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A note on the convex combinations of triangular norms
Fuzzy Sets and Systems 159 (2008) 77 – 80 www.elsevier.com/locate/fss
A note on the convex combinations of triangular norms Fabrizio Durante, Peter Sarkoci∗ Department of Knowledge-Based Mathematical Systems, Johannes Kepler University, A-4040 Linz, Austria Received 28 November 2006; received in revised form 2 July 2007; accepted 2 July 2007 Available online 26 July 2007
Abstract We consider a property on binary operations, which we call -migrativity, and we use it to obtain t-norms by means of convex combinations of two t-norms, one of them being discontinuous. © 2007 Elsevier B.V. All rights reserved. Keywords: Triangular norm; Associativity; Convex combination
1. Introduction Since their introduction in the framework of probabilistic metric spaces, triangular norms, t-norms for short, found important applications in many different fields, like fuzzy sets, fuzzy preference modelling, statistics and semigroup theory: see, for instance, [2,8,12] and the references therein. In the recent book [2] (see also [1]), the authors presented several open problems on triangular norms and related operations. In particular, they asked whether “the convex combination of two distinct t-norms is ever a t-norm”, and they conjectured that “except in trivial cases, the answer is never”. This conjecture was analyzed in the recent paper by Jenei [7], who presented some conditions that ensure that the convex combination of two distinct left-continuous t-norms is never a t-norm. Moreover, in [7] the author clarified also that several counterexamples can be given when considering convex combinations of t-norms that are both discontinuous. In this paper, we consider a property of binary operations, which we call -migrativity, and we use it to obtain t-norms by means of convex combinations of two t-norms, one of them being discontinuous. 2. Definitions and properties Let us recall that a t-norm is any binary operation T : [0, 1]2 → [0, 1] such that, for all x, y, z in [0, 1], the following properties hold: (T1) (T2) (T3) (T4)
T (x, y) = T (y, x), T (x, T (y, z)) = T (T (x, y), z), T (x, y) T (x, z) whenever y z, T (x, 1) = x.
∗ Corresponding author. Tel./fax: +43 732 2468.
E-mail addresses: [email protected] (F. Durante), [email protected] (P. Sarkoci). 0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.07.005
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F. Durante, P. Sarkoci / Fuzzy Sets and Systems 159 (2008) 77 – 80
Among others, important examples of t-norms are the minimum TM , the product TP , and the drastic t-norm TD given by TM (x, y) = min(x, y), TP (x, y) = xy, xy if max(x, y) = 1, TD (x, y) = 0 otherwise. A t-norm T is said to be strict if it is continuous and strictly monotone. In particular, it is well-known that T is strict if, and only if, there exists a decreasing function t: [0, 1] → [0, ∞], t (0) = ∞, such that for any x, y ∈ [0, 1] T (x, y) = t −1 (t (x) + t (y)). The function t is known as an additive generator of T. Another example of a t-norm is given by the ordinal sum construction. Let (]ai , bi [)i∈I be a family of non-empty, pairwise disjoint, open subintervals of [0, 1] and let (Ti )i∈I be a family of t-norms. Then the function ⎧ x − a i y − ai ⎨ if (x, y) ∈]ai , bi [2 , ai + (bi − ai ) Ti , T (x, y) = bi − ai bi − ai ⎩ min(x, y) otherwise is a t-norm, called ordinal sum of the summands (ai , bi , Ti )i∈I . We shall write T = (ai , bi , Ti )i∈I . For a general introduction to t-norms and their properties, including historical comments on the above notions, we refer to [2,8,12]. A modification of the axiom (T4) leads to the notion of t-subnorm [6]. Specifically, a t-subnorm is a mapping F : [0, 1]2 → [0, 1] that satisfies (T1)–(T3) and F (x, y) TM (x, y) for every (x, y) ∈ [0, 1]2 . Given a t-subnorm F, the mapping T : [0, 1]2 → [0, 1] given by min(x, y) if max(x, y) = 1, T (x, y) = (1) F (x, y) otherwise is a t-norm (see [5] and [8, Corollary 1.8]). The following property will be useful for our purpose. Definition 1. Let be in ]0, 1[. A binary operation T : [0, 1]2 → [0, 1] is said to be -migrative if, for every x, y in [0, 1], T (x, y) = T (x, y) holds. This property was considered for the first time in [9, Problem 8b]—anyway, notice that the term -migrative is introduced here—where the authors asked to find examples of -migrative t-norms different from the t-norms Ta given, for each a ∈]0, 1[, by min(x, y) if max(x, y) = 1, Ta (x, y) = axy otherwise. This problem was solved by Budinˇcevi`c and Kurili`c [3], who provided a class of -migrative t-norms that are discontinuous at each point of a dense subset of [0, 1]2 . The next result characterizes additive generators of -migrative strict t-norms. Proposition 2. Let be in ]0, 1[ and let T be a strict t-norm with additive generator t. Then T is -migrative if, and only if, for all x, y in [0, 1] t (x) − t (x) = t (y) − t (y).
(2)
Proof. The -migrativity of T means that t −1 (t (x) + t (y)) = t −1 (t (x) + t (y)) holds for all x, y in [0, 1]. As t is bijective, this property is equivalent to (2).
F. Durante, P. Sarkoci / Fuzzy Sets and Systems 159 (2008) 77 – 80
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Notice that condition (2) implies that t (x) = t () + t (x) for each x ∈ [0, 1]. Example 3. Let be in ]0, 1[ and let : [0, a] → [0, b] be an increasing bijection, where a = − log and b > 0. Define : [0, ∞[→ [0, ∞[ by x
x +b , (x) = x − a a a where x/a denotes the integer part of x/a. Now, put t (x) = (− log x). Notice that, regardless the choice of , the function t possesses the property (2), and, hence, it generates an -migrative t-norm. It can be shown that such a t-norm is strict. In particular, for b = a and = id[0,a] , we obtain t (x) = − log x, which is an additive generator of TP . 3. Main results The set of all t-norms is a subset of the functional space of all real functions defined over [0, 1]2 . Thus, it makes sense to inspect properties such as convexity, maximal convex subsets or betweenness of this set. It is well known that the set of all t-norms is not convex which is exemplified by any nontrivial convex combination of TM and TP . One can see immediately that any convex combination of two t-norms is a binary operation on [0, 1] which satisfies (T1), (T3) and (T4). Thus the only property that has to be checked in order to resolve whether the convex combination of t-norms is again a t-norm is associativity. The following lemma will be essential for the sequel. Lemma 4. Let be in ]0, 1[ and let T be an -migrative t-subnorm. Then the mapping T : [0, 1]2 → [0, 1] given by T (x, y) = T (x, y) is a t-subnorm. Proof. It is immediate to check that T satisfies (T1) and (T3) and it is bounded from above by TM . Thus, we have only to check that T is associative. For sake of simplicity, we will use the algebraic notation for T and T . Specifically, for every x, y in [0, 1], x y denotes T (x, y) and x y denotes T (x, y). For any x, y, z ∈ [0, 1], we have the following chain of equalities: (x y) z = ((x y) z) = ((x y) z) = ((x y) z) = (x (y z)) = (x (y z)) = ((x y) z) = ((x y) z) = (x (y z)) = (x (y z)) = (x (y z)) = x (y z). The first two equalities are expansions of defining expressions for , later equalities follow either from the associativity or from the -migrativity of the operation . In particular, the first and the last expression together give the associativity of the operation . Thanks to this result, we give now two examples of convex combinations of t-norms that are t-norms again. Theorem 5. Let be in ]0, 1[. For any -migrative t-norm T and for every n ∈ N, the mapping Tn : [0, 1]2 → [0, 1] given by Tn (x, y) = n T (x, y) + (1 − n )TD (x, y)
(3)
is again a t-norm. Proof. It is clear that, for each n in N, the operation Tn satisfies (T1), (T3) and (T4). The only which has to be proved is the associativity of Tn . By simple computations we have that min(x, y) if max(x, y) = 1, Tn (x, y) = n T (x, y) otherwise.
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Thus, in order to carry out the proof, by means of (1) it is sufficient to prove that n T is a t-subnorm. But, it can be shown easily that, for each n in N, the binary operation n T is -migrative, and by induction, Lemma 4 ensures that n T is also associative, and hence a t-subnorm. Theorem 6. Let (]ai , bi [)i∈I be a family of non-empty, pairwise disjoint, open subintervals of [0, 1] and let (Ti )i∈I be a family of -migrative t-norms. Let T1 be the ordinal sum t-norm given by T1 = (ai , bi , Ti )i∈I and let T2 be the ordinal sum t-norm given by T2 = (ai , bi , TD )i∈I . Then T = T1 + (1 − )T2 is again a t-norm. Proof. By means of some calculations, it is immediate to show that T is an ordinal sum of the type T = (ai , bi , Ti∗ )i∈I , where min(x, y) if max(x, y) = 1, ∗ Ti (x, y) = Ti (x, y) otherwise is a t-norm, because Ti is -migrative. Therefore T is itself a t-norm.
Final remarks: Since this work was submitted, new results have been obtained by other authors on the same topic. Firstly, J. Fodor and I.J. Rudas completed the investigation on the class of continuous t-norms that are -migrative [4]. Moreover, the present manuscript, together with [4], can be used in order to re-obtain (in a different way) some of the results by Ouyang et al. [11]. Finally, some comments about the convex combinations of continuous t-norms have been also provided in [10]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
C. Alsina, M.J. Frank, B. Schweizer, Problems on associative functions, Aequationes Math. 66 (2003) 128–140. C. Alsina, M.J. Frank, B. Schweizer, Associative Functions. Triangular Norms and Copulas, World Scientific, Hackensack, 2006. M. Budinˇcevi`c, M.S. Kurili`c, A family of strict and discontinuous triangular norms, Fuzzy Sets and Systems 95 (1998) 381–384. J. Fodor, I.J. Rudas, On continuous triangular norms that are migrative, Fuzzy Sets and Systems 158 (2007) 1692–1697. S. Jenei, Fibred triangular norms, Fuzzy Sets and Systems 103 (1999) 67–82. S. Jenei, Structure of left-continuous triangular norms with strong induced negations. (II) Rotation–annihilation construction, J. Appl. NonClassical Logics 11 (2001) 351–366. S. Jenei, On the convex combination of left-continuous t-norms, Aequationes Math. 72 (2006) 47–59. E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Publishers, Dordrecht, 2000. R. Mesiar, V. Novák (Eds.), Open problems, Tatra Mount. Math. Publ. 6 (1995) 12–22. Y. Ouyang, J. Fang, Some observations about the convex combination of continuous triangular norms, Nonlinear Anal., in press. Y. Ouyang, J. Fang, G. Li, On the convex combination of TD and continuous triangular norms, Inform. Sci. 177 (2007) 2945–2953. B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland, Amsterdam 1983; second ed., Dover Publications, Mineola, NY, 2006.
A note on the convex combinations of triangular norms Fabrizio Durante, Peter Sarkoci∗ Department of Knowledge-Based Mathematical Systems, Johannes Kepler University, A-4040 Linz, Austria Received 28 November 2006; received in revised form 2 July 2007; accepted 2 July 2007 Available online 26 July 2007
Abstract We consider a property on binary operations, which we call -migrativity, and we use it to obtain t-norms by means of convex combinations of two t-norms, one of them being discontinuous. © 2007 Elsevier B.V. All rights reserved. Keywords: Triangular norm; Associativity; Convex combination
1. Introduction Since their introduction in the framework of probabilistic metric spaces, triangular norms, t-norms for short, found important applications in many different fields, like fuzzy sets, fuzzy preference modelling, statistics and semigroup theory: see, for instance, [2,8,12] and the references therein. In the recent book [2] (see also [1]), the authors presented several open problems on triangular norms and related operations. In particular, they asked whether “the convex combination of two distinct t-norms is ever a t-norm”, and they conjectured that “except in trivial cases, the answer is never”. This conjecture was analyzed in the recent paper by Jenei [7], who presented some conditions that ensure that the convex combination of two distinct left-continuous t-norms is never a t-norm. Moreover, in [7] the author clarified also that several counterexamples can be given when considering convex combinations of t-norms that are both discontinuous. In this paper, we consider a property of binary operations, which we call -migrativity, and we use it to obtain t-norms by means of convex combinations of two t-norms, one of them being discontinuous. 2. Definitions and properties Let us recall that a t-norm is any binary operation T : [0, 1]2 → [0, 1] such that, for all x, y, z in [0, 1], the following properties hold: (T1) (T2) (T3) (T4)
T (x, y) = T (y, x), T (x, T (y, z)) = T (T (x, y), z), T (x, y) T (x, z) whenever y z, T (x, 1) = x.
∗ Corresponding author. Tel./fax: +43 732 2468.
E-mail addresses: [email protected] (F. Durante), [email protected] (P. Sarkoci). 0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.07.005
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F. Durante, P. Sarkoci / Fuzzy Sets and Systems 159 (2008) 77 – 80
Among others, important examples of t-norms are the minimum TM , the product TP , and the drastic t-norm TD given by TM (x, y) = min(x, y), TP (x, y) = xy, xy if max(x, y) = 1, TD (x, y) = 0 otherwise. A t-norm T is said to be strict if it is continuous and strictly monotone. In particular, it is well-known that T is strict if, and only if, there exists a decreasing function t: [0, 1] → [0, ∞], t (0) = ∞, such that for any x, y ∈ [0, 1] T (x, y) = t −1 (t (x) + t (y)). The function t is known as an additive generator of T. Another example of a t-norm is given by the ordinal sum construction. Let (]ai , bi [)i∈I be a family of non-empty, pairwise disjoint, open subintervals of [0, 1] and let (Ti )i∈I be a family of t-norms. Then the function ⎧ x − a i y − ai ⎨ if (x, y) ∈]ai , bi [2 , ai + (bi − ai ) Ti , T (x, y) = bi − ai bi − ai ⎩ min(x, y) otherwise is a t-norm, called ordinal sum of the summands (ai , bi , Ti )i∈I . We shall write T = (ai , bi , Ti )i∈I . For a general introduction to t-norms and their properties, including historical comments on the above notions, we refer to [2,8,12]. A modification of the axiom (T4) leads to the notion of t-subnorm [6]. Specifically, a t-subnorm is a mapping F : [0, 1]2 → [0, 1] that satisfies (T1)–(T3) and F (x, y) TM (x, y) for every (x, y) ∈ [0, 1]2 . Given a t-subnorm F, the mapping T : [0, 1]2 → [0, 1] given by min(x, y) if max(x, y) = 1, T (x, y) = (1) F (x, y) otherwise is a t-norm (see [5] and [8, Corollary 1.8]). The following property will be useful for our purpose. Definition 1. Let be in ]0, 1[. A binary operation T : [0, 1]2 → [0, 1] is said to be -migrative if, for every x, y in [0, 1], T (x, y) = T (x, y) holds. This property was considered for the first time in [9, Problem 8b]—anyway, notice that the term -migrative is introduced here—where the authors asked to find examples of -migrative t-norms different from the t-norms Ta given, for each a ∈]0, 1[, by min(x, y) if max(x, y) = 1, Ta (x, y) = axy otherwise. This problem was solved by Budinˇcevi`c and Kurili`c [3], who provided a class of -migrative t-norms that are discontinuous at each point of a dense subset of [0, 1]2 . The next result characterizes additive generators of -migrative strict t-norms. Proposition 2. Let be in ]0, 1[ and let T be a strict t-norm with additive generator t. Then T is -migrative if, and only if, for all x, y in [0, 1] t (x) − t (x) = t (y) − t (y).
(2)
Proof. The -migrativity of T means that t −1 (t (x) + t (y)) = t −1 (t (x) + t (y)) holds for all x, y in [0, 1]. As t is bijective, this property is equivalent to (2).
F. Durante, P. Sarkoci / Fuzzy Sets and Systems 159 (2008) 77 – 80
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Notice that condition (2) implies that t (x) = t () + t (x) for each x ∈ [0, 1]. Example 3. Let be in ]0, 1[ and let : [0, a] → [0, b] be an increasing bijection, where a = − log and b > 0. Define : [0, ∞[→ [0, ∞[ by x
x +b , (x) = x − a a a where x/a denotes the integer part of x/a. Now, put t (x) = (− log x). Notice that, regardless the choice of , the function t possesses the property (2), and, hence, it generates an -migrative t-norm. It can be shown that such a t-norm is strict. In particular, for b = a and = id[0,a] , we obtain t (x) = − log x, which is an additive generator of TP . 3. Main results The set of all t-norms is a subset of the functional space of all real functions defined over [0, 1]2 . Thus, it makes sense to inspect properties such as convexity, maximal convex subsets or betweenness of this set. It is well known that the set of all t-norms is not convex which is exemplified by any nontrivial convex combination of TM and TP . One can see immediately that any convex combination of two t-norms is a binary operation on [0, 1] which satisfies (T1), (T3) and (T4). Thus the only property that has to be checked in order to resolve whether the convex combination of t-norms is again a t-norm is associativity. The following lemma will be essential for the sequel. Lemma 4. Let be in ]0, 1[ and let T be an -migrative t-subnorm. Then the mapping T : [0, 1]2 → [0, 1] given by T (x, y) = T (x, y) is a t-subnorm. Proof. It is immediate to check that T satisfies (T1) and (T3) and it is bounded from above by TM . Thus, we have only to check that T is associative. For sake of simplicity, we will use the algebraic notation for T and T . Specifically, for every x, y in [0, 1], x y denotes T (x, y) and x y denotes T (x, y). For any x, y, z ∈ [0, 1], we have the following chain of equalities: (x y) z = ((x y) z) = ((x y) z) = ((x y) z) = (x (y z)) = (x (y z)) = ((x y) z) = ((x y) z) = (x (y z)) = (x (y z)) = (x (y z)) = x (y z). The first two equalities are expansions of defining expressions for , later equalities follow either from the associativity or from the -migrativity of the operation . In particular, the first and the last expression together give the associativity of the operation . Thanks to this result, we give now two examples of convex combinations of t-norms that are t-norms again. Theorem 5. Let be in ]0, 1[. For any -migrative t-norm T and for every n ∈ N, the mapping Tn : [0, 1]2 → [0, 1] given by Tn (x, y) = n T (x, y) + (1 − n )TD (x, y)
(3)
is again a t-norm. Proof. It is clear that, for each n in N, the operation Tn satisfies (T1), (T3) and (T4). The only which has to be proved is the associativity of Tn . By simple computations we have that min(x, y) if max(x, y) = 1, Tn (x, y) = n T (x, y) otherwise.
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F. Durante, P. Sarkoci / Fuzzy Sets and Systems 159 (2008) 77 – 80
Thus, in order to carry out the proof, by means of (1) it is sufficient to prove that n T is a t-subnorm. But, it can be shown easily that, for each n in N, the binary operation n T is -migrative, and by induction, Lemma 4 ensures that n T is also associative, and hence a t-subnorm. Theorem 6. Let (]ai , bi [)i∈I be a family of non-empty, pairwise disjoint, open subintervals of [0, 1] and let (Ti )i∈I be a family of -migrative t-norms. Let T1 be the ordinal sum t-norm given by T1 = (ai , bi , Ti )i∈I and let T2 be the ordinal sum t-norm given by T2 = (ai , bi , TD )i∈I . Then T = T1 + (1 − )T2 is again a t-norm. Proof. By means of some calculations, it is immediate to show that T is an ordinal sum of the type T = (ai , bi , Ti∗ )i∈I , where min(x, y) if max(x, y) = 1, ∗ Ti (x, y) = Ti (x, y) otherwise is a t-norm, because Ti is -migrative. Therefore T is itself a t-norm.
Final remarks: Since this work was submitted, new results have been obtained by other authors on the same topic. Firstly, J. Fodor and I.J. Rudas completed the investigation on the class of continuous t-norms that are -migrative [4]. Moreover, the present manuscript, together with [4], can be used in order to re-obtain (in a different way) some of the results by Ouyang et al. [11]. Finally, some comments about the convex combinations of continuous t-norms have been also provided in [10]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
C. Alsina, M.J. Frank, B. Schweizer, Problems on associative functions, Aequationes Math. 66 (2003) 128–140. C. Alsina, M.J. Frank, B. Schweizer, Associative Functions. Triangular Norms and Copulas, World Scientific, Hackensack, 2006. M. Budinˇcevi`c, M.S. Kurili`c, A family of strict and discontinuous triangular norms, Fuzzy Sets and Systems 95 (1998) 381–384. J. Fodor, I.J. Rudas, On continuous triangular norms that are migrative, Fuzzy Sets and Systems 158 (2007) 1692–1697. S. Jenei, Fibred triangular norms, Fuzzy Sets and Systems 103 (1999) 67–82. S. Jenei, Structure of left-continuous triangular norms with strong induced negations. (II) Rotation–annihilation construction, J. Appl. NonClassical Logics 11 (2001) 351–366. S. Jenei, On the convex combination of left-continuous t-norms, Aequationes Math. 72 (2006) 47–59. E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Publishers, Dordrecht, 2000. R. Mesiar, V. Novák (Eds.), Open problems, Tatra Mount. Math. Publ. 6 (1995) 12–22. Y. Ouyang, J. Fang, Some observations about the convex combination of continuous triangular norms, Nonlinear Anal., in press. Y. Ouyang, J. Fang, G. Li, On the convex combination of TD and continuous triangular norms, Inform. Sci. 177 (2007) 2945–2953. B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland, Amsterdam 1983; second ed., Dover Publications, Mineola, NY, 2006.