Migrativity properties of overlap functions over uninorms

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Fuzzy Sets and Systems ••• (••••) •••–•••

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www.elsevier.com/locate/fss

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Migrativity properties of overlap functions over uninorms

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Hongjun Zhou ∗ , Xinxin Yan

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College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China

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Received 20 April 2019; received in revised form 14 November 2019; accepted 15 November 2019

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Abstract

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The migrativity of aggregation functions is an interesting property from both theoretic and applied points of view in fuzzy set theory, and many profound results on the migrative functional equation for different combinations of some specific subclasses of aggregation functions have been achieved in the last decades. The present paper chooses to study the migrativity properties of overlap functions over uninorms, where overlap functions are a newly-born subclass of nonassociative aggregation functions for modeling overlap problems in decision making and image processing. The main results are characterizations of solutions to some migrative functional equations for overlap functions over uninorms in the usual classes such as uninorms continuous on the boundary except at their neutral elements, representable uninorms, idempotent uninorms and uninorms continuous in the open unit square, together with plenty of supporting examples for positive solutions and illustrating remarks for negative results. © 2019 Elsevier B.V. All rights reserved.

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Keywords: Migrativity; Overlap function; Uninorm; Nullnorm

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1. Introduction

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1.1. Importance of migrativity

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The notion of α-migrativity of t-norms appeared first in an open problem on whether there exist strictly monotone t-norm solutions to the migrative functional equation [46,47], and then was formally named by Durante and Sarkoci [21] with the intent of constructing new t-norms by means of convex combinations of t-norms with this property and the drastic product t-norm TD . More precisely, a t-norm T : [0, 1]2 → [0, 1] is said to be α-migrative for a given α ∈ [0, 1] if the migrative functional equation T (αx, y) = T (x, αy) holds for all x, y ∈ [0, 1], which states that the effect of scalar multiplication for one of the arguments of the t-norm remains the same no matter which argument is multiplied by the same weight. The migrativity has aroused great interests of researchers in related communities of aggregation functions. The interest in migrativity is supported by its importance in the following four aspects:

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This work was supported by the National Natural Science Foundation of China (Grant No. 61473336) and the Youth Science and Technology Program of Shaanxi Province (Grant No. 2016KJXX-24). * Corresponding author. E-mail addresses: [email protected] (H. Zhou), [email protected] (X. Yan). https://doi.org/10.1016/j.fss.2019.11.011 0165-0114/© 2019 Elsevier B.V. All rights reserved.

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(1) The solution of functional equations is one of the oldest but very active topics all along in mathematical analysis. The migrative functional equation as well its various extensions involving other aggregation functions (an overview will be given in Subsection 1.2) builds a bridge between Aczél’s generalized associativity equation [1,2,4,41] and recursiveness equation [15,17]. Since its birth in the setting of t-norms, the migrative functional equation received considerable attention [7,33,12,62]. (2) The migrativity property plays an important role in constructions of new aggregation functions by convex combinations of given ones [33,51,21,52,38]. For example, for an α-migrative t-norm T , the convex combination α n T + (1 − α n )TD is again a t-norm for every natural number n [21], and for a continuous t-norm T , αT + (1 − α)TD is a t-norm if and only if T is α-migrative [52]. (3) The migrativity property is closely related to other algebraic properties such as Lipschitz property, associativity, bisymmetry, additive generators, neutral elements and absorbing elements [12,9]. From the theoretic point of view, it is interesting to characterize aggregation functions satisfying each of these algebraic properties and/or to clarify the relationships between these algebraic properties. (4) From the point of view of applications, the migrativity property is particularly interesting too. As pointed out by Mesiar et al. [45], the concept of migrativity captures the idea that variations in the value of some functions caused by considering just a given fraction of one of the input variables is independent of the actual choice of variable, which is essential for some image processing applications and decision-making problems. The study on migrativity will potentially produce useful decision aid techniques for decision-makers in many applications involving decision-making problems of knowledge management [48,59,49,10,35]. We provide here one more example of migrativity for possible applications on health risk assessment. According to the report on global health risks from The World Health Organization in 2009 [61], the global and regional mortality and disease burden are identified with 24 health risk factors, ranging from environmental risks such as exposure to smoke from indoor solid fuel use, to metabolic risks such as high blood pressure. Understanding and then aggregating the effects of these health risks are vital to designing and targeting prevention efforts. However, these risk factors are not all equally amenable to interventions, e.g., effective policies are known to interventions on micronutrient deficiencies rather than on high body mass index. Assume now xi denotes the quantitative assessment of harms caused by the i-th measurable risk factor where i = 1, . . . , n for some n ≤ 24, and assume the n-ary aggregation function A(x1 , . . . , xn ) denotes one kind of healthy indices of populations in some regions. Assume further that the aggregation function A is α-migrative in each argument with respect to the respective intervention functions Bi for some risk reduction level α ∈ (0, 1), then a natural and effective intervention strategy is to carry out harm reductions B(α, xi ) of some suitable level α on those risk factors xi such as micronutrient deficiencies that are more amenable to interventions rather than others under the premise that the final health index A(x1, . . . , B(α, xi ), . . .) satisfies the international or regional healthy criteria. 1.2. Overview on the development of migrativity

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The importance of migrativity at both sides of theory and application promotes its rapid development in the last decades. We provide now an overview on the development of migrativity properties of aggregation functions. We limit ourselves to the case of binary aggregation functions. So far, this property has been mainly generalized in the following two ways: one is to study the α-migrativity of more general aggregation operators such as copulas [45,20], uninorms [43,63–65], overlap functions [57] and aggregation functions [12,27] by releasing the t-norms T in the original definition to be aforementioned general aggregation operators, and the second way is to replace the usual product αx with other aggregation operators B(α, x) such as t-norms [25,68], t-operators [64], uninorms [63, 65], overlap functions [57,55] and aggregation functions [9]. These two ways to generalization of α-migrativity are usually considered simultaneously for a pair of aggregation functions. So the most general setting for α-migrativity of aggregation functions in the literature is the functional equation A(B(α, x), y) = A(x, C(α, y)) for aggregation functions A, B and C, which establishes also a link between Durante-Sarkoci’s migrativity and a particular case of Aczel’s associativity equation [9], and in this case, A is called (α, B, C)-migrative. The well-studied setting of (α, B, C)-migrativity in literature is the special case where B = C, and then the (α, B)-migrativity of A is also called that A is α-migrative over B. Apart from the above mainstream on α-migrativity, by exchanging the order of A and B at one side of the migrativity equation we then get the notion of cross-migrativity [24,39,71,70], where cross-migrative

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t-norms and uninorms were characterized, respectively. Note further that two other generalizations of migrativity by imposing reductions on both arguments of a binary aggregation function has also been studied in [50,40]. Next, we briefly review main results on α-migrativity of aggregation functions existing in the literature.

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(1) Fodor and Rudas [26], and Ouyang et al. [52] independently characterized all continuous t-norms that are α-migrative, and the results show that only strict t-norms have this property. Fodor and Rudas [25] generalized the α-migrativity of t-norms by replacing the product t-norm TP with a fixed arbitrary t-norm T0 and characterized the continuous α-migrative t-norms over the minimum t-norm TM and the Łukasiewicz t-norm TL , respectively. They then extended in [28] the characterizations of continuous α-migrative t-norms over an arbitrary continuous t-norm as an ordinal sum. Wu and Ouyang [68] characterized the α-migrativity of t-subnorms over the respective prototype t-norms TM , TP and TL . (2) Mas et al. [43] introduced the notion of α-migrativity of uninorms over a fixed uninorm U0 with the same neutral element e ∈ [0, 1] and characterized (α, U0 )-migrativity for all usual classes of uninorms including uninorms in Uemin and Uemax , idempotent uninorms, representable uninorms and uninorms continuous in the open unit square. Mas et al. [44] studied the α-migrativity of uninorms and nullnorms over t-norms and t-conorms, respectively, and characterized all solutions of the migrativity equation for all possible combinations of U and T and for all possible combinations of U and S. As an addendum to [44], Zong et al. [74] investigated the α-migrativity of t-norms and t-conorms over uninorms and nullnorms. Su et al. [64] considered the mutual α-migrativity of uninorms and semi t-operators over each other and characterized all solutions of the migrativity equations for all possible combinations of semi t-operators and uninorms. Su et al. [63,65,62] characterized the migrativity equation for two uninorms with different neutral elements. Zong et al. [73] characterized the migrative functional equations for all possible combinations of nullnorms. (3) Bustince et al. [12,9] studied the α-migrativity of aggregation functions. The main results of [12] show that migrative aggregation functions are distortions of the algebraic product, and the k-Lipschitz migrative aggregation functions are also characterized. Some partial structures of (α, B)-migrative aggregation functions as well as influences of neutral elements and absorbing elements on migrativity were investigated in [9]. (4) Qiao and Hu [57] studied the α-migrativity for the usual classes of uninorms as well as nullnorms over overlap functions and characterized the (α, O)-migrativity equation for a large family of uninorms. Qiao and Hu [55] continued to study the (α, O)-migrativity equation of the most general form for overlap functions and obtained the characterizations for different possible combinations of overlap functions. 1.3. Motivation of the present paper The motivation of the present paper comes from the following three aspects of consideration.

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(1) As discussed above, the study on migrative functional equations is a research topic with important theoretical significance and application value. That is why it has been the subject of investigation in recent years, and hence it deserves our further pursuit. (2) Although many interesting results, as listed in Subsection 1.2, have been obtained, the investigation on migrativity for different combinations of some specific aggregation functions is far being complete. The study on the migrativity of uninorms over overlap functions was rather complete [57], however, there are few results on the migrativity properties of overlap functions over uninorms. Notice that the (α, U )-migrativity of an overlap function O is vastly different from the (α, O)-migrativity of a uninorm U even for some trivial α. For example, every uninorm U is (1, O)-migrative for all overlap functions O with the neutral element 1 [57, Corollary 3.1], but none of overlap functions are (1, U )-migrative for any proper uninorm U (see Remark 3.12 in this paper). Hence, the migrative functional equation for each pair of aggregation functions, in particular, the equation of an overlap function over a uninorm deserves a detailed analysis. (3) As stated in Subsection 1.1 (3), the migrativity can find interesting applications in image processing and decision making. For instance, it is sometimes of interest to darken a certain part of an image, and in decision-making, the information on invariance of aggregating effect under a reduction interchangeability between arguments of the aggregation function is very important for decision makers, even though the ordering of inputs may be relevant. And on the other hand, overlap functions provide a mathematical model for some practical problems in image

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processing, decision making and classification [10,8,13]. Then, the study of migrativity properties of overlap functions is very necessary, at least in the fields of image processing and decision making. In addition, the special interest in overlap functions lies not only in their applications in decision making and image processing but also in their close relations to t-norms where the associativity is not strongly required yet. Therefore, studies on migrativity properties of overlap functions over other aggregation functions are more amazing research topics.

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Based on aforementioned consideration, the present paper continues the studies in [57] and considers instead the other direction, i.e., the migrativity properties of overlap functions over uninorms which have not been studied in details in the literature. The main results of the paper are characterizations of solutions to the migrative functional equation for several possible combinations of overlap functions over t-norms, nullnorms and uninorms in the usual rep classes Uemin , Uemax , Ue , Ueid and Ucos , respectively. The remaining of the paper is arranged as follows: Section 2 recalls the basic notions that are necessary to understand the paper, including the concepts of t-norms, t-conorms, uninorms, nullnorms, overlap functions and their related properties. In Section 3 the notion and basic properties of α-migrativity of overlap functions over uninorms in the usual classes are studied. Section 4 concludes the paper with our final remarks, and outlines some further works. 2. Preliminaries

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In this section, we recall some fundamental definitions and notations which are necessary for the reader to follow the paper. A function A : [0, 1]2 −→ [0, 1] is called a binary aggregation function [36,6,31,58] if A is non-decreasing in each variable, and A(0, 0) = 0 and A(1, 1) = 1. The following are several well-studied classes of binary aggregation functions.

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Definition 2.1. ([36]) A binary aggregation function T associative, and has 1 as the neutral element.

−→ [0, 1] is called a t-norm if it is commutative and

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as a real function of two variables if and only if it is continuous in each variable. A t-norm is continuous on So far, a full structural characterization of continuous t-norms has been obtained, which states that each continuous t-norm can be uniquely written as an ordinal sum of continuous Archimedean t-norms, while each continuous Archimedean t-norm is isomorphic to either the product t-norm TP or the Łukasiewicz t-norm TL whose expression will be given below.

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Proposition 2.2. ([36]) A t-norm T is continuous if and only if T is uniquely representable as an ordinal sum of continuous Archimedean t-norms {Tγ }γ ∈A , more precisely, there exist a uniquely determined finite or countably infinite index set A, a family of uniquely determined continuous Archimedean t-norms {Tγ }γ ∈A and a family of uniquely determined non-empty pairwise disjoint open subintervals {(aγ , eγ )}γ ∈A of [0, 1] such that    x−a y−a aγ + (eγ − aγ )Tγ eγ −aγγ , eγ −aγγ , (x, y) ∈ [aγ , eγ ]2 , T (x, y) = min{x, y}, otherwise.

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In this case, denote T more precisely by T = (aγ , eγ , Tγ )γ ∈A .

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Example 2.3. ([36]) The following are typical examples of t-norms.

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Minimum: TM (x, y) = min{x, y}; Product: TP (x, y) = xy; Łukasiewicz: TL (x, y) = max{x + y − 1, 0}; Drastic product:  0, (x, y) ∈ [0, 1)2 , TD (x, y) = min{x, y}, otherwise.

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A binary aggregation function S : [0, 1]2 −→ [0, 1] is called a t-conorm if it satisfies commutativity, associativity and S(x, 0) = x for all x ∈ [0, 1]. There is a standard duality between t-norms T and t-conorms S: S(x, y) = 1 − T (1 − x, 1 − y)

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which shows that S is a t-conorm if and only if T is a t-norm. Such a duality allows easy transformations between tnorms and t-conorms for their construction methods and algebraic properties. For example, by duality each continuous t-conorm can be uniquely represented as an ordinal sum of continuous Archimedean t-conorms. Next we recall the notion of overlap function introduced by Bustince et al. in [11,10]. Overlap functions arise actually from practical problems for modeling the degree to which an object is simultaneously supported by two different classes in decision making and image processing based on fuzzy preference relations, where the associativity property of separation operators is not strongly required [10,5,19,34]. Definition 2.4. ([11,10]) A binary aggregation function O : [0, 1]2 −→ [0, 1] is called an overlap function if, for all x, y ∈ [0, 1], it satisfies: (O1) (O2) (O3) (O4) (O5)

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O(x, y) = O(y, x); O(x, y) = 0 if and only if xy = 0; O(x, y) = 1 if and only if xy = 1; O is increasing; O is continuous.

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By definition, overlap functions are closely related to t-norms. Indeed, it has been shown that associative overlap functions coincide with continuous and positive t-norms [10]. And on the other hand, the class of overlap functions is closed under the convex combinations while the latter is not. Overlap functions’ construction approaches [5,6,13, 19,30,34], analytical properties such as migrativity, homogeneity, idempotency, Lipschitzianity and distributivity [10, 5,11,19,18,34,56], and practical applications in decision making [23,22] and image processing [10,5,8,34] have been extensively investigated.

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Example 2.5. ([5]) The following are some examples of overlap functions that are not t-norms.

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(i) The OmM is a non-associative overlap function with 1 as the neutral element, where OmM (x, y) = min{x, y} max{x , y }, x, y ∈ [0, 1]. 2

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(ii) The following functions are examples of overlap functions that neither are associative nor have 1 as the neutral element, where p > 0, p = 1: • Op (x, y) = x p y p , • Omp (x, y) = min{x p , y p }, • OMp (x, y) = 1 − max{(1 − x)p , (1 − y)p },  2xy , x + y = 0, • ODB (x, y) = x+y 0, otherwise.

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Analogous to the ordinal sum representations of t-norms, we have the following slightly weaker version for overlap functions: Theorem 2.6. ([18]) Every overlap function is representable as an ordinal sum of overlap functions {Oγ }γ ∈A . More precisely, for each overlap function O, there exist a finite or countably infinite index set A, a family of overlap functions {Oγ }γ ∈A and a family of non-empty pairwise disjoint open subintervals {(aγ , eγ )}γ ∈A of [0, 1] such that    x−a y−a aγ + (eγ − aγ )Oγ eγ −aγγ , eγ −aγγ , (x, y) ∈ [aγ , eγ ]2 , O(x, y) = otherwise, min{fA (x), fA (y)},

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where fA : [0, 1] → [0, 1] is given by    x−a aγ + (eγ − aγ )Oγ eγ −aγγ , 1 , x ∈ [aγ , eγ ], fA (x) = x, otherwise.

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In this case, denote O more precisely by O = (aγ , eγ , Oγ )γ ∈A . Notice that fA (x) = x for all x ∈ [0, 1] whenever all Oγ have 1 as neutral element.

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Another direction on generalizing the notions of t-norm and t-conorm is to allow the neutral element of an aggregation operator to be any element of [0, 1], and the resulting aggregation operators are called uninorms, which were first introduced by Yager and Rybalov in [69].

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Definition 2.7. ([69]) A binary aggregation function U : [0, 1]2 −→ [0, 1] is called a uninorm if it is commutative and associative, and has a neutral element e ∈ [0, 1].

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By Ue we denote the class of all uninorms with the neutral element e ∈ [0, 1]. In [29] Fodor et al. showed that uninorms can be built up from t-norms and t-conorms by using an ordinal sum structure. In the following, we recall only several important classes of uninorms whose structures will be cited in sequential sections. For detailed structural characterizations of uninorms we refer to the recent paper [37] and references therein.

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Proposition 2.8. ([29]) Let e ∈ (0, 1) and U ∈ Ue . Then, the functions x → U (x, 1) and x → U (x, 0) are continuous except perhaps at e, if and only if U is given by one of the following formulas:

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(i) If U (1, 0) = 0, then ⎧

eT x , y , ⎪ ⎪ ⎨ U e e   U (x, y) = e + (1 − e)SU x−e , y−e , 1−e 1−e ⎪ ⎪ ⎩ min{x, y},

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(ii) If U (1, 0) = 1, then ⎧

eT x , y , ⎪ ⎪ ⎨ U e e   U (x, y) = e + (1 − e)SU x−e , y−e , 1−e 1−e ⎪ ⎪ ⎩ max{x, y},

(x, y) ∈ [e, 1]2 ,

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otherwise.

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(x, y) ∈ [0, e]2 ,

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otherwise.

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In both formulas, TU is a t-norm and SU is a t-conorm.

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By Uemin we denote the class of uninorms with neutral element e ∈ (0, 1) as in Proposition 2.8 (i) and by Uemax the class of uninorms with neutral element e ∈ (0, 1) as in Proposition 2.8 (ii).

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Definition 2.9. ([29]) A uninorm U : [0, 1]2 → [0, 1] is called representable if there exists a strictly increasing continuous function h : [0, 1] → [−∞, +∞] with h(0) = −∞, h(e) = 0 and h(1) = +∞ such that −1

U (x, y) = h

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(h(x) + h(y))

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for all (x, y) ∈ [0, 1]2 \{(0, 1), (1, 0)} and U (0, 1) = U (1, 0) ∈ {0, 1}.

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The function h is usually called an additive generator of U . By Ue uninorms with neutral element e ∈ (0, 1).

we denote the class of the representable

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Proposition 2.10. ([16]) Let U ∈ Ue . Then U is idempotent if and only if there exists a decreasing function g : [0, 1] → [0, 1], which is symmetric with respect to the main diagonal of [0, 1]2 and satisfies g(e) = e, such that

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⎧ ⎪ y < g(x) or (y = g(x) and x < g(g(x))), ⎨min{x, y}, U (x, y) = max{x, y}, y > g(x) or (y = g(x) and x > g(g(x))), ⎪ ⎩ min{x, y} or max{x, y}, y = g(x) and x = g(g(x)).

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Moreover, U is commutative on the set {(x, y) | y = g(x) and x = g(g(x))}.

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By

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Proposition 2.11. ([32,60]) A uninorm U with neutral element e ∈ (0, 1) is continuous in the following cases is satisfied: (i) There exist u ∈ [0, e), λ ∈ [0, u], two continuous t-norms

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and a representable uninorm R such that

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(x, y) ∈ [0, λ]2 , λTU1 xλ , yλ , ⎪ ⎪ ⎪   ⎪ ⎪ y−λ ⎪ 2 ⎪ λ + (u − λ)TU2 x−λ ⎪ u−λ , u−λ , (x, y) ∈ [λ, u] , ⎪   ⎪ ⎪ ⎨u + (1 − u)R x−u , y−u , (x, y) ∈ (u, 1)2 , 1−u 1−u U (x, y) = ⎪ ⎪ 1, min{x, y} ∈ (λ, 1] and max{x, y} = 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ λ or 1, (x, y) ∈ {(λ, 1), (1, λ)}, ⎪ ⎪ ⎪ ⎩ min{x, y}, otherwise.

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(ii) There exist ν ∈ (e, 1], w ∈ [ν, 1], two continuous t-norms SU1 and SU2 and a representable uninorm R such that ⎧ x y

(x, y) ∈ (0, ν)2 , ⎪ ⎪νR ν , ν , ⎪   ⎪ ⎪ x−ν y−ν ⎪ ⎪ , w−ν , (x, y) ∈ [ν, w]2 , ν + (w − ν)SU1 w−ν ⎪ ⎪   ⎪ ⎪ ⎨w + (1 − w)S 2 x−w , y−w , (x, y) ∈ [w, 1]2 , U 1−w 1−w U (x, y) = ⎪0, ⎪ max{x, y} ∈ [0, w) and min{x, y} = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (x, y) ∈ {(0, w), (w, 0)}, ⎪w or 1, ⎪ ⎪ ⎩ max{x, y}, otherwise. By Ucos we denote the class of all uninorms continuous in (0, 1)2 . A uninorm as in Proposition 2.11 (i) will be denoted by U ≡ TU1 , λ, TU2 , u, (R, e)cos,min and the class of all uninorms of this form will be denoted by Ucos,min . Analogously, a uninorm as in Proposition 2.11 (ii) will be denoted by U ≡ SU1 , ν, SU2 , w, (R, e)cos,max and the class of all uninorms of this form will be denoted by Ucos,max . Nullnorms were introduced by Calvo et al. in [14] as solutions F of the generalized version of Frank functional equation U (x, y) + F (x, y) = x + y in case where U is a uninorm, which turn out to be an equivalent description of t-operators [42]. The formalization of nullnorm is as follows:

45

Definition 2.12. ([14]) A binary aggregation function F : [0, 1]2 −→ [0, 1] is called a nullnorm if it is commutative and associative, and has an absorbing element k ∈ [0, 1], i.e., F (x, k) = k for all x ∈ [0, 1], such that

48

(i) F (x, 0) = x for all x ∈ [0, k]; (ii) F (x, 1) = x for all x ∈ [k, 1].

51 52

13 14 15 16 17 18 19 20 21 22

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

44 45

47 48 49

49 50

12

46

46 47

11

43

43 44

10

23

23 24

9

By Fk we denote the class of all nullnorms with absorbing element k ∈ [0, 1]. It follows from the above items (i) and (ii) that F (0, 1) = k. Obviously, F is a t-norm if k = 0, and a t-conorm if k = 1. For the case of 0 < k < 1, we have the following characterization:

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8

1 2 3 4 5 6 7

Proposition 2.13. ([14,42]) Let F ∈ Fk and F (0, 1) = k ∈ / {0, 1}. Then ⎧ x y

2 ⎪ ⎪ ⎨kSF k , k ,   (x, y) ∈ [0, k] , y−k 2 F (x, y) = k + (1 − k)TF x−k 1−k , 1−k , (x, y) ∈ [k, 1] , ⎪ ⎪ ⎩k, otherwise, where SF is a t-conorm and TF is a t-norm.

12

15 16

19 20 21

2

32

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

52

16

18 19 20 21

23

25

29 30

Proposition 3.3. Let O be an overlap function with neutral element 1, T a t-norm and α ∈ (0, 1). Then O is (α, T )-migrative if and only if T (α, x) = O(α, x) for all x ∈ [0, 1] and O is (α, O)-migrative.

31 32 33

Proof. (⇒) Let O be (α, T )-migrative and x, y ∈ [0, 1]. Then O(T (α, x), y) = O(x, T (α, y)). Letting y = 1 in the preceding equation, we then have T (α, x) = O(x, α) = O(α, x). Moreover, O(O(α, x), y) = O(T (α, x), y) = O(x, T (α, y)) = O(x, O(α, y)). Hence, O is (α, O)-migrative. (⇐) Under the assumption, it follows immediately that, for all x, y ∈ [0, 1], O(T (α, x), y) = O(O(α, x), y) = O(x, O(α, y)) = O(x, T (α, y)) which shows the (α, T )-migrativity of O.

34 35 36 37 38 39 40 41

2

42

To have an insight into structures of (α, T )-migrative overlap functions, we characterize first the idempotent elements of overlap functions with neutral element 1.

43

Lemma 3.4. For each set E with {0, 1} ⊆ E ⊆ [0, 1], E is the set of all idempotent elements of an overlap function O with neutral element 1 if and only if there exist a finite or  countably infinite index set A and a family {(aγ , bγ )}γ ∈A of (aγ , bγ ) = [0, 1] \ E. pairwise disjoint open subintervals of [0, 1] such that

46

γ ∈A

44 45

47 48 49 50

50 51

15

28

For α ∈ (0, 1), we have the following results.

33 34

14

27

30 31

12

26

Proof. Obvious.

28 29

11

24

Proposition 3.2. Let O be an overlap function, T a t-norm and α ∈ {0, 1}. Then O is α-migrative over T .

26 27

9

22

3.1. Migrativity properties of overlap functions over t-norms and t-conorms

24 25

6

17

The migrativity properties of uninorms and nullnorms over overlap functions have been investigated in [57]. In this section, we focus on the inverse direction, i.e., the migrativities of overlap functions over uninorms and nullnorms. As seen from Section 2, the basic structures underlying uninorms and nullnorms are t-norms and t-conorms. To do so, we first study migrativity properties of overlap functions over t-norms and t-conorms, respectively.

22 23

5

13

Definition 3.1. ([9]) Let A, B : [0, 1]2 −→ [0, 1] be two commutative binary aggregation functions and α ∈ [0, 1]. Then A is called α-migrative over B, or equivalently, (α, B)-migrative, if A(B(α, x), y) = A(x, B(α, y)) for all x, y ∈ [0, 1].

17 18

4

10

The notion of α-migrativity has been generalized to the framework of aggregation functions by Bustince et al. in [9] as follows:

13 14

3

8

3. Migrativity properties of overlap functions over uninorms and nullnorms

10 11

2

7

8 9

1

Proof. Let O be an overlap function with neutral element 1 and E the set of all idempotent elements of O. Fix x  ∈ [0, 1] \ E, i.e., O(x  , x  ) < x  , and put

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1 2 3 4

9

ax  = sup{x ∈ [0, x  ) | O(x, x) = x},

1

bx  = inf{x ∈ (x  , 1] | O(x, x) = x}.

2 3

Then we have:

4 5

5 6

Claim. ax  and bx  both are idempotent elements of O.

8 9 10

• ax  is an idempotent element of O because of O(ax  , ax  ) ≥ sup{O(x, x) | x ∈ [0, x  ), O(x, x) = x}

11

= sup{x ∈ [0, x  ) | O(x, x) = x}

12

= ax  .

13 14 15 16 17

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

n→∞

n→∞

Since {x  } ⊆ (ax  , bx  ) ⊆ [0, 1] \ E, we have (ax  , bx  ) ⊆ [0, 1] \ E ⊆ (ax  , bx  ) x  ∈[0,1]\E

38

γ ∈A

disjoint open intervals of [0, 1] and A is a finite or countably infinite index set, then it is not difficult to check that the function O : [0, 1]2 −→ [0, 1], defined by  1 2 2 aγ + (b −a (x, y) ∈ [aγ , bγ ]2 , 2 (min{x − aγ , y − aγ } · max{(x − aγ ) , (y − aγ ) }), γ γ) O(x, y) = min{x, y}, otherwise, is an overlap function with neutral element 1 whose set of idempotent elements coincides with E. 2

41

46

49 50 51 52

20 21 22 23

25 26 27 28 29 30 31 32 33 34 35

37 38

40 41

43 44

Theorem 3.6. Let O be an overlap function with neutral element 1 such that O(x, y) = b implies x = b or y = b for all x, y ∈ [b, 1] whenever b is an idempotent element of O, T a t-norm and α ∈ (0, 1). Then:

45 46 47

47 48

19

42

then there exist two overlap functions O1 and O2 such that O = (0, α, O1 , α, 1, O2 ).

44 45

17

39

(i) O(α, x) = min{α, x} for all x ∈ [0, 1], (ii) O(x, y) = α implies x = α or y = α for all x, y ∈ [α, 1],

42 43

14

36

Lemma 3.5. ([55]) Let O be an overlap function with 1 as its neutral element, and α ∈ (0, 1) a non-trivial idempotent element of O. If the following hold:

39 40

13

24

Finally, from [36, Theorem 2.7], we know that A is a finite or countably infinite subset of [0, 1] \ E satisfying  (aγ , bγ ) = [0, 1] \ E. γ ∈A  Conversely, if E is a subset of [0, 1] satisfying (aγ , bγ ) = [0, 1] \ E where {(aγ , bγ )}γ ∈A is a family of pairwise

36 37

11

18

This shows that bx  is an idempotent element of O.

x  ∈[0,1]\E

10

16

= O( lim bn , lim bn ) = O(bx  , bx  ). n→∞

9

15

bx  = lim bn = lim O(bn , bn ) n→∞

8

12

• If (bn )n∈N is a non-increasing sequence of idempotent elements of O which converges to bx  ∈ (x  , 1], then

18 19

6 7

7

(i) If T (α, α) = α, then O is α-migrative over T if and only if there exist two overlap functions O1 and O2 such that O = (0, α, O1 , α, 1, O2 ), and T (α, x) = min{α, x} for all x ∈ [0, 1]. (ii) If T (α, α) < α and O(b, x) = x for all x ∈ [0, b] whenever b is an idempotent element of O, then O is α-migrative over T if and only if there exist overlap functions O3 , O4 and O5 such that O = (0, β, O3 , β, η, O4 , η, 1, O5 ),

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10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

⎧ ⎪ ⎪ ⎨x,   x−β T (α, x) = β + (η − β)O4 α−β , η−β η−β , ⎪ ⎪ ⎩α, 



x ∈ [0, β),

1

x ∈ [β, η],

(3.1)

3

x ∈ (η, 1],

4

and O4 is α−β η−β , O4 -migrative, where α ∈ (β, η), β = sup{x ∈ [0, α) | O(x, x) = x}, η = inf{x ∈ (α, 1] | O(x, x) = x}. Proof. (i) (⇒) Let T (α, α) = α and O be α-migrative over T , then for all x, y ∈ [0, 1], it follows from Proposition 3.3 that O(α, x) = T (α, x) on [0, 1], in particular, O(α, α) = T (α, α) = α, and hence α is an idempotent element of O.

Indeed, if x ∈ [α, 1], then it follows from the fact that 1 is the neutral element and α is an idempotent element of O that α = O(α, α) ≤ O(α, x) ≤ O(α, 1) = α. Let x ∈ [0, α]. Since O(α, 0) = 0 and O(α, α) = α, the continuity of O implies that there exists y ∈ [0, α] such that O(α, y) = x. It then follows from the (α, T )-migrativity of O and T (α, α) = α that O(α, x) = O(α, O(α, y)) = O(α, T (α, y)) = O(T (α, α), y) = O(α, y) = x. It follows that T (α, x) = O(α, x) = min{α, x} for all x ∈ [0, 1]. Moreover, by the assumption we have that O(x, y) = α implies x = α or y = α for all x, y ∈ [α, 1]. It now follows from Lemma 3.5 that there exist two overlap functions O1 and O2 such that O = (0, α, O1 , α, 1, O2 ). (⇐) The sufficiency follows from a routine verification on the four cases x, y ∈ [0, α], x, y ∈ (α, 1], (x, y) ∈ [0, α] × (α, 1] and (x, y) ∈ (α, 1] × [0, α].

(3.3)

33 34 35 36 37 38 39 40 41 42 43

It follows from Lemma 3.4 that β and η are idempotent elements of O. By assumption, O(β, x) = x for all x ∈ [0, β] and O(η, x) = x for all x ∈ [0, η]. It follows from Lemma 3.5 that there exist overlap functions O3 , O4 and O5 such that O = (0, β, O3 , β, η, O4 , η, 1, O5 ), where α ∈ (β, η). Next, we show the expression of T (α, x) as follows.

45

48 49 50 51 52

14 15 16 17 18 19 20 21 22

26 27 28 29 30 31

34 35 36 37 38 39 40 41

= x.

42

Then T (α, x) = x for all x ∈ [0, β).

43



α−β x−β η−β , η−β



44

.

45 46

46 47

13

33

T (α, x) = O(α, x) = min{fA (α), fA (x)}    

α−β x , 1 , βO3 ,1 = min β + (η − β)O4 η−β β   x = βO3 ,1 β = O(x, β)

Case 2: If x ∈ [β, η], then T (α, x) = O(α, x) = β + (η − β)O4

10

32

Case 1: If x ∈ [0, β), then

44

9

25

η = inf{x ∈ (α, 1] | O(x, x) = x}.

32

8

24

27

31

7

23

(ii) Suppose that T (α, α) < α and O(b, x) = x for all x ∈ [0, b] whenever b is an idempotent element of O. (⇒) Let O be α-migrative over T . Since O(α, α) = T (α, α) < α, then put (3.2)

30

6

12

β = sup{x ∈ [0, α) | O(x, x) = x},

29

5

11

Claim. O(α, x) = min{α, x} for all x ∈ [0, 1].

26

28

2

Case 3: If x ∈ (η, 1], then T (α, x) = O(α, x) = min{fA (α), fA (x)}   α−β ,1 = fA (α) = β + (η − β)O4 η−β = O(α, η) = α.

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11

       α−β Finally, we prove that O4 O4 α−β for all x, y ∈ [0, 1]. Since O is (α, T )-migrative, η−β , x , y = O4 x, O4 η−β , y then, for all x1 , y1 ∈ [β, η], it follows from the equation (3.1) and ordinal sum expression of O that     α − β x1 − β O(T (α, x1 ), y1 ) = O β + (η − β)O4 , , y1 η−β η−β     y1 − β α − β x1 − β = β + (η − β)O4 O4 , , η−β η−β η−β    α − β y1 − β O(x1 , T (α, y1 )) = O x1 , β + (η − β)O4 , η−β η−β    x1 − β α − β y1 − β = β + (η − β)O4 , O4 , η−β η−β η−β          α−β implying O4 O4 α−β for all x, y ∈ [0, 1]. Therefore O4 is α−β η−β , x , y = O4 x, O4 η−β , y η−β , O4 -migrative. (⇐) It is not difficult to check the (α, T )-migrativity of O by a routine verification. 2

39 40

(i) Let α = 12 , T = TM , O1 = OmM and O2 = TP , and let O = (0, α, O1  , α, 1, O2 ): ⎧ 2  ⎪ 4 min{x, y} max{x 2 , y 2 }, (x, y) ∈ 0, 12 , ⎪ ⎨   O(x, y) = 1 + 1 (2x − 1)(2y − 1), (x, y) ∈ 1 , 1 2 , 2 2 2 ⎪ ⎪ ⎩ min{x, y}, otherwise. ( 12 , T )-migrativity.

Then O and T satisfy the condition (i) of Theorem 3.6, and thus O satisfies

     

(ii) Suppose that α ∈ 13 , 34 , and let O = 0, 13 , O3 , 13 , 34 , O4 , 34 , 1, O5 , where O3 = OmM , O4 = TP and O5 = TM , i.e., ⎧  2 ⎪ 9 min{x, y} max{x 2 , y 2 }, (x, y) ∈ 0, 13 , ⎪ ⎨



  O(x, y) = 1 + 12 x − 1 y − 1 , (x, y) ∈ 1 , 3 2 , 3 5 3 3 3 4 ⎪ ⎪ ⎩ min{x, y}, otherwise, ⎧ 1 ⎪ + ⎪ ⎨3

12 5



x − 13 )(y − 13  + y − 1, 34 ,

 T (x, y) = max x ⎪ ⎪ ⎩ min{x, y},



,

1

2 (x, y) ∈ 3 , 34 , 2  (x, y) ∈ 34 , 1 , otherwise.

51 52

9 10 11 12 13 14 15 16 17

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

36 37 38 39 40

42

44 45

Proposition 3.8. Let O be an overlap function and S a t-conorm. Then O is 0-migrative over S.

46 47

Proof. Obvious.

2

48 49

49 50

7

43

Finally, we consider the migrativity of an overlap function O over a t-conorm S.

47 48

6

41

Then O and T satisfy the condition (ii) of Theorem 3.6, and thus O satisfies (α, T )-migrativity.

45 46

5

35

and

43 44

4

19

41 42

3

18

Example 3.7.

37 38

2

8

and

35 36

1

Remark 3.9. Observe that for α ∈ (0, 1], none of overlap functions O satisfy (α, S)-migrativity for any t-conorm S. Suppose O is an arbitrary overlap function and S is a t-conorm, it follows from the definition of S and O that O(S(α, 0), 1) = O(α, 1) = 0 = O(0, 1) = O(0, S(α, 1)). Then the migrativity of O over S is not valid for α ∈ (0, 1].

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1

3.2. Migrativity properties of overlap functions over uninorms

3 4

Proposition 3.10. Let O be an overlap function and U a uninorm with neutral element e ∈ (0, 1). Then O is (0, U )-migrative if and only if U is conjunctive. Proof. An easy verification.

7 8 9 10 11 12 13 14 15 16 17 18 19 20

2

23 24

27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

48 49 50 51 52

7

Proof. For all x, y ∈ [0, 1], O(U (e, x), y) = O(x, y) = O(x, U (e, y)), which shows the e-migrativity of O over U. 2

11

Remark 3.12. None of overlap functions O satisfy 1-migrativity over uninorms U with neutral element e ∈ (0, 1). In fact, let O be an arbitrary overlap function and U a uninorm with neutral element e ∈ (0, 1). Since O(U (1, e), 1) = O(1, 1) = 1 = O(e, 1) = O(e, U (1, 1)), O is not (1, U )-migrative. Next, we turn to study the migrativity of O over U from the usual classes Uemin , Uemax , Ueid , Ucos,min of uninorms for α ∈ (0, e) ∪ (e, 1).

9 10

12 13 14 15 16 17 18 19 20 21

Theorem 3.13. Suppose that O is an overlap function with neutral element 1 satisfying O(x, y) = b implies x = b or y = b for all x, y ∈ [b, 1] whenever b is an idempotent element of O, U ∈ Uemin is a uninorm with neutral element e ∈ (0, 1) and α ∈ (0, e). Then the following statements hold:

22 23 24 25

(i) If U (α, α) = α, then O is α-migrative over U if and only if there exist two overlap functions O1 and O2 such that O = (0, α, O1 , α, 1, O2 ), and U (α, x) = min{α, x} for all x ∈ [0, 1]. (ii) If U (α, α) < α and O(b, x) = x for all x ∈ [0, b] whenever b is an idempotent element of O, then O is α-migrative over U if and only if there exist overlap functions O3 , O4 and O5 such that O = (0, β, O3 , β, η, O4 , η, 1, O5 ), ⎧ ⎪ ⎪ ⎨x,   x ∈ [0, β), α−β x−β U (α, x) = β + (η − β)O4 η−β , η−β , x ∈ [β, min{e, η}] , (3.4) ⎪ ⎪ ⎩α, x ∈ (min{e, η}, 1] , 

α−β η−β , O4

and O4 is O(x, x) = x}.



-migrative, where α ∈ (β, η), β = sup{x ∈ [0, α) | O(x, x) = x}, η = inf{x ∈ (α, 1] |

Proof. (i) (⇒) Let U (α, α) = α and O be α-migrative over U . Note first that U (α, 1) = α since α ∈ (0, e) and U ∈ Uemin . Then for all x ∈ [0, 1], U (α, x) = O(U (α, x), 1) = O(x, U (α, 1)) = O(x, α) = O(α, x). A similar proof based on continuity and migrativity of O as done in the proof of Theorem 3.6 (i) shows that O(α, x) = min{α, x} for all x ∈ [0, 1], and consequently, U (α, x) = min{α, x} for all x ∈ [0, 1]. Then it follows from the assumption and Lemma 3.5 that there exist two overlap functions O1 and O2 such that O = (0, α, O1 , α, 1, O2 ). (⇐) It is routine to check that O is α-migrative over U , as done in the proof of Theorem 3.6 (i).

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

46 47

6

8

25 26

4

Proposition 3.11. Let O be an overlap function and U a uninorm with neutral element e ∈ (0, 1). Then O is (e, U )-migrative.

21 22

3

5

5 6

1 2

2

(ii) Let U (α, α) < α and O(b, x) = x for all x ∈ [0, b] whenever b is an idempotent element of O. (⇒) Suppose that O is α-migrative over U . As in (i) we still have U (α, x) = O(α, x) for all x ∈ [0, 1], in particular, O(α, α) = U (α, α) < α. Let β and η be defined by (3.2) and (3.3), respectively. It follows from Lemma 3.4 that β and η are idempotent elements of O. By assumption we have O(β, x) = min{β, x} and O(η, x) = min{η, x} for all x ∈ [0, 1]. Then, by Lemma 3.5, there exist overlap functions O3 , O4 and O5 such that O = (0, β, O3 , β, η, O4 , η, 1, O5 ), where α ∈ (β, η). We consider the following two cases:

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1 2 3 4 5 6

13

Case 1: η ≤ e. In this case, a similar proof to that Theorem 3.6 (ii) shows that ⎧ x, x ∈ [0, β), ⎪ ⎪ ⎨   x−β U (α, x) = β + (η − β)O4 α−β η−β , η−β , x ∈ [β, η] , ⎪ ⎪ ⎩ α, x ∈ (η, 1] .

1 2 3 4 5 6 7

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Case 2: e < η. Since O(α, β) = β, O(α, η) = α and O is continuous, then, for every x ∈ [β, α], there exists y ∈ [β, η] such that x = O(α, y), and consequently, x = O(α, y) = O(U (α, e), y) = O(e, U (α, y)) = O(e, O(α, y)) = O(e, x). Let (x, y) ∈ [β, α] × [e, η], then x = O(x, e) ≤ O(x, y) ≤ O(x, η) = x, and thus we have O(x, y) = x = min{x, y} on [β, α] × [e, η]. Hence, it follows from the commutativity of O that O(x, y) = min{x, y} for all (x, y) with β ≤ min{x, y} ≤ α and e ≤ max{x, y} ≤ η. The remaining proof is routine to check the form of U (α, x): ⎧ x, x ∈ [0, β), ⎪ ⎪ ⎨   α−β x−β U (α, x) = β + (η − β)O4 η−β , η−β , x ∈ [β, e], ⎪ ⎪ ⎩ α, x ∈ (e, 1]. The fact that O4 is ( α−β η−β , O4 )-migrative follows immediately from the (α, U )-migrativity of O and the analytic expression of U (α, x). (⇐) It is routine to check that O is α-migrative over U with the structures of O and U (α, ·). 2

24 25 26 27 28 29 30 31

Example 3.14.

and

34 35 36

39 40 41 42 43 44 45 46 47 48 49 50 51 52



11 12 13 14 15 16 17 18 19 20 21

24

(i) Let α = 12 , e = 34 , O = 0, 12 , OmM ,  12 , 1, TP : ⎧  2 ⎪ 4 min{x, y} max{x 2 , y 2 }, (x, y) ∈ 0, 12 , ⎪ ⎨



 1 2 O(x, y) = 1 + 2 x − 1 y − 1 , (x, y) ∈ 2 2 2 2,1 , ⎪ ⎪ ⎩ min{x, y}, otherwise,

33

38

10

23



32

37

9

22

22 23

8

25 26 27 28 29 30 31

⎧ 2    ⎪ max x + y − 12 , 0 , (x, y) ∈ 0, 12 , ⎪ ⎨ 2  U (x, y) = max{x, y}, (x, y) ∈ 34 , 1 , ⎪ ⎪ ⎩ min{x, y}, otherwise.

32 33 34 35 36

( 12 , U )-migrativity.

Then O and U satisfy the condition (i) of Theorem 3.13, and thus O satisfies      

(ii) Let α ∈ ( 13 , 23 ), e = 23 , O = 0, 13 , TM , 13 , 23 , TP , 23 , 1, OmM : ⎧

 2 1 ⎪ (x, y) ∈ 13 , 23 , + 3 x − 13 )(y − 13 , ⎪ 3 ⎨ 





 O(x, y) = 2 + 9 x − 2 y − 2 max{x, y} − 2 , (x, y) ∈ 2 , 1 2 , 3 3 3 3 3 ⎪ ⎪ ⎩ min{x, y}, otherwise, and

⎧

1 ⎪ + 3 x − 13 )(y − 13 , ⎪ 3 ⎨ U (x, y) = max{x, y}, ⎪ ⎪ ⎩ min{x, y},

37 38 39 40 41 42 43 44 45

1

2 (x, y) ∈ 3 , 23 , 2  (x, y) ∈ 23 , 1 , otherwise.

Then O and U satisfy the condition (ii) of Theorem 3.13, and thus O satisfies (α, U )-migrativity, where e = η = 2 3.

46 47 48 49 50 51 52

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H. Zhou, X. Yan / Fuzzy Sets and Systems ••• (••••) •••–•••

14

1



, e = 12 , ⎧ 2  ⎪ (x, y) ∈ 0, 14 , 16 min{x, y} max{x 2 , y 2 }, ⎪ ⎪ ⎪



       

⎪ 1 ⎪ (x, y) ∈ 14 , 38 × 14 , 12 ∪ 38 , 12 × 14 , 38 , + 4 x − 14 y − 14 , ⎪ ⎪ 4 ⎨   2  O(x, y) = 14 + 12 max x − 14 , y − 14 , (x, y) ∈ 38 , 12 , ⎪  ⎪  1 3 1 3 3 1   ⎪ ⎪ 3 1 1 3 2 ⎪ × ∪ × , , , , ] ∪ [ , , A(x, y), (x, y) ∈ ⎪ 8 2 2 4 2 4 8 2 2 4 ⎪ ⎪ ⎩ min{x, y}, otherwise,



where A(x, y) = 38 + 4 min{x, y} − 38 max{x, y} − 12 , and ⎧



 2 1 ⎪ + 4 x − 14 y − 14 , (x, y) ∈ 14 , 12 , ⎪ ⎨4 2  U (x, y) = max{x, y}, (x, y) ∈ 12 , 1 , ⎪ ⎪ ⎩ min{x, y}, otherwise.

(iii) Let α ∈

3 4, 8

Then O and U satisfy the condition (ii) of Theorem 3.13, and thus O satisfies (α, U )-migrativity, where e = η = 34 .

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2

<

50

19 20

(i) O(x, y) = O(α, α) for all (x, y) ∈ [e, α]2 ; (ii) O(x, y) = O(min{x, y}, α) for all 0 ≤ min{x, y} ≤ e ≤ max{x, y} ≤ α; (iii) O(x, y) = O(α, max{x, y}) for all e ≤ min{x, y} ≤ α ≤ max{x, y} ≤ 1.

24

Proof. Under assumptions, we prove first that U (α, x) = max{α, x} for all x ∈ [e, 1]. Since U is continuous on [e, 1]2 , U (α, α) = α and U (α, 1) = 1, then for every x ∈ [α, 1], there exists y ∈ [α, 1] such that x = U (α, y), and consequently we have x = U (α, y) = U (U (α, α), y) = U (α, U (α, y)) = U (α, x). For x ∈ [e, α], α = U (α, e) ≤ U (α, x) ≤ U (α, α) = α implies U (α, x) = α. Thus, we conclude that U (α, x) = max{α, x} for all x ∈ [e, 1]. Let further O be (α, U )-migrative, then for all x, y ∈ [0, 1], O(U (α, x), y) = O(x, U (α, y)). Letting x = e and y = α, we have O(α, α) = O(U (α, e), α) = O(e, U (α, α)) = O(e, α). Since O is continuous, it follows from the structure of U that

28

O(e, e) = lim O(e, x) = lim O(e, min{α, x}) x→e−

x→e−

= lim O(e, U (α, x)) = lim O(U (α, e), x) x→e−

x→e−

= O(α, e). So, for all (x, y) ∈ [e, α]2 , we have O(e, e) ≤ O(x, y) ≤ O(α, α), which implies O(x, y) = O(α, α). If x ∈ [0, e) and y ∈ [e, α], we have O(x, y) = O(min{α, x}, y) = O(U (α, x), y) = O(x, U (α, y)) = O(x, max{α, y}) = O(x, α). Hence, it follows from item (O1) and (O5) of Definition 2.4 and the proof above that O(x, y) = O(α, y) for all (x, y) ∈ [e, α] × [0, e]. By the commutativity of O, one gets item (ii). If x ∈ [e, α] and y ∈ [α, 1], we have O(x, y)=O(x, max{α, y})=O(x, U (α, y))=O(U (α, x), y) = O(max{α, x}, y) = O(α, y). It follows from the commutativity of O that the item (iii) is true. Conversely, we prove the (α, U )-migrativity of O by considering the following cases. Let x, y ∈ [0, 1].

21 22 23

25 26 27

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Case 1: If x, y ∈ [0, e), then it follows from the structure of U that O(U (α, x), y) = O(min{α, x}, y) = O(x, y) = O(x, min{α, y}) = O(x, U (α, y)).

49 50 51

51 52

18

Theorem 3.15. Let O be an overlap function and U ∈ Uemin a uninorm with the underlying continuous t-conorm SU . If there exists α ∈ (e, 1) such that U (α, α) = α, then O is (α, U )-migrative if and only if the following statements hold:

48 49

17

Case 2: If x, y ∈ [e, α), then it follows from item (i) that O(U (α, x), y) = O(α, y) = O(x, α) = O(x, U (α, y)).

52

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H. Zhou, X. Yan / Fuzzy Sets and Systems ••• (••••) •••–•••

1

15

Case 3: If x, y ∈ [α, 1], then O(U (α, x), y) = O(x, y) = O(x, U (α, y)).

3 4 5

Case 4: If (x, y) ∈ [0, e) × [e, α), then it follows from item (ii) that O(U (α, x), y) = O(x, y) = O(x, α) = O(x, U (α, y)). Similarly, if (x, y) ∈ [e, α) × [0, e), then O(U (α, x), y) = O(x, U (α, y)).

8

Case 5: If (x, y) ∈ [0, e) × [α, 1], then O(U (α, x), y) = O(x, y) = O(x, U (α, y)). And, if (x, y) ∈ [α, 1] × [0, e), then O(U (α, x), y) = O(x, U (α, y)) can be shown in a similar way as in Case 5.

11 12 13

Case 6: If (x, y) ∈ [e, α) × [α, 1], then it follows from item (iii) that O(U (α, x), y) = O(α, y) = O(x, y) = O(x, U (α, y)). Further, if (x, y) ∈ [α, 1] × [e, α), then a proof similar to that of Case 6 shows O(U (α, x), y) = O(x, U (α, y)). Therefore O satisfies (α, U )-migrativity. 2

16 17 18 19 20 21 22

∈ ( 12 , 23 ]

Example 3.16. Suppose α and e = and consider the overlap function ⎧ 2 1 ⎪ min{x, y}, (x, y) ∈ [0, 1]2 \ 12 , 1 , ⎪ ⎨2 2  O(x, y) = B(x, y), (x, y) ∈ 23 , 1 , ⎪ ⎪ ⎩1 otherwise, 4, where B(x, y) =

23 24 25

U (x, y) =

1 4

1 2,

+ 34 min{3x − 2, 3y − 2} max{(3x − 2)2 , (3y − 2)2 }, and the uninorm

  2 max{x, y}, (x, y) ∈ 12 , 1 ,

28 29 30 31 32 33 34 35 36 37 38 39 40

43 44 45 46 47 48 49 50 51 52

8

10 11 12 13

min{x, y},

15 16 17 18 19 20 21 22

24 25

otherwise.

26

Then, from Theorem 3.15, we have that O satisfies (α, U )-migrativity. Theorem 3.17. Let O be an overlap function and U ∈ Uemin a uninorm with the underlying continuous nonArchimedean t-conorm SU . If there exists α ∈ (e, 1) such that U (α, α) > α, then O is (α, U )-migrative if and only if there exist t-conorm SU2 , two continuous t-conorms SU1 and SU3 such that SU =    a continuous Archimedean   1 −e 0, σ1−e , SU1 ,

σ1 −e σ2 −e 2 1−e , 1−e , SU

,

σ2 −e 3 1−e , 1, SU

, where σ1 < α < σ2 < 1 and the following statements hold:

27 28 29 30 31 32 33

(i) O(x, y) = O(α, α) for all (x, y) ∈ [e, σ2 ]2 ; (ii) O(x, y) = O(min{x, y}, α) for all 0 ≤ min{x, y} ≤ e ≤ max{x, y} ≤ σ2 ; (iii) O(x, y) = O(α, max{x, y}) for all e ≤ min{x, y} ≤ σ2 ≤ max{x, y} ≤ 1.

34

α−e α−e Proof. If U (α, α) > α, then SU ( α−e 1−e , 1−e ) > 1−e . Since SU t-conorm SU2 , two continuous t-conorms SU1 and SU3 such that

38



41 42

7

23

26 27

5

14

14 15

4

9

9 10

3

6

6 7

1 2

2

SU =

is continuous, there exist a continuous Archimedean

     σ1 − e 1 σ 1 − e σ2 − e 2 σ2 − e 0, , SU , , , SU , , 1, SU3 , 1−e 1−e 1−e 1−e

where α ∈ (σ1 , σ2 ). Let O be (α, U )-migrative, then for all x, y ∈ [0, 1], O(U (α, x), y) = O(x, U (α, y)).

35 36 37

39 40 41 42 43 44 45 46

• Since O is continuous and SU2 is continuous Archimedean, it follows from the structure of O that

47

O(e, e) = lim O(e, x) = lim O(e, U (α, x))

48

= lim O(U (α, e), x) = lim O(α, x)

50

x→e− x→e−

x→e−

x→e−

= lim O(α, U (α, x)) = lim O(U (α, α), x) x→e−

x→e−

49

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H. Zhou, X. Yan / Fuzzy Sets and Systems ••• (••••) •••–•••

16

    (2) (2) = lim O αU , x = lim O αU , U (α, x) x→e− x→e−       (2) (3) = lim O U α, αU , x = lim O αU , x x→e− x→e−   (n) = · · · = lim O αU , x (∀n ∈ N) x→e−   (n) = lim O(σ2 , x) αU = σ2 from some n ∈ N on

1 2 3 4 5 6 7

x→e−

8

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

2 3 4 5 6 7 8

= O(σ2 , e),

9

1

9

and

10

O(e, σ2 ) = O(e, max{α, σ2 }) = O(e, U (α, σ2 ))

11

= O(U (α, e), σ2 ) = O(α, σ2 ) = O(α, U (α, σ2 ))   (2) = O(U (α, α), σ2 ) = O αU , σ2       (2) (2) = O αU , U (α, σ2 ) = O U αU , α , σ2   (3) = O αU , σ2 = · · ·   (n) = O αU , σ2 (∀n ∈ N)   (n) = σ2 from some n ∈ N on . = O(σ2 , σ2 ) αU ]2 .

The monotonicity implies that O is a constant on [e, σ2 Note here that σ2 < 1. Indeed, if it was the case σ2 = 1, then it follows from the above results that O(e, 1) = O(1, 1) = 1, which implies e = 1 by the item (O3) of Definition 2.4, a contradiction. • If x ∈ [0, e) and y ∈ [e, σ1 ], then we have O(x, y) = O(min{α, x}, y) = O(U (α, x), y) = O(x, max{α, y})  = O(x, α)  U (α,y)) = O(x,  (2) x, αU

(∀n ∈ N)   (n) = O(x, σ2 ) αU = σ2 from some n ∈ N on .

=O

= ··· = O

(n) x, αU

Hence, it follows from item (O1) and (O5) of Definition 2.4 and the fact above that O(x, y) = O(min{x, y}, α) for all (x, y) ∈ [0, 1]2 with 0 ≤ min{x, y} ≤ e ≤ max{x, y} ≤ σ2 . • If x ∈ [e, σ1 ] and y ∈ [σ2 , 1], then we have O(x, y) = O(x, max{α, y}) = O(x, U (α, y)) = O(U (α, x), y) = O(max{α, x}, y) = O(α, y)   (n) = · · · = O αU , y (∀n ∈ N)   (n) = O (σ2 , y) αU = σ2 from some n ∈ N on . Thus, O(x, y) = O(α, max{x, y}) for all (x, y) with e ≤ min{x, y} ≤ σ2 ≤ max{x, y} ≤ 1. The sufficiency follows from a routine verification. 2

Example 3.18. Let α ∈ 12 , 34 and e = 13 , and consider the overlap function ⎧ 2 1 ⎪ min{x, y}, (x, y) ∈ [0, 1]2 \ 13 , 1 , ⎪ 3 ⎨



  O(x, y) = 1 + 128 x − 3 y − 3 , (x, y) ∈ 3 , 1 2 , 9 9 4 4 4 ⎪ ⎪ ⎩1 otherwise, 9,

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

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H. Zhou, X. Yan / Fuzzy Sets and Systems ••• (••••) •••–•••

1 2 3 4 5 6 7

17

and the uninorm

⎧



   min{x, y}, (x, y) ∈ 0, 13 × [0, 1] ∪ 13 , 1 × 0, 13 , ⎪ ⎪ ⎨ 2  U (x, y) = 3x + 3y − 4xy − 32 , (x, y) ∈ 12 , 34 , ⎪ ⎪ ⎩ max{x, y}, otherwise.

Then, from Theorem 3.17, we have that O satisfies (α, U )-migrativity.

10 11 12 13 14 15 16 17 18 19 20 21

Remark 3.19. It follows from σ2 < 1 in Theorem 3.17 that, for a uninorm U ∈ Uemin with a underlying continuous Archimedean t-conorm SU such that U (α, α) > α for some α ∈ (e, 1), none of overlap functions O satisfy the (α, U )-migrativity. Analogously, we can obtain the results on migrativity of overlap functions O over uninorms U ∈ Uemax . Proposition 3.20. Let O be an overlap function and U ∈ Uemax a uninorm with the underlying continuous t-norm TU . If there exists α ∈ (0, e) such that U (α, α) = α, then O is (α, U )-migrative if and only if the following statements hold: (i) O(x, y) = O(α, α) for all (x, y) ∈ [α, e]2 ; (ii) O(x, y) = O(min{x, y}, α) for all 0 ≤ min{x, y} ≤ α ≤ max{x, y} ≤ e; (iii) O(x, y) = O(α, max{x, y}) for all α ≤ min{x, y} ≤ e ≤ max{x, y} ≤ 1.

22 23 24 25 26 27 28

Proof. The proof is analogous to that of Theorem 3.15.

31 32

Proposition 3.21. Let O be an overlap function and U ∈ Uemax a uninorm with the underlying continuous nonArchimedean t-norm TU . If there exists α ∈ (0, e) such that U (α, α) < α, then O is (α, U )-migrative if and only if there exist a continuous Archimedean t-norm TU2 , two continuous t-norms TU1 and TU3 such that TU =  σ1 1   σ1 σ2 2   σ2 

0, e , TU , e , e , TU , e , 1, TU3 , where 0 < σ1 < α < σ2 , and the following statements hold:

37 38 39 40 41 42 43 44 45

6 7

9 10 11 12 13 14 15 16 17 18 19 20 21

24 25 26 27 28

30 31 32 33

Proof. The proof is analogous to that of Theorem 3.17.

2

34 35

Remark 3.22. Observe that for α ∈ (e, 1), none of overlap functions O satisfy (α, U )-migrativity when U ∈ Uemax . Indeed, suppose O is an arbitrary overlap function and U ∈ Uemax , it follows from the structure of U that O(U (α, 0), 1) = O(α, 1) = O(0, 1) = O(0, U (α, 1)). So, O is not (α, U )-migrative. rep

Remark 3.23. Observe that for U ∈ Ue and α ∈ (0, e) ∪ (e, 1), then none of overlap functions O satisfy the rep (α, U )-migrativity. In fact, let O be an overlap function and U ∈ Ue a representable uninorm with its additive generator h, and suppose that O is (α, U )-migrative. Then O(U (α, x), y) = O(x, U (α, y)) for all x, y ∈ [0, 1]. • If α ∈ (0, e), then h(α) < 0. Take x = 1 and y = α in the preceding migrativity equation, we then have O(1, α) = O(U (α, 1), α)

46 47

= O(1, U (α, α)) = O(1, h−1 (2h(α)))

48

= ··· = O(1, h

49 50

5

29

(i) O(x, y) = O(α, α) for all (x, y) ∈ [σ1 , e]2 ; (ii) O(x, y) = O(min{x, y}, α) for all 0 ≤ min{x, y} ≤ σ1 ≤ max{x, y} ≤ e; (iii) O(x, y) = O(α, max{x, y}) for all σ1 ≤ min{x, y} ≤ e ≤ max{x, y} ≤ 1.

35 36

4

23

33 34

3

22

2

29 30

2

8

8 9

1

36 37 38 39 40 41 42 43 44 45 46 47 48

−1

(nh(α))) (∀n ∈ N).

49

51

It follows from the continuity of O and h that O(1, α) = lim O(1, h−1 (nh(α))) = O(1, 0) = 0, which is a

52

contradiction.

n→∞

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H. Zhou, X. Yan / Fuzzy Sets and Systems ••• (••••) •••–•••

18

1 2

• If α ∈ (e, 1), then a similar argument shows O(1, α) = lim O(1, h−1 (nh(α))) = O(1, 1) = 1, which is also a n→∞ contradiction.

4

Therefore, O is not (α, U )-migrative.

5

5

6

6 7

In the following, we turn to study the migrativity properties of overlap functions O over idempotent uninorms U .

10 11

Theorem 3.24. Let O be an overlap function, U ∈ Ueid an idempotent uninorm with neutral element e ∈ (0, 1) and with the generator function g, and α ∈ (0, e). Then O is (α, U )-migrative if and only if g(α) < 1 and the following statements hold:

14 15

(i) O(x, y) = O(α, α) for all (x, y) ∈ [α, g(α)]2 ; (ii) O(x, y) = O(min{x, y}, α) for all 0 ≤ min{x, y} ≤ α ≤ max{x, y} ≤ g(α); (iii) O(x, y) = O(α, max{x, y}) for all α ≤ min{x, y} ≤ g(α) ≤ max{x, y} ≤ 1.

18

Proof. Since 0 < α < e, we have g(α) ≥ e > α and it then follows that, besides possessing general structure given in Proposition 2.10, U (α, ·) has the following partial structure:

19

22 23 24 25 26 27 28 29

U (α, x) =

34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

14 15

17 18

21 22

To show the necessity, there are three cases to be distinguished:

23 24 25

• α < g(g(α)). Since O is (α, U )-migrative, we have

26

O(α, α) = O(min{α, g(α)}, α) = O(U (α, g(α)), α) = O(g(α), U (α, α)) = O(g(α), α).

27 28 29 30

On the other hand, it follows from the continuity of O that

31 32

32 33

13

20

α, x ∈ (α, g(α)), x, x ∈ [0, α] ∪ (g(α), 1].

30 31

11

19



20 21

10

16

16 17

9

12

12 13

7 8

8 9

2 3

3 4

1

O(g(α), α) = = = =

lim

O(x, α)

lim

O(x, U (α, g(α)))

lim

O(U (α, x), g(α))

lim

O(x, g(α))

x→g(α)+ x→g(α)+ x→g(α)+ x→g(α)+

33 34 35 36 37 38 39 40

= O(g(α), g(α)).

41

[α, g(α)]2 ,

Then, we have O(x, y) = O(α, α) for all (x, y) ∈ which shows the item (i) and also g(α) < 1. If x ∈ [0, α] and y ∈ [α, g(α)], we have O(x, y) = O(U (α, x), y) = O(x, U (α, y)) = O(x, α). And, if x ∈ [α, g(α)] and y ∈ (g(α), 1], then O(x, y) = O(x, U (α, y)) = O(U (α, x), y) = O(α, y). Hence, it follows from the continuity and commutativity of O that the items (ii) and (iii) are true. • α > g(g(α)). Since O is (α, U )-migrative and O is continuous, then O(g(α), α) = =

lim

O(x, α) =

lim

O(U (α, x), α) =

x→g(α)− x→g(α)−

= O(α, α),

lim

x→g(α)−

O(x, U (α, α)) lim

x→g(α)−

O(α, α)

42 43 44 45 46 47 48 49 50 51 52

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H. Zhou, X. Yan / Fuzzy Sets and Systems ••• (••••) •••–•••

1 2 3

19

and

1

O(g(α), g(α)) =

4

=

5

lim

O(g(α), x) =

lim

O(g(α), U (α, x)) =

x→g(α)− x→g(α)−

lim

x→g(α)−

O(U (α, g(α)), x) lim

x→g(α)−

O(g(α), α)

= O(g(α), α).

6

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

=

lim

O(x, y) =

lim

O(x, U (α, y)) =

y→g(α)− y→g(α)−

lim

y→g(α)−

O(U (α, x), y) lim

y→g(α)−

O(x, α)

= O(x, α),

=

lim

O(x, y) =

lim

O(U (α, x), y) =

x→g(α)− x→g(α)−

lim

x→g(α)−

O(x, U (α, y)) lim

x→g(α)−

O(α, y)

= O(α, y), we have O(x, y) = O(α, y) for all x ∈ [α, g(α)] and y ∈ [g(α), 1]. Therefore, in this case, it follows from the item (O1) and (O5) of Definition 2.4 that the items (ii) and (iii) are true. • α = g(g(α)). It follows from Proposition 2.10 that U (α, g(α)) ∈ {α, g(α)}. Then a slight modification to the proofs of Case 1 and Case 2 shows the items (i)-(iii) are still true in the cases U (α, g(α)) = α and U (α, g(α)) = g(α), which is omitted here.

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

8 9 10 11 12 13

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

30 31

5

14

we have O(x, y) = O(x, α) for all x ∈ [0, α] and y ∈ [α, g(α)]. For all y ∈ [g(α), 1], since O(g(α), y) =

4

7

Hence, we have O(x, y) = O(α, α) for all (x, y) ∈ [α, g(α)]2 . For all x ∈ [0, α], since O(x, g(α)) =

3

6

7 8

2

For the sufficiency, we consider the following six cases to check the (α, U )-migrativity of O: 1) If x, y ∈ [0, α], then O(U (α, x), y) = O(x, y) = O(x, U (α, y)). 2) If x, y ∈ (α, g(α)), then it follows from item (i) that O(U (α, x), y) = O(α, y) = O(α, α) = O(x, α) = O(x, U (α, y)). 3) If x, y ∈ [g(α), 1], in this case, we need to distinguish four subcases: 3.1) If x = y = g(α), then it is obvious that O(U (α, x), y) = O(x, U (α, y)). 3.2) If (x, y) ∈ {g(α)} × (g(α), 1], there are three subcases to be considered: 3.2.1) α < g(g(α)). It follows from item (iii) that O(U (α, g(α)), y) = O(α, y) = O(g(α), y) = O(g(α), U (α, y)). 3.2.2) α > g(g(α)). Then O(U (α, g(α)), y) = O(g(α), y) = O(g(α), U (α, y)). 3.2.3) α = g(g(α)). If U (α, g(α)) = min{α, g(α)} = α, then O(U (α, g(α)), y) = O(α, y) = O(g(α), y) = O(g(α), U (α, y)); if U (α, g(α)) = max{α, g(α)} = g(α), then O(U (α, g(α)), y) = O(g(α), y) = O(g(α), U (α, y)). 3.3) If (x, y) ∈ (g(α), 1] × {g(α)}, then O(U (α, x), y) = O(x, U (α, y)) follows from the commutativity of O and 3.2). 3.4) If (x, y) ∈ (g(α), 1]2 , then O(U (α, x), y) = O(x, y) = O(x, U (α, y)). 4) If (x, y) ∈ [0, α] × (α, g(α)), then it follows from item (ii) that O(U (α, x), y) = O(x, y) = O(x, α) = O(x, U (α, y)). Similarly, if (x, y) ∈ (α, g(α)) × [0, α], then O(U (α, x), y) = O(x, U (α, y)). 5) If (x, y) ∈ [0, α] × [g(α), 1], there are two subcases to be considered:

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20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

5.1) If (x, y) ∈ [0, α] × {g(α)}, then

1

O(U (α, x), g(α)) = O(x, g(α)) = O(x, α) = O(x, U (α, g(α))). 5.2) If (x, y) ∈ [0, α] × (g(α), 1], then O(U (α, x), y) = O(x, y) = O(x, U (α, y)). And, if (x, y) ∈ [g(α), 1] × [0, α], then O(U (α, x), y) = O(x, U (α, y)) can be shown in a similar way as the above 5). 6) If (x, y) ∈ (α, g(α)) × [g(α), 1], then there are two subcases to be considered: 6.1) If (x, y) ∈ (α, g(α)) × {g(α)}, then O(U (α, x), g(α)) = O(α, g(α)) = O(x, g(α)) = O(x, α) = O(x, U (α, g(α))). 6.2) If (x, y) ∈ (α, g(α)) × (g(α), 1], then it follows from item (iii) that O(U (α, x), y) = O(α, y) = O(x, y) = O(x, U (α, y)). Further, if (x, y) ∈ [g(α), 1] × (α, g(α)), then O(U (α, x), y) = O(x, U (α, y)) can be shown in a similar way as the above 6).

3 4 5 6 7 8 9 10 11 12 13 14 15

2

Therefore, O satisfies (α, U )-migrativity.

2

16

Example 3.25. Let α ∈ and e = and consider the idempotent uninorm U ∈ Ueid generated by the function g(x) = 1 − x on [0, 1], i.e.,  min{x, y} x + y ≤ 1, U (x, y) = max{x, y} otherwise,

17

and the overlap function ⎧ ⎪ 2 min{x, y}, ⎪ ⎨ O(x, y) = 23 , ⎪ ⎪ ⎩ min{x, y},

23

[ 13 , 12 )

1 2,

18 19 20 21 22



 

  0, 13 × [0, 1] ∪ 13 , 1 × 0, 13 ,   

     (x, y) ∈ 13 , 23 × 13 , 1 ∪ 23 , 1 × 13 , 23 ,

24

otherwise.

28

(x, y) ∈



Then it is easy to check O and U satisfy conditions of Theorem 3.24, and so O satisfies (α, U )-migrativity.

25 26 27

29 30

Proposition 3.26. Let O be an overlap function, U ∈ Ueid an idempotent uninorm with neutral element e ∈ (0, 1) and with the generator function g, and α ∈ (e, 1). Then O is (α, U )-migrative if and only if g(α) > 0 and the following statements hold:

31

(i) O(x, y) = O(α, α) for all (x, y) ∈ [g(α), α]2 ; (ii) O(x, y) = O(min{x, y}, α) for all 0 ≤ min{x, y} ≤ g(α) ≤ max{x, y} ≤ α; (iii) O(x, y) = O(α, max{x, y}) for all g(α) ≤ min{x, y} ≤ α ≤ max{x, y} ≤ 1.

35

Proof. The proof is analogous to that of Theorem 3.24.

2

Now, we study the cases of uninorms continuous in (0, 1)2 . Theorem 3.27. Suppose that U ≡ TU1 , λ, TU2 , u, (R, e)cos,min ∈ Ucos,min is a uninorm with neutral element e ∈ (0, 1), α ∈ (0, λ), and O is an overlap function with neutral element 1 satisfying O(x, y) = b implies x = b or y = b for all x, y ∈ [b, 1] whenever b is an idempotent element of O. Then the following statements hold: (i) If U (α, α) = α, then O is α-migrative over U if and only if there exist two overlap functions O1 and O2 such that O = (0, α, O1 , α, 1, O2 ), and U (α, x) = min{α, x} for all x ∈ [0, 1]. (ii) If U (α, α) < α and O(b, x) = x for all x ∈ [0, b] whenever b is an idempotent element of O, then O is α-migrative 1 1 over  U if and only  if O and TU are ordinal  sums of the form O = (0, β, O3 , β, η, O4 , η, 1, O5 ) and TU = 0, βλ , TU1−1 ,

β η 1−2 λ , λ , TU

,

η 1−3 λ , 1, TU

with TU1−2 being continuous Archimedean, respectively,

32 33 34

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⎧ ⎪ ⎪ ⎨x,   x ∈ [0, β), α−β x−β U (α, x) = β + (η − β)O4 η−β , η−β , x ∈ [β, η ], ⎪ ⎪ ⎩α, x ∈ (η , 1],

1 2 3 4



5 6 7 8

21

and O4

η −β 1−2 η−β TU



α−β x−β η −β , η −β





, y−β η−β = O4



x−β η −β 1−2 η−β , η−β TU

1

(3.5)

3 4



α−β y−β η −β , η −β



for all x, y ∈ [β, η ], where β <

α < η ≤ η, β = sup{x ∈ [0, α) | O(x, x) = x}, η = inf{x ∈ (α, 1] | O(x, x) = x}, η = inf{x ∈ (α, λ] | U (x, x) = x}.

11 12 13 14 15

Proof. (i) (⇒) Let U (α, α) = α and O be α-migrative over U . Then for all x ∈ [0, 1], U (α, x) = O(U (α, x), 1) = O(x, U (α, 1)) = O(x, α) = O(α, x). A similar proof based on continuity and migrativity of O as done in the proof of Theorem 3.6 (i) shows that O(α, x) = min{α, x} for all x ∈ [0, 1], and consequently, U (α, x) = min{α, x} for all x ∈ [0, 1]. Then it follows from the assumption and Lemma 3.5 that there exist two overlap functions O1 and O2 such that O = (0, α, O1 , α, 1, O2 ). (⇐) It is routine to check that O is α-migrative over U , as done in the proof of Theorem 3.6 (i).

18 19 20 21 22 23 24 25 26 27 28 29 30

(ii) Let U (α, α) < α and O(b, x) = x for all x ∈ [0, b] whenever b is an idempotent element of O. (⇒) Suppose that O is α-migrative over U . Since O(α, α) = U (α, α) < α, then let β and η be defined by (3.2) and (3.3), respectively. It follows from Lemma 3.4 that β and η are idempotent elements of O. By assumption we have O(β, x) = min{β, x} and O(η, x) = min{η, x} for all x ∈ [0, 1]. Then, by Lemma 3.5, there exist overlap functions O3 , O4 and O5 such that O = (0, β, O3 , β, η, O4 , η, 1, O 5 ), where α ∈ (β, η).

Next, we study the structure of TU1 . Since U (α, α) = λTU1 αλ , αλ < α, we have TU1 αλ , αλ < αλ . Since TU1 is continthen there exist t-norm TU1−2 and two continuous t-norms TU1−1 and TU1−3 uous on [0, 1]2 ,   a continuousArchimedean   such that

TU1

=

0,

33

42 43 44 45 46 47 48 49 50

8

10 11 12 13 14 15







β = O(α, β ) = O(α, U (α, β ))   (2) = O(U (α, α), β  ) = O αU , β        = O αU(2) , U (α, β  ) = O U α, αU(2) , β      (3) (n) = O αU , β  = · · · = O αU , β  (∀n ∈ N) 



= O(β , β )

(n) (αU

17 18 19 20 21 22 23 24 25 26

28

which contradicts U (α, β) = O(α, β) = β. • If β  > β, then



= β from some n ∈ N on).

By construction, O4 only has trivial idempotent elements, and so it is impossible that O(β , β  ) = β  .

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51

51 52

7

27

 α β , λ λ     β η − β  1−2 α − β  β − β  =λ , + TU λ λ η − β  η − β  < β,

37

41

, where α

∈ (β  , η ).



36

40

,

η 1−3 λ , 1, TU

U (α, β) = λTU1

35

39

,

β  η 1−2 λ , λ , TU

• If β  < β, then since TU1−2 is Archimedean, we have

34

38

β 1−1 λ , TU

Claim. β  = β and η ≤ η.

31 32

6

16

16 17

5

9

9 10

2

Therefore β = β  .

52

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22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Next, we prove η ≤ η by absurdum. Suppose that η > η. Since TU1−2 is Archimedean, α η U (α, η) = λTU1 ,  λ λ   β η − β 1−2 α − β η − β =λ + TU , λ λ η − β η − β < α, η

which contradicts U (α, η) = O(α, η) = α. Therefore ≤ η. And, a similar proof based on continuity and migrativity of O as done in the proof of e < η in Theorem 3.6 (ii) shows the equation (3.5). Finally, for all x, y ∈ [β, η ], it follows from the equation (3.5) and ordinal sum expression of O that     α−β x −β O(U (α, x), y) = O β + (η − β)TU1−2 , ,y η − β η − β      y−β η − β 1−2 α − β x − β = β + (η − β)O4 TU , , η−β η − β η − β η−β and

30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Example 3.28.

27

28 29

2

   α−β y −β O(x, U (α, y)) = O x, β + (η − β)TU1−2 , η − β η − β    x − β η − β 1−2 α − β y − β = β + (η − β)O4 , TU , η−β η−β η − β η − β         y−β −β 1−2 α−β x−β x−β η −β 1−2 α−β y−β which imply that O4 ηη−β TU for all x, y ∈ [β, η ]. η −β , η −β , η−β = O4 η−β , η−β TU η −β , η −β Conversely, it is routine to check that O is α-migrative over U . 2

26 27

1

(i) Let α =



 1 

, 4 , 1, OmM :

and λ = and let the overlap function O = 0,   2 C(x, y), (x, y) ∈ 14 , 1 , O(x, y) = min{x, y}, otherwise,



   2 x − 14 y − 14 max x − 14 , y − 14 + 14 for all (x, y) ∈ 14 , 1 and the uninorm U ≡ where C(x, y)= 16 9        

x 0, 14 , TL , 14 , 12 , TP , 12 , TM , 23 , (R, 56 ) with R generated by h(x) = ln 1−x : 1 4

1 2,

1 4 , TM

cos,min

⎧ 2    1 ⎪ (x, y) ∈ 0, 14 , ⎪max x + y − 4 , 0 , ⎪ ⎪



 2 ⎪ ⎪ 1 1 1 ⎪ (x, y) ∈ 14 , 12 , ⎪ ⎨4 + 4 x − 4 y − 4 ,

2 (3x−2)(3y−2) U (x, y) = 2 + 1 , (x, y) ∈ 23 , 1 , 3 3 (3−3x)(3−3y)+(3x−2)(3y−2) ⎪ ⎪   ⎪ ⎪1, ⎪ min{x, y} ∈ 12 , 1 and max{x, y} = 1, ⎪ ⎪ ⎪ ⎩ min{x, y}, otherwise. Then it is easy to verify that O and U satisfy conditions of Theorem 3.27 (i), and hence O satisfies ( 14 , U )-migrativity.

     

(ii) Let α ∈ 14 , 34 , and consider the overlap function O = 0, 14 , OmM , 14 , 34 , TP , 14 , 1, TM : ⎧ 2  ⎪ 16xy max{x, y}, (x, y) ∈ 0, 14 , ⎪ ⎨



  O(x, y) = 1 + 1 2x − 1 2y − 1 , (x, y) ∈ 1 , 3 2 , 4 2 2 2 4 4 ⎪ ⎪ ⎩ min{x, y}, otherwise,

19 20 21 22 23 24 25 26

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H. Zhou, X. Yan / Fuzzy Sets and Systems ••• (••••) •••–•••

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

23

and the uninorm

1

⎧ 1 1 1 1 ⎪ ⎪ 4 + 2 (2x − 2 )(2y − 2 ), ⎪ ⎪ ⎪ ⎪ ⎨5 1 (6x−5)(6y−5) + , U (x, y) = 6 6 (6−6x)(6−6y)+(6x−5)(6y−5) ⎪ ⎪ 1, ⎪ ⎪ ⎪ ⎪ ⎩ min{x, y},

 3 2 4, 4 ,  2 (x, y) ∈ 56 , 1 ,   min{x, y} ∈ 45 , 1 and max{x, y} = 1, (x, y) ∈

1

otherwise.

24 25 26 27 28 29 30 31 32 33

Then it is easy to verify that O and U satisfy conditions of Theorem 3.27 (ii) where η = η = 34 , and hence O satisfies (α, U )-migrativity.  5 (iii) Let α ∈ 13 , 12 , and consider the overlap function    ⎧   5 1   1 5  1 1 1 1 5 1 1 ⎪ ∪ 12 , 2 × 3 , 12 , + 6(x − )(y − ), (x, y) ∈ , , × ⎪ 3 3 3 3 12 3 2 ⎪ ⎪ ⎪   2 ⎪ ⎪1 1 5 1 ⎪ + 2 max{x − 13 , y − 13 }, (x, y) ∈ 12 ,2 , ⎪ 3 ⎪ ⎨      1 3   5 3  5 1 1 3 O(x, y) = D(x, y), (x, y) ∈ , , × , 12 2 2 4 ∪ 2 , 4 × 12 , 4 ⎪ ⎪ ⎪   ⎪ 2 ⎪ ⎪ E(x, y), (x, y) ∈ 34 , 1 , ⎪ ⎪ ⎪ ⎪ ⎩ min{x, y}, otherwise,

36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

4 5 6 7

9 10 11 12 13 14 15 16 17 18 19 20 21 22

5 5 5 1 1 3 3 3 12 +4 min{x − 12 , y − 12 } max{x − 2 , y − 2 } and E(x, y) = 4 +16 min{x − 4 , y − 4 } max{(x −

where D(x, y) = 3 2 3 2 4 ) , (y − 4 ) }, and the uninorm

23 24 25

⎧ 1 1 1 ⎪ ⎪ 3 + 6(x − 3 )(y − 3 ), ⎪ ⎪ ⎪ ⎪ ⎪ F (x, y), ⎪ ⎪ ⎨ U (x, y) = 2 , 3 ⎪ ⎪ ⎪ ⎪ ⎪ 1, ⎪ ⎪ ⎪ ⎪ ⎩ min{x, y},

(x, y) ∈ (x, y) ∈

1

 1 2

4

2

3, 2

5,1

27 28

,

2  , 1, 3 , 4  min{x, y} ∈ 5 , 1 and max{x, y} = 1,

29

otherwise,

33

(x, y) ∈

 2

26

,

3,1

30 31 32

34

34 35

3

8

22 23

2

where F (x, y) =

4 5

(5x−4)(5y−4) 1 5 (5−5x)(5−5y)+(5x−4)(5y−4) . Then it is easy to verify that O and where η = 34 , η = 12 , and hence O satisfies (α, U )-migrativity.

+

of Theorem 3.27 (ii)

U satisfy also conditions

35 36 37

Proposition 3.29. Let U ∈ Ucos,min and O be an overlap function with neutral element 1. If U (λ, ·) is continuous on [0, 1], then O is λ-migrative over U if and only if O(λ, x) = min{λ, x} for all x ∈ [0, 1].

38

Proof. Since U (λ, ·) is continuous, then U (λ, x) = min{λ, x} for all x ∈ [0, 1]. If O is λ-migrative over U , then O(λ, x) = O(U (λ, 1), x) = O(1, U (λ, x)) = U (λ, x) = min{λ, x} for all x ∈ [0, 1]. Conversely, it is not difficult to check the (λ, U )-migrativity of O by a routine verification. 2

41

≡ TU1 , λ, TU2 , u, (R, e)cos,min





Remark 3.30. (i) Let U ≡ TU1 , λ, TU2 , u, (R, e) cos,min ∈ Ucos,min and α ∈ (λ, e) ∪ (e, 1). Then none of overlap func  tions O satisfy (α, U )-migrativity. In fact, let O be an arbitrary overlap function, U ≡ TU1 , λ, TU2 , u, (R, e) cos,min with h as the generator function of R, and α ∈ (λ, e) ∪ (e, 1). Suppose that O is (α, U )-migrative, then for all x, y ∈ [0, 1], O(U (α, x), y) = O(x, U (α, y)). If α ∈ (λ, u], then by the continuity of O, there exists y ∈ (u, 1) such that O(1, y) > O(1, u). Thus, O(U (α, 1), y) = O(1, y) = O(1, α) = O(1, min{α, y}) = O(1, U (α, y)), which shows that O is not (α, U )-migrative.

39 40

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

If α ∈ (e, 1), then h

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

47 48 49 50 51 52



> 0. Take x = 1 and y = α in the migrativity equation, then we have

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

= 1,

24

which is a contradiction.   If α ∈ (u, e), then h α−u 1−u < 0. There are two cases to be considered: (−∞) + (+∞) = −∞ or (−∞) + (+∞) = +∞. If it was the case of (−∞) + (+∞) = −∞, then, for all y ∈ (u, 1), we have O(1, y) = O(U (α, 1), y) = O(1, U (α, y))       y −u α−u +h = O 1, u + (1 − u)h−1 h 1−u 1−u       y −u α − u = O U (α, 1), u + (1 − u)h−1 h +h 1−u 1−u        y−u α − u = O 1, U α, u + (1 − u)h−1 h +h 1−u 1−u       y − u α − u = O 1, u + (1 − u)h−1 2h +h 1−u 1−u = · · ·      y −u α−u −1 = O 1, u + (1 − u)h +h (∀n ∈ N). nh 1−u 1−u

lim O(1, y) y→1− ,n→∞ 

lim

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

43 44











y −u α−u +h 1−u 1−u y→1− ,n→∞       α−u y−u −1 = O 1, u + (1 − u)h + lim h lim nh n→∞ 1−u 1−u y→1−   = O 1, u + (1 − u)h−1 ((−∞) + (+∞))

=

25

42

It follows from the continuity of O and h that 1 = O(1, 1) =

1 2

Then it follows from the continuity of O and h that     α−u −1 O(1, α) = lim O 1, u + (1 − u)h nh n→∞ 1−u     α−u −1 = O 1, u + (1 − u) · lim h nh n→∞ 1−u = O(1, 1)

45 46

α−u 1−u

O(1, α) = O(U (α, 1), α) = O(1, U (α, α))     α−u = O 1, u + (1 − u)h−1 2h 1−u     α−u = O U (α, 1), u + (1 − u)h−1 2h 1−u      α−u = O 1, U α, u + (1 − u)h−1 2h 1−u     α−u = O 1, u + (1 − u)h−1 3h 1−u = · · ·    α−u −1 (∀n ∈ N). = O 1, u + (1 − u)h nh 1−u

24 25



O 1, u + (1 − u)h−1 nh

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  = O 1, u + (1 − u)h−1 (−∞)

1 2

= O(1, u),

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which is a contradiction. If it was the case that (−∞) + (+∞) = +∞, then, for all y ∈ (u, 1), we have

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O(α, y) = O(U (α, e), y) = O(e, U (α, y))       y−u α−u +h . = O e, u + (1 − u)h−1 h 1−u 1−u   Since h is a strictly increasing continuous function and h α−u 1−u < 0, then for each n ∈ N, there exists dn ∈ (e, 1)     n −u = −nh α−u such that h d1−u 1−u . Then it is obvious that {dn }n∈N is an increasing sequence with limit 1. Moreover, an easy computation by induction shows that e = U (α, d1 ), d1 = U (α, d2 ), . . . , dn−1 = U (α, dn ), . . .. Thus, the above equation becomes:

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     y−u α−u −1 O(α, y) = O e, u + (1 − u)h +h h 1−u 1−u       y−u α−u −1 +h h = O U (α, d1 ), u + (1 − u)h 1−u 1−u        y −u α−u +h = O d1 , U α, u + (1 − u)h−1 h 1−u 1−u       y −u α−u −1 +h 2h = O d1 , u + (1 − u)h 1−u 1−u       y −u α−u +h = O U (α, d2 ), u + (1 − u)h−1 2h 1−u 1−u       y −u α−u +h = O d2 , u + (1 − u)h−1 3h 1−u 1−u = ···       y−u α−u = O dn−1 , u + (1 − u)h−1 nh +h ∀n ∈ N. 1−u 1−u

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lim

y→1− ,n→∞

O(α, y)

38













y −u α−u = lim O dn−1 , u + (1 − u)h +h nh 1−u 1−u y→1− ,n→∞       y −u α−u −1 = O lim dn−1 , u + (1 − u)h + h lim lim nh n→∞ n→∞ 1−u y→1− 1 − u   = . . . = O 1, u + (1 − u)h−1 ((−∞) + (+∞))   = O 1, u + (1 − u)h−1 (+∞) = O(1, 1) = 1, −1

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It follows from the continuity of O and h that O(α, 1) =

8

which is also a contradiction. (ii) By duality, for U ≡ SU1 , ν, SU2 , w, (R, e)cos,max ∈ Ucos,max and for α ∈ (0, e) ∪(e, 1), none of overlap functions O satisfy (α, U )-migrativity.

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3.3. Migrativity properties of overlap functions over nullnorms

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Proposition 3.31. Let O be an overlap function and F a nullnorm with absorbing element k ∈ (0, 1). Then O is (0, F )-migrative if and only if

5



6 7 8

O(k, x) =

O(1, x), O(1, k),

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x ∈ [0, k], x ∈ (k, 1].

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Remark 3.33. None of overlap functions O satisfy (α, F )-migrativity for α ∈ (0, 1] and for nullnorms F with absorbing elements k ∈ (0, 1). Suppose that O is an arbitrary overlap function and F a nullnorm with absorbing element k ∈ (0, 1). If α ∈ (0, k], then it follows from the definition of O that O(F (α, 0), 1) = O(α, 1) = 0 = O(0, k) = O(0, F (α, 1)); if α ∈ (k, 1), then we have O(F (α, 0), 1) = O(k, 1) = 0 = O(0, α) = O(0, F (α, 1)). Consequently, O is not (α, F )-migrative.

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4. Concluding remarks

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• The migrativity properties between overlap functions and 2-uninorms [3,75] including uni-nullnorms [66,67] with the same or different 2-neutral elements, with emphasis on results that are not just structural block combinations of existing ones. • The cross-migrativity properties between two overlap functions, or between overlap functions and other aggregation functions such as t-norms, uninorms and nullnorms. • The bimigrativity properties for possible combinations of t-norms, uninorms, nullnorms and overlap functions. • Some migrativity-like or other analytic properties for the new theory of betweenness relation based aggregation functions proposed in [54,53].

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In this paper, we studied the migrativity properties of overlap functions over uninorms, and characterized in detail some migrativity equations of overlap functions over t-norms, nullnorms and uninorms from the usual classes rep Uemin , Uemax , Ue , Ueid and Ucos , respectively. Plenty of supporting examples were constructed for positive solutions and illustrating remarks were marked for negative results. Note that Lemma 3.5, which ensures the ordinal sum decompositions of overlap functions, plays an important role in some characterizations given in Theorems 3.6, 3.13, and 3.27. Ordinal sum decomposable overlap functions are an important subclass of overlap functions and numerous examples in the paper show also the universality of these characterizations for them although Lemma 3.5 seems very restrictive from the theoretic point of view. As future works, we will continue to study:

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7

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Corollary 3.32. Let O be an overlap function with neutral element 1 and F a nullnorm with absorbing element k ∈ (0, 1). Then O is (0, F )-migrative if and only if O(k, x) = min{k, x} for all x ∈ [0, 1].

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6

13

• If x, y ∈ [0, k], then O(F (0, x), y) = O(x, y) = O(x, F (0, y)). • If x, y ∈ (k, 1], then O(F (0, x), y) = O(k, y) = O(1, k) = O(x, k) = O(x, F (0, y)). • If x ∈ [0, k] and y ∈ (k, 1], then O(F (0, x), y) = O(x, y). Since O(1, x) = O(x, k) ≤ O(x, y) ≤ O(x, 1), then O(x, y) = O(x, 1). On the other hand, O(x, F (0, y)) = O(x, k) = O(1, x). Thus, O(F (0, x), y) = O(x, F (0, y)). • If x ∈ (k, 1] and y ∈ [0, k], then the proof is analogous to that of the third case. 2

20 21

4

9

Proof. Let O be (0, F )-migrative. If x ∈ [0, k], then it follows from Definition 2.12 that O(k, x) = O(F (0, 1), x) = O(1, F (0, x)) = O(1, x); if x ∈ (k, 1], then O(k, x) = O(F (0, 1), x) = O(1, F (0, x)) = O(1, k) by Proposition 2.13. Conversely, we consider the following four cases.

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3

5

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1 2

2

Final remarks: During this works was under review, some results on migrativity of overlap functions over uninorms in the cases of α ∈ {0, e, 1} (see Propositions 3.2, 3.3, 3.8, 3.10 and 3.11) have been obtained in [72].

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Acknowledgements

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The authors want to thank the Area Editor R. Mesiar and the anonymous referees for their helpful comments that substantially improved the paper. References

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[1] J. Aczél, Lectures on Functional Equations and Their Applications, Academic Press, New York, 1966. [2] J. Aczél, V.D. Belousov, M. Hosszú, Generalized associativity and bisymmetry on quasigroups, Acta Math. Acad. Sci. Hung. 11 (1960) 127–136. [3] P. Akella, Structure of n-uninorm, Fuzzy Sets Syst. 158 (2007) 1631–1651. [4] C. Alsina, M.J. Frank, B. Schweizer, Associative Functions: Triangular Norms and Copulas, World Scientific Publishing Co. Pte. Ltd., New Jersey, 2006. [5] B. Bedregal, G.P. Dimuro, H. Bustince, E. Barrenechea, New results on overlap and grouping functions, Inf. Sci. 249 (2013) 148–170. [6] G. Beliakov, A. Pradera, T. Calvo, Aggregation Functions: A Guide for Practitioners, Springer, Heidelberg, 2007. [7] M. Budincˇcevi´c, M.S. Kurili´c, A family of strict and discontinuous triangular norms, Fuzzy Sets Syst. 95 (1998) 381–384. [8] H. Bustince, E. Barrenechea, M. Pagola, Image thresholding using restricted equivalence functions and maximizing the measures of similarity, Fuzzy Sets Syst. 158 (2007) 496–516. [9] H. Bustince, B. De Baets, J. Fernandez, R. Mesiar, J. Montero, A generalization of the migrativity property of aggregation functions, Inf. Sci. 191 (2012) 76–85. [10] H. Bustince, J. Fernandez, R. Mesiar, J. Montero, R. Orduna, Overlap functions, Nonlinear Anal. 72 (2010) 1488–1499. [11] H. Bustince, J. Fernandez, R. Mesiar, J. Montero, R. Orduna, Overlap index, overlap functions and migrativity, in: Proc. IFSA-EUSFLAT Conf., 2009, pp. 300–305. [12] H. Bustince, J. Montero, R. Mesiar, Migrativity of aggregation functions, Fuzzy Sets Syst. 160 (2009) 766–777. [13] H. Bustince, M. Pagola, R. Mesiar, E. Hüllermeier, F. Herrera, Grouping, overlap, and generalized bientropic functions for fuzzy modeling of pairwise comparisons, IEEE Trans. Fuzzy Syst. 20 (3) (2012) 405–415. [14] T. Calvo, B. De Baets, J. Fodor, The functional equations of Frank and Alsina for uninorms and nullnorms, Fuzzy Sets Syst. 120 (2001) 385–394. [15] V. Cutello, J. Montero, Recursive connective rules, Int. J. Intell. Syst. 14 (1999) 3–20. [16] B. De Baets, Idempotent uninorms, Eur. J. Oper. Res. 118 (1999) 631–642. [17] A. Del Amo, J. Montero, E. Molina, Representation of consistent recursive rules, Eur. J. Oper. Res. 130 (2001) 29–53. [18] G.P. Dimuro, B. Bedregal, Archimedean overlap functions: the ordinal sum and the cancellation, idempotency and limiting properties, Fuzzy Sets Syst. 252 (2014) 39–54. [19] G.P. Dimuro, B. Bedregal, H. Bustince, M.J. Asiain, R. Mesiar, On additive generators of overlap functions, Fuzzy Sets Syst. 287 (2016) 76–96. [20] F. Durante, J. Fernández-Sánchez, J.J. Quesada-Molina, On the α-migrativity of multivariate semi-copulas, Inf. Sci. 187 (2012) 216–223. [21] F. Durante, P. Sarkoci, A note on the convex combinations of triangular norms, Fuzzy Sets Syst. 159 (2008) 77–80. [22] M. Elkano, M. Galar, J. Sanz, H. Bustince, Fuzzy rule-based classification systems for multi-class problems using binary decomposition strategies: on the influence of n-dimensional overlap functions in the fuzzy reasoning method, Inf. Sci. 332 (2016) 94–114. [23] M. Elkano, M. Galar, J. Sanz, A. Fernández, E. Barrenechea, F. Herrera, H. Bustince, Enhancing multiclass classification in FARC-HD fuzzy classifier: on the synergy between n-dimensional overlap functions and decomposition strategies, IEEE Trans. Fuzzy Syst. 23 (5) (2015) 1562–1580. [24] J. Fodor, E.P. Klement, R. Mesiar, Cross-migrative triangular norms, Int. J. Intell. Syst. 27 (2012) 411–428. [25] J. Fodor, I.J. Rudas, An extension of the migrative property for triangular norms, Fuzzy Sets Syst. 168 (2011) 70–80. [26] J. Fodor, I.J. Rudas, On continuous triangular norms that are migrative, Fuzzy Sets Syst. 158 (2007) 1692–1697. [27] J. Fodor, I.J. Rudas, On some classes of aggregation functions that are migrative, in: Proc. IFSA-EUSFLAT2009, Lisbon, Portugal, July 20-24, 2009, pp. 653–656. [28] J. Fodor, I.J. Rudas, Migrative t-norms with respect to continuous ordinal sums, Inf. Sci. 181 (2011) 4860–4866. [29] J. Fodor, R.R. Yager, A. Rybalov, Structure of uninorms, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 5 (1997) 411–427. [30] D. Gómez, J.T. Rodríguez, J. Montero, H. Bustince, E. Barrenechea, n-Dimensional overlap functions, Fuzzy Sets Syst. 287 (2016) 57–75. [31] M. Grabisch, J.L. Marichal, R. Mesiar, E. Pap, Aggregation Functions, Cambridge University Press, Cambridge, 2009. [32] S.K. Hu, Z.F. Li, The structure of continuous uninorms, Fuzzy Sets Syst. 124 (2001) 43–52. [33] S. Jenei, On the convex combination of left-continuous t-norms, Aequ. Math. 72 (2006) 47–59. [34] A. Jurio, H. Bustince, M. Pagola, A. Pradera, R. Yager, Some properties of overlap and grouping functions and their application to image thresholding, Fuzzy Sets Syst. 229 (2013) 69–90. [35] E.E. Kerre, M. Nachtegael, Fuzzy Techniques in Image Processing, Springer-Verlag, Berlin, 2000. [36] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Publishers, Dordrecht, 2000. [37] G. Li, H.-W. Liu, On properties of uninorms locally internal on the boundary, Fuzzy Sets Syst. 332 (2018) 116–128. [38] G. Li, H.-W. Liu, Some results on the convex combination of uninorms, Fuzzy Sets Syst. 372 (2019) 50–61. [39] S. Li, F. Qin, J. Fodor, On the cross-migrativity with respect to continuous t-norms, Int. J. Intell. Syst. 30 (2015) 550–562.

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[40] C. Lopez-Molina, B. De Bates, H. Bustince, E. Induráin, A. Stupˇnanová, R. Mesiar, Bimigrativity of binary aggregation functions, Inf. Sci. 274 (2014) 225–235. [41] G. Maksa, Quasisums and generalized associativity, Aequ. Math. 69 (2005) 6–27. [42] M. Mas, G. Mayor, J. Torrens, t-Operators, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 7 (1) (1999) 31–50. [43] M. Mas, M. Monserrat, D. Ruiz-Aguilera, J. Torrens, An extension of the migrative property for uninorms, Inf. Sci. 246 (2013) 191–198. [44] M. Mas, M. Monserrat, D. Ruiz-Aguilera, J. Torrens, Migrative uninorms and nullnorms over t-norms and t-conorms, Fuzzy Sets Syst. 261 (2015) 20–32. [45] R. Mesiar, H. Bustince, J. Fernandez, On the α-migrativity of semicopulas, quasi-copulas, and copulas, Inf. Sci. 180 (2010) 1967–1976. [46] R. Mesiar, V. Novák, Open problems, Tatra Mt. Math. Publ. 6 (1995) 195–204. [47] R. Mesiar, V. Novák, Open problems from the 2nd international conference on fuzzy sets theory and its applications, Fuzzy Sets Syst. 81 (1996) 185–190. [48] J. Montero, D. Gómez, S. Muñoz, Fuzzy information representation for decision aiding, in: Proc. IPMU Conf., Málaga, Spain, 2008, pp. 1425–1430. [49] J. Montero, V. López, D. Gómez, The role of fuzziness in decision making, in: D. Ruan, et al. (Eds.), Fuzzy Logic: A Spectrum of Applied and Theoretical Issues, Springer, 2007, pp. 337–349. [50] Y. Ouyang, Generalizing the migrativity of continuous t-norms, Fuzzy Sets Syst. 211 (2013) 73–83. [51] Y. Ouyang, J. Fang, Some observations about the convex combination of continuous triangular norms, Nonlinear Anal. 68 (2008) 3382–3387. [52] Y. Ouyang, J. Fang, G. Li, On the convex combination of TD and continuous triangular norms, Inf. Sci. 177 (2007) 2945–2953. [53] R. Pérez-Fernández, B. De Baets, On the role of monometrics in penalty-based data aggregation, IEEE Trans. Fuzzy Syst. 27 (7) (2019) 1456–1468. [54] R. Pérez-Fernández, B. De Baets, The role of betweenness relations, monometrics and penalty functions in data aggregation, in: Proc. the Joint 17th World Congress of the Int. Fuzzy Syst. Assoc. (IFSA) and the 9th Int. Conf. Soft Comput. Intell. Syst. (SCIS), Otsu, Japan, 2017. [55] J. Qiao, B.Q. Hu, On generalized migrativity property for overlap functions, Fuzzy Sets Syst. 357 (2019) 91–116. [56] J. Qiao, B.Q. Hu, On the distributivity laws of fuzzy implication functions over additively generated overlap and grouping functions, IEEE Trans. Fuzzy Syst. 26 (4) (2018) 2421–2433. [57] J. Qiao, B.Q. Hu, On the migrativity of uninorms and nullnorms over overlap and grouping functions, Fuzzy Sets Syst. 346 (2018) 1–54. [58] F. Qin, Aggregation Functions and Its Application, Science Press, Beijing, 2019 (in Chinese). [59] B. Roy, Decision science or decision-aid science, Eur. J. Oper. Res. 66 (1993) 184–203. [60] D. Ruiz-Aguilera, J. Torrens, Distributivity and conditional distributivity of a uninorm and a continuous t-conorm, IEEE Trans. Fuzzy Syst. 14 (2006) 180–190. [61] G. Stevens, M. Mascarenhas, C. Mathers, Global health risks: progress and challenges, Bull. World Health Organ. 87 (2009) 646. [62] Y. Su, H.-W. Liu, J.V. Riera, D. Ruiz-Aguilera, J. Torrens, The migrativity equation for uninorms revisited, Fuzzy Sets Syst. 323 (2017) 56–78. [63] Y. Su, W. Zong, H.-W. Liu, Migrativity property for uninorms, Fuzzy Sets Syst. 287 (2016) 213–226. [64] Y. Su, W. Zong, H.-W. Liu, P. Xue, Migrativity property for uninorms and semi t-operators, Inf. Sci. 325 (2015) 455–465. [65] Y. Su, W. Zong, H.-W. Liu, F. Zhang, On migrativity property for uninorms, Inf. Sci. 300 (2015) 114–123. [66] F. Sun, X.P. Wang, X.B. Qu, Uni-nullnorms and null-uninorms, J. Intell. Fuzzy Syst. 32 (2017) 1969–1981. [67] F. Sun, X.P. Wang, X.B. Qu, Characterization of uni-nullnorms with continuous Archimedean underlying t-norms and t-conorms, Fuzzy Sets Syst. 334 (2018) 24–35. [68] L. Wu, Y. Ouyang, On the migrativity of triangular subnorms, Fuzzy Sets Syst. 226 (2013) 89–98. [69] R.R. Yager, A. Rybalov, Uninorm aggregation operators, Fuzzy Sets Syst. 80 (1996) 111–120. [70] H. Zhan, H.-W. Liu, The cross-migrative property for uninorms, Aequ. Math. 90 (2016) 1219–1239. [71] H. Zhan, H.-W. Liu, Cross-migrative uninorms with different neutral elements, Int. J. Intell. Syst. 32 (2017) 1877–1889. [72] K.Y. Zhu, B.Q. Hu, Addendum to “On the migrativity of uninorms and nullnorms over overlap and grouping functions” [Fuzzy Sets Syst. 346 (2018) 1–54], Fuzzy Sets Syst. (2019), https://doi.org/10.1016/j.fss.2019.05.001, in press. [73] W. Zong, H.-W. Liu, Migrative property for nullnorms, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 22 (5) (2014) 749–759. [74] W. Zong, Y. Su, H.-W. Liu, An addendum to “migrative uninorms and nullnorms over t-norms and t-conorms”, Fuzzy Sets Syst. 299 (2016) 146–150. [75] W. Zong, Y. Su, H.-W. Liu, B. De Baets, On the structure of 2-uninorms, Inf. Sci. 467 (2018) 506–527.

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43

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45

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50

50

51

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