February 3, 2015

11:24

118-ijufks

S0218488515500051

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems Vol. 23, No. 1 (2015) 105–115 c World Scientific Publishing Company

DOI: 10.1142/S0218488515500051

Int. J. Unc. Fuzz. Knowl. Based Syst. 2015.23:105-115. Downloaded from www.worldscientific.com by 37.44.207.135 on 01/22/17. For personal use only.

On the α-Migrativity of Idempotent Uninorms

Feng Qin College of Mathematics and Information Science, Jiangxi Normal University, 330022 Nanchang, P. R. China [email protected]

Daniel Ruiz-Aguilera Department of Mathematics and Computer Science, University of the Balearic Islands, Cra. de Valldemossa, km 7.5, 07122 Palma,Illes Balears, Spain [email protected]

Received 28 October 2013 Revised 25 September 2014 The α-migrative property for uninorms with different neutral elements is presented, and some general results are given. The case for idempotent uninorms is studied, characterizing those uninorms (from any of the main classes of uninorms), which are α-migrative over an idempotent uninorm. The solutions obtained generalize the results where both uninorms have the same neutral elements. Keywords: Fuzzy connectives; migrativity; T-norm; T-conorm; uninorm.

1. Introduction For some image processing applications and decision-making problems, the concept of α-migrativity is of great importance.17 In fact the migrativity property ensures that variations of the value of a variable by a ratio do not vary the result in terms of the affected variable. This property is needed, for instance, to darken or to lighten a certain part of an image. In decision-making, the ordering of inputs may be relevant, even though the result of modifying one or another evaluation by a given ratio should be the same. In both applications, the α-migrativity model this idea. The α-migrativity property for t-norms was introduced by Durante and Sarkoci5 to investigate the convex combination of a continuous t-norm T and the drastic product TD (which is indeed a special form of the open problem of convex combination of t-norms18 ). Some generalizations have been studied in depth: for t-norms, t-subnorms, semicopulas, quasi-copulas, copulas and aggregation functions.2,4,12 However, one of the most influential and impressive results about this topic are probably given out by Fodor et al.6–9 105

page 105

February 3, 2015

Int. J. Unc. Fuzz. Knowl. Based Syst. 2015.23:105-115. Downloaded from www.worldscientific.com by 37.44.207.135 on 01/22/17. For personal use only.

106

11:24

118-ijufks

S0218488515500051

F. Qin & D. Ruiz-Aguilera

Recently, in the general case, Mas et al.15 introduced and characterized the notion of α-migrative uninorm U over a fixed uninorm U0 with the same neutral element. Following this study, in this paper, we investigate the same migrativity equation for two uninorms U and U0 with different neutral elements, in the case where U0 is idempotent and U belongs to any of the main classes of uninorms: idempotent, Umin , Umax or Ucos . In all cases, the properties required to ensure the αmigrativity for uninorms with different neutral element are similar to the properties required for uninorms with the same neutral element, obtained in Ref. 15. The main difference between the α-migrativity for t-norms, and the αmigrativity for uninorms resides on the sections that have to coincide the operators involved. Just recall that for the case of t-norms, the α-migrativity is determined by two sections T0 (α, x) and T (α, x). In the case of α-migrative uninorms it is completely determined by the sections U0 (α, x) and U (x, U0 (α, e)). This difference is stated and studied in the paper. The paper is organized as follows: in Sec. 2, we recall some results concerning uninorms, and the main classes that will be used throughout the paper. In Sec. 3, we introduce the (α, U0 )-migrativity and some properties, and in Sec. 4 we characterize the α-migrativity equation of two uninorms U and U0 with different neutral elements, whenever U0 is an idempotent uninorm. Finally, conclusions and future work are presented in Sec. 5. 2. Preliminaries Definition 1. (Yager-Rybalov21) A binary function U : [0, 1]2 → [0, 1] is called a uninorm if it is commutative, associative, non-decreasing in each variable and there exists some element e ∈ [0, 1], called neutral element, such that U (e, x) = x for all x ∈ [0, 1]. In Ref. 19, it is clear that a uninorm U becomes a t-norm when e = 1 while U becomes a t-conorm when e = 0. Hence, in this paper, we only consider the case e ∈]0, 1[. For any e ∈]0, 1[, a uninorm works as a t-norm in the square [0, e]2 and as a t-conorm in [e, 1]2 , and its values are between the minimum and the maximum in the set of points E given by E = [0, e[×]e, 1]∪]e, 1] × [0, e[.

(1)

We will denote a uninorm with neutral element e ∈]0, 1[ and a underlying t-norm TU and a underlying t-conorm SU by U ≡ hTU , e, SU i. In fact, it holds that TU (x, y) =

U (ex, ey) , e

SU (x, y) =

U (e + (1 − e)x, e + (1 − e)y) − e 1−e

(2)

for all x, y ∈ [0, 1]. For any uninorm we have U (0, 1) ∈ {0, 1}, and a uninorm is called conjunctive when U (0, 1) = 0 and disjunctive when U (0, 1) = 1.

page 106

February 3, 2015

11:24

118-ijufks

S0218488515500051

On the α-Migrativity of Idempotent Uninorms

107

Moreover, the most studied classes of uninorms are: • Uninorms in Umin (respectively Umax ), those given by minimum (respectively maximum) in E. • Idempotent uninorms, Uide , those that satisfy U (x, x) = x for all x ∈ [0, 1]. • Uninorms continuous in the open square ]0, 1[2 , Ucos .

Int. J. Unc. Fuzz. Knowl. Based Syst. 2015.23:105-115. Downloaded from www.worldscientific.com by 37.44.207.135 on 01/22/17. For personal use only.

Now we present the structure of each one of these classes of uninorms. Theorem 1. (Fodor-Yager-Rybalov10) Let U : [0, 1]2 → [0, 1] be a uninorm with neutral element e ∈]0, 1[. Then, the sections x 7→ U (x, 1) and x 7→ U (x, 0) are continuous at each point except perhaps at the point x = e if and only if U is given by one of the following formulas. (i) If U (0, 1) = 0, then   eTU ( xe , ye )    U (x, y) = e + (1 − e)SU ( x−e , y−e ) 1−e 1−e    min(x, y)

if (x, y) ∈ [0, e]2 , if (x, y) ∈ [e, 1]2 ,

(3)

otherwise.

(ii) If U (0, 1) = 1, then

U (x, y) =

  eTU ( xe , ye )   

if (x, y) ∈ [0, e]2 , x−e y−e

e + (1 − e)SU ( 1−e 1−e )    max(x, y)

if (x, y) ∈ [e, 1]2 ,

(4)

otherwise.

In the following, the set of uninorms as in Case (i) be denoted by Umin and the set of uninorms as in Case (ii) by Umax . Idempotent uninorms were characterized first in Ref. 3 for those with a lateral continuity and in Ref. 13 for the general case. An improvement of this last result was done in Ref. 20 as follows. Theorem 2. (Ruiz-Aguilera, et al.20 ) U is an idempotent uninorm with neutral element e ∈]0, 1[ if and only if there exists a decreasing function g: [0, 1] → [0, 1] symmetric with respect to the main diagonal and satisfying g(e) = e such that   min(x, y) if y < g(x) or (y = g(x) and x < g(g(x))),   U (x, y) = max(x, y) if y > g(x) or (y = g(x) and x > g(g(x))),    min(x, y) or max(x, y) if (y = g(x) and x = g(g(x))), (5) being commutative in the points (x, y) such that y = g(x) and x = g(g(x)). Any idempotent uninorm U with neutral element e and associated function g will be denoted by U ≡ hg, eiide , and the class of idempotent uninorms will be denoted by Uide . Obviously, for any of these uninorms, the underlying t-norm TU is

page 107

February 3, 2015

108

11:24

118-ijufks

S0218488515500051

F. Qin & D. Ruiz-Aguilera

min and the underlying t-conorm SU is max. From the previous theorem, a function g can be associated to more than one uninorm U .

Int. J. Unc. Fuzz. Knowl. Based Syst. 2015.23:105-115. Downloaded from www.worldscientific.com by 37.44.207.135 on 01/22/17. For personal use only.

Theorem 3. (Hu-Li11 ) Suppose U is a uninorm continuous in ]0, 1[2 with neutral element e ∈]0, 1[. Then one of the following two cases is satisfied: (i) There exist µ ∈ [0, e[, λ ∈ [0, µ], two continuous t-norms T1 and T2 and a continuous strictly increasing function h: [µ, 1] → [−∞, +∞] with h(µ) = −∞, h(e) = 0 and h(1) = +∞ such that U can be represented as  x y  if x, y ∈ [0, λ], λT1 ( λ , λ )     y−λ   λ + (µ − λ)T2 ( x−λ  µ−λ , µ−λ ) if x, y ∈ [λ, µ],     −1 h (h(x) + h(y)) if x, y ∈]µ, 1[, U (x, y) =   1 if min(x, y) ∈]λ, 1] and max(x, y) = 1,       min(x, y) or 1 if (x, y) ∈ {(λ, 1), (1, λ)},      min(x, y) otherwise. (6) (ii) There exist ν ∈ (e, 1], ω ∈ [ν, 1], two continuous t-conorms S1 and S2 and a continuous strictly increasing function h: [0, ν] → [−∞, +∞] with h(0) = −∞, h(e) = 0 and h(ν) = +∞ such that U can be represented as  x−ν y−ν  ν + (ω − ν)S1 ( ω−ν , ω−ν ) if x, y ∈ [ν, ω],      y−ω   ω + (1 − ω)S2 ( x−ω  1−ω , 1−ω ) if x, y ∈ [ω, 1],     −1 h (h(x) + h(y)) if x, y ∈]0, ν[, U (x, y) =   0 if max(x, y) ∈ [0, ω[ and min(x, y) = 0,       max(x, y) or 0 if (x, y) ∈ {(ω, 0), (0, ω)},      max(x, y) otherwise. (7) The class of all uninorms that are continuous in ]0, 1[2 will be denoted by Ucos . A uninorm as in the formula (6) will be denoted by U ≡ hT1 , λ, T2 , µ, hicos,min , and the class of all uninorms that are continuous in the open unit square of this form will be denoted by Ucos,min . Analogously, a uninorm as in the formula (7) will be denoted by U = hh, ν, S1 , ω, S2 icos,max , and the class of all uninorms that are continuous in the open unit square of this form will be denoted by Ucos,max . For any uninorm U ≡ hT1 , λ, T2 , µ, hicos,min , the underlying t-norm TU is given by an ordinal sum of three t-norms, T1 , T2 and a strict t-norm, whereas the underlying t-conorm SU is always strict. Similarly, for any uninorm U ≡ hh, ν, S1 , ω, S2 icos,max , the underlying t-norm TU is always strict, whereas the underlying t-conorm SU is given by an ordinal sum of three t-conorms, a strict t-conorm,

page 108

February 3, 2015

11:24

118-ijufks

S0218488515500051

On the α-Migrativity of Idempotent Uninorms

109

S1 and S2 . Note that a uninorm U in Ucos,min with µ = 0 (or in Ucos,max with ν = 1) is a representable uninorm, which is named in Ref. 10. In this paper, we do not involve them because they have been investigated in detail in Refs. 1 and 15. 3. Migrativity for Uninorms with Different Neutral Elements

Int. J. Unc. Fuzz. Knowl. Based Syst. 2015.23:105-115. Downloaded from www.worldscientific.com by 37.44.207.135 on 01/22/17. For personal use only.

Now we give the definition of migrativity for two uninorms with any neutral elements, which is a generalization of Definition 5 in Ref. 15. Definition 2. Consider α ∈ [0, 1], and let U0 be a uninorm with neutral element e0 . We will say that a uninorm U with neutral element e is α-migrative over U0 , or U is (α, U0 )-migrative if U (U0 (x, α), y) = U (x, U0 (α, y))

for all (x, y) ∈ [0, 1]2 .

(8)

From the previous definition we have some trivial cases when U = U0 or α = e0 . Proposition 1. Consider U0 a uninorm with neutral element e0 . Then the following items hold: (i) U0 is (α, U0 )-migrative for all α ∈ [0, 1]. (ii) Any uninorm U with neutral element e is (e0 , U0 )-migrative. In view of (ii) in the previous propostion, we will consider α 6= e0 . The following lemma is crucial to characterize the (α, U0 )-migrativity of uninorms. Lemma 1. Let U and U0 be two uninorms with neutral elements e and e0 , respectively. Then U is (α, U0 )-migrative if and only if U0 (x, α) = U (x, U0 (α, e))

for all x ∈ [0, 1].

(9)

Proof. If U is (α, U0 )-migrative, take y = e in Eq. (8) and, since e is the neutral element of U , we have U0 (x, α) = U (U0 (x, α), e) = U (x, U0 (α, e)). Conversely, if U and U0 satisfy Eq. (9), as U and U0 are associative and commutative, we have U (U0 (x, α), y) = U (U (x, U0 (α, e)), y) = U (x, U (U0 (α, e), y)) = U (x, U (y, U0 (α, e))) = U (x, U0 (y, α)) = U (x, U0 (α, y)). Remark 1. Several cases included in Definition 1 have been studied yet. If e = e0 , then it follows from Eq. (9) that U0 (x, α) = U (x, α). This case has been studied by Mas et al. in Ref. 15. Another case is whenever e = 0, e = 1, e0 = 0 or e0 = 1 (U or U0 are t-norms or t-conorms), cases studied by Mas et al. in Refs. 14 and 16. Therefore, we will assume e 6= e0 and e, e0 ∈]0, 1[. Remark 2. For the case e 6= e0 Eq. (9) shows that the (α, U0 )-migrativity of a uninorm U is determined whenever the α-section of U0 coincides with the U0 (α, e)section of U . It differs from the similar property for t-norms because, in that case, the (α, T )-migrativity of T0 is determined whenever their α-sections coincide.

page 109

February 3, 2015

Int. J. Unc. Fuzz. Knowl. Based Syst. 2015.23:105-115. Downloaded from www.worldscientific.com by 37.44.207.135 on 01/22/17. For personal use only.

110

11:24

118-ijufks

S0218488515500051

F. Qin & D. Ruiz-Aguilera

Remark 3. The case α = e in Definition 1 is reduced to the case e = e0 . Note that U0 is idempotent, then we know from Theorem 2 that U0 (α, e) = e or U0 (α, e) = α. Now we can claim that U0 (e, e) = e hold for α = e. In fact, suppose U0 (α, e) = e, then we have from α = e that U0 (e, e) = U0 (e, α) = e; suppose U0 (α, e) = α, then, taking x = e in Eq. (9), we obtain that U0 (e, e) = U0 (e, α) = U (e, U0 (α, e)) = U (e, e) = e and then U0 (e, e) = e. Furthermore, substituting x = e0 in Eq. (9) and using U0 (e, e) = e, we can know that e = U0 (e0 , e) = U (e0 , U0 (e, e)) = U (e0 , e) = e0 . Hence, from now on, we only consider the case α ∈ [0, 1], e, e0 ∈]0, 1[ and α, e, e0 pairwise different. 4. α-Migrative Uninorms Over an Idempotent Uninorm In this section, we will investigate Eq. (8) when U0 is idempotent. From Theorem 2, we have that U0 (α, e) ∈ {e, α}. Now we discuss the case whenever U0 (α, e) = e. Proposition 2. Consider α ∈ [0, 1], a uninorm U with neutral element e ∈]0, 1[, a uninorm U0 ∈ Uide with neutral element e0 ∈]0, 1[ such that U0 (α, e) = e, and α, e, e0 pairwise different. Then U is not α-migrative over U0 . Proof. Obviously, from U0 (α, e) = e, Eq. (9), and the neutral element e of U , we have that U0 (x, α) = x for all x ∈ [0, 1]. Taking x = e0 in the previous formula, we obtain that α = U0 (e0 , α) = e0 , which is a contradiction. Now we consider the case U0 (α, e) = α. In that case, Eq. (9) is reduced to U0 (x, α) = U (x, α), that is, U and U0 have the same α-section. To continue this process, we will distinguish several cases depending on the class of uninorms where U belongs to. We start with U ∈ Umin . Proposition 3. Consider α ∈ [0, 1], a uninorm U ∈ Umin with neutral element e ∈]0, 1[ and continuously underlying t-norm TU , a uninorm U0 ≡ hg0 , e0 iide , such that U0 (α, e) = α, and α, e, e0 pairwise different. Then U is α-migrative over U0 if and only if one of the following two cases hold: (i) If α < min(e, e0 ), then g0 (α) = 1, U (α, α) = α and U0 (α, 1) = α. (ii) If e < e0 < α, then g0 (α) = e and U (α, α) = α. Proof. (i) Suppose α < min(e, e0 ). Then for any x > e, we know from Theorem 1 (i) that U (x, U0 (α, e)) = U (x, α) = min(x, α). Next, applying Eq. (9), we imply that U0 (α, x) = min(x, α) for any x > e. Furthermore, by idempotency of U0 and Theorem 2, we have g0 (α) ≥ x for any x > e. Hence it follows from arbitrariness of x that g0 (α) = 1. Finally, take x = α and x = 1 in Eq. (9) respectively, we obtain U (α, α) = α and U0 (α, 1) = α because of idempotency of U0 and structure of U . Conversely, we check that Eq. (9) holds under the conditions given above. That is, we need to show that U0 (x, α) = U (x, α) holds for all x ∈ [0, 1] when α < min(e, e0 ), g0 (α) = 1, U (α, α) = α and U0 (α, 1) = α. By means of idempotency

page 110

February 3, 2015

11:24

118-ijufks

S0218488515500051

Int. J. Unc. Fuzz. Knowl. Based Syst. 2015.23:105-115. Downloaded from www.worldscientific.com by 37.44.207.135 on 01/22/17. For personal use only.

On the α-Migrativity of Idempotent Uninorms

111

of α, continuity of TU and U ∈ Umin , we know that U (x, α) = min(x, α) for all x ∈ [0, 1]. On the other hand, it obviously follows from g0 (α) = 1, structure of U0 and U0 (1, α) = α that U0 (x, α) = min(x, α) for all x ∈ [0, 1]. Thus we had checked that U0 (x, α) = min(x, α) = U (x, α) holds for all x ∈ [0, 1]. (ii) Suppose e < e0 < α. Then for any x < e, we know from Theorem 1 (i) that U (x, U0 (α, e)) = U (x, α) = min(x, α). Next, applying Eq. (9), it holds that U0 (α, x) = min(x, α) for any x < e. Furthermore, by idempotency of U0 and Theorem 2, we have g0 (α) ≥ x for any x < e. Hence it follows from arbitrariness of x that g0 (α) ≥ e. On the other hand, it obviously follows from idempotency of U0 and U0 (α, e) = α that g0 (α) ≤ e. Hence, it holds g0 (α) = e. Finally, take x = α in Eq. (9), we obtain U (α, α) = α since U0 is idempotent. Conversely, we check that Eq. (9) holds under the conditions given above. That is, we need to show that U0 (x, α) = U (x, α) holds for all x ∈ [0, 1] when e < e0 < α, g0 (α) = e, U (α, α) = α. By means of the idempotent element α of U , continuity of SU and U ∈ Umin , we know that U (x, α) = min(x, α) for all x ∈ [0, e) and max(x, α) for all x ∈ [e, 1]. On the other hand, it obviously follows from g0 (α) = e, structure of U0 and the hypothesis U0 (α, e) = α that U0 (x, α) = min(x, α) for all x ∈ [0, e) and max(x, α) for all x ∈ [e, 1]. Thus we had verified that U0 (x, α) = U (x, α) holds for all x ∈ [0, 1]. In the following, we prove that the remaining cases are impossible. Suppose that min(e, e0 ) < α < max(e, e0 ). Furthermore, we assume that e0 < e, then it holds e0 < α < e, which implies that U0 (α, e) = max(α, e) = e 6= α, this is a contradiction. It is completely similar to the case e < α < e0 . Suppose that e0 < e < α, then, take x0 ∈ (e0 , e), it follows from structures of U0 and U that U0 (x0 , α) = max(x0 , α) = α and U (x0 , α) = min(x0 , α) = x0 , this is also a contradiction. Remark 4. Proposition 10 in Ref. 15 is consistent with Proposition 3, if taking e0 = e. The result for U ∈ Umax can be deduced by dualizing the previous case. Proposition 4. Consider α ∈ [0, 1], a uninorm U ∈ Umax with neutral element e ∈]0, 1[ and continuously underlying t-conorm SU , a uninorm U0 ≡ hg0 , e0 iide with e0 ∈]0, 1[, satisfying U0 (α, e) = α and α, e, e0 pairwise different. Then U is α-migrative over U0 if and only if one of the following two cases hold: (i) If max(e, e0 ) < α, then g0 (α) = 0, U (α, α) = α and U0 (α, 0) = α. (ii) If α < e0 < e, then g0 (α) = e and U (α, α) = α. Next, we take into account the case when U is an idempotent uninorm. Proposition 5. Consider α ∈ [0, 1], U ≡ hg, eiide with e ∈]0, 1[, U0 ≡ hg0 , e0 iide with e0 ∈]0, 1[ satisfying U0 (α, e) = α, and α, e, e0 pairwise different. Then U is α-migrative over U0 if and only if g(α) = g0 (α) and U0 (g(α), α) = U (g(α), α).

page 111

February 3, 2015

11:24

112

118-ijufks

S0218488515500051

F. Qin & D. Ruiz-Aguilera

Proof. Let us suppose that U is α-migrative over U0 , and take x ∈ [0, 1]:

Int. J. Unc. Fuzz. Knowl. Based Syst. 2015.23:105-115. Downloaded from www.worldscientific.com by 37.44.207.135 on 01/22/17. For personal use only.

• If x is such that x < g(α), we know from Eq. (9) that min(x, α) = U (x, α) = U0 (x, α). Therefore, applying Theorem 2, x ≤ g0 (α). • If x is such that x > g(α), we know from Eq. (9) that max(x, α) = U (x, α) = U0 (x, α). Again applying Theorem 2, we have that x ≥ g0 (α). Therefore, we can conclude that g(α) = g0 (α). Finally, take x = g(α) in Eq. (9), then it holds that U (α, g(α)) = U0 (α, g(α)). Conversely, with conditions g(α) = g0 (α) and U0 (g(α), α) = U (g(α), α), it is straightforward to see that U (x, α) = U0 (x, α) for all x ∈ [0, 1]. Remark 5. The condition required over g and g0 in Proposition 5 is the same as Proposition 12 in Ref. 15, where e = e0 . Finally, we investigate Eq. (8) when U ∈ Ucos . To do this, we start by considering U ∈ Ucos,min. Proposition 6. Let α ∈ [0, 1], a uninorm U ∈ Ucos,min with neutral element e ∈ ]0, 1[, a uninorm U0 ≡ hg0 , e0 iide with e0 ∈]0, 1[, satisfying U0 (α, e) = α and α, e, e0 pairwise different. Then U is α-migrative over U if and only if one of the following three cases hold: (i) If α = 1 and U (1, 0) = U0 (1, 0) = 1, then λ = 0 and g0 (1) = 0. (ii) If α = 1 and U (1, 0) = U0 (1, 0) = 0, then g0 (1) = λ and U (λ, 1) = U0 (λ, 1). (iii) If α ≤ µ, then α < e0 , U (α, α) = α, g0 (α) = 1, and moreover (a) If α < λ, then U0 (α, 1) = α, (b) If α = λ, then U0 (α, 1) = U (α, 1), (c) If λ < α ≤ µ, then U0 (α, 1) = 1. Proof. Take x = α in Eq. (9), then we obtain U (α, α) = α, which implies from Theorem 3 (i) that α ≤ µ or α = e or α = 1. Obviously, by hypotheses, we know that the case α = e is impossible. Thus we only need to consider the remaining two cases: α ≤ µ and α = 1. Suppose that α = 1, then we know from Eq. (9) that U (x, 1) = U0 (x, 1) for all x ∈ [0, 1], specially, U (0, 1) = U0 (0, 1). According to value of U (0, 1), there are two cases to consider: U (0, 1) = U0 (0, 1) = 0 and U (0, 1) = U0 (0, 1) = 1. (i) Suppose that α = 1 and U (0, 1) = U0 (0, 1) = 1, then it follows from monotonicity of U that U (x, 1) = 1 for all x ∈ [0, 1], which means from Theorem 3 (i) that λ = 0. Obviously, we imply from 2 and U0 (0, 1) = 1 that g0 (1) = 0. Conversely, we check that Eq. (9) holds under the conditions given above. By means of monotonicity of U and U0 , U (x, 1) = U0 (1, x) = 1 holds for all x ∈ [0, 1]. (ii) Suppose that α = 1 and U (0, 1) = U0 (0, 1) = 0. Note that for any x < λ, we know from Theorem 3 that U (x, 1) = min(x, 1). Next, applying Eq. (9), it holds that U0 (1, x) = min(x, 1) for any x < λ. Furthermore, by idempotency of U0 and

page 112

February 3, 2015

11:24

118-ijufks

S0218488515500051

Int. J. Unc. Fuzz. Knowl. Based Syst. 2015.23:105-115. Downloaded from www.worldscientific.com by 37.44.207.135 on 01/22/17. For personal use only.

On the α-Migrativity of Idempotent Uninorms

113

Theorem 2, we have x ≤ g0 (1) for any x < λ, and then g0 (1) ≥ λ. For any x > λ, we know from Theorem 3 that U (x, 1) = max(x, 1). Applying again Eq. (9), it holds that U0 (1, x) = max(x, 1) for any x > λ. By idempotency of U0 and Theorem 2, we have x ≥ g0 (1) for any x > λ, and then g0 (1) ≤ λ. Therefore, it holds g0 (1) = λ. Finally, take x = λ in Eq. (9), we obtain U (1, λ) = U0 (1, λ). Conversely, we need to show that U0 (x, α) = U (x, α) holds for all x ∈ [0, 1] when α = 1, U (0, 1) = U0 (0, 1) = 0, g0 (1) = λ and U (λ, 1) = U0 (λ, 1). However, by means of structures of U and U0 , and Theorems 2 and 3, we obviously know U (x, 1) = U0 (1, x) for all x ∈ [0, 1]. (iii) Suppose that α ≤ µ. At first, we claim that α < e0 . Otherwise it follows from hypothesis that α > e0 , which means that α = U0 (e, α) ≥ U0 (e, e0 ) = e > µ ≥ α, this is a contradiction. Note that for any 1 > x > µ, we know that U (x, α) = min(x, α). Next, applying Eq. (9), it holds that U0 (α, x) = min(x, α) for any 1 > x > µ. Furthermore, by idempotency of U0 and Theorem 2, we have x ≤ g0 (α) for any 1 > x > µ, and then g0 (α) = 1. Finally, take x = 1 in Eq. (9), we obtain U (1, α) = U0 (1, α), and moreover, if α < λ or α = λ or λ < α ≤ µ, then U0 (α, 1) = α or U0 (α, 1) = U (α, 1) or U0 (α, 1) = 1 respectively. Conversely, we need to show that U0 (x, α) = U (x, α) holds for all x ∈ [0, 1] when α ≤ µ, α < e0 , U (α, α) = α, U0 (α, 1) = U (α, 1) and g0 (α) = 1. However, by structures of U and U0 , and Theorems 2 and 3, we easily know U (x, α) = U0 (α, x) for all x ∈ [0, 1]. We can deduce a similar result when U ∈ Ucos,max , dualizing the previous case. Proposition 7. Let α ∈ [0, 1], a uninorm U ∈ Ucos,max with neutral element e ∈]0, 1[, a uninorm U0 ≡ hg0 , e0 iide with neutral element e0 ∈]0, 1[, satisfying U0 (α, e) = α and α, e, e0 pairwise different. Then U is α-migrative over U0 if and only if one of the following three cases hold: (i) If α = 0 and U (1, 0) = U0 (1, 0) = 0, then ω = 1 and g0 (0) = 1. (ii) If α = 0 and U (1, 0) = U0 (1, 0) = 1, then g0 (0) = ω and U (ω, 0) = U0 (ω, 0). (iii) If α ≥ ν, then α > e0 , U (α, α) = α, g0 (α) = 0, and moreover (a) If α > ω, then U0 (α, 0) = α, (b) If α = ω, then U (α, 0) = U0 (α, 0), (c) If ν ≤ α < ω, then U0 (α, 0) = 0. Remark 6. The conditions required in Proposition 6 are consistent with the ones in Proposition 13 in Ref. 15 when e0 = e. Example 1. Take e0 = 0.5 and function g0 : [0, 1] → [0, 1] as follows g0 (x) = 1−x. Then we construct the left-continuous uninorm U0 ≡ hg0 , 0.5iide . On the other hand, take e = 0.4 and construct the idempotent uninorm in Umin : ( max(x, y) if x, y ∈ [0.4, 1], U (x, y) = min(x, y) otherwise.

page 113

February 3, 2015

114

11:24

118-ijufks

S0218488515500051

F. Qin & D. Ruiz-Aguilera

Furthermore, take α = 0.6, then it follows from Theorem 2 that U0 (0.6, 0.4) = min(0.4, 0.6) = 0.4, and we know from Proposition 2 that U is not (U0 , 0.6)migrative. However, for the above function g0 , we construct the right-continuous uninorm U1 ≡ hg0 , 0.5iide . Thus it follows from Theorem 2 that U1 (0.6, 0.4) = max(0.4, 0.6) = 0.6 and we know from Proposition 3 that U is (U1 , 0.6)- migrative.

Int. J. Unc. Fuzz. Knowl. Based Syst. 2015.23:105-115. Downloaded from www.worldscientific.com by 37.44.207.135 on 01/22/17. For personal use only.

5. Conclusions and Future Work In this paper, we have studied the α-migrativity for uninorms U and U0 with different neutral elements, where U0 is idempotent and U belongs to one of the main classes of uninorms. The results obtained are a generalization to the case when U and U0 have the same neutral element. Moreover, we have shown that there is an essential distinction between the α-migrativity for t-norms and the α-migrativity for uninorm. Although our investigations are only made under the condition U0 is idempotent, we can similarly obtain the results of the case that U is idempotent. In fact, suppose that U is idempotent, then, taking x = e0 in Eq. (9), we have α = U (e0 , U0 (α, e)), which implies from Theorem 2 that either U0 (α, e) = α or e0 = α. If e0 = α, then we are in Proposition 1. If U0 (α, e) = α, then Eq. (9) can be rewritten as U0 (x, α) = U (x, α), which means that U has the same situation as U0 . By replacing U0 by U in the results of the case that U0 is idempotent, we easily obtain those ones of the case that U is idempotent. Our future work will be devoted to deal with the same migrativity equation of uninorms with different neutral elements whenever both U0 and U are not idempotent. Acknowledgments This work has been supported by the National Natural Science Foundation of China (Nos. 61165014 and 11161023), A Foundation for the Author of National Excellent Doctoral Dissertation of P. R. China (No. 2007B14), Jiangxi Natural Science Foundation (No. 20122BAB201009)and the Scientific Research Foundation of Jiangxi Provincial Education Department (No. GJJ12176). This work has also been supported by the Spanish grants MTM-2009-10320 and TIN2013-42795-P, both with FEDER support. We are very grateful to the referees and editors for useful suggestions that helped to improve the paper. References 1. H. Bustince, B. De Baets, J. Fernandez, R. Mesiar and J. Montero, A generalization of the migrativity property of aggregation functions, Inform. Sci. 191 (2012) 76–85.

page 114

February 3, 2015

11:24

118-ijufks

S0218488515500051

Int. J. Unc. Fuzz. Knowl. Based Syst. 2015.23:105-115. Downloaded from www.worldscientific.com by 37.44.207.135 on 01/22/17. For personal use only.

On the α-Migrativity of Idempotent Uninorms

115

2. H. Bustince, J. Montero and R. Mesiar, Migrativity of aggregation function, Fuzzy Sets Syst. 160 (2009) 766-777. 3. B. De Baets, Idempotent uninorms, Eur. J. Oper. Res. 118 (1999) 631–642. 4. F. Durant and R. G. Ricci, Supermigrative semi-copulas and triangular norms, Inform. Sci. 179 (2009) 2689–2694. 5. F. Durant and P. Sarkoci, A note on convex combinations of triangular norms, Fuzzy Sets Syst. 159 (2008) 77–80. 6. J. Fodor and I. J. Rudas, On continuous triangular norms that are migrativity, Fuzzy Sets Syst. 158 (2007) 1692–1697. 7. J. Fodor and I. J. Rudas, On some classes of aggregation functions that are migrative, in Proc. IFSA-EUSFLAT 2009, pp. 653–656. 8. J. Fodor and I. J. Rudas, An extension of the migrative property for triangular norms, Fuzzy Sets Syst. 168 (2011) 70–80. 9. J. Fodor and I. J. Rudas, Migrative t-norms with respect to continuous ordinal sums, Inform. Sci. 181 (2011) 4860–4866. 10. J. Fodor, R. R. Yager and A. Rybalov, Structure of uninorms, Int. J. Uncertainty Fuzziness Knowledge-Based Syst. 5 (1997) 411–427. 11. S. K. Hu and Z. F. Li, The structure of continuous uninorms, Fuzzy Sets Syst. 124 (2001) 43–52. 12. C. L´ opez-Molina, B. De Baets, H. Bustince, E. Indur´ ain, A. Stupnanov´ a and R. Mesiar, Bimigrativity of binary aggregation functions, Inform. Sci. 274 (2014) 225–235. 13. J. Mart´ın, G. Mayor and J. Torrens, On locally internal monotonic operations, Fuzzy Sets Syst. 137 (2003) 27–42. 14. M. Mas, M. Monserrat, D. Ruiz-Aguilera and J. Torrens, On migrative t-conorms and uninorms, Communications in Computer and Information Science 299 (2012) 286– 295. 15. M. Mas, M. Monserrat, D. Ruiz-Aguilera and J. Torrens, An extension of the migrative property for uninorms, Inform. Sci. 246 (2013) 191–198. 16. M. Mas, M. Monserrat, D. Ruiz-Aguilera and J. Torrens, Migrative uninorms and nullnorms over t-norms and t-conorms, Fuzzy Sets Syst. 261(15) (2015) 20–32. 17. R. Mesiar, H. Bustince and J. Fernandez, On the α-migrativity of semicopulas, quasicopulas and copulas, Inform. Sci. 180 (2010) 1967–1976. 18. R. Mesiar and V. Nov´ ak, Open problems from the 2nd international conference of fuzzy sets theory and its applications, Fuzzy Sets Syst. 81 (1996) 185–190. 19. F. Qin and B. Zhao, The distributivity equations for idempotent uninorms and nullnorms, Fuzzy Sets Syst. 155 (2005) 446–458. 20. D. Ruiz-Aguilera, J. Torrens, B. De Baets and J. Fodor, Some remarks on the characterization of idempotent uninorms, in E. H¨ ullermeier, R. Kruse, F. Hoffmann (eds.), Lecture Notes in Artificial Intelligence, Vol. 6178, 2010, pp. 425–434. 21. R. R. Yager and A. Rybalov, Uninorm aggregation operators, Fuzzy Sets Syst. 80 (1996) 111–120.

page 115