- Email: [email protected]
On continuous triangular norms that are migrative
Fuzzy Sets and Systems 158 (2007) 1692 – 1697 www.elsevier.com/locate/fss
On continuous triangular norms that are migrative夡 J. Fodor∗ , I.J. Rudas Institute of Intelligent Engineering Systems, Budapest Tech, Bécsi út 96/b, H-1034 Budapest, Hungary Received 29 December 2006; received in revised form 23 February 2007; accepted 25 February 2007 Available online 12 March 2007
Abstract In this paper we completely describe all continuous migrative triangular norms. Since the migrative property excludes both idempotent and nilpotent classes, the characterization and construction is carried out by solving a functional equation for additive generators of strict t-norms. We also study cases when the construction results in a smooth generator. © 2007 Elsevier B.V. All rights reserved. Keywords: Continuous triangular norm; Migrative property; Additive generator; Functional equations
1. Introduction In [2] the authors introduced the new term—-migrative—for a class of binary operations having an old property, presented already in [4, Problem 1.8(b)], as follows. Definition 1. Let be in ]0, 1[. A binary operation T : [0, 1]2 → [0, 1] is said to be -migrative if we have T (x, y) = T (x, y)
for all x, y ∈ [0, 1].
(1)
Although this definition seems to be rather general, the paper [2] deals with triangular norms and subnorms. A triangular norm (t-norm for short) T : [0, 1]2 → [0, 1] is an associative, commutative, non-decreasing function such that T (1, x) = x for all x ∈ [0, 1]. Prototypes of t-norms are the minimum TM (x, y) = min(x, y), the product TP (x, y) = xy, and the Łukasiewicz t-norm TL (x, y) = max(x + y − 1, 0). As it is well known, each continuous Archimedean t-norm T can be represented by means of a continuous additive generator (see e.g., [3]), i.e., a strictly decreasing continuous function t: [0, 1] → [0, ∞] with t (1) = 0 such that T (x, y) = t (−1) (t (x) + t (y)),
夡
Supported in part by OTKA T046762.
∗ Corresponding author. Tel.: +361 6665617; fax: +361 6665625.
E-mail addresses: [email protected] (J. Fodor), [email protected] (I.J. Rudas). 0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.02.020
(2)
J. Fodor, I.J. Rudas / Fuzzy Sets and Systems 158 (2007) 1692 – 1697
1693
where t (−1) : [0, ∞] → [0, 1] is the pseudo-inverse of t, and is given by t (−1) (u) = t −1 (min(u, t (0))). Turning back to Definition 1, one can easily see that the following t-norm T is -migrative for any ∈]0, 1[: T (x, y) =
min(x, y) if max(x, y) = 1, xy otherwise,
(3)
where is an arbitrary number from [0, 1]. It was provided by Budinˇcevi´c [4] as a counterexample to the question: Is a strictly monotone t-norm always continuous? Concerning further solutions of (1), an infinite class of -migrative t-norms was constructed in [1], each of them being discontinuous at each point of a dense subset of [0, 1]2 . Surprisingly enough, the continuous case has not been studied yet. Obviously, the product t-norm TP is -migrative for any ∈]0, 1[. In 1996 E. Pap raised also the following question [4, Problem 1.8(b)]: Does there exist a strictly monotone t-norm which is a solution of Eq. (1) and differs from TP ? An affirmative answer is given by the non-continuous t-norm in (3). We provide all continuous strictly monotone solutions in Theorem 3 via their additive generator functions (for the particular case of smooth generators see Theorem 5 and Example 6). The main aim of the present paper is to determine all continuous t-norms that are -migrative. It turns out that a continuous t-norm can be migrative only if it is strict. Then the -migrative property can be expressed, in an equivalent way, as a functional equation for additive generators of strict t-norms. That equation is completely solved in the paper. The solution at the same time yields a construction method for additive generators of strict t-norms that are migrative. We also study the case when the resulting additive generator is smooth. 2. Continuous migrative t-norms According to our main aim, in this section we concentrate on continuous migrative t-norms; that is, on continuous t-norms that are -migrative with some ∈]0, 1[. Notice that for a triangular norm T Definition 1 implies, by choosing y = 1, the following equality: T (x, ) = x
for all x ∈ [0, 1].
(4)
In the next theorem we show that the migrative property is rather strong for a continuous t-norm: it implies that the t-norm cannot have idempotent elements, and cannot be nilpotent. Theorem 2. Let T be a continuous t-norm. If T is -migrative then T is strict. Proof. First we prove that, under the above conditions, T (x, x) < x for all x ∈]0, 1[. Suppose there exists an x0 ∈]0, 1[ such that T (x0 , x0 ) = x0 . Since T is continuous, we have T (x0 , y) = min(x0 , y) for all y ∈ [0, 1] (see e.g. [3, Proposition 2.3]). Let y = . Then, T (x0 , ) = min(x0 , ), which contradicts (4). Therefore, if a continuous t-norm T is -migrative then it is Archimedean: it has an additive generator t such that representation (2) holds: T (x, y) = t (−1) (t (x) + t (y)). Suppose, to the contrary of the statement, that T is nilpotent; that is, t (0) < +∞. We have, by choosing y = 1 and by equality (4), the following equation: x = t (−1) (t () + t (x)) for all x ∈ [0, 1].
(5)
Let x be such that t () + t (x ) = t (0). Then, since ∈]0, 1[ and t is strictly decreasing, we have x > 0 and T (, x ) = 0. At the same time x > 0, which is a contradiction.
1694
J. Fodor, I.J. Rudas / Fuzzy Sets and Systems 158 (2007) 1692 – 1697
It is easy to conclude [2] that a strict t-norm T with additive generator t is -migrative if and only if t (x) − t (x) = t (y) − t (y) for all x, y ∈ [0, 1].
(6)
Eq. (6) says that the difference t (x) − t (x) is independent of x. More exactly, if we chose y = 1 in (6), this independent difference can be obtained as t (x) − t (x) = t (). We write it as follows: t (x) = t () + t (x)
for all x ∈ [0, 1].
(7)
In the next theorem we provide the general solution of the functional equation (7). It is based on the important fact that the restriction of t to the interval [, 1] uniquely determines t on each subinterval ]k+1 , k ], progressing from right to left. Theorem 3. Suppose t is an additive generator of a strict t-norm. Then t satisfies the functional equation (7) if and only if there exists a continuous, strictly decreasing function t0 from [, 1] to the non-negative reals with t0 () < +∞ and t0 (1) = 0 such that x t (x) = k · t0 () + t0 k (8) if x ∈]k+1 , k ], where k is any non-negative integer. Proof. (i) Assume that t is an additive generator of a strict t-norm that satisfies (7). We prove the existence of a function t0 that satisfies the conditions above. Observe first that Eq. (7) implies t (n ) = n · t () for any non-negative integer n. This is clear by choosing x = in (7) and by applying mathematical induction, together with 0 = t (1) = t (0 ) = 0 · t (). We prove, by mathematical induction, that the function t0 = t|[,1] satisfies (8). Indeed, define t0 on [, 1] by t0 (x) = t (x)
for x ∈ [, 1].
Then we have t0 (1) = t (1) = 0 and t0 () = t () < +∞. First we extend t0 to ]2 , ]. Consider x ∈]2 , ]. Then there exists a unique y ∈], 1] such that x = y; indeed, y = x/ is the appropriate choice. Then we have x t (x) = t (y) = t () + t (y) = t0 () + t0 , which proves (8) for k = 1. We can proceed from “right to left”. Assume that (8) holds for some k 1. We prove that it holds also for k + 1. That is, we have x t (x) = k · t0 () + t0 k if x ∈]k+1 , k ]. If x ∈]k+2 , k+1 ] then there is a unique y ∈]k+1 , k ] such that x = y; again y = x/ is the appropriate choice. Thus we have x t (x) = t (y) = t () + t (y) = t0 () + t x/ = t0 () + k · t0 () + t0 k x = (k + 1) · t0 () + t0 k+1 , which proves (8) for k + 1. Consecutive powers of generate intervals ]k+1 , k ], which provide a partition of ]0, 1]. That is, for any x ∈]0, 1] there exists a unique non-negative integer k such that x ∈]k+1 , k ]. Therefore, Eq. (8) defines t unambiguously for any x ∈]0, 1].
J. Fodor, I.J. Rudas / Fuzzy Sets and Systems 158 (2007) 1692 – 1697
1695
It is obvious by (8) that t is strictly decreasing and limx→+0 t (x) = ∞. Finally, we have t (k ) = limx→+k t (x) for all k, so t is continuous, and therefore it is an additive generator of a strict t-norm. (ii) Consider a continuous, strictly decreasing function t0 from [, 1] to the non-negative reals with t0 (0) < +∞ and t0 (1) = 0. Assume that we define a function t by (8). We prove that thus defined t is an additive generator of a strict t-norm, which is -migrative. Obviously, t is continuous and decreasing with t (1) = 0, and limx→0+ t (x) = +∞. Thus, it is an additive generator of a strict t-norm T. It remains to show that t satisfies Eq. (7). Consider an x ∈]0, 1]. Then there exists a unique k such that x ∈]k , k−1 ], and therefore x ∈]k+1 , k ]. Then we have x t (x) = k · t0 () + t0 k x = t0 () + (k − 1)t0 () + t0 k−1 = t () + t (x), which proves the statement.
Unfortunately, none of the famous t-norm families (like Frank, Hamacher, Dombi, Alsina) are migrative, except the particular case of t (x) = − log x, or equivalently, T (x, y) = TP (x, y) = xy. This latest case can be derived directly from Eq. (1). Proposition 4. A continuous t-norm T is -migrative for all ∈]0, 1[ if and only if T = TP . 3. Migrative t-norms with smooth generators Theorem 3 provides a construction method for additive generators of migrative t-norms. Generally speaking, if we start with a differentiable, strictly decreasing function t0 from [, 1] to the non-negative reals with t0 () < +∞ and t0 (1) = 0 then the derivative of the resulting additive generator t constructed by (8) need not exist at points that are powers of . One obvious exception is t0 (x) = − log x, with x ∈], 1]. In this case (8) yields t (x) = − log x for all x ∈]0, 1]. Considering t0 (x) =
1−x 1−
for x ∈ [, 1],
we obtain a piecewise linear additive generator t by (8), so t is not differentiable at powers of . We formulate a simple theorem to guarantee that the constructed generator t be smooth (i.e., differentiable). Theorem 5. Let ∈]0, 1[ and consider a continuous, strictly decreasing function t0 from [, 1] to the non-negative reals with t0 () < +∞ and t0 (1) = 0. Assume that t0 is differentiable on ], 1[ and its respective one-sided derivatives, () and t (1), exist at the end-points. Define the function t by (8). Then t is differentiable on ]0, 1[ if denoted by t0+ 0− and only if · t0+ () = t0− (1).
(9)
Proof. First, suppose (9) holds. We prove that t is differentiable. In fact, we have to show that the derivative of t exist at points k for all positive integer k. The definition of t and differentiability of t0 imply the differentiability of t at any other points in ]0, 1[. Since one-sided derivatives of t0 exist, this guarantees that both left- and right-sided derivatives of t exists at each k . It suffices to show that these are equal. Consider k for a fixed positive integer k. By Eq. (8) we have (k ) = t0− (1) · t−
1 k
(10)
1696
J. Fodor, I.J. Rudas / Fuzzy Sets and Systems 158 (2007) 1692 – 1697
and (k ) = t0+ () · t+
1 k−1
.
(11)
(k ) = t (k ), whence t is differentiable on ]0, 1[. Then (9) implies that t− + (k ) = t (k ) Second, assume that t is differentiable. Then its left- and right-sided derivatives are equal, so we have t− + for all positive integer k. Using (10) and (11), we must have (1) · t0−
1 1 = t0+ () · k−1 , k
which proves the statement.
The following example illustrates the construction method, taking into account the smoothness of the resulting additive generator t. Example 6. Let ∈]0, 1[ be given. We want to use an appropriate second-order polynomial t0 (x) = ax 2 + bx + c on [, 1] such that condition (9) is also satisfied. Together with requirements from Theorem 3 the following simple equations can be obtained for the parameters a, b, c: (1) t0 (1) = 0 gives a + b + c = 0; () = t (1) yields 2a = (2a + b)1/. (2) · t0+ 0− The solution defines the following form: t0 (x) = ax 2 − 2a(1 + )x + a(1 + 2) for x ∈ [, 1],
(12)
where a is any positive real number. One can easily check that t0 () > 0. Now we have () = −2a t0+
and
(1) = −2a, t0−
so (9) holds, and therefore the additive generator function t constructed by (8) is smooth. If we choose a = 1 and = 21 , we get t0 (x) = x 2 − 3x + 2 and t (x) = 22k · x 2 − 3 · 2k · x + 2 +
3k 4
if x ∈
1
2
, k+1
1 . 2k
Notice that the continuous t-norm in this example gives an affirmative answer to the question of E. Pap [4, Problem 1.8(b)]. The t-norm generated by thus defined t is strict, migrative, and differs from the product TP . 4. Conclusion In this paper we have completely characterized continuous t-norms that are migrative. It turned out that only strict t-norms can own this property. Their characterization has been developed through solutions of a functional equation. Although Definition 1 is seemingly general, notice that it does not provide a meaningful notion for triangular conorms. Indeed, if S is a t-conorm then it is -migrative if and only if S(x, y) = S(x, y) holds for all x, y ∈ [0, 1]. If we choose y = 0 then we must have x = x for all x ∈ [0, 1], because S is -migrative. This is impossible when = 1. Similarly, if y = 1 then we must have S(x, ) = 1 for all x ∈ [0, 1], which is again impossible unless = 1. Therefore, even the correct definition of -migrative t-conorms needs special care.
J. Fodor, I.J. Rudas / Fuzzy Sets and Systems 158 (2007) 1692 – 1697
1697
References [1] M. Budinˇcevi´c, M.S. Kurili´c, A family of strict and discontinuous triangular norms, Fuzzy Sets and Systems 95 (1998) 381–384. [2] F. Durante, P. Sarkoci, A note on the convex combinations of triangular norms, Fuzzy Sets and Systems 2006, submitted for publication. [3] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Trends in Logic. Studia Logica Library, Vol. 8, Kluwer Academic Publishers, Dordrecht, 2000. [4] R. Mesiar, V. Novák, Open problems from the 2nd international conference on fuzzy sets theory and its applications, Fuzzy Sets and Systems 81 (1996) 185–190.
On continuous triangular norms that are migrative夡 J. Fodor∗ , I.J. Rudas Institute of Intelligent Engineering Systems, Budapest Tech, Bécsi út 96/b, H-1034 Budapest, Hungary Received 29 December 2006; received in revised form 23 February 2007; accepted 25 February 2007 Available online 12 March 2007
Abstract In this paper we completely describe all continuous migrative triangular norms. Since the migrative property excludes both idempotent and nilpotent classes, the characterization and construction is carried out by solving a functional equation for additive generators of strict t-norms. We also study cases when the construction results in a smooth generator. © 2007 Elsevier B.V. All rights reserved. Keywords: Continuous triangular norm; Migrative property; Additive generator; Functional equations
1. Introduction In [2] the authors introduced the new term—-migrative—for a class of binary operations having an old property, presented already in [4, Problem 1.8(b)], as follows. Definition 1. Let be in ]0, 1[. A binary operation T : [0, 1]2 → [0, 1] is said to be -migrative if we have T (x, y) = T (x, y)
for all x, y ∈ [0, 1].
(1)
Although this definition seems to be rather general, the paper [2] deals with triangular norms and subnorms. A triangular norm (t-norm for short) T : [0, 1]2 → [0, 1] is an associative, commutative, non-decreasing function such that T (1, x) = x for all x ∈ [0, 1]. Prototypes of t-norms are the minimum TM (x, y) = min(x, y), the product TP (x, y) = xy, and the Łukasiewicz t-norm TL (x, y) = max(x + y − 1, 0). As it is well known, each continuous Archimedean t-norm T can be represented by means of a continuous additive generator (see e.g., [3]), i.e., a strictly decreasing continuous function t: [0, 1] → [0, ∞] with t (1) = 0 such that T (x, y) = t (−1) (t (x) + t (y)),
夡
Supported in part by OTKA T046762.
∗ Corresponding author. Tel.: +361 6665617; fax: +361 6665625.
E-mail addresses: [email protected] (J. Fodor), [email protected] (I.J. Rudas). 0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.02.020
(2)
J. Fodor, I.J. Rudas / Fuzzy Sets and Systems 158 (2007) 1692 – 1697
1693
where t (−1) : [0, ∞] → [0, 1] is the pseudo-inverse of t, and is given by t (−1) (u) = t −1 (min(u, t (0))). Turning back to Definition 1, one can easily see that the following t-norm T is -migrative for any ∈]0, 1[: T (x, y) =
min(x, y) if max(x, y) = 1, xy otherwise,
(3)
where is an arbitrary number from [0, 1]. It was provided by Budinˇcevi´c [4] as a counterexample to the question: Is a strictly monotone t-norm always continuous? Concerning further solutions of (1), an infinite class of -migrative t-norms was constructed in [1], each of them being discontinuous at each point of a dense subset of [0, 1]2 . Surprisingly enough, the continuous case has not been studied yet. Obviously, the product t-norm TP is -migrative for any ∈]0, 1[. In 1996 E. Pap raised also the following question [4, Problem 1.8(b)]: Does there exist a strictly monotone t-norm which is a solution of Eq. (1) and differs from TP ? An affirmative answer is given by the non-continuous t-norm in (3). We provide all continuous strictly monotone solutions in Theorem 3 via their additive generator functions (for the particular case of smooth generators see Theorem 5 and Example 6). The main aim of the present paper is to determine all continuous t-norms that are -migrative. It turns out that a continuous t-norm can be migrative only if it is strict. Then the -migrative property can be expressed, in an equivalent way, as a functional equation for additive generators of strict t-norms. That equation is completely solved in the paper. The solution at the same time yields a construction method for additive generators of strict t-norms that are migrative. We also study the case when the resulting additive generator is smooth. 2. Continuous migrative t-norms According to our main aim, in this section we concentrate on continuous migrative t-norms; that is, on continuous t-norms that are -migrative with some ∈]0, 1[. Notice that for a triangular norm T Definition 1 implies, by choosing y = 1, the following equality: T (x, ) = x
for all x ∈ [0, 1].
(4)
In the next theorem we show that the migrative property is rather strong for a continuous t-norm: it implies that the t-norm cannot have idempotent elements, and cannot be nilpotent. Theorem 2. Let T be a continuous t-norm. If T is -migrative then T is strict. Proof. First we prove that, under the above conditions, T (x, x) < x for all x ∈]0, 1[. Suppose there exists an x0 ∈]0, 1[ such that T (x0 , x0 ) = x0 . Since T is continuous, we have T (x0 , y) = min(x0 , y) for all y ∈ [0, 1] (see e.g. [3, Proposition 2.3]). Let y = . Then, T (x0 , ) = min(x0 , ), which contradicts (4). Therefore, if a continuous t-norm T is -migrative then it is Archimedean: it has an additive generator t such that representation (2) holds: T (x, y) = t (−1) (t (x) + t (y)). Suppose, to the contrary of the statement, that T is nilpotent; that is, t (0) < +∞. We have, by choosing y = 1 and by equality (4), the following equation: x = t (−1) (t () + t (x)) for all x ∈ [0, 1].
(5)
Let x be such that t () + t (x ) = t (0). Then, since ∈]0, 1[ and t is strictly decreasing, we have x > 0 and T (, x ) = 0. At the same time x > 0, which is a contradiction.
1694
J. Fodor, I.J. Rudas / Fuzzy Sets and Systems 158 (2007) 1692 – 1697
It is easy to conclude [2] that a strict t-norm T with additive generator t is -migrative if and only if t (x) − t (x) = t (y) − t (y) for all x, y ∈ [0, 1].
(6)
Eq. (6) says that the difference t (x) − t (x) is independent of x. More exactly, if we chose y = 1 in (6), this independent difference can be obtained as t (x) − t (x) = t (). We write it as follows: t (x) = t () + t (x)
for all x ∈ [0, 1].
(7)
In the next theorem we provide the general solution of the functional equation (7). It is based on the important fact that the restriction of t to the interval [, 1] uniquely determines t on each subinterval ]k+1 , k ], progressing from right to left. Theorem 3. Suppose t is an additive generator of a strict t-norm. Then t satisfies the functional equation (7) if and only if there exists a continuous, strictly decreasing function t0 from [, 1] to the non-negative reals with t0 () < +∞ and t0 (1) = 0 such that x t (x) = k · t0 () + t0 k (8) if x ∈]k+1 , k ], where k is any non-negative integer. Proof. (i) Assume that t is an additive generator of a strict t-norm that satisfies (7). We prove the existence of a function t0 that satisfies the conditions above. Observe first that Eq. (7) implies t (n ) = n · t () for any non-negative integer n. This is clear by choosing x = in (7) and by applying mathematical induction, together with 0 = t (1) = t (0 ) = 0 · t (). We prove, by mathematical induction, that the function t0 = t|[,1] satisfies (8). Indeed, define t0 on [, 1] by t0 (x) = t (x)
for x ∈ [, 1].
Then we have t0 (1) = t (1) = 0 and t0 () = t () < +∞. First we extend t0 to ]2 , ]. Consider x ∈]2 , ]. Then there exists a unique y ∈], 1] such that x = y; indeed, y = x/ is the appropriate choice. Then we have x t (x) = t (y) = t () + t (y) = t0 () + t0 , which proves (8) for k = 1. We can proceed from “right to left”. Assume that (8) holds for some k 1. We prove that it holds also for k + 1. That is, we have x t (x) = k · t0 () + t0 k if x ∈]k+1 , k ]. If x ∈]k+2 , k+1 ] then there is a unique y ∈]k+1 , k ] such that x = y; again y = x/ is the appropriate choice. Thus we have x t (x) = t (y) = t () + t (y) = t0 () + t x/ = t0 () + k · t0 () + t0 k x = (k + 1) · t0 () + t0 k+1 , which proves (8) for k + 1. Consecutive powers of generate intervals ]k+1 , k ], which provide a partition of ]0, 1]. That is, for any x ∈]0, 1] there exists a unique non-negative integer k such that x ∈]k+1 , k ]. Therefore, Eq. (8) defines t unambiguously for any x ∈]0, 1].
J. Fodor, I.J. Rudas / Fuzzy Sets and Systems 158 (2007) 1692 – 1697
1695
It is obvious by (8) that t is strictly decreasing and limx→+0 t (x) = ∞. Finally, we have t (k ) = limx→+k t (x) for all k, so t is continuous, and therefore it is an additive generator of a strict t-norm. (ii) Consider a continuous, strictly decreasing function t0 from [, 1] to the non-negative reals with t0 (0) < +∞ and t0 (1) = 0. Assume that we define a function t by (8). We prove that thus defined t is an additive generator of a strict t-norm, which is -migrative. Obviously, t is continuous and decreasing with t (1) = 0, and limx→0+ t (x) = +∞. Thus, it is an additive generator of a strict t-norm T. It remains to show that t satisfies Eq. (7). Consider an x ∈]0, 1]. Then there exists a unique k such that x ∈]k , k−1 ], and therefore x ∈]k+1 , k ]. Then we have x t (x) = k · t0 () + t0 k x = t0 () + (k − 1)t0 () + t0 k−1 = t () + t (x), which proves the statement.
Unfortunately, none of the famous t-norm families (like Frank, Hamacher, Dombi, Alsina) are migrative, except the particular case of t (x) = − log x, or equivalently, T (x, y) = TP (x, y) = xy. This latest case can be derived directly from Eq. (1). Proposition 4. A continuous t-norm T is -migrative for all ∈]0, 1[ if and only if T = TP . 3. Migrative t-norms with smooth generators Theorem 3 provides a construction method for additive generators of migrative t-norms. Generally speaking, if we start with a differentiable, strictly decreasing function t0 from [, 1] to the non-negative reals with t0 () < +∞ and t0 (1) = 0 then the derivative of the resulting additive generator t constructed by (8) need not exist at points that are powers of . One obvious exception is t0 (x) = − log x, with x ∈], 1]. In this case (8) yields t (x) = − log x for all x ∈]0, 1]. Considering t0 (x) =
1−x 1−
for x ∈ [, 1],
we obtain a piecewise linear additive generator t by (8), so t is not differentiable at powers of . We formulate a simple theorem to guarantee that the constructed generator t be smooth (i.e., differentiable). Theorem 5. Let ∈]0, 1[ and consider a continuous, strictly decreasing function t0 from [, 1] to the non-negative reals with t0 () < +∞ and t0 (1) = 0. Assume that t0 is differentiable on ], 1[ and its respective one-sided derivatives, () and t (1), exist at the end-points. Define the function t by (8). Then t is differentiable on ]0, 1[ if denoted by t0+ 0− and only if · t0+ () = t0− (1).
(9)
Proof. First, suppose (9) holds. We prove that t is differentiable. In fact, we have to show that the derivative of t exist at points k for all positive integer k. The definition of t and differentiability of t0 imply the differentiability of t at any other points in ]0, 1[. Since one-sided derivatives of t0 exist, this guarantees that both left- and right-sided derivatives of t exists at each k . It suffices to show that these are equal. Consider k for a fixed positive integer k. By Eq. (8) we have (k ) = t0− (1) · t−
1 k
(10)
1696
J. Fodor, I.J. Rudas / Fuzzy Sets and Systems 158 (2007) 1692 – 1697
and (k ) = t0+ () · t+
1 k−1
.
(11)
(k ) = t (k ), whence t is differentiable on ]0, 1[. Then (9) implies that t− + (k ) = t (k ) Second, assume that t is differentiable. Then its left- and right-sided derivatives are equal, so we have t− + for all positive integer k. Using (10) and (11), we must have (1) · t0−
1 1 = t0+ () · k−1 , k
which proves the statement.
The following example illustrates the construction method, taking into account the smoothness of the resulting additive generator t. Example 6. Let ∈]0, 1[ be given. We want to use an appropriate second-order polynomial t0 (x) = ax 2 + bx + c on [, 1] such that condition (9) is also satisfied. Together with requirements from Theorem 3 the following simple equations can be obtained for the parameters a, b, c: (1) t0 (1) = 0 gives a + b + c = 0; () = t (1) yields 2a = (2a + b)1/. (2) · t0+ 0− The solution defines the following form: t0 (x) = ax 2 − 2a(1 + )x + a(1 + 2) for x ∈ [, 1],
(12)
where a is any positive real number. One can easily check that t0 () > 0. Now we have () = −2a t0+
and
(1) = −2a, t0−
so (9) holds, and therefore the additive generator function t constructed by (8) is smooth. If we choose a = 1 and = 21 , we get t0 (x) = x 2 − 3x + 2 and t (x) = 22k · x 2 − 3 · 2k · x + 2 +
3k 4
if x ∈
1
2
, k+1
1 . 2k
Notice that the continuous t-norm in this example gives an affirmative answer to the question of E. Pap [4, Problem 1.8(b)]. The t-norm generated by thus defined t is strict, migrative, and differs from the product TP . 4. Conclusion In this paper we have completely characterized continuous t-norms that are migrative. It turned out that only strict t-norms can own this property. Their characterization has been developed through solutions of a functional equation. Although Definition 1 is seemingly general, notice that it does not provide a meaningful notion for triangular conorms. Indeed, if S is a t-conorm then it is -migrative if and only if S(x, y) = S(x, y) holds for all x, y ∈ [0, 1]. If we choose y = 0 then we must have x = x for all x ∈ [0, 1], because S is -migrative. This is impossible when = 1. Similarly, if y = 1 then we must have S(x, ) = 1 for all x ∈ [0, 1], which is again impossible unless = 1. Therefore, even the correct definition of -migrative t-conorms needs special care.
J. Fodor, I.J. Rudas / Fuzzy Sets and Systems 158 (2007) 1692 – 1697
1697
References [1] M. Budinˇcevi´c, M.S. Kurili´c, A family of strict and discontinuous triangular norms, Fuzzy Sets and Systems 95 (1998) 381–384. [2] F. Durante, P. Sarkoci, A note on the convex combinations of triangular norms, Fuzzy Sets and Systems 2006, submitted for publication. [3] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Trends in Logic. Studia Logica Library, Vol. 8, Kluwer Academic Publishers, Dordrecht, 2000. [4] R. Mesiar, V. Novák, Open problems from the 2nd international conference on fuzzy sets theory and its applications, Fuzzy Sets and Systems 81 (1996) 185–190.