Necessary and sufficient conditions for the equality of interactive and non-interactive extensions of continuous functions

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

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Necessary and sufficient conditions for the equality of interactive and non-interactive extensions of continuous functions

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Lucian Coroianu , Robert Fullér

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a Department of Mathematics and Computer Science, University of Oradea, Romania b Department of Informatics, Széchenyi István University, Gy˝or, Hungary

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c Institute of Applied Mathematics, Óbuda University, Budapest, Hungary

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Received 15 July 2016; received in revised form 24 July 2017; accepted 24 July 2017

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Abstract In this contribution we find the class of n-dimensional joint possibility distributions with the property that the interactive extension principle coincides with the non-interactive extension principle as long as the interactive operations are determined by continuous functions strictly increasing in each argument. This result completes recent studies by the authors, where the particular case of interactive additions and multiplications versus non-interactive additions and multiplications were investigated. In addition, this time we propose results that also cover the cases when we know the fuzzy numbers only from their membership functions. It means that we eliminated the limitations that appear when we cannot pass from membership function representation to parametric representation of fuzzy numbers. As important new applications, we mention the study on the completely correlated fuzzy numbers. Also of note is that we propose two simple methods to extend bidimensional joint possibility distributions to n-dimensional joint possibility distributions. One method is based on an inductive construction while the other one is based on a pairwise construction. © 2017 Elsevier B.V. All rights reserved.

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Obviously, when we discuss about fuzzy arithmetic, the first thing that comes in our mind is Zadeh’s extension principle. On the other hand, starting with the 80’s, researchers tried other ways to obtain operations with fuzzy numbers. In order to avoid repetition, we refer to paper [4], where a detailed discussion on this issue can be found in the Introduction. What is certain, is that two important trends exist in the literature. One is based on the so-called triangular norm-based extension principle and the other one is called extension principle based on joint possibility distributions. Please note that both methods are also referred as interactive extension principles, in the sense that we have an interaction between fuzzy numbers through the triangular norm or the joint possibility distribution. Both methods, actually, generalize the extension principle since by a suitable choice of the triangular norm, respectively

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1. Introduction

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Keywords: Fuzzy number; Joint possibility distribution; Interactive extension principle

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E-mail addresses: [email protected] (L. Coroianu), [email protected] (R. Fullér). http://dx.doi.org/10.1016/j.fss.2017.07.023 0165-0114/© 2017 Elsevier B.V. All rights reserved.

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L. Coroianu, R. Fullér / Fuzzy Sets and Systems ••• (••••) •••–•••

of the joint possibility distribution, we get the standard extension principle. The 80’s and 90’s are the years where triangular norm-based extensions were a front line topic in fuzzy arithmetic (see e.g. [7], [10], [12], [13], [15] and see also [4] for a richer reference list). But this topic is still of interest as one can find contributions in the near past (see e.g. [14], [21], [19]). In 2004, Fullér and Majlender (see [11]) introduced so called joint possibility distributions of fuzzy numbers, which can be viewed as Possibility Theory alternatives of the joint probability distributions. Joint possibility distributions have been applied in statistical type problems (see [1], [2] and other references which can be found e.g. in [4]), multi-period portfolio selection (see [23], [24]), but also in fuzzy arithmetic. More precisely, in [1] an interactive extension principle is proposed based on a joint possibility distributions. Actually, this extension agrees with the triangular norm-based extensions since the later ones can be obtained from suitably chosen joint possibility distributions. Let us recall that an important application can be found in paper [1], where the authors found a joint possibility distribution, for which one fuzzy number is actually the additive inverse of the other, where both fuzzy numbers are not crisp. It is well known that such property does not hold with respect to non-interactive addition. What is common to all papers, is that although definitions are presented in general n-dimension, almost all theoretical results and definitely all applications (at least in fuzzy arithmetic) employ only the case of bidimensional joint possibility distributions. Therefore, this paper addresses a problem in the very general setting of n-dimensional joint possibility distributions. More exactly, we will find necessary and sufficient conditions for the equality between the interactive and non-interactive outputs of the extension principle. We believe it is important to find out the cases when the two extensions coincide. This also makes possible to choose such joint possibility distributions for which the output is really interactive as it does not coincide with the output given by the non-interactive extension principle. We will consider a setting in which operations are generated by a continuous function strictly increasing in each argument. It is important to mention that similarly to the case of extensions based on triangular norms, interactive extensions based on joint possibility distributions satisfy the property that each α-cut of the output is included in the α-cut of the output given by the non-interactive extension principle. Therefore, the equality of the two outputs should be regarded as a limiting behavior. Now, in the case of extensions based on triangular norms, there are important studies where analytical formulae are given for the α-cuts of the outputs (see [10], [20], [22]). In particular, from these results it is quite easy to check when the extensions given by triangular norms and those given by the standard extension principle will give the same output. Actually, from Corollary 5.1 in [22], it is quite easy to prove that the two types of extensions cannot coincide unless the triangular norm is the minimum operator. But of course, extensions generated by the minimum norm are nothing else but non-interactive extensions. Therefore, in the case of triangular norm-based extensions we have a trivial solution for this problem. When we consider extensions based on joint possibility distributions, we do have non-trivial solutions of this problem. Therefore, these types of extensions are more flexible. On the other hand, it would be really important to find analytical formulae for the α-cuts of the outputs obtained from extensions based on joint possibility distributions completing the results obtained in the special case of triangular norm-based extensions. We will address this problem in a future contribution. Now, returning to the present contribution, the idea is to compare the outputs obtained from extensions based on joint possibility distributions with the outputs given by the non-interactive extension principle. The first paper where such problem was addressed is [1]. The authors considered the particular case of addition and the question was under which conditions the interactive and non-interactive sums of two fuzzy numbers coincide. This problem was solved in paper [4], where necessary and sufficient conditions were proposed in order to obtain this equality. Similar results were obtained in case of multiplication (see [5]). Now, obviously addition of reals or multiplication of strictly positive reals, is nothing else but a particular continuous and strictly increasing in each argument binary operation. Therefore, taking into account all the facts discussed just above, a more general approach is when we consider an arbitrary function in n variables which is continuous and strictly increasing in each argument. This is what we will do in this paper, and so the context will be very general as we consider n-dimensional joint possibility distributions and functions in n variables, with the aforementioned properties. As an interesting remark, let us mention that comparison between interactive (based on joint possibility distributions) and non-interactive operations is discussed in paper [17] too, but in that paper the discussion is with respect to so called discrete fuzzy numbers, that is, fuzzy sets having a finite set as support. The paper is organized as follows. Section 2 presents generalities on fuzzy numbers and joint possibility distributions. Section 3 contains the main results, that is, necessary and sufficient conditions for the equality between non-interactive based operations and joint possibility distribution-based interactive operations determined by continuous and strictly increasing in each argument functions. We will study the problem locally, at some point of the domain, but also globally on the whole domain. Analyzing the main results, we arrive at the same conclusion as in the case of

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interactive additions, namely, in order to obtain interactive operations that will give different outputs than the extension principle-based operations, we need a joint possibility distribution which coincides as less as possible with the independent joint possibility distribution on the so called diagonal of the fuzzy numbers. In other words, extension principle-based operations on fuzzy numbers and their interactive extensions will give the same output, if and only if the restriction of the joint possibility distribution on the diagonal of the fuzzy numbers coincide with the minimum operator. Note that this diagonal was introduced in [4] and in this contribution we extend it for an arbitrary finite set of fuzzy numbers. An important novelty with respect to our previous work [4], is that we also cover the cases when fuzzy numbers are given in terms of membership function. These results are important especially when we cannot pass from membership function to parametric representation. In Section 4 we analyzes on concrete examples the theoretical results from Section 3. First, we discuss the case of binary operations. A very important special case is when we deal with so called completely correlated fuzzy numbers. Then, we will extend the study to the case of n-dimensional joint possibility distributions. Two approaches will be proposed. In the first one, we use a simple inductive generalization of some bidimensional joint possibility distribution. The second one, uses a pairwise construction. Obviously both methods can be easily generalized for the n-dimensional setting. The paper ends with conclusions, where the main results are summarized and future research directions are discussed.

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In this section, we present well-known facts about fuzzy numbers and joint possibility distributions. A fuzzy set A is called a fuzzy number, if its membership function (denoted with A too) A : R → [0, 1], satisfies the following requirements:

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(i) (ii) (iii) (iv)

A is upper semicontinuous; there exists at least one x0 ∈ R, such that A(x0 ) = 1 (i.e. A is normal); A(αx + (1 − α)y) ≥ min{A(x), A(y)}, for all α ∈ [0, 1] and x, y ∈ R (i.e. A is fuzzy convex); the set {x ∈ R : A(x) > 0}, is bounded in R.

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If the set {x ∈ R : A(x) = 1} has exactly one element, then most often A is called a unimodal fuzzy number. We denote by F(R) the set of all fuzzy numbers. Another very useful representation of a fuzzy number, uses the so called level sets. More precisely, for some γ ∈ (0, 1], the compact interval

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supp(A) = cl{x ∈ R : A(x) > 0},

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and here cl denotes the closure operator. By denoting supp(A) = A0 , we may call supp(A) as the 0-cut of A. Summarizing, we obtain a parametric representation of A, of the form

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Aγ = [A− (γ ), A+ (γ )], γ ∈ [0, 1],

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and it is well known that for some γ > 0, we have

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A− (γ ) = inf{x ∈ R : A(x) ≥ γ },

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A (γ ) = sup{x ∈ R : A(x) ≥ γ }.

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A(A (γ )) ≥ γ , ∀γ ∈ [0, 1].

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is called the γ -cut of A. For γ = 1, we get the core of A, which we denote core(A), hence core(A) = A1 . This also means that A is unimodal, if and only if core(A) has exactly one element. The support of A is defined as in classical topology on R, hence

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Aγ = {x ∈ R : A(x) ≥ γ },

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Let us recall now some facts which were discussed in [4] too. If we have supp(A) = [a, b] and core(B) = [c, d], then it is easily seen that A(x) = 0 for all x ∈ (−∞, a) ∪ (b, ∞) and 0 < A(x) < 1 = A(c), for all x ∈ (a, c), 0 < A(x) < 1 = A(d), for all x ∈ (d, b). Then, if A is continuous at a, then we have A(a) = 0, otherwise we have A(a) > 0. Similarly, if A is continuous at b, then we have A(b) = 0, otherwise we have A(b) > 0. The restriction of A to the interval [a, c], will be called the left side (branch) of the fuzzy number A and it will be denoted with lA . Similarly, the restriction of A to the interval [d, b], will be called the right side of the fuzzy number A and it will be denoted with rA . Obviously, lA is nondecreasing and upper semicontinuous, while, rA is nonincreasing and upper semicontinuous. Suppose now that lA is strictly increasing and suppose that x ∈ [a, c] is such that A(x) = γ . In this case we have x = A− (γ ). This is immediate from the properties of lA and A− respectively, however, a detailed proof can be found in [4]. Similarly, if rA is strictly decreasing and x ∈ [b, d] is such that A(x) = γ , it follows that x = A+ (γ ). The literature is so rich with important classes of fuzzy numbers. In this paper, we will use only unimodal fuzzy numbers with linear side functions, that is, the well-known class of triangular fuzzy numbers. A triangular fuzzy number A is completely determined by three parameters t1 , t2 , t3 ∈ R, t1 ≤ t2 ≤ t3 , where ⎧ 0 if x < t1 , ⎪ ⎪ ⎪ x−t1 ⎪ ⎪ ⎨ t2 −t1 if t1 < x ≤ t2 , 1 if x = t2 , A(x) = ⎪ ⎪ t −x 3 ⎪ if t2 < x ≤ t3 , ⎪ ⎪ ⎩ t3 −t2 0 if t3 < x.

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It can be proved that the above function is indeed the membership function of a fuzzy number. A very important characterization of the level sets of f (A) is given in [16].

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Theorem 1 (see Proposition 5.1 in [16]). Suppose that A = (A1 , ..., An ), where Ai , i ∈ {1, ..., n}, are fuzzy numbers and consider an arbitrary continuous function f : Rn → R. Then   f (A)γ = f (A1 )γ , ..., (An )γ , γ ∈ [0, 1]. Let us note that in [16], the author considered only the case of bivariate functions. But nowadays it is well known that this result holds in the general setting of the above theorem (see, e.g., [18]). From the above theorem, it is not hard at all to prove the following useful result.

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It is immediate that A− (γ ) = t1 + (t2 − t1 )γ and A+ (γ ) = t3 − (t3 − t2 )γ , for all γ ∈ [0, 1]. Then, supp(T ) = [t1 , t3 ] and core(T ) = {t2 }. We use the notation A = (t1 , t2 , t3 ) and, the whole class of triangular fuzzy numbers will be denoted with F (R). Operations on fuzzy numbers are obtained from the famous extension principle of Zadeh. In this way, operations between real numbers are extended in a natural way to operations between fuzzy numbers. Suppose that Ai , i ∈ {1, ..., n}, are fuzzy numbers and consider an arbitrary function f : Rn → R. To simplify the exposure we will use the notation A = (A1 , ..., An ). We denote with f (A1 , ..., An ) = f (A), the fuzzy number with membership function (obtained by applying the extension principle) given by  sup min{A1 (x1 ), ..., An (xn )}, if y ∈ f (Rn ), −1 (x ,x ,...,x )∈f (y) f (A)(y) = (3) n 1 2 0, otherwise.

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Corollary 2. Suppose that A = (A1 , ..., An ), where Ai , i ∈ {1, ..., n}, are fuzzy numbers and consider a continuous function f : Rn → R, with the property that the restriction of f to supp(A1 ) × ... × supp(An ) is strictly − + increasing in each argument. Then, for any γ ∈ [0, 1], we have f − (A)(γ ) = f (A− 1 (γ ), ..., An (γ )) and f (A)(γ ) = + + f (A1 (γ ), ..., An (γ )). In particular, the addition of two fuzzy numbers A and B, denoted with A + B, is given by (A + B)γ = Aγ + Bγ , γ ∈ [0, 1], and this easily implies that (A + B)

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We continue by recalling the basics about joint possibility distributions, introduced by Fullér and Majlender in [11].

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Definition 3 ([11]). We say that C : Rn → R is a joint possibility distribution of fuzzy numbers Ai , i ∈ {1, ..., n}, if it holds that Ai (xi ) =

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From the above definition, we observe that a joint possibility distribution is the counterpart in Possibility Theory of a joint probability distribution. This time, fuzzy numbers Ai , i ∈ {1, ..., n}, are called the marginal distributions of the joint possibility distribution C (see [11]). We also notice that C(x1 , ..., xn ) ≤ min{Ai (xi ) : i = 1, n}. This also suggests to call C := Cid , the independent distribution of A1 , A2 , ..., An , where Cid (x1 , ..., xn ) = min{A1 (x1 ), ..., An (xn )}, ∀xi ∈ R, i = 1, ..., n.

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It is immediate that in general we have C ≤ Cid , that is,

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C(x1 , ..., xn ) ≤ Cid (x1 , ..., xn ), ∀xi ∈ R, i = 1, ..., n.

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For this reason (see [11]), when C = Cid , fuzzy numbers A1 , A2 , ..., An are said to be interactive with respect to C, and otherwise they are said to be non-interactive. If C is upper semicontinuous on supp(A1 ) × ... × supp(An ), then we may use maximum instead of supremum in (4).

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Proposition 4 ([4], Proposition 3). Let us consider fuzzy numbers Ai , i ∈ {1, ..., n}, and let C denote a joint possibility distribution function of these fuzzy numbers. If C is upper semicontinuous on supp(A1 ) × ... × supp(An ), then Ai (xi ) =

max C(x1 , x2 , ..., xi , ..., xn ), ∀xi ∈ R, i = 1, ..., n.

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xj ∈R,j =i

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Moreover, there exists (x 1 , ..., x i−1 , x i+1 , ..., x n ) ∈ Rn−1 , such that

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Ai (xi ) = C(x 1 , ..., x i−1 , xi , x i+1 , ..., x n ) = min{min Aj (x j ), Ai (xi )}. j =i

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In what follows, we present a generalization of the extension principle, called (see [1]) extension principle for interactive fuzzy numbers.

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Definition 5 ([1], Definition 2.1). Let C be the joint possibility distribution of fuzzy numbers A1 , A2 , ..., An and let f : Rn → R be a continuous function. Then fC (A) ∈ F(R), given by fC (A)(y) =

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C(x1 , ..., xn ), ∀y ∈ Rn .

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Just above, we used exactly the definition in [1], and this means that we make the convention that fC (A)(y) = 0, for all y ∈ R such that f −1 (y) = ∅. Moreover, fC (A) is the membership function of a fuzzy number (see Lemma 2.1 in [1]). If C is upper semicontinuous on supp(A1 ) × ... × supp(An ), then we actually have max

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is the interactive extension of f with respect to C.

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Indeed, supposing that supp(Ai ) = [ai , bi ], first of all, we notice that by the definition of fC (A), we have fC (A) ≤ f (A), because for any (x1 , ..., xn ) ∈ f −1 (y), we have (see (5)) C(x1 , ..., xn ) ≤ min{A1 (x1 ), ..., An (xn )} ≤ f (A)(y), and taking the supremum over all (x1 , ..., xn ) we get fC (A)(y) ≤ f (A)(y). This easily implies that fC (A)(y) = 0 if f −1 (y) ∩ supp(A1 ) × ... × supp(An ) = ∅. On the other hand, if f −1 (y) ∩ supp(A1 ) × ... × supp(An ) = ∅, then

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in formula (6) we easily observe that the supremum is searched in the set f −1 (y) ∩ supp(A1 ) × ... × supp(An ). The continuity of f implies that f −1 (y) is a closed subset of Rn and therefore, f −1 (y) ∩ supp(A1 ) × ... × supp(An ) is a compact subset of Rn as being the intersection between a closed and a compact subset of Rn . Therefore, combining the upper semicontinuity of C with the extreme value theorem for upper semicontinuous functions, which states that on compact sets an upper semicontinuous function achieves its maximum value, we obtain that (7) holds.

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3 = {(x1 , ..., xn ) : xi = A+ i (γ ), γ ∈ [0, 1]}. Here, [A− (1), A+ (1)] is the line which connects the

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points A− (1) and A+ (1) in Rn .

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In general, if A and B both are unimodal fuzzy numbers, then (A, B) = 1 ∪ 3 . In all that follows, for some function f : A → B, I mf denotes the image of A, that is, I mf = {f (x) : x ∈ A} = f (A). We will arrive at our main objectives, by using some useful auxiliary results which we believe are important in general, not only for the purposes of the present contribution.

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Proposition 7. Let A1 , ..., An denote n fuzzy numbers, and consider a continuous function f : Rn → R, with the property that the restriction of f to supp(A1 ) × ... × supp(An ) is strictly increasing in each argument. Then, it holds that ∪ni=1 I m(lAi ) = I m(lf (A) ) and ∪ni=1 I m(rAi ) = I m(rf (A) ).

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1 = {(x1 , ..., xn ) : xi = A− i (γ ), γ ∈ [0, 1]},

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Definition 6. Suppose that A = (A1 , ..., An ), where Ai , i ∈ {1, ..., n}, are fuzzy numbers such that supp(Ai ) = [ai , bi ], core(Ai ) = [ci , di ]. We call the diagonal of (A1 , ..., An ), the subset of Rn , (A) = 1 ∪ 2 ∪ 3 , where

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Suppose we have a sample of n fuzzy numbers, A1 , A2 , ..., An . In the previous section, we called these fuzzy numbers non-interactive, as long as their joint possibility distribution is the independent distribution. But sometimes, we can say that A1 , A2 , ..., An are non-interactive even with respect to a joint possibility distribution which is not the independent distribution. More exactly, if C is a joint possibility distribution of the sample and f : Rn → R, then we say that A1 , A2 , ..., An are non-interactive with respect to the pair (C, f ) if fC (A) = f (A), where A = (A1 , ..., An ). Therefore, in this case we have the same output for both types of extension principle. This problem was addressed as an open question in [1], in the case when we deal with addition, and it was solved in a very general setting in paper [4]. Then, the same problem was investigated with respect to multiplication in paper [5]. Obviously, addition (or multiplication of strictly positive numbers) is actually a bivariate function, which is continuous and strictly increasing in each argument. Naturally, we can ask whether the results can be generalized by considering a function in n variables, which is continuous and strictly increasing in each argument. In this section, we will find necessary and sufficient conditions such that non-interactivity holds with respect to (C, f ). Besides the global equality fC (A) = f (A), we will also investigate the equality fC (A)(z) = f (A)(z), for some z ∈ R. We can say that if this equality holds then we have non-interactivity with respect to the triplet (C, f, z). In all that follows, we will often use some compressed notations. Besides the notation A = (A1 , ..., An ), we will − − − − also use A− = (A− 1 , ..., An ), A (γ ) = (A1 (γ ), ..., An (γ )), and similarly for the superscript +. In paper [4], we defined the so called diagonal of two fuzzy numbers. We need to extend this definition to the n-dimensional case.

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3. Necessary and sufficient conditions for the equality f (A) = fC (A)

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Proof. Due to similar reasoning, it suffices to prove only the first assertion. Let us suppose that supp(Ai ) = [ai , bi ], core(Ai ) = [ci , di ], i = 1, ..., n. Then, let us choose γ ∈ ∪ni=1 I m(lAi ). Without any loss of generality, we may assume − that γ ∈ I m(lA1 ). By Proposition 6 in [4] it results that A1 (A− 1 (γ )) = γ . Let us prove that f (A)(f (A (γ ))) = γ . To n − this end, let us choose arbitrary (x1 , ..., xn ) ∈ R , so that f (x1 , ..., xn ) = f (A (γ )). Without any loss of generality, we may assume that xi ∈ supp(Ai ), for all i ∈ {1, ..., n}. The fact that f restricted to supp(A1 ) × ... × supp(An ) is strictly increasing in each argument, implies that we cannot have xi > A− i (γ ) for all i ∈ {1, ..., n}. We have two cases: − (γ ) and B) x > A (γ ). A) x1 ≤ A− 1 1 1

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− A) We have A1 (x1 ) ≤ A1 (A− 1 (γ )) = γ . Since Ai (Ai (γ )) ≥ γ , i = 2, n (see (1)), it easily results that

min{Ai (xi ) : i ∈ {1, ..., n}} ≤ min{Ai (A− i (γ )) : i ∈ {1, ..., n}} = γ . f (A− (γ ))

B) In this case, from f (x1 , ..., xn ) = and from the monotonicity of f , it results that there exists i0 ∈ (γ ) (otherwise, we easily obtain the contradiction f (x1 , ..., xn ) > f (A− (γ ))). This {2, ..., n}, such that xi0 < A− i0 implies Ai0 (xi0 ) < γ . Thus, min{Ai (xi ) : i ∈ {1, ..., n}} < γ = min{Ai (A− i (γ )) : i ∈ {1, ..., n}}. Therefore, we obtain the same conclusion as in the above case A). Taking into account the inequalities obtained in cases A) and B), we easily get that f (A)(f (A− (γ ))) = min{Ai (A− i (γ )) : i ∈ {1, ..., n}} = γ . Denoting y = f (A− (γ )), by Corollary 2 it is immediate that y belongs to the domain of the function lf (A) , and since from the above relations it also results that lf (A) (y) = γ , we conclude that γ ∈ I m(lf (A) ). So, we have just proved that ∪ni=1 I m(lAi ) ⊆ I m(lf (A) ). Now, let us choose arbitrary γ ∈ I m(lf (A) ). Again, by Proposition 6 in [4], we get that f (A)(f − (A)(γ )) = γ , which by Corollary 2, implies f (A)(f (A− (γ ))) = γ . Since f (A)(f (A− (γ ))) ≥ min{Ai (A− i (γ )) : i ∈ {1, ..., n}} ≥ γ , n it necessarily results that min{Ai (A− i (γ )) : i ∈ {1, ..., n}} = γ . Clearly, this implies that γ ∈ ∪i=1 I m(lAi ), which n n gives I m(lf (A) ) ⊆ ∪i=1 I m(lAi ). By the double inclusion, we get that ∪i=1 I m(lAi ) = I m(lf (A) ), and now the proof is complete. 2

23 24 25

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 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

As an important side remark, let us note that the equalities in the previous proposition do not depend on f . It means that I m(lf (A) ) and I m(rf (A) ) are constant as functions in variable f .

26 27

1

24 25 26

Proposition 8. Suppose that we are under the same hypotheses as in Proposition 7. Then, lf (A) is strictly increasing, if and only if lAi is strictly increasing for all i ∈ {1, ..., n}. Similarly, rf (A) is strictly decreasing, if and only if rAi is strictly decreasing for all i ∈ {1, ..., n}. Proof. Again we prove only the first equivalence. Let us suppose that supp(Ai ) = [ai , bi ], core(Ai ) = [ci , di ], i = 1, ..., n. First, we prove the direct implication and therefore suppose that lf (A) is strictly increasing. By way of contradiction, let us suppose that lA1 is not strictly increasing. It results the existence of x1 , x2 ∈ [a1 , c1 ], x1 < x2 , such that A1 (x1 ) = A1 (x2 ) = γ . Let us choose arbitrary x ∈ [x1 , x2 ]. First, we notice that, since A1 (x) = γ and Ai (A− i (γ )) ≥ γ for all i ∈ {1, ..., n}, it results that − f (A)(f (x, A− 2 (γ ), ..., An (γ ))) ≥ min{A1 (x), min{A− i (γ ) : i ∈ {2, ..., n}}} = γ .

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

Then, let us consider the vector (u1 , ..., un ), such that

40

− f (u1 , ..., un ) = f (x, A− 2 (γ ), ..., An (γ )).

41

We have two cases: A) u1 ≤ x and B) u1 > x. A) Since u1 ≤ x ≤ c1 , it results that A1 (u1 ) ≤ A1 (x) = γ , and this implies that min{Ai (ui ) : i ∈ {1, ..., n}} ≤ γ . B) In this case, by the monotonicity of f , it results the existence of i ∈ {2, ..., n}, such that ui < A− i (γ ), because, otherwise we obtain − f (x, A− 2 (γ ), ..., An (γ )) < f (u1 , ..., un ),

42 43 44 45 46 47

ui < A− i (γ ),

which obviously is a contradiction. Thus, from {1, ..., n}} < γ . From the above cases A)–B), we conclude that − f (A)(f (x, A− 2 (γ ), ..., An (γ ))) = γ , x

27

∈ [x1 , x2 ].

we obtain Ai (ui ) < γ , and this implies min{Ai (ui ) : i ∈

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8

1

Consequently, f (A) is constant on the interval

2

5 6

2

− − − [f (x1 , A− 2 (γ ), ..., An (γ )), f (x2 , A2 (γ ), ..., An (γ ))],

3 4

1

which (by the monotonicity of f ) is a non-degenerated interval. Since A− 1 (0) ≤ x1

3

< x2 ≤ A− 1 (1), by the monotonicity

of f together with Corollary 2, it results that

8 9 10 11 12 13 14 15 16 17 18

γ

20

22 23 24 25 26 27 28 29

= f (A)(y1 ) ≥ min{Ai (A− i (γ )) : i

∈ {1, ..., n}} ≥ γ .

Therefore, we easily obtain that the equality holds. Without any loss of generality, let us suppose that A1(A− 1 (γ )) ≤ − − − − Ai (Ai (γ )) ≤ An (An (γ )), for all i ∈ {1, ..., n}. We have two cases: A) A1 (A1 (γ )) = An (An (γ )) and B) − A1 (A− 1 (γ )) < An (An (γ )). − − A) In this case, we have f (A)(y1 ) = A1 (A− 1 (γ )) = An (An (γ )) = γ , and since γ < 1, this implies Ai (γ ) < − − − Ai (1), for all i ∈ {1, ..., n}. Take ε > 0 sufficiently small, so that Ai (γ ) + ε < Ai (1), for all i ∈ {1, ..., n}, and f (A− (γ ) + (ε, ..., ε)) < y2 (we can find such ε because f is continuous). Since lAi is strictly increasing for all − − i ∈ {1, ..., n}, it results that lAi (A− i (γ )) < lAi (Ai (γ ) + ε), for all i ∈ {1, ..., n}. Thus, min{Ai (Ai (γ ) + ε) : i ∈ {1, ..., n}} > min{Ai (A− i (γ )) : i ∈ {1, ..., n}}. In conclusion, we obtain

30

f (A

−

31 32

− (γ )) < f (A− 1 (γ ) + ε, ..., An (γ ) + ε) < y2

and

33

39 40 41 42 43 44

− min{min{Ai (A− i (γ ) + ε) : i = 1, r}, min{Ai (Ai (γ )) : i = r + 1, n}}

46

> min{Ai (A− i (γ )) : i

47 48

and reasoning as in the previous case, we obtain −

50

−

f (A (γ )) < f (A (γ ) + εr ) < y2

51 52

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

35 36

which contradicts the fact that lf (A) is nondecreasing. − − B) Now, we necessarily have A− 1 (γ ) < A1 (1). Let us consider all the indices i ∈ {1, ..., n}, such that A1 (A1 (γ )) = − Ai (Ai (γ )). Without any loss of generality, let us suppose that these indices are {1, ..., r}, where 1 ≤ r < n. First, we − notice that A− i (γ ) < Ai (1), for all i ∈ {1, ..., r}. Reasoning as in the previous case, let us take ε > 0, such that − − Ai (γ ) + ε < Ai (1) for all i ∈ {1, ..., r} and f (A− (γ ) + εr ) < y2 , where εr is the n-dimensional vector which has the value ε on first r positions and on the remaining positions it takes the value 0. Therefore, we have lAi (A− i (γ )) < − − (γ ) + ε), for all i ∈ {1, ..., r}. Since l (A (γ )) < l (A (γ )), for all i ∈ {1, ..., r} and j ∈ {r + 1, ..., n}, it lAi (A− A A i j i i j easily results that

45

49

11

34

36

38

10

33

− = f (A)(y2 ) < f (A)(f (A− 1 (γ ) + ε, ..., An (γ ) + ε)),

35

9

32

f (A)(f (A− (γ )))

34

37

7 8

is included in the domain of lf (A) . Consequently, lf (A) is constant on a non-degenerated interval, and this contradicts the assumption that lf (A) is strictly increasing. Therefore, it results that lA1 is strictly increasing. By similar reasoning, we can easily prove that lAi is strictly increasing for all i ∈ {2, ..., n}. Now, let us prove the converse implication. Let us denote with [a, c] the domain of lf (A) . By Corollary 2, we have a = f (A− (0)) and c = f (A− (1)). By way of contradiction, let us suppose that lf (A) is not strictly increasing and therefore, let y1 , y2 ∈ [a, c], y1 < y2 , be such that lf (A) (y1 ) = lf (A) (y2 ) = γ . Without any loss of generality, we may assume that y1 = f − (A)(γ ) (otherwise, we take y1 = f − (A)(γ ), which gives lf (A) (y1 ) = γ , hence, lf (A) is constant on [y1 , y2 ], which means that we can replace y1 with y1 , if necessary). By Corollary 2, it results that y1 = f (A− (γ )). We also notice that the obvious inequality y1 < c, implies γ < 1. Next, we prove that f (A)(y1 ) = min{Ai (A− i (γ )) : i ∈ {1, ..., n}}. Indeed, we have

19

21

5 6

− − − [f (x1 , A− 2 (γ ), ..., An (γ )), f (x2 , A2 (γ ), ..., An (γ ))]

7

4

and

∈ {1, ..., n}},

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9

f (A)(f (A− (γ ))) = f (A)(y2 ) < f (A)(f (A− (γ ) + εr )),

1

which again contradicts the fact that lf (A) is nondecreasing. Since in each case we reached a contradiction, it results that indeed, we must have lf (A) (y1 ) < lf (A) (y2 ), and this finishes the proof of the converse implication. 2

= {(x1 , ..., xn ) : xi = A− i (γ ), γ = {(x1 , ..., xn ) : xi = A+ i (γ ), γ

∈ ∪ni=1 I m(lAi )}, ∈ ∪ni=1 I m(rAi )}.

14 15 16 17 18

(8) (9)

We are now in position to present the main results of the paper.

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29 30 31 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

13 14 15 16 17 18 19

Proof. Let y = f − (A)(γ ), which by Corollary 2, implies y = f (A− (γ )). In addition, we obviously have y ≤ y. Since γ ∈ I m(lf (A) ), it results that f (A)(y ) = γ . Reasoning as in the proof of Proposition 7, we get that f (A)(y ) = min{Ai (A− i (γ )) : i ∈ {1, ..., n}} = γ .

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Noting that fC (A)(y ) =

27 28

5

12

Theorem 9. Let A1 , ..., An be n fuzzy numbers, A = (A1 , ..., An ), and consider a continuous function f : Rn → R, with the property that the restriction of f to supp(A1 ) × ... × supp(An ) is strictly increasing in each argument. Then, suppose that C is a joint possibility distribution of A1 , ..., An . Let y be a point in the domain of lf (A) such that f (A)(y) = γ . If C(A− (γ )) = γ then f (A)(y) = fC (A)(y). If lAi is strictly increasing for all i ∈ {1, ..., n} (by Proposition 8 it is equivalent to say that lf (A) is strictly increasing) and C is upper semicontinuous, then the converse implication holds too.

19 20

4

7

12 13

3

6

Keeping the notations from Definition 6, we introduce 1 ⊆ 1 and 3 ⊆ 3 , where 1 3

2

max

f (x1 ,...,xn )=f (A− (γ ))

C(x1 , ..., xn ) ≥ C(A− (γ )),

then taking into account the hypothesis, it results that



fC (A)(y ) ≥ γ = f (A)(y ). On the other hand, since obviously we have fC (A)(y ) ≤ f (A)(y ), it results that fC (A)(y ) = f (A)(y ) = γ . By the monotonicity of fC (A) and f (A) on [y , y], and since fC (A) ≤ f (A), it results that

γ = fC (A)(y ) ≤ fC (A)(y) ≤ f (A)(y) = γ , and this implies that fC (A)(y) = f (A)(y). Let us suppose now that C is upper semicontinuous and that lAi is strictly increasing for all i ∈ {1, ..., n} and therefore, let us prove the converse implication in this case. So, let γ ∈ [0, 1] be such that fC (A)(y) = f (A)(y) = γ . First, let us discuss the case when γ ∈ (0, 1]. This implies that y = f − (A)(γ ) (if not, then denoting y = f − (A)(γ ), we would obtain that f − (A)(0) ≤ y < y ≤ f − (A)(1) and lf (A) is constant on [y , y], a contradiction). Reasoning as in the proof of Proposition 7, we get f (A)(y) = min{Ai (A− i (γ )) : i ∈ {1, ..., n}}. Then, by the definition of fC (A)(y), let x1 , ..., xn ∈ R be such that f (x1 , ..., xn ) = y and fC (A)(y) = C(x1 , ..., xn ) (recall, by the upper semicontinuity of C we can always use “max” instead of “sup”). By the definition of C, we easily obtain that C(x1 , ..., xn ) ≤ min{Ai (xi ) : i ∈ {1, ..., n}}. On the other hand, we must have min{Ai (xi ) : i ∈ {1, ..., n}} ≤ f (A)(y). But since f (A)(y) = fC (A)(y) = C(x1 , ..., xn ), it results that in fact we must have C(x1 , ..., xn ) = min{Ai (xi ) : i ∈ {1, ..., n}} = f (A)(y). It remains to prove that xi = A− i (γ ), for all i ∈ {1, ..., n}. By way of contradiction suppose − that x1 < A1 (γ ). This implies A(x1 ) < γ and hence min{Ai (xi ) : i ∈ {1, ..., n}} < γ , a contradiction. Therefore, it − results that x1 ≥ A− 1 (γ ) and by similar reasoning we get that xi ≥ Ai (γ ) for all i ∈ {1, ..., n}. On the other hand, − − since f (x1 , ..., xn ) = f (A)(γ ) = f (A (γ )), by the monotonicity of f and by the previous inequalities, it results − that xi = A− i (γ ) for all i ∈ {1, ..., n}. Hence, we obtain C(A (γ )) = γ and this finishes the proof in this case. Lastly, let us discuss the case γ = 0, thus f (A)(y) = 0. Suppose that A− (0) = (a1 , ..., an ). It means that y = f (a1 , ..., an ), where we used that f (a1 , ..., an ) is the lower bound of supp(f (A)) and hence f (A)(z) > 0 for any z in the domain of lf (A) that satisfies z > f (a1 , ..., an ). As f (A)(y) = 0, it easily follows that min{Ai (ai ) : i ∈ {1, ..., n}} = 0. Without

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− any loss of generality suppose that A1 (a1 ) = 0. As obviously a1 = A− 1 (0), we get A1 (A1 (0)) = 0. From here, we easily conclude that C(A− (0)) = 0. Now the proof is complete. 2

3 4

7 8 9 10 11

We omit the proof of the next theorem, because the reasoning is absolutely similar to the previous one.

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Theorem 10. Let A1 , ..., An denote n fuzzy numbers, A = (A1 , ..., An ), and consider a continuous function f : Rn → R, with the property that the restriction of f to supp(A1 ) × ... × supp(An ) is strictly increasing in each argument. Then, suppose that C is a joint possibility distribution of A1 , ..., An . Let y be a point in the domain of rf (A) , such that f (A)(y) = γ . If C(A+ (γ )) = γ , then f (A)(y) = fC (A)(y). If rAi is strictly decreasing for all i ∈ {1, ..., n} (by Proposition 8 it is equivalent to say that rf (A) is strictly decreasing) and C is upper semicontinuous, then the converse implication holds too.

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46

49 50 51 52

8 9 10 11

13

Theorem 11. Let A1 , ..., An denote n fuzzy numbers, A = (A1 , ..., An ), and suppose that C is a joint possibility distribution of A1 , ..., An . If C(A− (γ )) = γ , for all γ ∈ ∪ni=1 I m(lAi ) and C(A+ (γ )) = γ , for all γ ∈ ∪ni=1 I m(rAi ), then f (A) = fC (A) for any continuous function f : Rn → R, with the property that the restriction of f to supp(A1 ) × ... × supp(An ) is strictly increasing in each argument. If lAi and rAi are strictly monotone for all i ∈ {1, ..., n} and C is upper semicontinuous, then the converse implication holds too.

15

Proof. Since fC (A) ≤ f (A), it suffices to prove that fC (A)(y) = f (A)(y), for any y ∈ supp (f (A)). First, let y ∈ [f − (A)(0), f − (A)(1)] be arbitrarily chosen, then suppose that f (A)(y) = γ . Since obviously γ ∈ I m(lf (A) ) = ∪ni=1 I m(lAi ) (see Proposition 7), it results that C(A− (γ )) = γ and then, by Theorem 9, we obtain fC (A)(y) = f (A)(y). Next, if y ∈ [f + (A)(1), f + (A)(0)], then by Theorem 10, we obtain that fC (A)(y) = f (A)(y). It remains to prove that for any y ∈ [f − (A)(1), f + (A)(1)], we have fC (A)(y) = f (A)(y). Taking into account that lf (A) (f − (A)(1)) = 1, the hypothesis implies C(A− (1)) = 1 and by Theorem 9, we obtain fC (A)(f − (A)(1)) = f (A)(f − (A)(1)). Similarly, we get fC (A)(f + (A)(1)) = f (A)(f + (A)(1)). From here, we easily obtain that fC (A)(y) = f (A)(y), for any y ∈ [f − (A)(1), f + (A)(1)]. In conclusion, we obtain f (A) = fC (A). Suppose now that C is upper semicontinuous and that lAi and rAi are strictly monotone for all i ∈ {1, ..., n}, and let us prove the converse implication. Let us choose arbitrary γ ∈ ∪ni=1 I m(lAi ). Then, there exists y ∈ [f − (A)(0), f − (A)(1)], such that f (A)(y) = γ . The hypothesis implies f (A)(y) = fC (A)(y) and therefore, by Theorem 9 (the converse implication), it results that C(A− (γ )) = γ . Similarly, we obtain that C(A+ (γ )) = γ , for all γ ∈ ∪ni=1 I m(rAi ). 2

21

14

16 17 18 19 20

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

Obviously, the previous theorem implies the following result, written in terms of independent joint possibility distribution and the sets 1 and 3 , given in (8)–(9). Corollary 12. Let A1 , ..., An denote n fuzzy numbers, A = (A1 , ..., An ), and suppose that C is a joint possibility distribution of A1 , ..., An . If C(x1 , ..., xn ) = Cid (x1 , ..., xn ), for any (x1 , ..., xn ) ∈ 1 ∪ 3 , then f (A) = fC (A), for any continuous function f : Rn → R, with the property that the restriction of f to supp(A1 ) × ... × supp(An ) is strictly increasing in each argument. If lAi and rAi are strictly monotone for all i ∈ {1, ..., n} and C is upper semicontinuous, then the converse implication holds too.

36 37 38 39 40 41 42 43 44

In case of continuous fuzzy numbers with strictly monotone side functions, we can characterize the equality f (A) = fC (A), with the help of the diagonal of (A1 , ..., An ).

47 48

7

In view of Theorems 9–10, we can now present necessary and sufficient conditions for the equality fC (A) = f (A).

44 45

6

12

35 36

4 5

12 13

2 3

5 6

1

45 46 47

Corollary 13. Let A1 , ..., An denote n fuzzy numbers, A = (A1 , ..., An ), and consider a continuous function f : Rn → R, with the property that the restriction of f to supp(A1 ) × ... × supp(An ) is strictly increasing in each argument. Furthermore, suppose that f (A) is continuous and lf (A) and rf (A) both are strictly monotone functions (this holds if, for example, each Ai is continuous with strictly monotone sides), and suppose that C is an upper semicontinuous joint possibility distribution of A1 , ..., An . The following assertions are equivalent.

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11

(i) f (A) = fC (A); (ii) C(A− (γ )) = γ and C(A+ (γ )) = γ , ∀γ ∈ [0, 1]; (iii) C(x1 , ..., xn ) = Cid (x1 , ..., xn ), ∀(x1 , ...xn ) ∈ 1 ∪ 3 .

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

20 21 22 23 24 25

28

31 32 33 34 35 36

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

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19 20 21 22 23 24 25

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Theorem 15. Let A1 , ..., An be n fuzzy numbers, A = (A1 , ..., An ) and let J ⊆ {1, ..., n} be such that Ai is continuous on [A+ i (1), ∞) for any i ∈ J and for any i ∈ {1, ..., n} \ J the domain of rAi is degenerated. Then, consider a continuous function f : Rn → R, with the property that the restriction of f to supp(A1 ) × ... × supp(An ) is strictly increasing in each argument and suppose that C is a joint possibility distribution of A1 , ..., An . If for some y in the domain of rf (A) there exist xi in the domain of rAi , i = 1, n, such that f (x1 , ..., xn ) = y, Ai (xi ) = Aj (xj ) for any i, j ∈ J and C(x1 , ..., xn ) = Cid (x1 , ..., xn ), then f (A)(y) = fC (A)(y). If rAi is strictly decreasing for all i ∈ J and C is upper semicontinuous, then the converse implication holds too.

37 38

5

26

We omit the proof as the reasoning is very much the same as in the proof of Theorem 9. Similarly, we have an analogous result to the one given in Theorem 10.

29 30

3

18

Theorem 14. Let A1 , ..., An be n fuzzy numbers, A = (A1 , ..., An ) and let J ⊆ {1, ..., n} be such that Ai is continuous on (−∞, A− i (1)] for any i ∈ J and for any i ∈ {1, ..., n} \ J the domain of lAi is degenerated. Then, consider a continuous function f : Rn → R, with the property that the restriction of f to supp(A1 ) × ... × supp(An ) is strictly increasing in each argument and suppose that C is a joint possibility distribution of A1 , ..., An . If for some y in the domain of lf (A) there exist xi in the domain of lAi , i = 1, n, such that f (x1 , ..., xn ) = y, Ai (xi ) = Aj (xj ) for any i, j ∈ J and C(x1 , ..., xn ) = Cid (x1 , ..., xn ), then f (A)(y) = fC (A)(y). If lAi is strictly increasing for all i ∈ J and C is upper semicontinuous, then the converse implication holds too.

26 27

2

4

It is possible to have f (A) continuous, even if one of A1 , ..., An has discontinuity points. The simplest example is when we consider addition of fuzzy numbers, as this situation was also mentioned in [4]. Let us now discuss the practicality of the main results of this section. Suppose we have n fuzzy numbers A1 , ..., An , A = (A1 , ..., An ), f : Rn → R, where f is continuous and the restriction of f to supp(A1 ) × ... × supp(An ) is strictly increasing in each argument and suppose that C is an upper semicontinuous joint possibility distribution of A1 , ..., An . An issue is that all the main results of this section use the functions A− and A+ , respectively. Most often we easily obtain the parametric representation of fuzzy numbers given in terms of membership function. For example, we can always do that when we use triangular fuzzy numbers, fuzzy numbers with quadratic shapes, etc. But sometimes, given the membership function of a fuzzy number, we cannot precisely find its parametric representation. Therefore, it would be really useful to present equivalent theoretical results in terms of membership function. We will present only the local results as they can easily be extended for the global equality. In what follows, by degenerated domain of a function, we mean a domain reduced to a single value. If A is a fuzzy number then the domain of lA is degenerated if and only if A− (0) = A− (1) and the domain of rA is degenerated if and only if A+ (1) = A+ (0).

18 19

1

30 31 32 33 34 35 36 37

The previous two theorems cover also the cases when we do not have degenerated domains as in such cases we can take J = ∅. Either we look over the results where parametric representations of fuzzy numbers are used, or we look over the previous two theorems, there are situations where we have to face some limitations. These limitations are not because of incomplete results, but they are because of computational complexity. For example, in the previous theorem we need to solve the equation f (x1 , ..., xn ) = y in variables x1 , ..., xn . Obviously, it may happen that we cannot find precisely the solution of this equation. Now, considering the results that depend on the parametric representation of fuzzy numbers, the only limitation is that sometimes it is very difficult to apply numerically the extension principle. It may happen that we cannot find precisely the set where the minimum operator is applied. The same problem may occur when we consider the interactive extension principle instead. Therefore, the main results of this section are important especially when we cannot make direct computations. It even may happen that we can find the output based on the extension principle but we cannot do it using the interactive extension principle. Indeed, by Theorem 1 we can find the parametric representation of an output based on the extension principle. But, we do not have a similar result when we consider instead the interactive extension principle based on a joint possibility distribution. We believe such result is possible (at least when the operation is generated by a function with the properties assumed in this

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paper) and this will be the topic of a future research. It is important to mention that in the special case of interactive operations generated by triangular norms, parametric representation of the output has been obtained in paper [10] (see also [20]) and in paper [22] such result was obtained using a more general approach than the one based on the triangular norm. But, what is really important with respect to the present results, is that regardless of the fact that we can make direct computations or not, we can give a precise answer regarding the equality of the outputs given by the two types of extensions. We just need to compare the results over the diagonal of the fuzzy numbers. More precisely, if our intention is to produce interactive operations that do not coincide with those based on the extension principle, then we need to propose joint possibility distributions that coincide as less as possible with the independent joint possibility distribution over the diagonal of the fuzzy numbers. It is irrelevant whether we can or cannot compute the outputs, we know a priori that they will not coincide. We can do this with respect to the global equality but with respect to the local problem as well.

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(A ∗ B) (z) = sup {A(x) ∧ B(y)}

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x∗y=z

Taking into account Propositions 7 and 8, all the results obtained in the special case when f (x, y) = x + y can be generalized to the case when f is continuous and strictly increasing in each argument on supp(A) × supp(B). More precisely, A + B = A +C B if and only if f (A, B) = fC (A, B) for any function f satisfying the properties mentioned just above. The same is true considering the local problem when we compare f (A, B)(z) and fC (A, B)(z). For example, considering all the joint possibility distributions from paper [4] (see Example 18 there), then applying Theorems 9–10 and 11, respectively, for the study of the local problem and the global one, respectively, we get to the same conclusions when we replace A + B and A +C B with f (A, B) and fC (A, B), respectively. A new interesting application, is the one based on so called completely correlated fuzzy numbers. Two fuzzy numbers A and B are said to be completely correlated (see Definition 3.1 in [1]), if there exist q, r ∈ R, q = 0 and a joint possibility distribution C of A and B, such that C(x1 , x2 ) = A(x1 ) · χ{qu+r=v} (x1 , x2 )

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In this section, we will discuss the main results obtained so far, by using concrete examples of joint possibility distributions. First, we discuss the case when the joint possibility distribution is bidimensional. Although the theory considers the n-dimensional case, we have to notice that all the papers devoted to this topic, exclusively address the bidimensional setting in applications. Moreover, almost all papers consider only the case of interactive additions, omitting other important operations. There are some exceptions, such as paper [5], where interactive and non-interactive multiplications are discussed, or paper [9], where an arbitrary binary operation is considered. This is the starting point of our analysis too. Let us consider two fuzzy numbers A and B, and let C denote a joint possibility distribution of A and B. Then, consider the continuous function f : R2 → R, which is strictly increasing in each argument on supp(A) × supp(B). For simplicity, when there is no risk of confusion, we may denote f (x, y) = x ∗ y, f (A, B) = A ∗ B and fC (A, B) = A ∗C B. Therefore, for some z ∈ R we have

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Here, χ{qu+r=v} is the characteristic function of the set

(u, v) ∈ R2 : qu + r = v . If q > 0 (q < 0), then A and B are called completely positively (negatively) correlated. It is also important to note that, if A and B are completely correlated, then Bγ = qAγ + r, for all γ ∈ [0, 1]. Considering the case when q > 0, we get B − (γ ) = qA− (γ ) + r and B + (γ ) = qA+ (γ ) + r, for all γ ∈ [0, 1]. From the definition of C, it easily follows that

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C(A− (γ ), B − (γ ))

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= C(A− (γ ), qA− (γ ) + r) = A(A− (γ )),

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C(A+ (γ ), B + (γ )) = A(A+ (γ )).

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Therefore, reasoning as in the proof of Proposition 7, then taking into account Theorem 11, we easily conclude that f (A, B) = fC (A, B) (or, by the above notations, A ∗ B = A ∗C B). Thus, for completely positively correlated fuzzy numbers, the operations are non-interactive with respect to any function f satisfying the continuity and monotonicity assumptions considered in this paper. On the other hand, if q < 0, this is not necessarily true, as a counterexample can be found in paper [1] (see Remark 3.2 there), where actually the authors propose an example where A = B with A being a proper (not crisp) fuzzy number and the interactive addition between A and B is in fact 0, which in case of the standard non-interactive addition is possible only for crisp fuzzy numbers. Therefore, in this example we cannot have equality between the interactive and non-interactive sums of A and B. As we said, all the examples in the literature consider the case of bidimensional joint possibility distributions. The simplest method to obtain n-dimensional joint possibility distributions is to construct them inductively. For example, we can consider A = (A1 , ..., An ), A1 , ..., An ∈ F (R), A1 = ... = An = (0, 0, 1) and C : Rn → R, C(x) = (1 − (x1 + ... + xn ))χT (x), where χT is the characteristic function of the set T = {x ∈ Rn : x ≥ 0, x1 + ... + xn ≤ 1}. Now, consider the continuous function f : Rn → R, which is strictly increasing in each argument on supp(A1 ) × ... × supp(An ). In this example it is more convenient to use Theorems 14 and 15. So, we have to find all the n-tuples (x1 , ..., xn ), such that A1 (x1 ) = ... = An (xn ) and C(x1 , ..., xn ) = A1 (x1 ), where xi ∈ [0, 1], i = 1, n. It is easily seen that these equalities hold if and only if xi = 0, i = 1, n, or xi = 1, i = 1, n. It means that fC (A)(z) = f (A)(z), for z ∈ {f (0, ..., 0), f (1, ..., 1)} and fC (A)(z) < f (A)(z), for all z ∈ (f (0, ..., 0), f (1, ..., 1)). Note that it may happen that we cannot compute fC (A), yet still we know exactly the points where fC (A) and f (A) coincide, as well as the points where they do not coincide. Obviously, we can use the same trick for many other bidimensional joint possibility distributions. We propose now another approach, in the sense that if C1 is a joint possibility distribution of A1 and A2 , and C2 is a joint possibility distribution of A2 and A3 , respectively, then we construct a (pairwise) joint possibility distribution C of A1 , A2 , A3 . Although we believe the method works in general, from practical point of view, it suffices to assume that C1 and C2 are upper semicontinuous. This easily implies that for any x, y ∈ R, there exist x0 , y0 ∈ R, such that C1 (x, y0 ) = A1 (x) = A1 (x) ∧ A2 (y0 ) and C1 (x0 , y) = A2 (y) = A1 (x0 ) ∧ A2 (y). Similar property holds for C2 . We define C : R3 → R, C(x, y, z) = min {C1 (x, y), C2 (y, z)}. Let us prove that indeed, C is a joint possibility distribution of A1 , A2 and A3 , respectively. Suppose that x ∈ R is chosen arbitrarily. Obviously, we have C(x, y, z) ≤ A1 (x). Then, let y0 , z0 ∈ R be such that C1 (x, y0 ) = A1 (x) = A1 (x) ∧ A2 (y0 ) and C2 (y0 , z0 ) = A2 (y0 ) = A2 (y0 ) ∧ A3 (z0 ). This implies C(x, y0 , z0 ) = A1 (x) and hence we get sup C(x, y, z) = A1 (x). Similarly, we obtain sup C(x, y, z) = y,z∈R

x,y∈R

A3 (z). Finally, let us prove that sup C(x, y, z) = A2 (y). Again, it is easy to check that sup C(x, y, z) ≤ A2 (y). x,z∈R

x,z∈R

Then, let x0 , z0 ∈ R be such that C1 (x0 , y) = A2 (y) = A1 (x0 ) ∧ A2 (y) and C2 (y, z0 ) = A2 (y) = A2 (y) ∧ A3 (z0 ). From here, we easily conclude that sup C(x, y, z) = A2 (y). All these imply that C is a joint possibility distribution x,z∈R

of A1 , A2 , A3 . Let us construct a concrete example, by using two joint possibility distributions mentioned earlier. Hence, (with new notations for the fuzzy numbers) suppose that A1 , A2 ∈ F (R), A1 = A2 = (0, 0, 1) and C1 : R2 → R, C1 (x, y) = (1 − x − y)χT (x, y), T = {(x, y) ∈ R2 : x ≥ 0, y ≥ 0, x + y ≤ 1}. Then, consider A3 ∈ F (R), A3 = (0, 1, 1) and C2 : R2 → R, C2 (x, y) = (y − x)χS (x, y), S = {(x, y) ∈ R2 : x ≥ 0, y ≤ 1, y − x ≥ 0}. As we know, C1 is a joint possibility distribution of A1 and A2 , while C2 is a joint possibility distribution of A2 and A3 . Now, taking C(x, y, z) = min {C1 (x, y), C2 (y, z)}, we obtain a joint possibility distribution of A1 , A2 , A3 , where, after some simple calculations, we obtain ⎧ ⎪ ⎨ 1 − x − y, if x ≥ 0, y ≥ 0, z ≤ 1, x + y ≤ 1 ≤ x + z; z − y, if x ≥ 0, y ≥ 0, x + y ≤ x + z ≤ 1; C(x, y, z) = ⎪ ⎩ 0, otherwise.

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In general, it is very easy to notice that this pairwise construction of C induces non-interactive operations with respect to continuous and strictly increasing functions in each argument if and only if the same holds for C1 and C2 respectively. Finally, let us notice that in particular we may consider joint possibility distributions generated by triangular norms. However, the conclusion will be the same as in the particular case of interactive addition versus non-interactive addition (see [4]). More exactly, for upper semicontinuous triangular norms (including their inductive extensions to n variables) we will obtain interactive operations with respect to continuous and strictly increasing functions in each argument, if and only if the triangular norm is actually the strongest triangular norm.

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In this work we investigated on the equality of interactive and non-interactive operations on fuzzy numbers. We found necessary and sufficient conditions for this equality in the case when the operations are determined by continuous functions strictly increasing in each argument. In a forthcoming paper, we will propose sufficient conditions to obtain an Nguyen type theorem for the interactive extension principle, that is, an analogue of Theorem 1. In the case of triangular norm-based extension principle, as we already mentioned in the Introduction, such results already exist in the literature. The importance of such results (including the extensions based on joint possibility distributions) is underlined in [18]. Therefore, a further generalization for joint possibility distribution-based extension principle seems to be really important. There are some other interesting problems which are worth investigation in the future. For example, in paper [6], we investigated the additivity of the weighted possibilistic mean operator when addition is based on a joint possibility distribution. We found only particular examples and hence it is worth trying a more solid approach to the problem. An interesting problem was proposed in [9]. Here the authors found a sufficient condition such that a family of binary joint possibility distributions is invariant with respect to the counterimage of a fixed element from the domain of values. Then, in paper [8], for two given fuzzy numbers, a family of joint possibility distributions depending on one parameter is given, such that the interactive addition of the fuzzy numbers has an increasing Pompeiu–Hausdorff norm with respect to the parameter. This problem too can be further investigated.

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The authors are grateful to the anonymous reviewers for their comments and suggestions which improved significantly the quality of this contribution. The contribution of Lucian Coroianu was possible with the support of a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0861. Uncited references [3]

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