is Arcimedean (see Definition in [16]) if and only if Idemp T ¼f0; ng (see Remark (iii)). Smoot t-norms ave been caracterized as ordinal sums of Arcimedean ones as follows. Proposition 2. ([16], Proposition 7.3.7) Tere is one and only one Arcimedean smoot t-norm on L, denoted by T L, given by T L ðx; yþ ¼maxf0; x þ y ng for all x; y 2 L; ð8þ wic is known as te Łukasiewicz t-norm.

3 424 M. Mas et al. / International Journal of Approximate Reasoning 49 (2008) Proposition 3. ([16], Teorem 7.3.8) A t-norm T on L is smoot if and only if tere exists a natural number m wit 0 6 m 6 n and a subset of L J ¼f0 ¼ i 0 < i 1 < < i m 1 < i m ¼ ng suc tat T ¼ T J is given by: T J ðx; yþ ¼ maxfi k; x þ y i kþ1 g if x; y 2½i k ; i kþ1 Š; 0 6 k 6 m 1; minfx; yg oterwise: In tis case, te set of idempotent elements of T is precisely Idemp T ¼ J. Smoot t-conorms ave a classification teorem like te above one for t-norms wic can be easily deduced by N-duality. Te expression of te only Arcimedean smoot t-conorm on L is given by S L ðx; yþ ¼minfn; x þ yg for all x; y 2 L; ð10þ wic is also known as te Łukasiewicz t-conorm. In general, we ave Proposition 4. ([16], Teorem 7.3.9) A t-conorm S on L is smoot if and only if tere exists a natural number m wit 0 6 m 6 n and a subset of L J ¼f0 ¼ i 0 < i 1 < < i m 1 < i m ¼ ng suc tat S ¼ S J is given by: S J ðx; yþ ¼ minfi kþ1; x þ y i k g if x; y 2½i k ; i kþ1 Š; 0 6 k 6 m 1; maxfx; yg oterwise: In tis case, te set of idempotent elements of T is precisely Idemp S ¼ J. Note tat wit tese notations, te Łukasiewicz t-norm and t-conorm can be written, respectively as T L ¼ T f0;ng and S L ¼ S f0;ng. Te following result follows from te previous propositions. Proposition 5. ([16], Corollary ) Tere are exactly 2 n 1 different smoot t-norms (t-conorms) on L. One of te most accepted definitions of implications is te following, tat as been extracted from [5]. Definition 3. ([5], Definition 1.15) A binary operation I : L L! L is said to be a (discrete) implication if it satisfies: (I1) I is nonincreasing in te first variable and nondecreasing in te second one. (I2) Ið0; 0Þ ¼Iðn; nþ ¼n and Iðn; 0Þ ¼0: ð9þ ð11þ Note tat, from te definition, it follows tat Ið0; xþ ¼n and Iðx; nþ ¼n for all x 2 L. Te four ways to define fuzzy implications apply ere to define discrete implications. However, since te only strong negation on L is te one given by (7) and we deal wit a finite scale, in our case tey can be rewritten as follows: Iðx; yþ ¼maxfz 2 LjTðx; zþ 6 yg; x; y 2 L; ð12þ Iðx; yþ ¼Sðn x; yþ; x; y 2 L; ð13þ Iðx; yþ ¼Sðn x; Tðx; yþþ; x; y 2 L; ð14þ Iðx; yþ ¼SðTðn x; n yþ; yþ; x; y 2 L: ð15þ All tese classes of discrete operations ave been already studied: R and S-implications in [10], and QL and D-operations in [11]. Tus we refer to tese cited papers for details on tese kinds of discrete operations tat we will use in te paper. Altoug te non-smoot case is considered in tese references, in te present work we will mainly deal wit R, S, QL and D-operations derived from smoot t-norms and smoot t-conorms. Let us now deal wit (implication) operations on te finite cain L tat satisfy te modus ponens, te modus tollens or bot, wit respect to a t-norm T 1. Again, since we ave only one strong negation on L, in our case Eq. (6) can be rewritten depending only on te t-norm T 1 and tus, we can adopt te following definitions. 2 Definition 4. Let T 1 be a t-norm on L. A function I : L 2! L will be called: an MP-operation for T 1 wenever it satisfies T 1 ðx; Iðx; yþþ 6 y for all x; y 2 L: ð16þ an MT-operation for T 1 wenever it satisfies T 1 ðn y; Iðx; yþþ 6 n x for all x; y 2 L: ð17þ an MPT-operation for T 1 wenever it is bot, an MP and an MT-operation. 2 We give te definitions for a general (not necessarily smoot) t-norm T 1 since in Section 4 we will deal wit te non-smoot case, as well.

4 M. Mas et al. / International Journal of Approximate Reasoning 49 (2008) Moreover, we will say tat I is an MP-implication (or MT, or MPT-implication) if it is an implication and also an MPoperation (or MT or MPT-operation, respectively). Note tat wereas all R and S-operations (given by Eqs. (12) and (13), respectively) are always implications, tis is not te case for QL and D-operations (given by Eqs. (14) and (15), respectively), see for instance [11]. Tus, we will divide our study of properties MP, MT and MPT in tree subsections, te first one devoted to R-implications, te second one devoted to S- implications and, finally, te tird one devoted to QL and D-operations in general, including of course QL and D-implications. Before tis, let us begin wit two easy but important propositions related to te mentioned properties. Te first one deals wit MP-operations. Proposition 6. Let T 1 be a t-norm on L and I : L 2! L an MP-operation for T 1. Ten T 1 ðx; Iðx; 0ÞÞ ¼ 0 for all x 2 L: Proof. Taking y ¼ 0in(16) we obtain T 1 ðx; Iðx; 0ÞÞ 6 0 and ence te result. Te second one deals wit MT-operations. Proposition 7. Let T 1 be a t-norm on L and I : L 2! L an MT-operation for T 1. Ten T 1 ðn y; Iðn; yþþ ¼ 0 for all y 2 L: Proof. Just take in tis case x ¼ n in (17). 3. Main results wen T 1 is a smoot t-norm We divide our study in tree sections, te first devoted to R-implications, te second devoted to S-implications, and te last one devoted to QL and D-operations R-implications Given any discrete t-norm T on L we will denote by I T its residual implication, tat is, te operation given by Eq. (12): I T ðx; yþ ¼maxfz 2 LjTðx; zþ 6 yg: We deal in tis subsection wit implications I T : L 2! L, were T is a smoot t-norm. Let us recall ere te formulas for tese implications. Proposition 8. ([10], Proposition 10) Let T : L 2! L be a smoot t-norm wit te following set of idempotent elements Idemp T ¼f0 ¼ i 0 < i 1 < < i m 1 < i m ¼ ng: Ten its derived R-implication I T is given by 8 < n if x 6 y I T ðx; yþ ¼ i kþ1 þ y x if tere exists i k suc tat i k 6 y < x 6 i kþ1 ; : y oterwise: For an easier understanding we give also teir grapical structure tat can be found in Fig. 1. We begin wit te property of MP-implications. Recall first tat R-implications satisfy te so-called residuation property. Tat is, given any discrete t-norm T 1, it is verified: T 1 ðx; yþ 6 z () I T1 ðx; zþ P y for all x; y; z 2 L: ð18þ In tis case we can give a general caracterization witout te assumption of smootness. Proposition 9. Let T; T 1 be t-norms on L and I T : L 2! L te R-implication associated to T. Te following statements are equivalent: (i) I T is an MP-implication for T 1. (ii) I T 6 I T1. (iii) T 1 6 T. Proof. I T is an MP-implication for T 1 if and only if it satisfies T 1 ðx; I T ðx; yþþ 6 y for all x; y 2 L; wic is equivalent by Eq. (18) to I T1 ðx; yþ P I T ðx; yþ for all x; y 2 L; proving te equivalence between (i) and (ii).

5 426 M. Mas et al. / International Journal of Approximate Reasoning 49 (2008) Fig. 1. Te structure of te R-implication derived from T J were J ¼f0 ¼ i 0 < i 1 < < i m 1 < i m ¼ ng and I ikþ1 ðx; yþ ¼i kþ1 þ y x for k ¼ 0;...; m 1. To prove te equivalence between (ii) and (iii) we sow bot implications separately. Assuming tat T 1 6 T we ave I T1 ðx; zþ ¼maxfy 2 LjT 1 ðx; yþ 6 zg P maxfy 2 LjTðx; yþ 6 zg ¼I T ðx; zþ: Conversely, suppose tat I T 6 I T1 and tat tere exists ða; bþ 2L 2 suc tat T 1 ða; bþ > Tða; bþ. Ten I T1 ða; Tða; bþþ ¼ maxfy 2 LjT 1 ða; yþ 6 Tða; bþg < b; wereas I T ða; Tða; bþþ ¼ maxfy 2 LjTða; yþ 6 Tða; bþg P b; obtaining contradiction. Now, in te smoot case we ave te following results. Proposition 10. Let T; T 1 be smoot t-norms on L and I T : L 2! L te R-implication associated to T. Ten I T is an MP-implication for T 1 if and only if Idemp T1 Idemp T. Proof. Suppose first tat I T is an MP-implication for T 1 and let Idemp T ¼f0 ¼ j 0 < j 1 < < j r 1 < j r ¼ ng; ð19þ be te set of idempotents of T. Now let i k be an idempotent of T 1 and suppose tat i k is not an idempotent of T. Ten we will ave j l < i k < j lþ1 for some 0 6 l 6 r 1 and, using te formula given in te caption of Fig. 1, we obtain by one and I T ði k ; j l Þ¼j lþ1 þ j l i k > j l and by te oter and I T1 ði k ; j l Þ¼j l ; obtaining contradiction. Tus, any idempotent element of T 1 must be also an idempotent element of T. Conversely, suppose tat Idemp T is given again by (19) and tat Idemp T1 ¼f0 ¼ i 0 < i 1 < < i m 1 < i m ¼ ng Idemp T and we will prove tat I T ðx; yþ 6 I T1 ðx; yþ for all x; y 2 L. If x 6 y ten clearly I T ðx; yþ ¼n ¼ I T1 ðx; yþ and so we need to deal only wit te case x > y. In tis case we will ave i k 6 x 6 i kþ1 for some 0 6 k 6 m 1 and we distinguis two cases: If y < i k 6 x 6 i kþ1. Ten necessarily tere is an l suc tat 0 6 l 6 r 1 and y < i k 6 j l 6 x 6 j lþ1 6 i kþ1. But in tis case we ave I T ðx; yþ ¼I T1 ðx; yþ ¼y. If i k 6 y < x 6 i kþ1. In tis case, I T1 ðx; yþ ¼i kþ1 þ y x wereas we will ave some l suc tat i k 6 j l 6 x 6 j lþ1 6 i kþ1, allowing two possibilities for I T : If i k 6 j l 6 y < x 6 j lþ1 6 i kþ1, ten I T ðx; yþ ¼j lþ1 þ y x 6 i kþ1 þ y x ¼ I T1 ðx; yþ: If i k 6 y < j l 6 x 6 j lþ1 6 i kþ1, ten I T ðx; yþ ¼y 6 i kþ1 þ y x ¼ I T1 ðx; yþ: In all cases I T ðx; yþ 6 I T1 ðx; yþ and tus te implication is proved. Note in particular tat taking T ¼ T 1 te modus ponens is always satisfied, a partial result tat was directly proved in [10]. On te oter and, it is clear tat we can deduce te following particular cases.

6 M. Mas et al. / International Journal of Approximate Reasoning 49 (2008) Corollary 1. Let T; T 1 be smoot t-norms on L and I T : L 2! L te R-implication associated to T. Ten (i) I min is an MP-implication for any smoot t-norm T 1. (ii) I TL is an MP-implication for T 1 if and only if T 1 ¼ T L. (iii) I T is an MP-implication for T L for any smoot t-norm T. (iv) I T is an MP-implication for te t-norm min if and only if T ¼ min. Wit respect to MT-implications we obtain solutions for smoot t-norms only wen T 1 ¼ T L is te Łukasiewicz t-norm and in tis case I T is an MT-implication for T L for any smoot t-norm T. Tis can be proved troug te following two propositions. Proposition 11. Let T; T 1 be t-norms on L and let I T : L 2! L be te R-implication associated to T. If I T is an MT-implication for T 1 ten necessarily T 1 ðx; n xþ ¼0 for all x 2 L. Proof. For any discrete t-norm T, it is obvious tat I T ðn; yþ ¼y for all y 2 L. Ten applying Proposition 7, we ave 0 ¼ T 1 ðn y; I T ðn; yþþ ¼ T 1 ðn y; yþ ¼T 1 ðy; n yþ for all y 2 L and te proposition is proved. Remark 1. It is proved in Lemma 1 of [10] tat, for smoot t-norms, te previous condition is equivalent to T 1 be te Łukasiewicz t-norm. Tat is, for smoot t-norms T 1 ðx; n xþ ¼0 for all x 2 L () T 1 ¼ T L : Proposition 12. Let T; T 1 be smoot t-norms on L and I T : L 2! L te R-implication associated to T. Ten I T is an MT-implication for T 1 if and only if T 1 ¼ T L. Proof. If I T is an MT-implication for T 1 by te previous proposition and Remark 1 we clearly obtain T 1 ¼ T L : Conversely, if T 1 ¼ T L we need to sow tat T L ðn y; I T ðx; yþþ 6 n x for all x; y 2 L for any smoot t-norm T. To do it, suppose again tat Idemp T ¼f0 ¼ i 0 < i 1 < < i m 1 < i m ¼ ng: Wen x 6 y te result is trivial since ten T L ðn y; I T ðx; yþþ 6 n y 6 n x: Tus, we can suppose y < x. Moreover, tere will exist 0 6 k 6 m 1 suc tat i k 6 x 6 i kþ1 and we divide our reasoning in two cases: If i k 6 y < x 6 i kþ1, ten I T ðx; yþ ¼i kþ1 þ y x and so T L ðn y; I T ðx; yþþ ¼ T L ðn y; i kþ1 þ y xþ ¼i kþ1 x 6 n x: If y < i k 6 x 6 i kþ1, ten I T ðx; yþ ¼y and so T L ðn y; I T ðx; yþþ ¼ T L ðn y; yþ ¼0 6 n x: Now, joining te obtained results for MP and for MT-implications we obtain te following corollary. Corollary 2. Let T; T 1 be smoot t-norms on L and I T : L 2! L te R-implication associated to T. Ten I T is an MPT-implication for T 1 if and only if T 1 ¼ T L S-implications Let us now deal wit S-implications. Given any t-conorm S we will denote by I S te corresponding S-implication given by Eq. (13). Tat is I S ðx; yþ ¼Sðn x; yþ for all x; y 2 L: Te structure of S-implications can be viewed in Fig. 2 and teir expression is given in te following proposition.

7 428 M. Mas et al. / International Journal of Approximate Reasoning 49 (2008) Fig. 2. Te structure of te S-implication derived from te t-conorm S J, were J ¼f0 ¼ n i m < n i m 1 < < n i 1 < n i 0 ¼ ng; I maxðx; yþ ¼maxðn x; yþ and I ikþ1 ðx; yþ ¼minðn i k ; i kþ1 þ y xþ for k ¼ 0;...; m 1. Proposition 13. ([10], Proposition 5)Let S : L 2! L be a smoot t-conorm wit te following set of idempotent elements Idemp S ¼f0 ¼ n i m < n i m 1 < < n i 1 < n i 0 ¼ ng: Ten its derived S-implication I S is given by 8 >< minðn i k ; i kþ1 x þ yþ I S ðx; yþ ¼ >: maxðn x; yþ if tere is some i k suc tat i k 6 x; n y 6 i kþ1 oterwise: For S-implications, te study of te modus ponens for smoot t-norms is easy and it is solved troug te following two propositions. Proposition 14. Let T 1 be a t-norm, S a t-conorm on L and I S : L 2! L te corresponding S-implication. If I S is an MP-implication for T 1 ten necessarily T 1 ðx; n xþ ¼0 for all x 2 L. Proof. Given any t-conorm S on L we always ave I S ðx; 0Þ ¼n x for all x 2 L and ten by Proposition 6 we directly obtain T 1 ðx; n xþ ¼0 for all x 2 L. Recall tat a particular solution of te modus ponens for S-implications was already given in [10]. Specifically, wen T 1 ¼ T L and S ¼ S L it was proved tere tat I SL is an MP-implication for T L, tat is, T L ðx; S L ðn x; yþþ 6 y for all x; y 2 L: We will use tis result to caracterize all solutions as follows. Proposition 15. Let T 1 be a smoot t-norm and S a smoot t-conorm on L and I S : L 2! L te corresponding S-implication. Ten I S is an MP-implication for T 1 if and only if T 1 ¼ T L. Proof. If I S is an MP-implication for T 1 ten T 1 ðx; n xþ ¼0 for all x 2 L and, because T 1 is smoot, we ave T 1 ¼ T L. Conversely, wen T 1 ¼ T L we know tat S 6 S L for any smoot t-conorm S (see Remark in [16]) and ten T L ðx; I S ðx; yþþ ¼ T L ðx; Sðn x; yþþ 6 T L ðx; S L ðn x; yþþ 6 y: In tis case, te study of modus tollens gives exactly te same solutions. Specifically Proposition 16. Let T 1 be a t-norm and S a smoot t-conorm on L and I S : L 2! L te corresponding S-implication. If I S is an MTimplication for T 1 ten necessarily T 1 ðx; n xþ ¼0 for all x 2 L. Proof. Tis is obvious just taking x ¼ n in (17) because I S ðn; yþ ¼y for all y 2 L. Proposition 17. Let T 1 be a smoot t-norm and S a smoot t-conorm on L and I S : L 2! L te corresponding S-implication. Ten I S is an MT-implication for T 1 if and only if T 1 ¼ T L. Proof. Te necessity is due to te same reasoning as in Proposition 15. Conversely, since S is smoot we ave T L ðn y; I S ðx; yþþ ¼ T L ðn y; Sðn x; yþþ 6 T L ðn y; S L ðn x; yþþ 6 T L ðn y; n x þ yþ ¼maxð0; n xþ ¼n x:

8 M. Mas et al. / International Journal of Approximate Reasoning 49 (2008) Fig. 3. Te structure of te QL-implication derived from T J were J ¼f0 ¼ i 0 < i 1 < < i m 1 < i m ¼ ng: For j ¼ 0;...; m 1, eac I ij is given by I ij ðx; yþ ¼maxfn x þ i j; n þ y i jþ1g for all x; y 2½i j; i jþ1š. Te same structure corresponds to te D-implication derived from T NðJÞ. Remark 2. Te result for modus tollens can be also derived from modus ponens and contraposition in a similar way as we will do in te non-smoot case (see te next section). However, we ave coosen te direct proof wit te aim of stressing te fact tat T 1 ðx; n xþ ¼0 for all x 2 L is again a necessary condition. From te results wit modus ponens and modus tollens we ave: Corollary 3. Let T 1 be a smoot t-norm and S a smoot t-conorm on L and I S : L 2! L te corresponding S-implication. Ten I S is an MPT-implication for T 1 if and only if T 1 ¼ T L QL and D-operations In tis subsection we deal wit QL and D-operations given by Eqs. (14) and (15), respectively. As we ave commented, not all of tem are implications in te sense of Definition 3. In fact, it is proved in [11] tat tey are implications in te smoot case (for bot QL and D) 3 if and only if S ¼ S L. However, we will study te properties MP, MT and MPT in general for all QL and D-operations, weter tey are or tey are not implications. We can begin again wit te similar proposition as before. Proposition 18. Let T 1 be a t-norm on L, I QL a QL-operation and I D a D-operation. Ten: (i) If I QL (I D ) is an MP-operation for T 1 ten necessarily T 1 ðx; n xþ ¼0 for all x 2 L. (ii) If I QL (I D ) is an MT-operation for T 1 ten necessarily T 1 ðx; n xþ ¼0 for all x 2 L. Proof. To prove (i) it is enoug to take y ¼ 0 in Eq. (14) for QL-operations or in Eq. (15) for D-operations. Similarly, taking x ¼ n in te same equations we easily obtain (ii). Wen we deal wit QL and D-implications, te smoot t-conorm S in Eqs. (14) and (15) must be S ¼ S L as we ave already commented at te beginning of tis section. Tus tey are derived simply from a smoot t-norm T as follows: I QL ðx; yþ ¼S L ðn x; Tðx; yþþ ¼ n x þ Tðx; yþ ð20þ for all x; y 2 L, and I D ðx; yþ ¼S L ðtðn x; n yþ; yþ ¼y þ Tðn x; n yþ ð21þ for all x; y 2 L, respectively. Moreover, it is proved in [11] tat te set of QL-implications and te set of D-implications coincide wen we derive tem from smoot t-norms and t-conorms, and consequently we can study bot kind of implications at te same time. Te structure of QL and D-implications can be found in Fig. 3 and teir formulas are recalled in te following proposition. Proposition 19. ([11], Proposition 17 and 29) Let T be te only smoot t-norm wit set of idempotent elements Idemp T ¼f0 ¼ i 0 < i 1 < < i m 1 < i m ¼ ng: 3 Note tat D-implications are called NQL-implications in [11].

9 430 M. Mas et al. / International Journal of Approximate Reasoning 49 (2008) (i) Te QL-implication I QL derived from T is given by 8 maxfn x þ i k ; n þ y i kþ1 g if tere is some i k >< suc tat i k 6 x; y 6 i kþ1 I QL ðx; yþ ¼ n x þ y if tere is some i k suc tat y 6 i k 6 x >: n oterwise: (ii) Te D-implication I D derived from T is given by 8 maxfy þ i k ; 2n x i kþ1 g if tere is some i k >< suc tat n i kþ1 6 x; y 6 n i k I D ðx; yþ ¼ n x þ y if tere is some i k suc tat y 6 n i k 6 x >: n oterwise: It is known (see [11], Teorem 32) tat te D-implication derived from a smoot t-norm T wit te set of idempotents given by Idemp T ¼f0 ¼ i 0 < i 1 < < i m 1 < i m ¼ ng coincides wit te QL-implication derived from te t-norm wit set of idempotents given by NðIdemp T Þ¼f0 ¼ n i m < n i m 1 <...< n i 1 < n i 0 ¼ ng: In te study of QL and D-implications tat are MP, MT and MPT for smoot T 1 we obtain always te same result. Any of tese conditions is satisfied if and only if T 1 ¼ T L. However, tis result is true not only for implications but also for QL-operations and D-operations in general. Tus we will prove te general case in spite of studying bot kinds of operations separately. In all cases we ave: Proposition 20. Let T 1 be a smoot t-norm, I QL a QL-operation and I D a D-operation generated from a t-norm T and a smoot t- conorm S. Ten te following statements are equivalent: (i) I QL (I D ) is an MP-operation for T 1. (ii) I QL (I D ) is an MT-operation for T 1. (iii) I QL (I D ) is an MPT-operation for T 1. (iv) T 1 ¼ T L. Proof. Let us prove first te equivalence between (i) and (iv) for QL-operations. If I QL is an MP-operation for T 1 we know from Proposition 18 tat T 1 ðx; n xþ ¼0 and ten T 1 ¼ T L. Conversely, for all x; y 2 L we ave T L ðx; Sðn x; Tðx; yþþþ 6 T L ðx; S L ðn x; Tðx; yþþþ ¼ T L ðx; n x þ Tðx; yþþ ¼ maxð0; Tðx; yþþ 6 y: Te same equivalence for D-operations follows similarly and moreover, te equivalence between (ii) and (iv) in bot cases is also a straigtforward computation. Finally, te equivalence between (iii) and (iv) follows directly from definition and te proposition is proved. Remark 3. In tis case again te results concerning te modus tollens could be also derived from te modus ponens by contraposition in a similar way as it is done in te next section for te non-smoot case. A table summarizing all results in tis section can be viewed in Table Main results wen T 1 is non-smoot In our previous study we ave seen tat, given any t-norm T 1 on L and any binary operation I : L 2! L belonging to any of te four cases (I R ; I S ; I QL ; I D ), for I to be an MP, MT, or MPT-operation, te condition T 1 ðx; n xþ ¼0 for all x 2 L ð22þ Table 1 Caracterization of R, S, QL and D-operations tat are MP, MT and MPT-operations for a smoot t-norm T 1 MP for T 1 MT for T 1 MPT for T 1 Idemp R-implications, I T1 T T Idemp 1 ¼ T L T 1 ¼ T L S-implications, I S T 1 ¼ T L T 1 ¼ T L T 1 ¼ T L QL-operations, I QL T 1 ¼ T L T 1 ¼ T L T 1 ¼ T L D-operations, I D T 1 ¼ T L T 1 ¼ T L T 1 ¼ T L

10 M. Mas et al. / International Journal of Approximate Reasoning 49 (2008) is necessary in almost all cases. From Remark 1 we know tat in te smoot case tis is equivalent to T 1 be te Łukasiewicz t- norm. However, in te non-smoot case we ave many oter t-norms on L satisfying tis condition. For example, for any k 2 L suc tat n k 6 k we ave te following indexed family of t-norms T k given by 8 >< 0 if x þ y 6 n T k ðx; yþ ¼ x þ y k if x þ y > n and n k 6 x; y 6 k ð23þ >: minfx; yg oterwise: Eac t-norm of tis family satisfies Eq. (22), see for instance [10]. In fact, tese t-norms T k are te discrete counterpart of te family of t-norms introduced by Fodor in [3] (see also [7]) were e studies contrapositive symmetry (genuine property of S- implications) for R-implications. For eac t-norm of tis family te corresponding R and S-implications coincide (see also [8], Jenei family of t-norms in pp ). Tis fact is also true for our family T k in te discrete case (see [10]). Note tat T k is non-smoot except for te case k ¼ n and in tis extreme case T n coincides wit te Łukasiewicz t-norm T L. Note also tat te nilpotent minimum (see again [10]) is obtained in te oter extreme case given by n k ¼bn=2c were bn=2c means te floor of n=2, tat is, te greatest integer wic is smaller tan or equal to n=2. Remark 4. Note tat, in te particular case wen n k ¼ k, n must be an even number and k ¼ n=2. Moreover, in tis case bot T k and T kþ1 coincide wit te nilpotent minimum. Tus, from now on, we will consider only te cases wen n k < k witout any loose of generality. Moreover, we will only deal wit T k wit n k < k < n since te case k ¼ n corresponds to te Łukasiewicz t-norm wic as been studied in te previous section. We will consider also n P 3 since for te cases n ¼ 1; 2 it is always T k ¼ T L. Te indexed family of t-norms T k can be viewed in Fig. 4. Tus, since eac T k satisfies te necessary condition (22), we can study wic R, S, QL and D-implications derived from smoot t-norms are MP, MT or MPT-implications for eac T k. In tis case we ave te following results wit respect to te modus ponens. We begin wit R-implications. Proposition 21. Let T be a smoot t-norm and I T its corresponding R-implication. Ten I T is an MP-implication for T k if and only if Idemp T contains te set ½k; nš ¼fx 2 L j k 6 xg. Proof. For te necessity, let suppose tat tere is some non-idempotent element a P k. Tere exist two consecutive idempotent elements of T, i r ; i rþ1 suc tat i r < a < i rþ1 and since n k < k 6 a we can take an element b 2 L suc tat maxðn k; i r Þ 6 b < a < i rþ1. Ten I T ða; bþ ¼i rþ1 þ b a P i rþ1 þ n k a > n a and consequently T k ða; I T ða; bþþ ¼ minða; i rþ1 þ b aþ > b obtaining contradiction wit te modus ponens. Conversely, suppose tat ½k; nš Idemp T. Since modus ponens is always satisfied for x 6 y, we only need to ceck it for te values y < x and we will do it by considering tree cases: If x P k. In tis case x is an idempotent element of T and consequently I T ðx; yþ ¼y and T k ðx; I T ðx; yþþ ¼ T k ðx; yþ 6 y. If x < k and y 6 n k. In tis case, from Proposition 8 we ave I T ðx; yþ 6 k þ y x 6 n x and ten T k ðx; I T ðx; yþþ ¼ 0 6 y. If x < k and y > n k. In tis case, n k < y < x < k and we ave also k > k þ y x > n x > n k. Now, applying again Proposition 8 we ave I T ðx; yþ 6 k þ y x and consequently T k ðx; I T ðx; yþþ 6 T k ðx; k þ y xþ ¼x þðk þ y xþ k ¼ y: Fig. 4. Te structure of te t-norm T k.

11 432 M. Mas et al. / International Journal of Approximate Reasoning 49 (2008) Te case of S-implications is solved in te following proposition. Proposition 22. Let S be a smoot t-conorm and I S its corresponding S-implication. Ten I S is an MP-implication for T k if and only if Idemp S contains te set ½0; n kš. Proof. Suppose tat tere exists some a 6 n k suc tat Sða; aþ > a. Ten we ave a 6 n k < k 6 n a and consequently T k ðn a; I S ðn a; aþþ ¼ T k ðn a; Sða; aþþ ¼ minðn a; Sða; aþþ > a; wic proves tat I S is not an MP-implication for T k. Conversely, if ½0; n kš Idemp S we want to prove tat I S is an MP-implication for T k. Since Eq. (16) always olds for x 6 y we only need to prove it for y < x. Let us consider tree cases: If x P k, ten n x 6 n k and T k ðx; Sðn x; yþþ ¼ T k ðx; maxðn x; yþþ ¼ maxð0; T k ðx; yþþ 6 y: If x < k and y 6 n k, ten it is also y < n x and T k ðx; Sðn x; yþþ ¼ T k ðx; maxðn x; yþþ ¼ T k ðx; n xþ ¼0 6 y: If x < k and y > n k, ten n k < y < x < k. Since for smoot t-conorms it is S 6 S L, te greatest smoot t-conorm wit ½0; n kš Idemp S will be te t-conorm wit Idemp T ¼½0; n kš[fng. Ten, using Proposition 4, we obtain for all x; y suc tat n k < y < x < k Sðn x; yþ 6 minðn; n x þ y n þ kþ ¼minðn; y x þ kþ: Now, we ave T k ðx; Sðn x; yþþ 6 T k ðx; minðn; y x þ kþþ ¼ minðx; T k ðx; y x þ kþþ ¼ y: Example 1. From Propositions 3 and 4 we ave a smoot t-norm (t-conorm) wit set of idempotents given by any subset of L containing f0; ng. Tus, given any k suc tat n k < k < n, taking te set J ¼f0g[½k; nš and te smoot t-norm T wit Idemp T ¼ J (but also any smoot t-norm T 0 wit J Idemp T 0) we obtain a smoot t-norm suc tat I T is an MP-implication for T k. Specifically, suc t-norm is given by maxf0; x þ y kg if x; y 6 k Tðx; yþ ¼ minfx; yg oterwise: Note tat tis family of t-norms were caracterized in te framework of ½0; 1Š in [15] and also studied as a special family of t-norms in [8]. On te oter and, we can similarly obtain a smoot t-conorm suc tat I S is an MP-implication for T k just taking J ¼½0; n kš[fng and S suc tat Idemp S ¼ J (but also taking any smoot t-conorm S 0 wit J Idemp S 0). Te case of QL and D-implications is also easy. Since tey are derived from S L and a smoot t-norm T, tey are given by Eqs. (20) and (21), respectively. Proposition 23. Let T be a smoot t-norm and I QL and I D te corresponding QL and D-implications derived from T. Ten te following statements are equivalent: (i) I QL is an MP-implication for T k. (ii) I D is an MP-implication for T k. (iii) T is te Łukasiewicz t-norm T L. (iv) I QL ¼ I D ¼ I is te Kleene Dienes implication given by Iðx; yþ ¼maxðn x; yþ: Proof. We divide te proof in several steps. ðiþ )ðiiiþ Let I QL be an MP-implication for T k and suppose tat tere is an element a suc tat 0 < a < n wit Tða; aþ ¼a. Taking 0 < y < x < n wit y 6 a 6 x we ave using (9) tat Tðx; yþ is given by te minimum. Tat is, Tðx; yþ ¼y and ten T k ðx; I QL ðx; yþþ ¼ T k ðx; n x þ Tðx; yþþ ¼ T k ðx; n x þ yþ: Now, from Eq. (23), since n x þ y > n x tis value is positive and it can be given by minðx; n x þ yþ or x þ n x þ y k ¼ n þ y k:

12 M. Mas et al. / International Journal of Approximate Reasoning 49 (2008) Table 2 Caracterization of MP-implications for te t-norm T k wit n k < k < n MP-implication for T k R-implication from T S-implication from S QL-implication from T D-implication from T ½k; nš Idemp T ½0; n kš Idemp S T ¼ T L T ¼ T L In any case te value is greater tan y wic contradicts te modus ponens. 4 Tus, te only idempotents of T are 0; n and T must be te Łukasiewicz t-norm T L. ðiiiþ )ðivþ If T ¼ T L ten I QL ðx; yþ ¼n x þ T L ðx; yþ ¼n x þ maxð0; x þ y nþ ¼maxðn x; yþ: ðivþ )ðiþ Tis is clear because T k ðx; I QL ðx; yþþ ¼ T k ðx; maxðn x; yþþ 6 y: Te previous steps prove te equivalence among (i), (iii) and (iv). Finally, te equivalence wit (ii) follows from Proposition 19-ii) taking into account tat for T ¼ T L it is verified tat I QL ¼ I D. In Table 2 we can see summarization of all results in tis section concerning MP-implications for te t-norm T k. On te oter and, te general case of QL and D-operations is not so easy. Tat is, wen tese operations are derived from a smoot t-conorm S different from S L. In tis case, we can give only some partial results, as follows: Wen S ¼ max, for any smoot t-norm T, it is a straigtforward computation to sow tat te QL-operation given by I QL ðx; yþ ¼maxðn x; Tðx; yþþ and te D-operation given by I D ðx; yþ ¼maxðTðn x; n yþ; yþ are MP-operations for T k (n k 6 k 6 n). Wen Sðx; xþ ¼x for all x 6 n k, we ave as in te proof of Proposition 22 tat minfn; x þ y n þ kg if x; y P n k; Sðx; yþ 6 Sðx; yþ ¼ maxfx; yg oterwise: Tus, a simple calculation sows tat T k ðx; Sðn x; Tðx; yþþþ 6 T k ðx; Sðn x; Tðx; yþþþ 6 y and similarly T k ðx; SðTðn x; n yþ; yþþ 6 T k ðx; SðTðn x; n yþ; yþþ 6 y: Tat is, te QL and te D-operation derived from S and any smoot t-norm T, are MP-operations for T k. Finally, we want to discuss te MT-property for T k. From te duality between MP and MT, we will be able to derive identical results for te case of modus tollens, to te ones obtained for te modus ponens, just by contraposition. Te only exception is for R-implications. In tis case te results can not be derived from contraposition and we need to study MT independently of MP. However, we also obtain an identical result to te one obtained for modus ponens. Proposition 24. Let T be a smoot t-norm and I T its corresponding R-implication. Ten I T is an MT-implication for T k if and only if Idemp T contains te set ½k; nš. Proof. For te necessity, suppose tat tere is some a P k suc tat Tða; aþ < a and take i r 2 Idemp T suc tat i r < a < i rþ1. Ten i r 6 a 1 < a < i rþ1 and 4 Recall tat from Remark 4 we ave reduced our study to te case k < n.

13 434 M. Mas et al. / International Journal of Approximate Reasoning 49 (2008) T k ðn ða 1Þ; I T ða; a 1ÞÞ ¼ T k ðn a þ 1; i rþ1 1Þ: Since i rþ1 1 þðn a þ 1Þ ¼n a þ i rþ1 > n tis last value is positive and from (23) it can be given by n a þ i rþ1 k or minðn a þ 1; i rþ1 1Þ: Te result is in any case greater tan n a (note tat since k 6 a < i rþ1, we ave n a 6 n k < k 6 i rþ1 1) wic contradicts te modus tollens. Conversely, suppose tat ½k; nš Idemp T. Since modus tollens is always satisfied for x 6 y we only need to prove T k ðn y; I T ðx; yþþ 6 n x for all x; y 2 L wit y < x: ð24þ Note tat wen x P k, x is idempotent and I T ðx; yþ ¼y. Tus T k ðn y; I T ðx; yþþ ¼ T k ðn y; yþ ¼0 and (24) is satisfied. On te oter and, wen x < k we ave y < x < k and applying Proposition 8 we obtain I T ðx; yþ 6 k þ y x. Now, we can distinguis two cases: If n k < y. Ten we ave n k < n y; k þ y x < k and T k ðn y; I T ðx; yþþ 6 T k ðn y; k þ y xþ ¼ðn yþþðk þ y xþ k ¼ n x: If n k P y. Ten T k ðn y; I T ðx; yþþ 6 T k ðn y; k þ y xþ 6 T k ðn y; n xþ 6 n x: In bot cases (24) is again satisfied and te proposition is proved. In all remaining cases te results can be derived from contraposition. First of all, note tat S-implications always satisfy contraposition wit respect to te unique negation NðxÞ ¼n x, tat is I S ðx; yþ ¼I S ðn y; n xþ for all x; y 2 L: ð25þ On te oter and, D-operations are te contraposition (wit respect to NðxÞ ¼n x) of QL-operations and vice versa, tat is I QL ðx; yþ ¼I D ðn y; n xþ for all x; y 2 L: ð26þ Using tese facts and te duality between MP and MT we can easily prove te results concerning MT. Proposition 25. Let S be a smoot t-conorm and I S its corresponding S-implication. Ten te following statements are equivalent: (i) I S is an MP-implication for T k. (ii) I S is an MT-implication for T k. (iii) I S is an MPT-implication for T k. (iv) Idemp S contains te set ½0; n kš. Proof. Taking into account te previous results it is enoug to prove te equivalence between (i) and (ii). But tis is clear because using Eq. (25) we ave te following equivalences: T k ðx; I S ðx; yþþ 6 y () T k ðx; I S ðn y; n xþþ 6 y and canging x ¼ n b and y ¼ n a, tis is equivalent to T k ðn b; I S ða; bþþ 6 n a for all a; b 2 L. Proposition 26. Let T be a smoot t-norm and I QL and I D te corresponding QL and D-implications derived from T. Ten te following statements are equivalent: (i) I QL (and I D ) is an MP-implication for T k. (ii) I QL (and I D ) is an MT-implication for T k. (iii) I QL (and I D ) is an MPT-implication for T k. (iv) T is te Łukasiewicz t-norm T L. (v) I QL ¼ I D ¼ I is te Kleene Dienes implication. Proof. Again all reduce to prove te equivalence between (i) and (ii). And again te result is clear from te following equivalences were we use in tis case Eq. (26): T k ðx; I QL ðx; yþþ 6 y () T k ðx; I D ðn y; n xþþ 6 y

14 M. Mas et al. / International Journal of Approximate Reasoning 49 (2008) Table 3 Caracterization of MT-implications for te t-norm T k wit n k < k < n MT-implication for T k R-implication from T S-implication from S QL-implication from T D-implication from T ½k; nš Idemp T ½0; n kš Idemp S T ¼ T L T ¼ T L and canging x ¼ n b and y ¼ n a, tis is equivalent to T k ðn b; I D ða; bþþ 6 n a for all a; b 2 L. Te results for MT are summarized now in Table 3. To finis, note tat te general case of modus ponens for QL and D-operations can be translated also for modus tollens via duality, obtaining again exactly te same results. Since te caracterizations of MP and MT conditions coincide in all four cases, it is clear tat in eac case te corresponding caracterization also works for te MPT-condition. 5. Conclusion Te two main inference rules, modus ponens (MP) and modus tollens (MT), are studied for te four most usual classes of discrete implications: R, S, QL and D-implications. A caracterization of MP and a caracterization of MT is given for all tese kinds of implications, obtaining in te majority of cases te condition T 1 ðx; n xþ ¼0, wic determines (in te smoot case) te Łukasiewicz t-norm. For tis reason, te non-smoot case is also studied for a general class of discrete t-norms T 1 tat satisfy te condition above. In tis study, a lot of new solutions among R, S, QL and D-implications, derived from smoot t-norms, is obtained for bot properties MP and MT. Acknowledgements Te autors want to tank te referees for teir valuable comments, specially for te inclusion of condition (iii) in Proposition 9 tat was suggested by one of tem. Tis paper as been partially supported by te Spanis Grant MTM and te Government of te Balearic Islands Grant PCTIB-2005GC1-07. References [1] B. De Baets, J.C. Fodor, Residual operators of uninorms, Soft Computing 3 (1999) [2] B. De Baets, R. Mesiar, Triangular norms on product lattices, Fuzzy Sets and Systems 104 (1999) [3] J.C. Fodor, Contrapositive symmetry on fuzzy implications, Fuzzy Sets and Systems 69 (1995) [4] J.C. Fodor, Smoot associative operations on finite ordinal scales, IEEE Transactions on Fuzzy Systems 8 (2000) [5] J.C. Fodor, M. Roubens, Fuzzy Preference Modelling and Multicriteria decision Support, Teory and Decision Library, Series D: System teory, Knowledge Engineering and Problem Solving, Kluwer Academic Publisers, Dordrect, [6] L. Godo, C. Sierra, A new approac to connective generation in te framework of expert systems using fuzzy logic, in: Proceedings of te XVIIIt ISMVL, Palma de Mallorca, 1988, pp [7] S. Jenei, New family of triangular norms via contrapositive symmetrization of residuated implications, Fuzzy Sets and Systems 110 (2000) [8] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Publisers, Dordrect, [9] M. Mas, G. Mayor, J. Torrens, t-operators and uninorms on a finite totally ordered set, International Journal of Intelligent Systems 14 (1999) [10] M. Mas, M. Monserrat, J. Torrens, S-implications and R-implications on a finite cain, Kybernetika 40 (2004) [11] M. Mas, M. Monserrat, J. Torrens, On two types of discrete implications, International Journal of Approximate Reasoning 40 (2005) [12] M. Mas, M. Monserrat, J. Torrens, Two types of implications derived from uninorms, Fuzzy Sets and Systems 158 (2007) [13] M. Mas, M. Monserrat, J. Torrens, MP and MT-implications on a finite scale, in: Proceedings of EUSFLAT-2007, Ostrava, Czec Republic, 2007, pp [14] M. Mas, M. Monserrat, J. Torrens, E. Trillas, A survey on fuzzy implication functions, IEEE Transactions on Fuzzy Systems 15 (6) (2007) [15] G. Mayor, J. Torrens, On a family of t-norms, Fuzzy Sets and Systems 41 (1991) [16] G. Mayor, J. Torrens, Triangular norms in discrete settings, in: E.P. Klement, R. Mesiar (Eds.), Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms, Elsevier, Amsterdam, 2005, pp [17] D. Ruiz, J. Torrens, Residual implications and co-implications from idempotent uninorms, Kybernetika 40 (2004) [18] E. Trillas, C. Alsina, A. Pradera, On MPT-implication functions for fuzzy logic, Revista de la Real Academia de Ciencias, Serie A, Matemáticas 98 (2004)