Subsethood measures for interval-valued fuzzy sets based on the aggregation of interval fuzzy implications

Accepted Manuscript Subsethood measures for interval-valued fuzzy sets based on the aggregation of interval fuzzy implications Zdenko Takáˇc

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S0165-0114(15)00171-2 http://dx.doi.org/10.1016/j.fss.2015.03.022 FSS 6781

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Fuzzy Sets and Systems

Received date: 11 May 2012 Revised date: 28 March 2015 Accepted date: 31 March 2015

Please cite this article in press as: Z. Takáˇc, Subsethood measures for interval-valued fuzzy sets based on the aggregation of interval fuzzy implications, Fuzzy Sets and Systems (2015), http://dx.doi.org/10.1016/j.fss.2015.03.022

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Subsethood measures for interval-valued fuzzy sets based on the aggregation of interval fuzzy implications Zdenko Tak´aˇc Institute of Information Engineering, Automation and Mathematics Faculty of Chemical and Food Technology, Slovak University of Technology in Bratislava Radlinsk´eho 9, 812 37 Bratislava, Slovak Republic e-mail: [email protected], fax: +421252495177

Abstract The connection between fuzzy subsethood measures and fuzzy implications straightforwardly follows from the definition of inclusion for (crisp) sets. In accordance with this connection we introduce a new constructive method for (weak) fuzzy subsethood measures for interval-valued fuzzy sets based on the aggregation of fuzzy interval implications. For this purpose we generalize aggregation functions and propose a technique for aggregation of intervals. The benefit of our approach lies in the wide range of obtained fuzzy subsethood measures for interval-valued fuzzy sets. Moreover, the method gives the possibility to construct new fuzzy subsethood measures with properties that are appropriate to the situation under investigation. Another advantage of the method is its clear semantic interpretation. Keywords: Subsethood measure, Interval valued fuzzy sets, Interval aggregation, Aggregation function, Fuzzy implication, Inclusion indicator, Lipschitz, Best interval representation 1. Introduction An inclusion for fuzzy sets A, B in a set U was defined by Zadeh [1] as follows A⊆B iff (A(x) ≤ B(x) , ∀x ∈ U ) . (1) This is a binary relation, fuzzy set A is completely contained within B or it is not (A is subset or is not subset of B). In fuzzy logic, it is more natural to consider an indicator of degree to which A is subset of B. In general, such Preprint submitted to Fuzzy Sets and Systems

April 2, 2015

an indicator, called a fuzzy subsethood measure (or inclusion indicator ), is a mapping σ : F S(U ) × F S(U ) → [0, 1], where F S(U ) denotes the class of fuzzy sets in a universe U . Pioneers of inclusion indicators are Bandler and Kohout [2]; since then subsethood measures have been considered by many researchers, e.g. [3], [4], [5], [6], [7]. One of the most important generalizations of fuzzy sets (FSs, in abbreviation) are interval-valued fuzzy sets (IVFSs) introduced by Zadeh [8]. There are two different visions of IVFSs. They are fuzzy sets whose membership grades are intervals; or they are fuzzy sets whose membership grades are precise values but our incomplete knowledge of these values is represented by intervals. See the discussion in [9]. The foundations of this paper follows from the second vision. The authors of [10] propose axiomatic requirements a measure of subsethood for IVFSs should comply with, and some similar approach is used in [11] and [12]. In these papers, the subsethood measure for IVFSs is defined as a mapping σ : IV F S(U ) × IV F S(U ) → [0, 1], where IV F S(U ) denotes the class of IVFSs in a universe U , i.e., the subsethood is only one number from [0, 1]. Bustince [13] investigates in detail axioms for subsethood measures for interval-valued fuzzy sets, and concludes that the measure should be rather interval, i.e., the subsethood measure should be a mapping σ : IV F S(U ) × IV F S(U ) → D, where D denotes the set of closed subintervals of [0, 1]. Our approach is based on this conclusion; moreover, for reasons stated in [7], we adjust the axioms of Young [14] (which were proposed for FSs) to IVFSs. A different view of the issue is presented in [15], where the proposed subsethood measure takes values in a Boolean lattice, in other words, the subsethood is viewed as an L-fuzzy valued relation between fuzzy sets. Subsethood measures of fuzzy sets have been used in many applications, e.g. approximate reasoning [13], classification [16], computing with words [17] and control [18]. Recently many fuzzy sets applications are generalized to interval-valued fuzzy sets (or interval type-2 fuzzy sets or Atanassov’s intuitionistic fuzzy sets). Hence it is important to deeply study theoretical aspects of subsethood measures in the settings of IVFSs and to propose some appropriate formulas for the measures. Moreover, our results can be easily adapted to Atanassov’s intuitionistic fuzzy sets settings (see [19], [20] for relations between the two generalizations of fuzzy sets). In this paper, we propose a novel constructive method for fuzzy subsethood measures for IVFSs. We apply so-called best interval representation, 2

which is a new approach to subsethood measures, and a similar approach can be used to fuzzy entropy, distance measure and similarity measure (that are three basic concepts of fuzzy sets theory [4]). Our method was inspired by work of Bustince et al. [21] who defined so-called DI-subsethood measures constructed as an aggregation of fuzzy implications. We adapt their work to IVFSs. For this purpose we generalize aggregation functions and propose a technique for aggregation of intervals. Then we introduce a new constructive method for fuzzy subsethood measures for IVFSs based on the interval aggregation of fuzzy interval implications. Note that our approach takes into account the specific nature of intervals: B  computed via our method is the resulting subsethood measure of IVFSs A, an interval whose midpoint may not be equal to the DI-subsethood measure of FSs A, B given by midpoints of intervals representing membership grades  B.  of A, The connection between fuzzy subsethood measures and fuzzy implications is clear and very intuitive. It follows from the definition of inclusion for (crisp) sets A, B: A⊆B

iff

(∀x)(x ∈ A → x ∈ B) .

(2)

So our method has clear semantic interpretation and our approach is in accordance with an intuitive understanding of subsethood measures. This gives reasons to believe that it will be appropriate also from the point of view of some real-world applications. Moreover, the method gives the possibility to construct new fuzzy subsethood measures with properties that are appropriate to the situation under investigation. The properties of obtained fuzzy subsethood measure depend on the properties of chosen fuzzy implication and aggregation function which are well-known. The paper is organized as follows. Section 2 contains basic definitions and notations that are used in the remaining parts of the paper. Section 3 presents the notion of interval representation of a real function. Section 4 describes interval fuzzy negations and Section 5 interval fuzzy implications. Section 6 introduces interval aggregation functions. Section 7 discusses the notions of fuzzy subsethood measures for IVFSs. In Section 8, we propose a new constructive method for fuzzy subsethood measures for IVFSs based on the interval aggregation of fuzzy interval implications. Some examples are given in Section 9; and finally, in Section 10 we show that our approach takes into account the specific nature of intervals. The conclusions are discussed in Section 11. 3

2. Preliminaries In this section we recall some well-known concepts that will be useful for subsequent developments in the paper. 2.1. Fuzzy sets and interval-valued fuzzy sets Let U be a crisp set. A mapping A : U → [0, 1] is called a fuzzy set in a set U . A(x) is called a membership function and, for each x ∈ U , the value A(x) represents the membership grade of x. The class of all FSs in U will be denoted by F S(U ). Let D be the set of closed subintervals of unit interval, i.e., D = {[a, b] | 0 ≤ a ≤ b ≤ 1}. For a given interval D ∈ D, endpoints of D are denoted by D and D, i.e., D = [D, D]. We will consider the following partial order on D: D1 ≤ D2

iff

D1 ≤ D2 and D1 ≤ D2

(3)

which one get by extending the min and max connectives and the latticeinduced ordering to intervals. There are also another possible orders for intervals. For instance, the inclusion order (i.e. D1 ≥ D2 and D1 ≤ D2 ) or the class of linear orders described in [22], which encompasses also linear order of Xu and Yager [23]. However, we will not deal with them (we explain why the linear orders from [22] are not appropriate for our purposes in example, page 12). We apply just (3) and the following Fishburn’s interval ordering [24]: D1 ≤N D2 iff D1 ≤ D2 . (4) in this paper. The reason is that the two orderings are very natural in the settings of inclusion of IVFSs, moreover, (3) defines a commonly used and many times appropriate order between intervals. Note that (4) gives too strong conditions for our purpose, hence we will concentrate on (3), and (4) apply only a few times in case of need of some stronger order.  : U → D is called an interval-valued Let U be a crisp set. A mapping A  − (x), A + (x)] = fuzzy set in a set U . For each x ∈ U , the value A(x) = [A   [A(x), A(x)] ⊆ [0, 1] represents our limited knowledge of the degree of mem-

 Ordinary fuzzy sets A − and A + are called bership of an element x to IVFS A.  and an upper fuzzy set of A,  respectively. Obviously, if a lower fuzzy set of A − +    A (x) = A (x), for all x ∈ U , then A reduces to an ordinary FS. The class of all IVFSs in U will be denoted by IV F S(U ). 4

The definition of inclusion for IVFSs is based on the definition of inclusion for ordinary fuzzy sets (1): ⊆B  A

iff

  (A(x) ≤ B(x) ,

∀x ∈ U )

(5)

whereas ≤ denotes the partial order of intervals given by (3), i.e., ⊆B  A

iff

− (x) ≤ B  − (x) and A + (x) ≤ B  + (x) , (A

∀x ∈ U ) .

This is a standard definition of inclusion for IVFSs based on the order given by (3). Grzegorzewski in [25] studied the notions of inclusion and subsethood measure for intuitionistic fuzzy sets (see [26]) and suggested approach to this problem based on the well-known notions of necessity and possibility. We adapt his results and, based on the correspondence between IVFSs and intuitionistic fuzzy sets (see [19], [20]), we define necessary inclusion relations for IVFSs as follows:  ⊆N B  A that is

 ⊆N B  A

iff

iff

  (A(x) ≤N B(x) ,

∀x ∈ U )

 − (x) , + (x) ≤ B (A

∀x ∈ U ) .

(6)

2.2. Fuzzy negation A mapping N : [0, 1] → [0, 1] is said to be a fuzzy negation if it satisfies the following axioms: (N1) N (0) = 1 and N (1) = 0. (N2) If x ≤ y then N (x) ≥ N (y) for all x, y ∈ [0, 1]. A fuzzy negation is called strong if it satisfies the involutive property, i.e., (N3) N (N (x)) = x for all x ∈ [0, 1]. Let A be a fuzzy set in U . A fuzzy set Ac is said to be a complementary of A associated to a strong negation N if Ac (x) = N (A(x)) for all x ∈ U . An element e ∈ (0, 1) is called an equilibrium point if N (e) = e. If N is a strong negation then N has a unique equilibrium point.

5

2.3. Fuzzy implication A mapping I : [0, 1] × [0, 1] → [0, 1] is called a fuzzy implication if it satisfies the boundary conditions: (BC1) I(0, 0) = I(0, 1) = I(1, 1) = 1. (BC2) I(1, 0) = 0. This is the weakest definition of fuzzy implication. However, there is no standard definition in literature. We propose the following list of potential axioms: (I1) x ≤ y implies I(x, z) ≥ I(y, z) for all z ∈ [0, 1] (the first place antitonicity). (I2) x ≤ y implies I(z, x) ≤ I(z, y) for all z ∈ [0, 1] (the second place isotonicity). (I3) I(0, x) = 1 for all x ∈ [0, 1] (dominance of falsity of antecedent). (I4) I(x, 1) = 1 for all x ∈ [0, 1] (dominance of truth of consequent). (I5) I(x, y) = I(x, I(x, y)) (iterative boolean-like law). (I6) I(1, x) = x for all x ∈ [0, 1] (neutrality of truth). (I7) I(x, I(y, z)) = I(y, I(x, z)) (exchange property). (I8) I(x, y) = 1 if and only if x ≤ y (ordering principle). (I9) the mapping N defined as (∀x ∈ [0, 1]) N (x) = I(x, 0) is a strong fuzzy negation (strong fuzzy negation principle). (I10) I(x, y) ≥ y (consequent boundary). (I11) I(x, x) = 1 (identity). (I12) I(x, y) = I(N (y), N (x)) being N a strong fuzzy negation (contrapositive principle). (I13) I(x, N (x)) = N (x) being N a strong fuzzy negation. (I14) I(x, y) = 0 if and only if x = 1 and y = 0 (strong boundary condition). See [27] for deeper discussion and relationships between these axioms. The last two properties were stated and discussed in [5].

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2.4. Fuzzy subsethood measure A fuzzy subsethood measure (also called an inclusion indicator ) is a mapping σ : F S(U ) × F S(U ) → [0, 1] satisfying special properties. In fuzzy literature, three most accepted axiomatizations have been given for fuzzy subsethood measures. The first one was given by Kitainik [6] in 1987, the second by Sinha and Dougherty [3] in 1993, and the third by Young [14] in 1996. For comparison of the axiomatizations see [7]. We will consider the Young’s axiomatization: Definition 1 ([14]). A mapping σ : F S(U )×F S(U ) → [0, 1] is called a fuzzy subsethood measure, if σ satisfies the following properties (for all A, B, C ∈ F S(U )): (Y1) σ(A, B) = 1 if and only if A ⊆ B (i.e., A(x) ≤ B(x) for all x ∈ U ). (Y2) If A(x) ≥ eˇ(x) for all x ∈ U then, σ(A, Ac ) = 0 if and only if A = ˇ1. (Where e is the equilibrium point of the considered strong negation and eˇ, ˇ1 are fuzzy sets with eˇ(x) = e, ˇ1(x) = 1 for all x ∈ U .) (Y3) If A ⊆ B ⊆ C then σ(C, A) ≤ σ(B, A); and if A ⊆ B then σ(C, A) ≤ σ(C, B). We will consider also two alternative definitions of fuzzy subsethood measure. Fan’s axioms [4]: (F1) = (Y1) (F2) = (Y2) (F3) If A ⊆ B ⊆ C then σ(C, A) ≤ σ(B, A) and σ(C, A) ≤ σ(C, B). and DI-fuzzy subsethood measure - axioms of Bustince et al. [21]: (DI1) = (Y1) = (F1) (DI2) σ(A, Ac ) = 0 if and only if A = ˇ1. (DI3) If A ⊆ B then σ(A, C) ≥ σ(B, C) and σ(C, A) ≤ σ(C, B). Proposition 1 ([21]). Every fuzzy subsethood measure in the sense of Young is a fuzzy subsethood measure in the sense of Fan. Every DI-subsethood measure is a fuzzy subsethood measure in the sense of Young (and in the sense of Fan).

7

In some circumstances these axioms are too strong. Young [14] proposed weak fuzzy subsethood measure as a mapping satisfying axioms (Y2) and (Y3). Fan et al. [4] stated the following axioms (note that ˇ0 is a fuzzy set with ˇ0(x) = 0 for all x ∈ U ): (F1’) σ(ˇ0, ˇ0) = σ(ˇ0, ˇ1) = σ(ˇ1, ˇ1) = 1. ˇ 0) ˇ = 0. (F2’) σ(1, (F3’) = (F3) Note that the first two axioms correspond to the boundary conditions for fuzzy implications so, the Fan’s weak fuzzy subsethood measure takes an important role in this paper. Galar et al. [5] modified Fan’s work and proposed Class 1 of weak fuzzy subsethood measures characterized by axioms (F1’), (F3’) and (G2): (G2) σ(A, B) = 0 if and only if A = ˇ1 and B = ˇ0. and Class 2 is characterized by axioms: (F1), (G2) and (F3). 2.5. Aggregation functions Klir and Folger [28] defined an aggregation function as a mapping M : [0, 1]n → [0, 1], where n ∈ {2, 3, 4, . . .}, satisfying the following axioms: (KF1) M (0, . . . , 0) = 0. (KF2) M (1, . . . , 1) = 1. (KF3) If xi ≤ yi for all i ∈ {1, . . . , n}, then M (x1 , . . . , xn ) ≤ M (y1 , . . . , yn ). We are going to use stronger conditions for aggregation functions, so we will consider also alternative definition characterized by the following axioms: (A1) M (x1 , . . . , xn ) = 0 if and only if xi = 0 for all i ∈ {1, . . . , n}. (A2) M (x1 , . . . , xn ) = 1 if and only if xi = 1 for all i ∈ {1, . . . , n}. (A3) = (KF3). (A4) M is a symmetric function in all its arguments, i.e., M (x1 , . . . , xn ) = M (xp(1) , . . . , xp(n) ) for any permutation p on {1, . . . , n}.

8

2.6. Fuzzy subsethood measures by aggregation of implications Bustince et al. [21] proposed a construction method for (weak) fuzzy subsethood measures based on the aggregation of fuzzy implications in the following way: n σ(A, B) = Mi=1 I(A(xi ), B(xi )) (7) where A, B are fuzzy sets in U = {x1 , . . . , xn }, M is an aggregation function and I is a fuzzy implication. In this section we recall some of their results. Proposition 2 ([21], page 3220). Let N be a strong negation and let σ : F S(U ) × F S(U ) → [0, 1] be given by (7), where M : [0, 1]n → [0, 1] is a mapping that satisfies (A1), (A2) and (A3), and I : [0, 1]2 → [0, 1] is a mapping that fulfills (I1), (I2), (I6) and (I8). In these conditions σ is a fuzzy DI-subsethood measure on U (i.e., σ satisfies (DI1), (DI2) and (DI3)). Proposition 3 ([5], page 151). Let N be a strong negation and let σ : F S(U ) × F S(U ) → [0, 1] be given by (7), where M : [0, 1]n → [0, 1] is a mapping that satisfies (KF1), (KF2) and (KF3), and I : [0, 1]2 → [0, 1] is a mapping that satisfies (I1), (I2) and (BC1). Then σ is a weak fuzzy subsethood measure on U in the sense of Fan (i.e., σ satisfies (F1’), (F2’) and (F3’)). Remark. From (7) it is clear that the subsethood measure of fuzzy sets A, B depends not only on the membership functions of A and B, but also on the universe U . For example: Let A, B be fuzzy sets in U , let U  be a (crisp) set with U ⊂ U  , and let A , B  be fuzzy sets in U  such that   A(x) , if x ∈ U B(x) , if x ∈ U   , B (x) = A (x) =  0 , if x ∈ U − U 0 , if x ∈ U  − U Then A and A (also B and B  ) are the same on their supports, however, σ(A , B  ) = σ(A, B) for some aggregation functions (e.g. arithmetic mean, quadratic mean). The reason is that elements xi such that A(xi ) = B(xi ) = 0 contribute to the resulting subsethood measures. So, in some circumstances, it could be reasonable: • to aggregate only values of implications for elements xi such that A(xi ) = 0 or B(xi ) = 0, i.e., values xi such that xi ∈ supp(A) ∪ supp(B) , or • to aggregate only values of implications for elements xi such that A(xi ) = 0, i.e., values xi such that xi ∈ supp(A)} . 9

Based on the previous consideration, formula (7) can be generalized: σ(A, B) = Mi∈J I(A(xi ), B(xi )) where J = {1, . . . , n} or J ⊆ {1, . . . , n} is an index set such that supp(A) ∪ supp(B) = {xi | i ∈ J} (or alternatively supp(A) = {xi | i ∈ J}). The choice of index set J depends on the particular situation. We will discuss this fact later in Section 8 and use the three various index sets J in examples of Section 9. 3. Interval representation of a real function In this section we recall the concept of interval representation of a real function, and the best (canonical) representation which will be used throughout the paper. An interval X is called an interval representation for a real number x if x ∈ X. Considering two interval representations X and Y for a real number x, X is called a better interval representation of x than Y , denoted by Y  X, if X ⊆ Y . Definition 2 ([29]). A function F : Dn → D is called an interval representation for a real function f : [0, 1]n → [0, 1] if, for each (X1 , . . . , Xn ) ∈ Dn and all (x1 , . . . , xn ) ∈ [0, 1]n such that xi ∈ Xi for all i = 1, . . . , n, it holds f (x1 , . . . , xn ) ∈ F (X1 , . . . , Xn ). F is also said to be correct with respect to f . Let F, G be interval representations for a real function f . Then F is said to be a better interval representation than G, denoted by G  F , if F (X1 , . . . , Xn ) ⊆ G(X1 , . . . , Xn ), for each (X1 , . . . , Xn ) ∈ Dn . Definition 3 ([30]). The best (canonical) interval representation for a real function f : [0, 1]n → [0, 1] is the interval function f : Dn → D defined by f(X1 , . . . , Xn ) = [inf{f (X1 , . . . , Xn )}, sup{f (X1 , . . . , Xn )}] where f (X1 , . . . , Xn ) = {f (x1 , . . . , xn ) | xi ∈ Xi for all i = 1, 2, . . . , n} . The function f is well defined and it is clearly interval representation of f . For any other interval representation F of f , it holds F  f which means that f returns a narrower interval than any other interval representation of f . So, f has the optimality property of interval algorithms [31]. If the real function f is continuous in the usual sense then the best interval representation f of a real function f coincides with its range - see [29] and [30]. 10

4. Interval fuzzy negation A mapping N : D → D is said to be an interval fuzzy negation if (IN1) N([0, 0]) = [1, 1] and N([1, 1]) = [0, 0]. (IN2) If X ≤ Y then N(X) ≥ N(Y ) for all X, Y ∈ D. An interval fuzzy negation is called strong if it satisfies the involutive property, i.e., (IN3) N(N(X)) = X for all X ∈ D.  be an IVFS in a set U . An IVFS A c is said to be a complementary Let A c    of A associated to a strong negation N if A (x) = N(A(x)) for all x ∈ U . An interval E ∈ D is called an equilibrium interval of an interval fuzzy negation N if N(E) = E. Clearly, [0, 1] is an equilibrium interval. So, equilibrium interval E such that E = [0, 1] is called a non-trivial equilibrium interval. If N is a strong fuzzy negation then N has a unique degenerate equilibrium interval (see [32], Proposition 6.7 and [33] Proposition 15). Detailed information on interval fuzzy negations can be found in [32]. 5. Interval fuzzy implication We generalize the notion of fuzzy implication to intervals. A mapping I : D2 → D is said to be an interval fuzzy implication if it satisfies the boundary conditions: (IBC1) I([1, 1], [1, 1]) = I([0, 0], [0, 0]) = I([0, 0], [1, 1]) = [1, 1]. (IBC2) I([1, 1], [0, 0]) = [0, 0]. We propose the following list of other potential axioms: (II1) X ≤ Z implies I(X, Y ) ≥ I(Z, Y ) for all Y ∈ D. (II2) Y ≤ Z implies I(X, Y ) ≤ I(X, Z) for all X ∈ D. (II3) I([0, 0], X) = [1, 1] for all X ∈ D. (II4) I(X, [1, 1]) = [1, 1] for all X ∈ D. (II5) I(X, Y ) = I(X, I(X, Y )). (II6) I([1, 1], X) = X for all X ∈ D. (II7) I(X, I(Y, Z)) = I(Y, I(X, Z)) . 11

(II8) I(X, Y ) = [1, 1] if and only if X ≤ Y . (II9) The mapping N defined as (∀X ∈ D) N(X) = I(X, [0, 0]) is a strong interval fuzzy negation. (II10) I(X, Y ) ≥ Y . (II11) I(X, X) = [1, 1] . (II12) I(X, Y ) = I(N(Y ), N(X)) being N a strong interval fuzzy negation. (II13) I(X, N(X)) = N(X) being N a strong interval fuzzy negation. (II14) I(X, Y ) = [0, 0] if and only if X = [1, 1] and Y = [0, 0]. It is easy to see that for fuzzy implications I satisfying (I1) and (I2) the best interval representation I can be expressed in the following way (see [33], Proposition 21):  Y ) = [I(X, Y ), I(X, Y )] . (8) I(X, Bedregal et al. [33] proved that the best interval representation I preserves many properties that are satisfied by the corresponding fuzzy implication. Proposition 4 ([33], Proposition 16). If I is a fuzzy implication (i.e., I satisfies the boundary conditions) then I is an interval fuzzy implication (i.e., I satisfies the boundary conditions). Proposition 5 ([33], Theorem 17). Let I be a fuzzy implication satisfying (I1) and (I2). If I satisfies (Ii), for i ∈ {1, . . . , 7, 10, 12, 13}, then I satisfies (IIi), for i ∈ {1, . . . , 7, 10, 12, 13}. Example. Now we show that using some linear orders of intervals given in [22] (so-called generated admissible order) the previous proposition does not hold, so this order (in general) is not appropriate for our purposes. Bustince et al. [22] defined admissible order on D as linear order , which refines order (3). Admissible order is called generated if there exist two continuous functions f, g such that for all [a, b], [c, d] ∈ D, [a, b] [c, d] iff

(f (a, b) ≤ f (c, d) or f (a, b) = f (c, d), g(a, b) ≤ g(c, d)) .

Functions f, g can be replaced by two continuous aggregation functions A, B on [0, 1], such that A(x, y) = A(u, v) and B(x, y) = B(u, v) can only hold if (a, b) = (u, v) (see Proposition 3.2 of [22]). 12

We show that I does not always satisfy (II2) if I satisfy (I2), for some generated admissible order . We leave as an open problem whether there are some conditions for f and g (or A and B) under which Proposition 5 holds. Let A = a+b , B = a+2b , X = [0.6, 0.9], Y = [0.4, 0.6], Z = [0.2, 0.8]. Then 2 3 A(0.4, 0.6) = 0.5 = A(0.2, 0.8) and B(0.4, 0.6) = 1.6 < 0.6 = B(0.2, 0.8), so 3 Y Z. Moreover,  I(X, Y ) = [I(X, Y ), I(X, Y )] = [I(0.9, 0.4), I(0.6, 0.6)],  I(X, Z) = [I(X, Z), I(X, Z)] = [I(0.9, 0.2), I(0.6, 0.8)], and

I(0.9, 0.4) + I(0.6, 0.6) (9) 2 I(0.9, 0.2) + I(0.6, 0.8) A(I(0.9, 0.2), I(0.6, 0.8)) = (10) 2 If I is Lukasiewicz implication, then (9) is equal to 0.75 and (10) is equal to   0.65, hence I(X, Y ) I(X, Z) does not hold, and finally I does not satisfy (II2). If I is Reichenbach implication, then (9) is equal to 0.61 and (10) is equal to 0.58 and the conclusion is the same. A(I(0.9, 0.4), I(0.6, 0.6)) =

Now we are going to study the properties (IIi) for i ∈ {8, 9, 11, 14}. In the next theorem we show that I preserves properties (II9) and (II14). Theorem 1. Let I : [0, 1]2 → [0, 1] be a mapping satisfying (Ii), for i ∈ {9, 14}. Then I satisfies (IIi), for i ∈ {9, 14}. Proof. (II9) A mapping N is clearly well defined. Moreover  0], [0, 0]) = [I(0, 0), I(0, 0)] = [N (0), N (0)] = [1, 1] N([0, 0]) = I([0,  1], [0, 0]) = [I(1, 0), I(1, 0)] = [N (1), N (1)] = [0, 0] N([1, 1]) = I([1, Now let X ≤ Y . Then  I(X, [0, 0]) = [inf{I(x, 0) | x ∈ X}, sup{I(x, 0) | x ∈ X}] = [inf{N (x) | x ∈ X}, sup{N (x) | x ∈ X}] ≥ [inf{N (y) | y ∈ Y }, sup{N (y) | y ∈ Y }] = [inf{I(y, 0) | y ∈ Y }, sup{I(y, 0) | y ∈ Y }] =  [0, 0]) = N(Y ) = I(Y,

N(X) = = ≥ =

13

And finally  I(X,  N(N(X)) = I( [0, 0]), [0, 0]) =     = inf{I(t, 0) | t ∈ I(X, [0, 0])}, sup{I(t, 0) | t ∈ I(X, [0, 0])} =   = inf{I(t, 0) | t ∈ [inf{I(x, 0) | x ∈ X}, sup{. . .}]}, sup{. . .} =   = inf{N (t) | t ∈ [inf{N (x) | x ∈ X}, sup{. . .}]}, sup{. . .} =   = inf{N (t) | t ∈ [N (X), N (X)]}, sup{. . .} =     = N (N (X)), N (N (X)) = X, X = X  (II14) Let I(X, Y ) = [0, 0]. Then infimum and supremum of the set {I(x, y) | x ∈ X, y ∈ Y } are both equal to 0, so, I(x, y) = 0 for all x ∈ X, y ∈ Y . From (I14) it follows x = 1, y = 0, and finally X = [1, 1], Y = [0, 0]. The proof in opposite direction is obvious. Let us consider the last two axioms, namely (II8) and (II11): (II8) Let I satisfy axiom (I8). Let X ≤ Y , i.e., X ≤ Y and X ≤ Y . However, it is possible that X > Y . Then, using (I8), I(X, Y ) < 1 and  consequently also inf{I(x, y) | x ∈ X, y ∈ Y } < 1. Thus, I(X, Y ) = [1, 1]. So axiom does not hold in this direction. If we need to use some version of the axiom we should modify it in one of the following way: (nII8) I(X, Y ) = [1, 1] if and only if X ≤N Y for all X, Y ∈ D, where X ≤N Y denotes the Fishburn’s interval ordering (4). (II8’) I(X, Y ) = [1, 1] if and only if X ≤ Y for all X, Y being degenerate intervals in D (i.e., X = [x, x], Y = [y, y] for some x, y ∈ [0, 1]). (II11) Let I satisfy axiom (I11). Let X be a non-degenerate interval in D, i.e., X < X. Then it is possible that I(X, X) < 1, and consequently  X) = [1, 1]. So axiom also inf{I(x1 , x2 ) | x1 , x2 ∈} < 1. Thus, I(X, does not hold. Again, if we need to use some version of the axiom we should modify it in one of the following way: (II11’) I(X, X) = [1, 1] for all X being degenerate interval in D. (II11”) I(X, X) = 1 for all X ∈ D. 14

The axiom (II8) will be of special significance for our purpose. Although  we prove that its it is not preserved by the best interval representation I, weaker form (nII8) is preserved. Theorem 2. Let I : [0, 1]2 → [0, 1] be a mapping satisfying (I8). Then I satisfies (nII8). Proof. Let I satisfy axiom (I8). Let X ≤N Y , i.e., X ≤ Y . Then, using (I8), I(X, Y ) = 1 and consequently also inf{I(x, y) | x ∈ X, y ∈ Y } = 1. Thus,  I(X, Y ) = [1, 1]. The proof in the opposite direction is similar. Based on the previous results we can conclude that the best interval representation is appropriate for our purposes, because it preserves majority of properties (in fact all that we need), moreover, it involves all the possible pairs (or n-tuples) of the elements and gives the narrower interval as the result. 6. Interval aggregation functions In this section, we generalize aggregation functions to intervals. First, we propose axioms that correspond to axioms stated in Section 2.5. Definition 4. A mapping M : Dn → D, for some n ≥ 2, is called an interval aggregation function if it satisfies the following axioms: (IKF1) M([0, 0], . . . , [0, 0]) = [0, 0]. (IKF2) M([1, 1], . . . , [1, 1]) = [1, 1]. (IKF3) If Xi ≤ Yi for all i = 1, . . . , n, then M(X1 , . . . , Xn ) ≤ M(Y1 , . . . , Yn ). An alternative definition demanding stronger conditions for interval aggregation functions is characterized by the following axioms: (IA1) M(X1 , . . . , Xn ) = [0, 0] if and only if Xi = [0, 0] for all i = 1, . . . , n. (IA2) M(X1 , . . . , Xn ) = [1, 1] if and only if Xi = [1, 1] for all i = 1, . . . , n. (IA3) = (IKF3). (IA4) M is a symmetric function in all its arguments, i.e., M(X1 , . . . , Xn ) = M(Xp(1) , . . . , Xp(n) ) for any permutation p on {1, . . . , n}.

15

is defined: Recall that the best interval representation M   (X1 , . . . , Xn ) = inf{M (x1 , . . . , xn ) | xi ∈ Xi }, sup{M (x1 , . . . , xn ) | xi ∈ Xi } M preserves all properties satisfied by the Now we are going to show that M corresponding mapping M . Theorem 3. Let M : [0, 1]n → [0, 1] be a mapping satisfying property (KFi), satisfies property (IKFi), for i ∈ {1, 2, 3}. for i ∈ {1, 2, 3}. Then M Proof. (IKF1) and (IKF2) are obvious. (IKF3): Let Xi ≤ Yi for all i = 1, . . . , n. Then inf{M (x1 , . . . , xn ) | xi ∈ Xi } = = sup{M (x1 , . . . , xn ) | xi ∈ Xi } = =

M (X1 , . . . , Xn ) ≤ M (Y1 , . . . , Yn ) = inf{M (y1 , . . . , yn ) | yi ∈ Yi } M (X1 , . . . , Xn ) ≤ M (Y1 , . . . , Yn ) = sup{M (y1 , . . . , yn ) | yi ∈ Yi }

(X1 , . . . , Xn ) ≤ M (Y1 , . . . , Yn ). Thus, M Based on the fact that monotonicity (axiom (IKF3)) is an essential property of aggregation function M , we can express the best interval representa of M via the endpoints of intervals X1 , . . . , Xn : tion M Corollary 4. Let M : [0, 1]n → [0, 1] be a mapping satisfying property (KF3). Then (X1 , . . . , Xn ) = [M (X1 , . . . , Xn ), M (X1 , . . . , Xn )] M Proof. Immediately follows from the item (IKF3) of the Theorem 3. Theorem 5. Let M : [0, 1]n → [0, 1] be a mapping satisfying property (Ai), satisfies property (IAi), for i ∈ {1, 2, 3, 4}. for i ∈ {1, 2, 3, 4}. Then M (X1 , . . . , Xn ) = [0, 0]. Then Proof. (IA1) Let M inf{M (x1 , . . . , xn ) | xi ∈ Xi } = sup{M (x1 , . . . , xn ) | xi ∈ Xi } = 0 and consequently {M (x1 , . . . , xn ) | xi ∈ Xi } = {0}. From (A1) it follows x1 = x2 = . . . = xn = 0 and finally X1 = X2 = . . . = Xn = [0, 0]. For the proof in opposite direction see the proof of Theorem 3 - (IKF1). The proof of (IA2) is similar to the proof of (IA1). Axiom (IA3) is equal to (IKF3). The proof of (IA4) immediately follows from the symmetry of infimum and supremum. 16

7. Fuzzy subsethood measures for IVFSs In this section we generalize the notion of fuzzy subsethood measures to IVFSs. We recapitulate axioms from Section 2.4 and propose their interval version. Definition 5. A mapping σ : IV F S(U ) × IV F S(U ) → D is called a fuzzy subsethood measure for IVFSs (in the sense of Young), if σ satisfies the  B,  C  ∈ IV F S(U )): following properties (for all A,  B)  = [1, 1] if and only if A  ⊆ B  (i.e., A(x)   (IY1) σ(A, ≤ B(x) for all x ∈ U ).   A c ) = [0, 0] if and only if (IY2) If A(x) ≥ E(x) for all x ∈ U then, σ(A, = A 1. (Where E is a degenerate equilibrium interval of the considered    strong interval negation, and E, 1 are IVFSs with E(x) = E,  1(x) = [1, 1] for all x ∈ U .) ⊆B ⊆C  then σ(C,  A)  ≤ σ(B,  A);  and if A ⊆B  then σ(C,  A)  ≤ (IY3) If A  B).  σ(C, We will consider also two alternative definitions of fuzzy subsethood measure for IVFSs. The generalization of Fan’s axioms: (IF1) = (IY1) (IF2) = (IY2)  A)  ≤ σ(C,  B).  ⊆B ⊆C  then σ(C,  A)  ≤ σ(B,  A)  and σ(C, (IF3) If A and the generalization of DI-fuzzy subsethood measure: (IDI1) = (IY1) = (IF1) = A c ) = [0, 0] if and only if A 1. (IDI2) σ(A,        A)  ≤ σ(C,  B).  (IDI3) If A ⊆ B then σ(A, C) ≥ σ(B, C) and σ(C, The relationships between these three kinds of subsethood measures for IVFSs are very similar to the case of ordinary fuzzy sets. Theorem 6. Every fuzzy subsethood measure for IVFSs in the sense of Young is a fuzzy subsethood measure for IVFSs in the sense of Fan. Every DI-subsethood measure for IVFSs is a fuzzy subsethood measure for IVFSs in the sense of Young (and in the sense of Fan). Proof. Immediately follows from the axioms (IY1)-(IY3), (IF1)-(IF3) and (IDI1)-(IDI3). 17

7.1. Weak fuzzy subsethood measures for IVFSs Now we generalize the notion of weak fuzzy subsethood measure. A mapping σ : IV F S(U )×IV F S(U ) → D is said to be a weak fuzzy subsethood measure for IVFSs in the sense of Young if it satisfies axioms (IY2) and (IY3). Similarly, σ is a weak fuzzy subsethood measure for IVFSs in the sense of Fan if it satisfies the following axioms (note that  0 is an IVFS with  0(x) = [0, 0] for all x ∈ U ): (IF1’) σ( 0,  0) = σ( 0,  1) = σ( 1,  1) = [1, 1]. (IF2’) σ( 1,  0) = [0, 0]. (IF3’) = (IF3) Note that the first two axioms correspond to the boundary conditions for interval fuzzy implications so, the Fan’s weak fuzzy subsethood measure takes an important role in this paper. Finally, Class 1 of weak fuzzy subsethood measures for IVFSs is characterized by axioms (IF1’), (IF3’) and (IG2):  B)  = [0, 0] if and only if A = = (IG2) σ(A, 1 and B 0. and Class 2 is characterized by axioms: (IF1), (IG2) and (IF3). 8. Subsethood measures for IVFSs by aggregation of interval implications We propose a new constructive method for (weak) fuzzy subsethood measures for IVFSs based on the aggregation of fuzzy interval implications. Our constructive method is very easy to use, it is sufficient to choose some fuzzy implication (with suitable properties) and some aggregation function (with suitable properties) and to derive (weak) subsethood measure for IVFSs. We study properties of obtained (weak) fuzzy subsethood measures and give some examples. The constructive method is based on the following formula:  B)  = Mi∈J I(A(x  i ), B(x  i )) σ(A,

(11)

B  are IVFSs in U = where σ is a fuzzy subsethood measure for IVFSs, A, {x1 , . . . , xn }, J = {1, . . . , n} or J ⊆ {1, . . . , n} is an index set such that

18

Table 1: Summary of the relationships between the axioms of subsethood measures for IVFSs, interval implications and interval aggregation functions.

σ (IF1’) (IF2’) (IF3’)=(IF3) (IG2) (IDI1)=(IY1)=(IF1) (IDI2) (IDI3) (IY2)=(IF2) (IY3)

I (IBC1) (IBC2) (II1), (II2) (II14) (II8) (II14) (II1), (II2) (II14) (II1), (II2)

M (IKF2) (IKF1) (IKF3) (IA1) (IA2) (IA1), (IA2) (IKF3) (IA1), (IA2) (IKF3)

 ∪ supp(B)  = {xi | i ∈ J} (or alternatively supp(A)  = {xi | i ∈ J})1 , supp(A) M is an interval aggregation function and I is a fuzzy interval implication. Now we are going to study σ defined by (11) with respect to the axioms of (weak) subsethood measures for IVFSs. Theorem 7. Let σ : IV F S(U )×IV F S(U ) → D be a mapping given by (11). Let M : Dn → D be a mapping satisfying axioms specified in the third column of the Table 1, and I : D2 → D be a mapping satisfying axioms specified in the second column of the Table 1. Then σ satisfies axioms specified in the first column of the Table 1. Proof. (IF1’) σ( 0,  0) = M(I([0, 0], [0, 0]), . . . , I([0, 0], [0, 0])) = = M([1, 1], . . . , [1, 1]) = [1, 1] The cases σ( 0,  1) and σ( 1,  1) are similar. (IF2’) The proof is very similar to (IF1’).  ⊆ C.  Then for all i ∈ J it holds B(x  i ) ≤ C(x  i ), and (IF3’)=(IF3) Let B     from axiom (II1) it follows I(C(xi ), A(xi )) ≤ I(B(xi ), A(xi )). Then, by axiom (IKF3): 1

See Remark, Section 2.6 of this paper.

19

 A)  = Mi∈J (I(C(x  i ), A(x  i ))) ≤ Mi∈J (I(B(x  i ), A(x  i ))) = σ(B,  A)  σ(C, The proof of the second part is similar.  B)  = Mi∈J (I(A(x  i ), B(x  i ))) = [0, 0]. Then, by axiom (IA1) (IG2) Let σ(A,  i )) = [0, 0] for all i ∈ J. Finally, by axiom (II14)  i ), B(x it follows I(A(x  i ) = [0, 0], so A = =  i ) = [1, 1] and B(x 1 and B 0. we have A(x The proof in the opposite direction follows from (IF2’).  i ), B(x  i ))) = [1, 1]. Then,  B)  = Mi∈J (I(A(x (IDI1)=(IY1)=(IF1) Let σ(A,  i )) = [1, 1] for all i ∈ J. Thus,  i ), B(x by axiom (IA2) it follows I(A(x   by axiom (II8) we have A(xi ) ≤ B(xi ) for all i ∈ J, and consequently  ⊆ B.  A The proof in the opposite direction is similar. A c ) = Mi∈J (I(A(x  i ), A c (xi ))) = [0, 0]. From axiom (IA1) (IDI2) Let σ(A, c (xi )) = [0, 0], for all i ∈ J, and from axiom (II14)  i ), A we have I(A(x  i ) = [1, 1], so A = it follows A(x 1. The proof in the opposite direction is obvious. (IDI3) See the proof of (IF3’). (IY2)=(IF2) See the proof of (IDI2). (IY3) See the proof of (IF3’). From the previous theorem we have conditions for the interval aggregation function M and the interval fuzzy implication I under which a mapping σ given by (11) is a fuzzy subsethood measure. Corollary 8. Let σ : IV F S(U ) × IV F S(U ) → D be a mapping given by (11). Let M : Dn → D be a mapping satisfying axioms (IA1)-(IA3), and I : D2 → D be a mapping satisfying axioms (II1), (II2), (II8) and (II14). Then • σ is a DI-subsethood measure for IVFSs, • σ is a fuzzy subsethood measure for IVFSs in the sense of Young, and • σ is a fuzzy subsethood measure for IVFSs in the sense of Fan. Note that from our point of view the axioms (II8) and (II14) are of special significance. Next corollary relates to weak fuzzy subsethood measures. 20

Corollary 9. Let σ : IV F S(U ) × IV F S(U ) → D be a mapping given by (11). Let M : Dn → D and I : D2 → D be mappings satisfying axioms • M: (IKF1)-(IKF3), I: (IBC1), (IBC2), (II1), (II2), then σ is a weak fuzzy subsethood measure for IVFSs in the sense of Fan, • M: (IA1)-(IA3), I: (II1), (II2), (II14), then σ is a weak fuzzy subsethood measure for IVFSs in the sense of Young, • M: (IA1), (IKF2), (IKF3), I: (IBC1), (II1), (II2), (II14), then σ is a weak fuzzy subsethood measure for IVFSs of Class 1, • M: (IA1)-(IA3), I: (II1), (II2), (II8), (II14), then σ is a weak fuzzy subsethood measure for IVFSs of Class 2. 8.1. Construction of new (weak) subsethood measures for IVFSs Our method for construction of new (weak) fuzzy subsethood measures is based on the best interval representation. We choose some fuzzy implication I and some aggregation function M , and apply their best interval , respectively, to the formula (11): representations, I and M  B)  =M i∈J I(  i )) .  A(x  i ), B(x σ(A,

(12)

Recall that in Section 6 (Theorems 3, 5) we showed that the best interval representation of an aggregation function preserves all the relevant axioms (IKF1)-(IKF3) and (IA1)-(IA4). Similarly, in Section 5 (Propositions 4, 5 and Theorem 1) we showed that the best interval representation of a fuzzy implication preserves all the relevant axioms (IBC1)-(IBC2) and (II1)-(II14), except of (II8) and (II11). So, all the results of Theorem 7 and Corollaries 8, 9 that do not relate to axioms (II8) and (II11) can be reformulated in the terms of the best interval representation. However, from the Table 1 it is obvious that (II8) relates to the axiom (IY1)=(IF1)=(IDI1), so it relates to all three considered kinds of subsethood measures and to weak subsethood measure of Class 2. Thus, our method allows us to construct only new weak fuzzy subsethood measures in the sense of Young, Fan and of Class 1: Theorem 10. Let σ : IV F S(U ) × IV F S(U ) → D be a mapping given by (12). Let M : [0, 1]n → [0, 1] and I : [0, 1]2 → [0, 1] be mappings satisfying axioms • M : (KF1)-(KF3), I: (BC1), (BC2), (I1), (I2), then σ is a weak fuzzy subsethood measure for IVFSs in the sense of Fan, 21

• M : (A1)-(A3), I: (I1), (I2), (I14), then σ is a weak fuzzy subsethood measure for IVFSs in the sense of Young, • M : (A1), (KF2), (KF3), I: (BC1), (I1), (I2), (I14), then σ is a weak fuzzy subsethood measure for IVFSs of Class 1. Proof. Immediately follows from the Corollary 9, Propositions 4, 5 and Theorems 1, 3, 5. Remark. In Section 5, we formulated weaker form of axiom (II8): (nII8)

I(X, Y ) = [1, 1] iff X ≤N Y for all X, Y ∈ D

where ≤N is defined: X ≤N Y if and only if X ≤ Y . This weaker form is satisfied by the best interval representation I of a fuzzy implication I satisfying (I8) - see Theorem 2. Recall that necessary inclusion for IVFSs (6) is based on the ordering of intervals ≤N :  ⊆N B  A

  A(x) ≤N B(x) ∀x ∈ U .

iff

This leads us to weaken the axiom (IY1)=(IF1)=(IDI1) in the following way:  B)  = [1, 1] if and only if A  ⊆N B  (i.e., A(x)   (nIY1) σ(A, ≤N B(x) for all x ∈ U ). If we replace (IY1)=(IF1)=(IDI1) by the axiom (nIY1), we will refer to emerged subsethood measures as w-subethood measure in the sense of Young, w-subsethood measure in the sense of Fan and wDI-subsethood measure, respectively. Definition 6. A mapping σ : IV F S(U ) × IV F S(U ) → D is called • w-subsethood measure for IVFSs in the sense of Young, if σ satisfies (nIY1), (IY2) and (IY3), • w-subsethood measure for IVFSs in the sense of Fan, if σ satisfies (nIY1), (IF2) and (IF3), • wDI-subsethood measure for IVFSs, if σ satisfies (nIY1), (IDI2) and (IDI3),

22

Now, our method allows us to construct not only new weak fuzzy subsethood measures in the sense of Young, Fan and of Class 1 (Theorem 10), but, also new w-subsethood mesures in the sense of Young, new w-subsethood measures in the sense of Fan and new wDI-subsethood measures for IVFS: Theorem 11. Let σ : IV F S(U ) × IV F S(U ) → D be a mapping given by (12). Let M : [0, 1]n → [0, 1] be a mapping satisfying axioms (A1)-(A3), and I : [0, 1]2 → [0, 1] be a mapping satisfying axioms (I1), (I2), (I8) and (I14). Then • σ is wDI-subsethood measure for IVFSs, • σ is w-subsethood measure for IVFSs in the sense of Young, and • σ is w-subsethood measure for IVFSs in the sense of Fan.  by Theorem Proof. If I satisfies (I8) then, its best interval representation I, 2 satisfies (nII8). We need to prove that then also σ satisfies (nIY1):  i ), B(x  i )) = [1, 1]. Then, by axiom (IA2) it  B)  = Mi∈J I(A(x Let σ(A,  i ), B(x  i )) = [1, 1] for all i ∈ J. Thus, by axiom (nII8) we have follows I(A(x  ⊆N B.  i ) for all i ∈ J, and consequently A  The proof in the  i ) ≤N B(x A(x opposite direction is similar. The rest immediately follows from the Corollary 8, Propositions 4, 5 and Theorems 1, 3, 5. 9. Examples  and B  be IVFSs The data in all of the examples are as following. Let A in U = {x1 , . . . , x10 }:  = { x1 , [0, 0], x2 , [0, 0], x3 , [0, 0], x4 , [0, 0], x5 , [0, 1 ], x6 , [ 1 , 2 ], A 3 2 3

x7 , [1, 1], x8 , [0, 12 ], x9 , [0, 0], x10 , [0, 0]},  = { x1 , [0, 0], x2 , [0, 0], x3 , [0, 1 ], x4 , [ 1 , 2 ], x5 , [1, 1], x6 , [ 3 , 4 ], B 3 2 3 4 5

x7 , [ 12 , 34 ], x8 , [ 14 , 25 ], x9 , [0, 15 ], x10 , [0, 0]}. Example 1. Let I be a Lukasiewicz implication, and let aggregation function M be an arithmetic mean, i.e., 1

I(x, y) = min(1, 1 − x + y) , Mi∈J (xi ) = xi |J| i∈J where |J| denotes cardinality of J. Then, using (8) we have I = [I(X, Y ), I(X, Y )] = [min(1, 1 − X + Y ), min(1, 1 − X + Y )] 23

and from Corollary 4 we have



1 1 i∈J (Xi ) = [Mi∈J (Xi ), Mi∈J (Xi )] = Xi , Xi . M |J| i∈J |J| i∈J 

B  is given by So, subsethood measure for IVFSs A,  B)  = M i∈J ([min(1, 1 − A(x  i )), min(1, 1 − A(x  i ) + B(x  i ) + B(x  i ))] σ1 (A, 

1 1

 i )),  i ) + B(x  i ) + B(x  i )) min(1, 1 − A(x min(1, 1 − A(x = |J| i∈J |J| i∈J Because Lukasiewicz implication satisfies all the considered axioms (BC1), (BC2), (I1), (I2), (I8), (I14), and arithmetic mean satisfies all the axioms (A1)-(A3), σ1 is weak subsethood measure in the sense of Young, in the sense of Fan and also of Class1 (Theorem 10), moreover, σ1 is wDI-subsethood measure, w-subsethood measure in the sense of Young and in the sense of Fan (Theorem 11).  B)  for our IVFSs A, B  for three various index sets J The values of σ1 (A, are in the Table 2. Recall that if J = {1, . . . , 10} we aggregate implications  for all x ∈ U ; if J = {3, . . . , 9} we aggregate implications for all x ∈ supp(A)∪  supp(B); and if J = {5, . . . , 8} we aggregate implications only for all x ∈  supp(A). Example 2. Let I be a Lukasiewicz implication as in previous example, and let aggregation function M be minimum. Then, using Corollary 4 we have   i∈J (Xi ) = [Mi∈J (Xi ), Mi∈J (Xi )] = min{Xi | i ∈ J}, min{Xi | i ∈ J} . M B  is given by So, subsethood measure for IVFSs A,  B)  = M i∈J ([min(1, 1 − A(x  i )), min(1, 1 − A(x  i ) + B(x  i ) + B(x  i ))] σ2 (A,   i )) | i ∈ J},  i ) + B(x = min{min(1, 1 − A(x   i ) + B(x  i )) | i ∈ J} = min{min(1, 1 − A(x   i )) | i ∈ J},  i ) + B(x = min(1, min{1 − A(x   i ) + B(x  i )) | i ∈ J} min(1, min{1 − A(x 24

Table 2: Summary of the Examples 1, 2, 3 and 4.

I(x, y) = min(1,

1 − x + y) 1 Mi∈J (xi ) = |J| i∈J xi I(x, y) = min(1, 1 − x + y) Mi∈J = min{xi | i ∈ J} I(x, y) = 1 − x + xy 

Mi∈J (xi ) =

i∈J

x2i

|J|

I(x, y) = 1 − x + xy Mi∈J (xi ) = i∈J xi

J {1, . . . , 10} {3, . . . , 9} {5, . . . , 8} {1, . . . , 10} {3, . . . , 9} {5, . . . , 8} {1, . . . , 10} {3, . . . , 9} {5, . . . , 8} {1, . . . , 10} {3, . . . , 9} {5, . . . , 8}

 B)  σ(A, [0.925, 0.975] [0.893, 0.964] [0.8125, 0.9375] [0.5, 0.75] [0.5, 0.75] [0.5, 0.75] [0.913, 0.968] [0.873, 0.954] [0.764, 0.918] [0.26, 0.675] [0.26, 0.675] [0.26, 0.675]

Because min satisfies the axioms (KF1)-(KF3) and does not satisfy the axiom (A1), σ2 is weak subsethood measure in the sense of Fan, but it is not weak subsethood measure in the sense of Young and of Class1 (Theorem 10), moreover, σ2 is neither wDI-subsethood measure nor w-subsethood measure in the sense of Young nor in the sense of Fan (Theorem 11).  B)  for our IVFSs A, B  for three various index sets J The values of σ2 (A, are in the Table 2. Example 3. Let I be a Reichenbach implication and let aggregation function M be quadratic mean, i.e.,    x2i  i∈J I(x, y) = 1 − x + xy , Mi∈J (xi ) = |J| B  is given by Then subsethood measure for IVFSs A,    B)  = M i∈J 1 − A(x  i ), 1 − A(x  i ) + A(x  i ) · B(x  i ) + A(x  i ) · B(x  i) σ3 (A,  ⎤ ⎡    (1 − A(x  2 2       (1 − A(xi ) + A(xi ) · B(xi )) ⎥ i ) + A(xi ) · B(xi ))  ⎢  i∈J ⎥ ⎢ i∈J , =⎢ ⎥ |J| |J| ⎦ ⎣

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Reichenbach implication satisfies (BC1), (BC2), (I1), (I2), (I14), but it does not satisfy (I8); quadratic mean satisfies all the axioms (A1)-(A3). So, σ3 is weak subsethood measure in the sense of Young, in the sense of Fan and also of Class1 (Theorem 10), however, σ3 is neither wDI-subsethood measure nor w-subsethood measure in the sense of Young nor in the sense of Fan (Theorem 11).  B)  for our IVFSs A, B  for three various index sets J The values of σ3 (A, are in the Table 2. Example 4. Let I be a Reichenbach implication as in previous example and B  is given by let M be product. Then subsethood measure for IVFSs A,    B)  = M i∈J 1 − A(x  i ), 1 − A(x  i ) + A(x  i ) · B(x  i ) + A(x  i ) · B(x  i) σ4 (A,  =



 i )),  i ) + A(x  i ) · B(x (1 − A(x

i∈J



 i ) + A(x  i ) · B(x  i )) (1 − A(x

i∈J

Product satisfies (KF1)-(KF3), but it does not satisfy (A1). So, σ4 is weak subsethood measure in the sense of Fan, however, σ4 is neither weak subsethood measure in the sense of Young nor weak subsethood measure of Class1 (Theorem 10). Obviously, σ4 is neither wDI-subsethood measure nor wsubsethood measure in the sense of Young nor in the sense of Fan (Theorem 11).  B)  for our IVFSs A, B  for three various index sets J The values of σ4 (A, are in the Table 2. 10. The specific nature of intervals 10.1. Comparison of subsethood measure for IVFSs and corresponding subsethood measure for FSs B  computed via our method The resulting subsethood measure of IVFSs A,  B)  whose midpoint may not be equal to the given by (12) is an interval σ(A, DI-subsethood measure [21] of FSs A, B given by midpoints of intervals rep B.  resenting membership grades of A,   More precisely, let A, B be IVFSs in U , let A, B be FSs in U given by A(x) =

 + A(x)  A(x) 2

,

B(x) = 26

  B(x) + B(x) 2

,

for all x ∈ U . Let σ(A, B) = Mi∈J I(A(xi ), B(xi )) be subsethood measure of FSs induced by some aggregation function M and implication I. Let  B)  =M i∈J I(  i ))  A(x  i ), B(x σ(A, be subsethood measure of IVFSs induced by the best interval representations of M and I. Then the following equality does not hold in general:  B)  + σ(A,  B)  σ(A, 2

= σ(A, B).

Example 5. Let FSs A, B in U = {x1 , . . . , x10 } be given by midpoints of B  from Examples 1-4 intervals representing membership grades of IVFSs A, (see Section 9): 5 , x7 , 1, x8 , 14 , A = { x1 , 0, x2 , 0, x3 , 0, x4 , 0, x5 , 16 , x6 , 12

x9 , 0, x10 , 0}, 5 B = { x1 , 0, x2 , 0, x3 , 16 , x4 , 12 , x5 , 1, x6 , 31 , x7 , 58 , x8 , 13 , 40 40 1

x9 , 10 , x10 , 0}. The values σ1 (A, B), . . . , σ4 (A, B) induced by M and I from Examples 1-4 are in the third column of Table 3. In the fourth column are midpoints  B),  . . . , σ4 (A,  B)  computed in Examples 1-4, i.e. the midpoints of of σ1 (A, intervals in the last column of Table 2. We can see that the results are not  B,  the results the same (except of σ2 , but also in this case, for some other A, need not be the same). 10.2. The width of the resulting subsethood measure The width of intervals, w(X) = X − X, is the crucial characteristic which represents the uncertainty linked to the construction of the intervals. Now we  B)  show that, under some conditions, the width of subsedhood measure σ(A, given by (12) depends on the widths of intervals representing membership  and B.  grades of IVFSs A Recall that an interval representation F : Dn → D of a real function f : [0, 1]n → [0, 1] is said to be Lipschitz if there is a constant L such that w(F (X)) ≤ L w(X) for every X ∈ Dn (e.g. [34]).

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Table 3: Comparison of the results of subsethood measure for IVFSs and FSs.

σ1

σ2

σ3

σ4

J {1, . . . , 10} {3, . . . , 9} {5, . . . , 8} {1, . . . , 10} {3, . . . , 9} {5, . . . , 8} {1, . . . , 10} {3, . . . , 9} {5, . . . , 8} {1, . . . , 10} {3, . . . , 9} {5, . . . , 8}

σ(A, B) 0.9625 0.9464 0.9062 0.625 0.625 0.625 0.94 0.8337 0.709 0.4513 0.4513 0.4513

 B)+σ(   B)  σ(A, A, 2

0.95 0.9285 0.875 0.625 0.625 0.625 0.9405 0.9135 0.841 0.4675 0.4675 0.4675

Proposition 6. If a real-valued function f satisfies an ordinary Lipschitz condition in [0, 1]n for some L ∈]0, ∞[, i.e., |f (x) − f (y)| ≤ L|x − y| ,

for all x, y ∈ [0, 1]n ,

then the united extension of f is a Lipschitz interval extension. Proof. See [34], Lemma 6.2. Note that the notion ’united extension’ of f encompasses also our best interval representation f. In the following theorem we describe the link  B)  and the widths of intervals between the width of subsethood measure σ(A,  B.  Recall that, for intervals representing membership grades of IVFSs A, X1 , . . . , Xn , we have w(X1 , . . . , Xn ) = max(w(X1 ), . . . , w(XN )). Theorem 12. Let σ be a subsethood measure for IVFSs given by (12), where I be a Lipschitz implication and M be a Lipschitz n-ary aggregation function. B  on the universe Then there exists a constant L such that, for all IVFSs A, {x1 , . . . , xn }, it holds    1 ), . . . , A(x  n ), B(x  n) .  B))  ≤ L w A(x  1 ), B(x w(σ(A,

28

Proof.

   i )) ≤  B))  =w M i∈J I(  A(x  i ), B(x w(σ(A,    1 )), . . . , I(  A(x  n ), B(x  n )) =  A(x  1 ), B(x ≤ L1 w I(         = L1 max w(I(A(x1 ), B(x1 ))), . . . , w(I(A(xn ), B(xn ))) ≤    1 ), B(x  1 )), . . . , L2 w(A(x  n ), B(x  n )) = ≤ L1 max L2 w(A(x    1 )), w(B(x  1 ))), . . . , max(w(A(x  n )), w(B(x  n ))) = = L1 L2 max max(w(A(x    1 )), w(B(x  1 )), . . . , w(A(x  n )), w(B(x  n )) = = L1 L2 max w(A(x    1 ), B(x  1 ), . . . , A(x  n ), B(x  n) . = L1 L2 w A(x

11. Conclusion We generalized aggregation functions and proposed a technique for aggregation of intervals. Then we used the technique and introduced a new constructive method for (weak) fuzzy subsethood measures for interval-valued fuzzy sets based on the aggregation of fuzzy interval implications. For this purpose the notion of the best interval representation of a real function was implemented. The benefit of our approach lies in the wide range of obtained fuzzy subsethood measures for interval-valued fuzzy sets. Moreover, the method gives the possibility to construct new fuzzy subsethood measures with properties that are appropriate to the situation under investigation. The properties of obtained fuzzy subsethood measure depend on the properties of chosen fuzzy implication and aggregation function which are well-known. Using our method, one can obtain new weak fuzzy subsethood measures for intervalvalued fuzzy sets in the sense of Fan [4], in the sense of Young [14] and of Class 1 [21]. Moreover, we proposed three (re)new classes of subsethood measures: w-subsethood measures in the sense of Fan, in the sense of Young and wDI-subsethood measures; and, our method allows us to obtain new fuzzy subsethood measures of each of the three classes.

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