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A characterization for some type-2 fuzzy strong negations
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A characterization for some type-2 fuzzy strong negations✩ ∗
S. Cubillo a , , C. Torres-Blanc a , P. Hernández-Varela b a
Universidad Politécnica de Madrid (UPM), DMATIC, 28660 Boadilla del Monte, Madrid, Spain Universidad Mayor, Facultad de Estudios Interdisciplinarios, Núcleo Matemáticas, Física & Estadística, Manuel Montt 318, Providencia, Santiago, Chile b
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info
Article history: Received 14 November 2018 Received in revised form 19 November 2019 Accepted 25 November 2019 Available online xxxx Keywords: Type-2 fuzzy sets Normal and convex functions Negation Strong negation Order automorphism
a b s t r a c t P. Hernandez et al. in 2014 established the axioms that an operation must fulfill in order to be a negation on a bounded poset (partially ordered set). In this work, we focus on the set L of the membership degrees of the type-2 fuzzy sets which are normal and convex functions in [0,1]. This set has a bounded and complete lattice structure, thank to which negations and strong negations have been constructed by the authors applying the Zadeh’s Extension Principle. In addition, the authors showed new negations on L that are different from the negations presented in 2014 applying the Zadeh’s Extension Principle. In this work, the authors obtain a characterization of the strong negations on L that leave the constant function 1 fixed. © 2019 Published by Elsevier B.V.
1. Introduction Type-2 fuzzy sets (T2FSs) were introduced by L.A. Zadeh in 1975 [1] as an extension of type-1 fuzzy sets (T1FSs) [2]. While the membership degree of an element in a T1FS is a value in the interval [0, 1], the membership degree of an element in a T2FS is a fuzzy set in [0, 1]. That is, the degree in which an element belongs to the set is just a label of the linguistic variable ‘‘TRUTH’’. In this way, a T2FS is determined by a membership function µ : X → M, where M = Map([0, 1], [0, 1]) is the set of the functions from [0, 1] on [0, 1] (see [3–5]). Because the membership degrees of T2FSs are fuzzy, they are better able to model uncertainty than FSs [3,6]. In the last decade, many researchers have studied type2 fuzzy set theory and its application to several fields of science (see, for example, [7–14]). In this paper, we will focus on T2FSs with degrees of membership in the subset L of normal and convex functions of M. Since the introduction of T2FSs, lots of the operations, properties and results obtained in the T1FSs have been extended to T2FSs throughout the Zadeh’s Extension Principle (see for example, [3–5,15]). This is the case of the negation operator, used to obtain, for example, complement of a set, dual of a t-norm or a t-conorm, entropies, implications, as well as to study the ✩ No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.knosys. 2019.105281. ∗ Corresponding author. E-mail addresses: [email protected] (S. Cubillo), [email protected] (C. Torres-Blanc), [email protected] (P. Hernández-Varela).
possible contradiction appearing in a fuzzy system. For example, strong negations on Atanassov’s intuitionistic fuzzy sets (A-IFSs) and on T2FSs have been applied in order to study the possible contradiction appearing in a fuzzy system (see [16,17]). Thus, the study of strong negations is essential for both fuzzy sets and their extensions, and particularly for type-2 fuzzy sets. In [15] the authors presented a deep study about the negations on the T2FSs. Particularly, they proposed the axioms for negations and strong negations on a bounded partially ordered set (bounded poset). Also, using the Zadeh’s Extension Principle, they obtained families of negations and strong negations on L. Moreover, in [18] for the first time negations on M were presented, as well as new negations and strong negations on L. Trillas in [19] studied and characterized strong negations in [0, 1] through order automorphisms in [0, 1]. Bustince et al. in [20] introduced intuitionistic generators in order to build negations in A-IFSs; and Deschrijver et al. in [21] characterized strong intuitionistic negations based on strong negations in [0, 1]. Then, in a similar way, to characterize the strong negations in the T2FS, or equivalently in the set of its membership degrees M, is a good challenge. But, as the algebra of these sets is much more complicated, this objective is not easy. On the other hand, usually, the linguistic labels used in the real situations are convex and normal, that is, they belong to L. More, as the label 1 (1(x) = 1 for all x in [0, 1]) means lack of information, it seems suitable that its negation be also the label 1. That is, the negation should leave the function constantly 1 fixed. For example, if a biologist must determine to what extent ‘‘the species X is appropriate for an environment’’, he or she could say that this assertion is ‘‘very true’’, ‘‘more or less true’’, ‘‘absolutely
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Please cite this article as: S. Cubillo, C. Torres-Blanc and P. Hernández-Varela, A characterization for some type-2 fuzzy strong negations, Knowledge-Based Systems (2019) 105281, https://doi.org/10.1016/j.knosys.2019.105281.
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false’’, . . . , or, if the expert does not have an opinion, he or she could associate the label 1. Then, in this case, the biologist should also associate to the statement ‘‘the species X is NOT appropriate for an environment’’ the linguistic label 1. According to the previous comments, the purpose of this paper is to characterize the negations in L that leave the label 1 fixed. That is, to find out all strong negations satisfying this condition. The paper is organized as follows. In Section 2 we review some definitions, operations and properties on FSs and T2FSs reported in the literature (see [5,15] and [18]) and needed to understand the rest of the work. Also in this Section 2 the axioms of the negation in [0, 1] and in a bounded poset are exposed, as well as some results about negations in L obtained by the authors in previous papers. Section 3 is devoted to obtaining some properties of the strong negations on L that leave the constant function 1 fixed. These properties allow us, given a such strong negation on L, to construct both a strong negation and an order automorphism on [0, 1] associated with it. In Section 4, a significant result (Theorem 5) is achieved, from which, and together with the results of Section 3, we are going to characterize all strong negations on L that leave the constant function 1 fixed. Finally, Section 5 contains some conclusions.
Fig. 1. Example of a T2FS.
2. Preliminaries Throughout the paper, X will denote a non-empty set which will represent the universe of discourse. Additionally, ≤ will denote the usual order relation in the lattice of real numbers, and ∧ the minimum operator and ∨ the maximum operator.
Fig. 2. Example of f L and f R .
2.1. Definitions and properties of FSs and T2FSs
Definition 3. A function f ∈ M is normal if sup{f (x) : x ∈ [0, 1]} = 1 and a function f ∈ M is convex, if for any x ≤ y ≤ z, it holds that f (y) ≥ f (x) ∧ f (z).
Definition 1 ([2]). A type-1 fuzzy set (FS), A, is characterized by a membership function µA ,
The following definition and theorem were given in previous papers in order to facilitate the operations in the set M:
µA : X → [0, 1], where µA (x) is the membership degree of an element x ∈ X in the set A. Definition 2 ([4]). A type-2 fuzzy set (T2FS), A, is characterized by a membership function:
µA : X → M = [0, 1][0,1] = Map ([0, 1], [0, 1]), that is, µA (x) is a type-1 fuzzy set in the interval [0, 1] and also the membership degree of the element x ∈ X in the set A. Therefore,
µA (x) = fx , where fx : [0, 1] → [0, 1]. Let T 2FS(X ) = Map(X , M) denotes the set of all type-2 fuzzy sets on X. Fig. 1 shows an example of a type-2 fuzzy set on the finite universe of discourse T = {0, 1, 2, 3, 4}. Walker and Walker justify in [5] that the operations on Map(X , M) can be defined naturally from the operations on M and have the same properties. Thus, in the same way as in the case of the FSs, where the definitions and properties are given on ([0, 1], ≤), in this paper we will work on M, as all the results are easily and directly extensible to Map(X , M). Moreover, we restrict to type-2 fuzzy sets in the special case in which the membership degrees are in L, the set of normal and convex functions of M. There are several reasons to do so. First, membership degrees represent linguistic labels of the TRUTH variable, so it is not uncommon to require them to be convex and normal. Furthermore, it has been pointed out that this set L will contain a bounded and complete lattice structure, and as consequence t-norms, t-conorms, aggregation operators and specifically negations can be properly constructed (see [15,22, 23]).
Definition 4 ([5,24,25]). If f ∈ M, we define f L , f R ∈ M as f L (x) = sup{f (y) : y ≤ x}, f R (x) = sup{f (y) : y ≥ x}. Note that f L and f R are increasing and decreasing, respectively (see Fig. 2), f ≤ f L , f ≤ f R , for all f ∈ M [5], where ≤ is the usual order in the set of functions (f ≤ g if and only if f (x) ≤ g(x), for all x). In previous papers two partial orders on M (⊑ and ⪯) were defined [4,5]. Generally, these partial orders do not coincide [4,5] and M does not have a lattice structure with either of the two orders [5,26]. However, in L these partial orders are equivalent, that is, ⊑ ≡ ⪯; and (L, ⊑) is a bounded and complete lattice (see [5,25]). Let us observe that the lattice structure is necessary for the following reasons. Firstly, a negation is always decreasing respect to an order. Furthermore, in the proof of some results, we will need to obtain the supremum and the infimum of two functions, and so it is necessary to have a lattice structure. The following characterization of the partial order ⊑ on L will be useful for establishing the results of this work. Theorem 1 ([25]). Let f , g ∈ L (see Fig. 3). f ⊑ g if and only if g L ≤ f L and f R ≤ g R . In the following proposition we will remember some properties of the functions in the lattice (L, ⊑), given by Walker and Walker. Proposition 1 ([5]). Let f , g ∈ L. 1. f is increasing ⇔ f = f L ⇔ f (1) = 1, and f is decreasing ⇔ f = f R ⇔ f (0) = 1. 2. If f , g ∈ L are increasing, it holds that, f ⊑ g ⇔ g ≤ f .
Please cite this article as: S. Cubillo, C. Torres-Blanc and P. Hernández-Varela, A characterization for some type-2 fuzzy strong negations, Knowledge-Based Systems (2019) 105281, https://doi.org/10.1016/j.knosys.2019.105281.
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3. Strong negations on L that leave the function 1 fixed In the previous section, we have indicated that, given n and
Fig. 3. Example where f ⊑ g.
3. If f , g ∈ L are decreasing, it holds that, f ⊑ g ⇔ f ≤ g. 4. f = f L ∧ f R , and f L ∨ f R = 1. 2.2. Negations on L This subsection is devoted to review the study carried out in [15] and [18] on the negations in the framework of the T2FSs with membership degrees in L. Firstly, let us recall the definition of negation in ([0, 1], ≤). Definition 5. A function n : [0, 1] → [0, 1] is a negation if it is decreasing respect to the order ≤ and satisfies n(0) = 1 and n(1) = 0. If, in addition, n(n(x)) = x for all x ∈ [0, 1], then it is said to be a strong negation. Definition 5 suggests us an extension to any partially ordered set (poset) with minimum and maximum elements (bounded). In this sense, Hernández et al. [15] introduced negations in this algebraic structure and gave some negations in L. Definition 6 ([15]). Let A be a set and ≤A be a partial order in A such that (A, ≤A ) has a minimum element 0≤A and a maximum element 1≤A . A negation in (A, ≤A ) is a decreasing function N : A → A such that N(0≤A ) = 1≤A and N(1≤A ) = 0≤A . If, additionally, N(N(x)) = x holds for all x ∈ A, it is said to be a strong negation. Proposition 2. Let (A, ≤A ) be a bounded lattice. Given a strong negation N : A → A, for any a, b ∈ A it is N(inf{a, b}) = sup{N(a), N(b)} and N(sup{a, b}) = inf{N(a), N(b)}. Definition 7 ([5]). Let [a, b] ⊆ [0, 1]. The characteristic function of [a, b] is [a, b] : [0, 1] → [0, 1], where
[a, b](x) =
1, 0,
{
if x ∈ [a, b], if x ∈ / [a, b],
and the characteristic function of a number a ∈ [0, 1] or singleton is a¯ = [a, a]. Note that 0¯ and 1¯ are the minimum and maximum, respectively, of the bounded lattice (L, ⊑). In a similar way we define the characteristic functions of the intervals [a, b), (a, b], or (a, b). Moreover, [a, b] ⊑ [c , d] if and only if a ≤ c, b ≤ d. In [18] we have obtained a family of negations on (L, ⊑) that transform singletons into singletons and are closed on the set of characteristic functions of closed intervals. Theorem 2 ([18]). Let n be a strong negation in ([0, 1], ≤) and let α be an order automorphism in ([0, 1], ≤). And the operation Nn,α : L → L defined as Nn,α (f ) = (α ◦ f R ◦ n) ∧ (α −1 ◦ f L ◦ n). Then (a) Nn,α is a strong negation in L. (b) Nn,α (a¯ ) = n(a) and Nn,α ([a, b]) = [n(b), n(a)]. (c) Nn,α (p) = p and Nn,α (1) = 1, being p the fixed point of the strong negation, n, in [0, 1] and 1 the constant function 1.
α strong negation and automorphism, respectively, on ([0, 1], ≤), the operations Nn,α are strong negations on (L, ⊑). Moreover, for all a, b ∈ [0, 1], with a ≤ b, they transform the function [a, b] on the function [n(b), n(a)], and, in particular, they transform the function 1 = [0, 1] into the same function 1 = [n(1), n(0)] = [0, 1]. That is, those strong negations leave the function 1 fixed. In order to characterize the strong negations on (L, ⊑) satisfying this condition, the present section is devoted to study the properties of these negations. In particular, we will present some previous results that, step by step, will allow us to get a final theorem. A first lemma, will characterize increasing and decreasing functions in L. Let us denote Lc = {f ∈ L : f increasing } and Ld = {f ∈ L : f decreasing }. Lemma 1. Given f ∈ L, we have: f ∈ Ld ⇔ f ⊑ 1, f ∈ Lc ⇔ 1 ⊑ f , f L = f R ⇔ f = 1. Proof. f ∈ Ld ⇔ f R = f ≤ 1 = 1R and f L = 1 = 1L (by Proposition 1) ⇔ f ⊑ 1 (by Theorem 1). In a similar way, we have the second equivalence, and the third one is trivial. □ Proposition 3. Let N : L → L be a strong negation on L, with N(1) = 1. Then f ∈ Ld ⇔ N(f ) ∈ Lc , f ∈ Lc ⇔ N(f ) ∈ Ld . Proof. By the previous Lemma, f ∈ Ld ⇔ f ⊑ 1. Then, as N is a strong negation with N(1) = 1, it is f ⊑ 1 ⇔ N(1) = 1 ⊑ N(f ) ⇔ N(f ) ∈ Lc . The proof of the second equivalence is analogous. □ Remark 1. Now, it is easy to proof that, if N : L → L is a strong negation with N(1) = 1, then N(Lc ) = Ld and N(Ld ) = Lc . Before obtaining new properties for the strong negations with N(1) = 1, we need the following lemmas and corollary on (L, ⊑). Lemma 2. If f , g ∈ L then f ⊑ g if and only if f L ⊑ g L and f R ⊑ g R . Proof. Straight from Theorem 1 and Proposition 1. □ Lemma 3. If f ∈ L, then f ⊑ f L and f R ⊑ f . Proof. By Lemma 2, it is f ⊑ f L ⇔ f L ⊑ (f L )L = f L and f R ⊑ (f L )R = 1 that is always true, as f R is decreasing (see Fig. 4). f R ⊑ f ⇔ (f R )L = 1 ⊑ f L , that is always true as f L is increasing, and (f R )R = f R ⊑ f R (see Fig. 4). □ Corollary 1. R
If f ∈ L, then,
f = max⊑ {g ∈ Ld : g ⊑ f }
and
f L = min⊑ {g ∈ Lc : f ⊑ g }.
Proof. As f R is decreasing, by Lemma 3, we have f R ∈ {g ∈ Ld : g ⊑ f }. Moreover, for all g ∈ Ld such that g ⊑ f , it is g = g R ⊑ f R , and so f R is an upper bound of the set {g ∈ Ld : g ⊑ f } and belongs to it. Then f R is the maximum of that set. The proof of the second equality is similar. □
Please cite this article as: S. Cubillo, C. Torres-Blanc and P. Hernández-Varela, A characterization for some type-2 fuzzy strong negations, Knowledge-Based Systems (2019) 105281, https://doi.org/10.1016/j.knosys.2019.105281.
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Moreover, N(f R )
N(g R ), and so
L
a = (N(f ))L = N(f R )
N(g R ) = (N(g))L
L
R
that is, (N(g))L ≨ a . Then a ⊑ N(g) with (N(g))R = a . Also, as a is a singleton, according to Lemma 4 it is N(g) = a, and N(a) = g, attaining a contradiction. In a similar way, that (ii) is not possible can be proved. Therefore, N(a) have to be a singleton. □ Fig. 4. Example of f ∈ L, f L and f R showing the result of Lemma 3.
Remark 2. Let (A, ≤A ) be a partially ordered set and D : A → A a decreasing function (respectively C : A → A increasing). Given a subset F ⊂ A, if there exists the maximum of F then D(max F ) = min D(F ) (respectively C (max F ) = max C (F )). And if there exists the minimum of F then D(min F ) = max D(F ) (respectively C (min F ) = min C (F )). Proposition 4. Let N : L → L be a strong negation with N(1) = 1. Then, for all f ∈ L it is (N(f ))L = N(f R ) and (N(f ))R = N(f L ). Proof. Direct from Corollary 1, and Remarks 1 and 2. □ This Proposition allows us, given a strong negation N on L with N(1) = 1, to obtain the negation of any f ∈ L as the minimum of the negation of a decreasing function and the negation of an increasing function. In fact, by Proposition 1, N(f ) = (N(f ))L ∧ (N(f ))R = N(f R ) ∧ N(f L )
Now, the following results are aimed at proving that the image of characteristic functions of intervals are also characteristic functions of intervals. To do this, firstly we will define a strong negation n on [0, 1] from the strong negation N on L. Theorem 3. If N : L → L is a negation such that the image of any singleton is also a singleton, then the function n : [0, 1] → [0, 1], ¯ is a negation on [0, 1]. Moreover, if given by n(a) = b if N(a¯ ) = b, N is strong, then n is also a strong negation. Proof. Trivial according to definition of n. □ From now on, nN will designate the negation in [0, 1] defined ¯ in this Theorem 3. That is, nN (a) = b if N(a¯ ) = b. Proposition 6. ( ) If N : L → L is a strong negation with N(1) = 1, then N [a, b] = [nN (b), nN (a)]. Proof.
( ) ( ( ))R ( ) L L [a, b] = a¯ L ⇒ N [a, b] = N [a, b] = N a¯ L = (N(a¯ ))R , ( ) ( ( ))L ( ) R R ¯ L. [a, b] = b¯ R ⇒ N [a, b] = N [a, b] = N b¯ R = (N(b))
Therefore, the negation N is determined by the images of Ld and Lc . Besides, as N is involutive, it also is determined by N(Ld ) or N(Lc ). The following results are aimed at proving that the image of a singleton is also a singleton.
By Proposition 5, N transforms singletons into singletons. Let ¯ = d. ¯ As a ≤ b, it is a¯ ⊑ b¯ us suppose that N(a¯ ) = c¯ and N(b) ¯ ⊑ N(a¯ ) = c¯ . So, d ≤ c. Moreover, because of and d¯ = N(b) ( ( ))R ( ( ))L N [a, b] = c¯ R and N [a, b] = d¯ L , it is that
Lemma 4. Let f ∈ L and let a be a singleton. Then,
N [a, b] = d¯ L ∧ c¯ R = [d, c ] = [nN (b), nN (a)]. □
(
L
1. If f ⊑ a such that f L = a ⇒ f = a R 2. If a ⊑ f such that f R = a ⇒ f = a L
)
Lemma 5. Let N be a strong negation in (L, ⊑) such that N(1) = 1. Then, R
Proof. Let f ⊑ a such that f L = a . f ⊑ a implies f R ≤ a , and R L so f (x) ≤ f R (x) ≤ a (x) = 0, ∀ x > a. On the other hand, f L = a L L implies f (x) ≤ f (x) = a (x) = 0, ∀ x < a. Then f (x) = 0, ∀ x ̸ = a, and since f is normal, f (a) = 1. Therefore f = a. The proof of the second property is analogous. □ Remark 3. If f ∈ L is not a singleton, there exists g ∈ L such that g ⊑ f with g L = f L and g R ≨ f R (see Fig. 5) or there exists g ∈ L such that f ⊑ g with g R = f R and g L ≨ f L (see Fig. 6). Proposition 5. Let N : L → L be a strong negation such that N(1) = 1. Then the image of a singleton by N is also a singleton.
N [0, a) = (nN (a), 1], ∀a ∈ (0, 1],
(
)
N (a, 1] = [0, nN (a)), ∀a ∈ [0, 1).
(
)
Proof. By the previous Proposition, N [0, a] = [nN (a), 1] and ( ) N [a, 1] = [0, nN (a)].
(
)
Let H = {f ∈ Ld : [0, b] ⊑ f ⊑ [0, a], ∀b < a}
= {f ∈ Ld : f (x) = 1, ∀x < a, f (x) = 0, ∀x > a and f (a) ∈ [0, 1]} (see Fig. 7a). Therefore, N(H) = {N(f ) ∈ Lc : f ∈ H }
Proof. Let a be a singleton. Let us suppose that N(a) = f ∈ L is not a singleton. By the previous Remark, there are only two possible cases: (i) There exists g ∈ L such that g ⊑ f with g L = f L and g R ≨ f R , that is, g L = f L and g R f R (g R ⊑ f R and g R ̸ = f R ). See Fig. 5. (ii) There exists g ∈ L such that f ⊑ g with g R = f R and g L ≨ f L (that is, g R = f R and f L g L ). See Fig. 6. In the case (i), N(g L ) = N(f L ) and by Proposition 4, (N(g))R = N(g L ) = N(f L ) = (N(f ))R = a
R
= {N(f ) ∈ Lc : [nN (a), 1] ⊑ N(f ) ⊑ [nN (b), 1], ∀nN (b) > nN (a)} = {g ∈ Lc : [nN (a), 1] ⊑ g ⊑ [nN (b), 1], ∀nN (b) > nN (a)} = {g ∈ Lc : g(x) = 0, ∀x < nN (a), g(x) = 1, ∀x > nN (a) and g(nN (a)) ∈ [0, 1]} (see Fig. 7b). Moreover, given f1 , f2 ∈ H, f1 ⊑ f2 if and only if f1 (a) ≤ f2 (a), as f1 , f2 are decreasing (see Fig. 7a). Therefore, min⊑ (H) = [0, a).
Please cite this article as: S. Cubillo, C. Torres-Blanc and P. Hernández-Varela, A characterization for some type-2 fuzzy strong negations, Knowledge-Based Systems (2019) 105281, https://doi.org/10.1016/j.knosys.2019.105281.
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Fig. 5. Function f is not a singleton, g
f with g L = f L and g R ≨ f R .
Fig. 6. Function f is not a singleton, f
g with f R = g R and g L ≨ f L .
5
Fig. 9. Example of N(fa,1 ) = f b,0 . Fig. 7. (a) Functions of H in the proof of Lemma 5, (b) Functions of N(H).
that [0, 1) ⊑ f ⊑ 1, then f = fa,1 for some a ∈ [0, 1] as [0, 1) ⊑ f ⊑ 1 ⇔ [0, 1) ≤ f ≤ 1, because they are decreasing. In a similar way, given f ∈ L such that 1 ⊑ f ⊑ (0, 1], then f = f a,0 for some a ∈ [0, 1]. Proposition 7. Let N : L → L be a strong negation with N(1) = 1. Then for each a ∈ [0, 1] there exists a unique b ∈ [0, 1] such that N(fa,1 ) = f b,0 (see Fig. 9), and for each c ∈ [0, 1] there exists a unique d ∈ [0, 1] such that N(f c ,0 ) = fd,1 . Fig. 8. (a) Function fa,1 , (b) function f a,0 .
Proof. Any function fa,1 satisfies that [0, 1) ⊑ fa,1 ⊑ 1. Then 1 = N(1) ⊑ N(fa,1 ) ⊑ N([0, 1)) = (0, 1] by Lemma 5.
And given g1 , g2 ∈ N(H), g1 ⊑ g2 if and only if g2 (nN (a)) ≤ g1 (nN (a)), as g1 , g2 increasing (see Fig. 7b). Therefore, max⊑ (N(H)) = (nN (a), 1]. ( ) Finally, by Remark 2, N [0, a) = N(min⊑ (H)) = max⊑ (N(H)) = (nN (a), 1]. ( ) In a similar way, we can prove that N (a, 1] = [0, nN (a)). □ The following functions will allow us to construct an order automorphism on ([0, 1], ≤), that, together with the negations obtained in Theorem 3, will lead us to obtain all the strong negations leaving the function 1 fixed. These functions are constantly 1, except in the 0 or in the 1 (see Fig. 8), and will be denoted as: 1, a,
{ fa,1 (x) =
if x ̸ = 1, if x = 1,
f
a,0
{ (x) =
a, 1,
if x = 0, if x ̸ = 0.
Remark 4. Let us observe that, [0, 1) ⊑ fa,1 ⊑ 1 and 1 ⊑ f a,0 ⊑ (0, 1] for any a ∈ [0, 1]. Moreover, given f ∈ L such
So, according to Remark 4, it is N(fa,1 ) = f b,0 for some b ∈ [0, 1]. ′ If there exists b′ ∈ [0, 1] such that N(fa,1 ) = f b ,0 , we have that ′ ′ f b,0 = f b ,0 and b = f b,0 (0) = f b ,0 (0) = b′ . Then b is unique. In a similar way, we can prove that N(f c ,0 ) = fd,1 for a unique d ∈ [0, 1]. □ The following remark characterizes the order on the set of the functions fa,1 and f a,0 throughout the order ≤ of the real numbers. Remark 5. Given a, a′ ∈ [0, 1], it is:
• fa,1 ⊑ fa′ ,1 ⇐⇒ a ≤ a′ . Consequently, min⊑ {fa,1 : a ∈ [0, 1]} = f0,1 = [0, 1), max⊑ {fa,1 : a ∈ [0, 1]} = f1,1 = 1. ′ • f a ,0 ⊑ f a,0 ⇐⇒ a ≤ a′ . Consequently, min⊑ {f a,0 : a ∈ [0, 1]} = f 1,0 = 1, max⊑ {f a,0 : a ∈ [0, 1]} = f 0,0 = (0, 1]. Theorem 4. If N : L → L is a strong negation with N(1) = 1, the function α : [0, 1] → [0, 1], given by α (a) = b if N(fa,1 ) = f b,0
Please cite this article as: S. Cubillo, C. Torres-Blanc and P. Hernández-Varela, A characterization for some type-2 fuzzy strong negations, Knowledge-Based Systems (2019) 105281, https://doi.org/10.1016/j.knosys.2019.105281.
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Fig. 10. Functions f , g ∈ Ld with f (a) = g(a) and (N(f ))(nN (a)) ̸ = (N(g))(nN (a)).
is an order automorphism on [0, 1]. Moreover, if N(f c ,0 ) = fd,1 we have α −1 (c) = d. Proof. (i) By Proposition 7, the function α is well defined. (ii) α is injective. Given a, a′ ∈ [0, 1] such that α (a) = α (a′ ), it is N(fa,1 ) = N(fa′ ,1 ) and since N is involutive, fa,1 = N(N(fa,1 )) = N(N(fa′ ,1 )) = fa′ ,1 . Consequently, a = fa,1 (1) = fa′ ,1 (1) = a′ , and α es injective. (iii) α is surjective. Let b ∈ [0, 1], according to Proposition 7, N(f b,0 ) = fb′ ,1 for some b′ ∈ [0, 1]. Then, N(fb′ ,1 ) = N(N(f b,0 )) = f b,0 and α (b′ ) = b. Therefore, α is surjective. (iv) α is increasing. If a ≤ a′ , it is fa,1 ≤ fa′ ,1 and fa,1 ⊑ fa′ ,1 . ′ ′ Then f b ,0 = N(fa′ ,1 ) ⊑ N(fa,1 ) = f b,0 and f b,0 ≤ f b ,0 . So, ′ α (a) = b = f b,0 (0) ≤ f b ,0 (0) = b′ = α (a′ ). (v) N(f1,1 ) = N(1) = 1 = f 1,0 , and then α (1) = 1. (vi) According to Lemma 5 and Remark 2, N(f0,1 ) = N(min{fa,1 : a ∈ [0, 1]})
Fig. 11. Infimum and supremum of two decreasing functions and their negations.
According to Proposition 2, N(I) = N(inf{f , g }) = sup{N(f ), N(g)}, ⊑
⊑
N(S) = N(sup{f , g }) = inf{N(f ), N(g)}. ⊑
⊑
Also, as N(f ) and N(g) are increasing, (N(I))(x) = min{(N(f ))(x), (N(g))(x)} and (N(S))(x) = max{(N(f ))(x), (N(g))(x)}, ∀x ∈ [0, 1] (see Fig. 11). 2. Now, let us consider M = inf⊑ {S , [0, a]} and P = sup⊑ {I , [0, a)}. As they are decreasing, we have M(x) = min{S(x), [0, a](x)}
{ =
S(x) = max{f (x), g(x)}, 0,
if x ≤ a, if x > a.
P(x) = max{I(x), [0, a)(x)}
{
⊑
=
= max{N(fa,1 ) : a ∈ [0, 1]}
1, I(x) = min{f (x), g(x)},
if x < a, if x ≥ a,
⊑
= max{f a ,0 : a′ ∈ [0, 1]} = f 0,0 , ′
⊑
and so α (0) = 0. (vii) Finally, if N(f c ,0 ) = fd,1 then f c ,0 = N(N(f c ,0 )) = N(fd,1 ). It is α (d) = c and, as α is bijective, α −1 (c) = d. □ This automorphism on [0, 1] will be denoted, in the rest of the paper, by αN .
see Fig. 12. Therefore N(M) = N(inf{S , [0, a]}) = sup{N(S), N [0, a] }
(
⊑
)
⊑
= sup{N(S), [nN (a), 1]}, ⊑
N(P) = N(sup{I , [0, a)}) = inf{N(I), N [0, a) }
(
⊑
)
⊑
= inf{N(I), (nN (a), 1]}. ⊑
4. Characterization of strong negations on L that leave 1 fixed
As these functions are increasing, The following theorem is a significant result that will allow us to build a method to pointwise obtaining the image of a function f ∈ L throughout a strong negation N in L that leave the constant function 1 fixed. Specifically, we will be able to know the image of the functions that take the value 1 in the interval [0, m) or (m, 1], and a constant value in the rest of the interval [0, 1]. The image of these functions together with the strong negation nN and the order automorphism αN in [0, 1] associated to N, will allow us to obtain N(f ). Finally, this result will lead us to characterize the strong negations in L with fixed point the function 1. All this will be obtained from Theorem 5. Theorem 5. Let N be a strong negation in (L, ⊑) such that N(1) = 1. Let f , g ∈ Ld and a ∈ [0, 1]. If f (a) = g(a), then (N(f ))(nN (a)) = (N(g))(nN (a)). Moreover, if f , g ∈ Lc with f (a) = g(a) then, (N(f ))(nN (a)) = (N(g))(nN (a)).
(N(M))(x) = min{(N(S))(x), [nN (a), 1](x)}
{ =
0, (N(S))(x) = max{(N(f ))(x), (N(g))(x)},
if x < nN (a), if x ≥ nN (a).
(N(P))(x) = max{(N(I))(x), (nN (a), 1](x)} (N(I))(x) = min{(N(f ))(x), (N(g))(x)}, 1,
{ =
if x ≤ nN (a), if x > nN (a),
see Fig. 12. 3. Let us denote m = min{(N(f ))(nN (a)), (N(g))(nN (a))}, p = max{(N(f ))(nN (a)), (N(g))(nN (a))}. By hypothesis, m < p. Now, let us consider the following function µ (see Fig. 13a)).
Proof. The proof will be done by reduction to the absurd. Let us suppose that there exists a ∈ [0, 1] and f , g ∈ Ld such that f (a) = g(a), but (N(f ))(nN (a)) ̸ = (N(g))(nN (a)) (see Fig. 10).
⎧ ⎨max{f (x), g(x)}, f (a) = g(a), µ(x) = ⎩ min{f (x), g(x)},
si x < a,
1. Let us consider I = inf⊑ {f , g } and S = sup⊑ {f , g }. Taking into account that f and g are decreasing I(x) = min{f (x), g(x)} and S(x) = max{f (x), g(x)}, ∀x ∈ [0, 1] (see Fig. 11).
it is easy to note that M ⊑ µ ⊑ P (see Fig. 13a)). Then N(P) ⊑ N(µ) ⊑ N(M), that is equivalent to (N(M))(x) ≤ (N(µ))(x) ≤ (N(P))(x) (see Fig. 13b)), as the functions are increasing.
si x = a, si x > a,
Please cite this article as: S. Cubillo, C. Torres-Blanc and P. Hernández-Varela, A characterization for some type-2 fuzzy strong negations, Knowledge-Based Systems (2019) 105281, https://doi.org/10.1016/j.knosys.2019.105281.
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Fig. 12. Functions infimum and supremum of decreasing and increasing functions.
Fig. 13. (a) Function µ in the proof of Theorem 5, (b) Impossible construction of N(µ) in the proof of Theorem 5.
≤
(N(µ))(nN (a))
≤
(N(P))
(N(M))(nN (a)) = max{(N(f ))(nN (a)), (N(g))(nN (a))} = p, (N(P))(nN (a)) = min{(N(f ))(nN (a)), (N(g))(nN (a))} = m, so p ≤ (N(µ))(nN (a)) ≤ m, that is impossible, as p > m by hypothesis. Therefore, it have to be (N(f ))(n(a)) = (N(g))(n(a)). In a similar way, we can prove that if f , g ∈ Lc with f (a) = g(a) then, (N(f ))(nN (a)) = (N(g))(nN (a)). □ In the following, we will build some new functions, including as a particular case the functions given in Fig. 8. They will lead us to give a method to pointwise obtain the image of the functions in Lc or in Ld throughout the negation N in (L, ⊑). For any a ∈ [0, 1] and for any m ∈ (0, 1], fa,m will denote the following function (see Fig. 14a)), 1, a,
{ fa,m (x) =
if x < m, if x ≥ m,
and for any a ∈ [0, 1] and any m ∈ [0, 1), f a,m will denote the function (see Fig. 14b)), f a,m (x) =
{
a, 1,
Fig. 15. Decreasing function f with f (1) = a and (N(f ))(0) = αN (a).
Fig. 16. Example of N(fa,m ) = f αN (a),nN (m) .
Fig. 17. (a) Function f ∈ Ld with f (1) = a and [0, m) ⊑ f , (b) function g ∈ Lc with g(0) = αN (a) and g ⊑ (nN (m), 1].
Fig. 14. (a) Function fa,m , (b) function f a,m .
In particular, (N(M))(nN (a)) (nN (a)). But,
7
if x ≤ m, if x > m.
Let us remember that, according to Proposition 7, for each a ∈ [0, 1] there is a unique b ∈ [0, 1] such that N(fa,1 ) = f b,0 .
Moreover, we have built nN : [0, 1] → [0, 1] the strong negation associated with N (nN (x) = y if and only if N(x¯ ) = y¯ ) and αN : [0, 1] → [0, 1] the order automorphism associated with N (αN (a) = b if and only if N(fa,1 ) = f b,0 ). Lemma 6. Let N be a strong negation in (L, ⊑) with N(1) = 1. If f ∈ Ld such that f (1) = a then (N(f ))(0) = αN (a) (see Fig. 15). Proof. As f (1) = fa,1 (1), then, by Theorem 5, (N(f ))(nN (1)) = (N(fa,1 ))(nN (1)) = (N(fa,1 ))(0)
= (f αN (a),0 )(0) = αN (a). Therefore, (N(f ))(0) = αN (a).
□
Lemma 7. Let N be a strong negation in (L, ⊑) with N(1) = 1. Then, ∀a ∈ [0, 1] and ∀m ∈ (0, 1] it holds that, N(fa,m ) = f αN (a),nN (m) (see Fig. 16). Proof. N(fa,m ) = N(min⊑ {f ∈ Ld : f (1) = a and [0, m) ⊑ f }) = max⊑ {g ∈ Lc : g(0) = αN (a) and g ⊑ (nN (m), 1]} = f αN (a),nN (m) (see Fig. 17). □ Theorem 6. Let N be a strong negation in (L, ⊑) with N(1) = 1. Then, there are an order automorphism, α , in [0, 1] and a strong negation, n, in [0, 1] such that ∀x ∈ [0, 1] it holds (N(f ))(x) = α (f (n(x))), ∀f ∈ Ld , and
Please cite this article as: S. Cubillo, C. Torres-Blanc and P. Hernández-Varela, A characterization for some type-2 fuzzy strong negations, Knowledge-Based Systems (2019) 105281, https://doi.org/10.1016/j.knosys.2019.105281.
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Fig. 18. Example f (x) = ff (x),x (x) and so (N(f ))(nN (x)) = (f αN (f (x)),nN (x) )(nN (x)).
(N(f ))(x) = α −1 (f (n(x))), ∀f ∈ Lc . Proof. Let f ∈ Ld and an x ∈ [0, 1]. Let us consider the function ff (x),x that satisfies ff (x),x (x) = f (x) (see Fig. 18). Therefore, by Theorem 5 and Lemma 7 it holds (N(f ))(nN (x)) = (N(ff (x),x ))(nN (x)) = (f αN (f (x)),nN (x) )(nN (x))
= αN (f (x)). Then, (N(f ))(nN (x)) = αN (f (x)), ∀x ∈ [0, 1], and so, (N(f ))(x) = αN (f (nN (x))), ∀x ∈ [0, 1]. Now, if g ∈ Lc then N(g) ∈ Ld and so, g = N(N(g)) = αN ◦ N(g) ◦ nN . Then, N(g) = αN−1 ◦ g ◦ nN . □ Theorem 7. N : L → L is a strong negation in (L, ⊑) such that N(1) = 1 if and only if there exists a strong negation n and an order automorphism α in ([0, 1], ≤), such that ∀x ∈ [0, 1], (N(f ))(x) = α (f R (n(x))) ∧ α −1 (f L (n(x))). Proof. ⇒) Let f ∈ L, by Proposition 4, N(f )L = N(f R ), N(f )R = N(f L ), and N(f ) = N(f )L ∧ N(f )R . Therefore, N(f ) = N(f R ) ∧ N(f L ) = (αN ◦ f R ◦ nN ) ∧ (αN−1 ◦ f L ◦ nN ).
⇐) This result is included in Theorem 2. □ Remark 6.
• When α = Id and n = 1 − Id we obtain (N(f ))(x) = f (1 − x), operation in M proposed by Walker and Walker [5]. Nevertheless, while they did not studied if it was a negation, we proved in [15] that it is not a negation in M, but a strong negation in L. • When α = Id and n is any strong negation in [0, 1], it results (N(f ))(x) = f (n(x)), strong negations in L presented by the authors in [15]. • Finally, if α is any automorphism in [0, 1] and n is any strong negation in [0,1], it turns out (Nn,α (f ))(x) = α (f R (n(x))) ∧ α −1 (f L (n(x))), strong negations in L presented by the authors in [18]. However, until now, the problem of whether there were more strong negations in L, apart from those determined by the previous formula, remained open. And, in particular, strong negations that leave the label 1 fixed. This work answers this question by characterizing the negations that satisfy these conditions and by proving that those obtained by said formula are all that exist, there is no other. 5. Conclusions Through literature about fuzzy sets, and in particular, regarding negations, a lot of results have been obtained. For example,
the characterization of strong negations in [0, 1] by Trillas in [19] is well known. Also, the negations in the membership degrees of Atanassov’s intuitionistic fuzzy sets, have been characterized by Deschrijver in [21]. However, in the case of the type-2 fuzzy sets and their membership degrees, the situation is not the same, there are still many gaps to be filled. This is because in this case the study entails much greater complexity. In this regard, this paper has showed a lot of new results. All of them have been directed, in an orderly manner and step by step, to obtain a characterization of a particular type of negations in the membership degrees of type-2 fuzzy sets: those negations in the set of convex and normal membership degrees that furthermore, leave the constant function 1 fixed. Of course, many questions remain open, since there are many other negations that do not verify these conditions, and that will have to be studied in future work. CRediT authorship contribution statement S. Cubillo: Conceptualization, Methodology, Writing - original draft, Investigation, Writing - review & editing. C. Torres-Blanc: Conceptualization, Methodology, Writing - original draft, Investigation, Writing - review & editing. P. Hernández-Varela: Conceptualization, Methodology, Writing - original draft, Investigation, Writing - review & editing. Acknowledgments This paper was partially supported by MCIU (Spain) project PGC2018-096509-B-100, Universidad Politécnica de Madrid (Spain) and Universidad Mayor (Chile). References [1] L. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-I, Inform. Sci. 8 (1975) 199–249. [2] L. Zadeh, Fuzzy sets, Inf. Control 20 (1965) 301–312. [3] J. Mendel, R. Jhon, Type-2 fuzzy sets made simple, IEEE Trans. Fuzzy Syst. 10 (2) (2002) 117–127. [4] M. Mizumoto, K. Tanaka, Some properties of fuzzy sets of type-2, Inf. Control 31 (1976) 312–340. [5] C. Walker, E. Walker, The algebra of fuzzy truth values, Fuzzy Sets and Systems 149 (2005) 309–347. [6] O. Castillo, M.A. Sánchez, C.I. González, G.E. Martínez, Review of recent type-2 fuzzy image processing applications, Information 8 (3) (2017) 97. [7] O. Castillo, L. Amador-Angulo, A generalized type-2 fuzzy logic approach for dynamic parameter adaptation in bee colony optimization applied to fuzzy controller design, Inf. Sci. 460–461 (2018) 476–496. [8] D.S. Comas, J.I. Pastore, A. Bouchet, V.L. Ballarin, G.J. Meschino, Interpretable interval type-2 fuzzy predicates for data clustering: A new automatic generation method based on self-organizing maps, Knowl.-Based Syst. 133 (2017) 234–254. [9] T.E. Dalkili, S. Sairkaya, Parameter prediction based on type-2 fuzzy clustering, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 26 (6) (2018) 877–892. [10] A. Doostparast Torshizi, M.H. Fazel Zarandi, A new cluster validity measure based on general type-2 fuzzy sets: Application in gene expression data clustering, Knowl-Based Syst. 64 (2014) 81–93. [11] S. Greenfield, F. Chiclana, Type-Reduced Set structure and the truncated type-2 fuzzy set, Fuzzy Sets and Systems 352 (2018) 119–141. [12] B.Q. Hu, C.K. Kwong, On type-2 fuzzy sets and their t-norm operations, Inform. Sci. 255 (2014) 58–81. [13] M. Najariyan, M. Mazandarani, R. John, Type-2 fuzzy linear systems, Granul. Comput. 2 (2017) 175–186. [14] T. Wu, X.W. Liu, An interval type-2 fuzzy clustering solution for large-scale multiple-criteria group decision-making problems, Knowl.-Based Syst. 114 (2016) 118–127. [15] P. Hernández, S. Cubillo, C. Torres-Blanc, Negations on type-2 fuzzy sets, Fuzzy Sets and Systems 252 (2014) 111–124. [16] E.E. Castiñeira, C. Torres-Blanc, S. Cubillo, Measuring contradiction on A-IFS defined in finite universes, Knowl.-Based Syst. 24 (2011) 1297–1309.
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A characterization for some type-2 fuzzy strong negations✩ ∗
S. Cubillo a , , C. Torres-Blanc a , P. Hernández-Varela b a
Universidad Politécnica de Madrid (UPM), DMATIC, 28660 Boadilla del Monte, Madrid, Spain Universidad Mayor, Facultad de Estudios Interdisciplinarios, Núcleo Matemáticas, Física & Estadística, Manuel Montt 318, Providencia, Santiago, Chile b
article
info
Article history: Received 14 November 2018 Received in revised form 19 November 2019 Accepted 25 November 2019 Available online xxxx Keywords: Type-2 fuzzy sets Normal and convex functions Negation Strong negation Order automorphism
a b s t r a c t P. Hernandez et al. in 2014 established the axioms that an operation must fulfill in order to be a negation on a bounded poset (partially ordered set). In this work, we focus on the set L of the membership degrees of the type-2 fuzzy sets which are normal and convex functions in [0,1]. This set has a bounded and complete lattice structure, thank to which negations and strong negations have been constructed by the authors applying the Zadeh’s Extension Principle. In addition, the authors showed new negations on L that are different from the negations presented in 2014 applying the Zadeh’s Extension Principle. In this work, the authors obtain a characterization of the strong negations on L that leave the constant function 1 fixed. © 2019 Published by Elsevier B.V.
1. Introduction Type-2 fuzzy sets (T2FSs) were introduced by L.A. Zadeh in 1975 [1] as an extension of type-1 fuzzy sets (T1FSs) [2]. While the membership degree of an element in a T1FS is a value in the interval [0, 1], the membership degree of an element in a T2FS is a fuzzy set in [0, 1]. That is, the degree in which an element belongs to the set is just a label of the linguistic variable ‘‘TRUTH’’. In this way, a T2FS is determined by a membership function µ : X → M, where M = Map([0, 1], [0, 1]) is the set of the functions from [0, 1] on [0, 1] (see [3–5]). Because the membership degrees of T2FSs are fuzzy, they are better able to model uncertainty than FSs [3,6]. In the last decade, many researchers have studied type2 fuzzy set theory and its application to several fields of science (see, for example, [7–14]). In this paper, we will focus on T2FSs with degrees of membership in the subset L of normal and convex functions of M. Since the introduction of T2FSs, lots of the operations, properties and results obtained in the T1FSs have been extended to T2FSs throughout the Zadeh’s Extension Principle (see for example, [3–5,15]). This is the case of the negation operator, used to obtain, for example, complement of a set, dual of a t-norm or a t-conorm, entropies, implications, as well as to study the ✩ No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.knosys. 2019.105281. ∗ Corresponding author. E-mail addresses: [email protected] (S. Cubillo), [email protected] (C. Torres-Blanc), [email protected] (P. Hernández-Varela).
possible contradiction appearing in a fuzzy system. For example, strong negations on Atanassov’s intuitionistic fuzzy sets (A-IFSs) and on T2FSs have been applied in order to study the possible contradiction appearing in a fuzzy system (see [16,17]). Thus, the study of strong negations is essential for both fuzzy sets and their extensions, and particularly for type-2 fuzzy sets. In [15] the authors presented a deep study about the negations on the T2FSs. Particularly, they proposed the axioms for negations and strong negations on a bounded partially ordered set (bounded poset). Also, using the Zadeh’s Extension Principle, they obtained families of negations and strong negations on L. Moreover, in [18] for the first time negations on M were presented, as well as new negations and strong negations on L. Trillas in [19] studied and characterized strong negations in [0, 1] through order automorphisms in [0, 1]. Bustince et al. in [20] introduced intuitionistic generators in order to build negations in A-IFSs; and Deschrijver et al. in [21] characterized strong intuitionistic negations based on strong negations in [0, 1]. Then, in a similar way, to characterize the strong negations in the T2FS, or equivalently in the set of its membership degrees M, is a good challenge. But, as the algebra of these sets is much more complicated, this objective is not easy. On the other hand, usually, the linguistic labels used in the real situations are convex and normal, that is, they belong to L. More, as the label 1 (1(x) = 1 for all x in [0, 1]) means lack of information, it seems suitable that its negation be also the label 1. That is, the negation should leave the function constantly 1 fixed. For example, if a biologist must determine to what extent ‘‘the species X is appropriate for an environment’’, he or she could say that this assertion is ‘‘very true’’, ‘‘more or less true’’, ‘‘absolutely
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false’’, . . . , or, if the expert does not have an opinion, he or she could associate the label 1. Then, in this case, the biologist should also associate to the statement ‘‘the species X is NOT appropriate for an environment’’ the linguistic label 1. According to the previous comments, the purpose of this paper is to characterize the negations in L that leave the label 1 fixed. That is, to find out all strong negations satisfying this condition. The paper is organized as follows. In Section 2 we review some definitions, operations and properties on FSs and T2FSs reported in the literature (see [5,15] and [18]) and needed to understand the rest of the work. Also in this Section 2 the axioms of the negation in [0, 1] and in a bounded poset are exposed, as well as some results about negations in L obtained by the authors in previous papers. Section 3 is devoted to obtaining some properties of the strong negations on L that leave the constant function 1 fixed. These properties allow us, given a such strong negation on L, to construct both a strong negation and an order automorphism on [0, 1] associated with it. In Section 4, a significant result (Theorem 5) is achieved, from which, and together with the results of Section 3, we are going to characterize all strong negations on L that leave the constant function 1 fixed. Finally, Section 5 contains some conclusions.
Fig. 1. Example of a T2FS.
2. Preliminaries Throughout the paper, X will denote a non-empty set which will represent the universe of discourse. Additionally, ≤ will denote the usual order relation in the lattice of real numbers, and ∧ the minimum operator and ∨ the maximum operator.
Fig. 2. Example of f L and f R .
2.1. Definitions and properties of FSs and T2FSs
Definition 3. A function f ∈ M is normal if sup{f (x) : x ∈ [0, 1]} = 1 and a function f ∈ M is convex, if for any x ≤ y ≤ z, it holds that f (y) ≥ f (x) ∧ f (z).
Definition 1 ([2]). A type-1 fuzzy set (FS), A, is characterized by a membership function µA ,
The following definition and theorem were given in previous papers in order to facilitate the operations in the set M:
µA : X → [0, 1], where µA (x) is the membership degree of an element x ∈ X in the set A. Definition 2 ([4]). A type-2 fuzzy set (T2FS), A, is characterized by a membership function:
µA : X → M = [0, 1][0,1] = Map ([0, 1], [0, 1]), that is, µA (x) is a type-1 fuzzy set in the interval [0, 1] and also the membership degree of the element x ∈ X in the set A. Therefore,
µA (x) = fx , where fx : [0, 1] → [0, 1]. Let T 2FS(X ) = Map(X , M) denotes the set of all type-2 fuzzy sets on X. Fig. 1 shows an example of a type-2 fuzzy set on the finite universe of discourse T = {0, 1, 2, 3, 4}. Walker and Walker justify in [5] that the operations on Map(X , M) can be defined naturally from the operations on M and have the same properties. Thus, in the same way as in the case of the FSs, where the definitions and properties are given on ([0, 1], ≤), in this paper we will work on M, as all the results are easily and directly extensible to Map(X , M). Moreover, we restrict to type-2 fuzzy sets in the special case in which the membership degrees are in L, the set of normal and convex functions of M. There are several reasons to do so. First, membership degrees represent linguistic labels of the TRUTH variable, so it is not uncommon to require them to be convex and normal. Furthermore, it has been pointed out that this set L will contain a bounded and complete lattice structure, and as consequence t-norms, t-conorms, aggregation operators and specifically negations can be properly constructed (see [15,22, 23]).
Definition 4 ([5,24,25]). If f ∈ M, we define f L , f R ∈ M as f L (x) = sup{f (y) : y ≤ x}, f R (x) = sup{f (y) : y ≥ x}. Note that f L and f R are increasing and decreasing, respectively (see Fig. 2), f ≤ f L , f ≤ f R , for all f ∈ M [5], where ≤ is the usual order in the set of functions (f ≤ g if and only if f (x) ≤ g(x), for all x). In previous papers two partial orders on M (⊑ and ⪯) were defined [4,5]. Generally, these partial orders do not coincide [4,5] and M does not have a lattice structure with either of the two orders [5,26]. However, in L these partial orders are equivalent, that is, ⊑ ≡ ⪯; and (L, ⊑) is a bounded and complete lattice (see [5,25]). Let us observe that the lattice structure is necessary for the following reasons. Firstly, a negation is always decreasing respect to an order. Furthermore, in the proof of some results, we will need to obtain the supremum and the infimum of two functions, and so it is necessary to have a lattice structure. The following characterization of the partial order ⊑ on L will be useful for establishing the results of this work. Theorem 1 ([25]). Let f , g ∈ L (see Fig. 3). f ⊑ g if and only if g L ≤ f L and f R ≤ g R . In the following proposition we will remember some properties of the functions in the lattice (L, ⊑), given by Walker and Walker. Proposition 1 ([5]). Let f , g ∈ L. 1. f is increasing ⇔ f = f L ⇔ f (1) = 1, and f is decreasing ⇔ f = f R ⇔ f (0) = 1. 2. If f , g ∈ L are increasing, it holds that, f ⊑ g ⇔ g ≤ f .
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3. Strong negations on L that leave the function 1 fixed In the previous section, we have indicated that, given n and
Fig. 3. Example where f ⊑ g.
3. If f , g ∈ L are decreasing, it holds that, f ⊑ g ⇔ f ≤ g. 4. f = f L ∧ f R , and f L ∨ f R = 1. 2.2. Negations on L This subsection is devoted to review the study carried out in [15] and [18] on the negations in the framework of the T2FSs with membership degrees in L. Firstly, let us recall the definition of negation in ([0, 1], ≤). Definition 5. A function n : [0, 1] → [0, 1] is a negation if it is decreasing respect to the order ≤ and satisfies n(0) = 1 and n(1) = 0. If, in addition, n(n(x)) = x for all x ∈ [0, 1], then it is said to be a strong negation. Definition 5 suggests us an extension to any partially ordered set (poset) with minimum and maximum elements (bounded). In this sense, Hernández et al. [15] introduced negations in this algebraic structure and gave some negations in L. Definition 6 ([15]). Let A be a set and ≤A be a partial order in A such that (A, ≤A ) has a minimum element 0≤A and a maximum element 1≤A . A negation in (A, ≤A ) is a decreasing function N : A → A such that N(0≤A ) = 1≤A and N(1≤A ) = 0≤A . If, additionally, N(N(x)) = x holds for all x ∈ A, it is said to be a strong negation. Proposition 2. Let (A, ≤A ) be a bounded lattice. Given a strong negation N : A → A, for any a, b ∈ A it is N(inf{a, b}) = sup{N(a), N(b)} and N(sup{a, b}) = inf{N(a), N(b)}. Definition 7 ([5]). Let [a, b] ⊆ [0, 1]. The characteristic function of [a, b] is [a, b] : [0, 1] → [0, 1], where
[a, b](x) =
1, 0,
{
if x ∈ [a, b], if x ∈ / [a, b],
and the characteristic function of a number a ∈ [0, 1] or singleton is a¯ = [a, a]. Note that 0¯ and 1¯ are the minimum and maximum, respectively, of the bounded lattice (L, ⊑). In a similar way we define the characteristic functions of the intervals [a, b), (a, b], or (a, b). Moreover, [a, b] ⊑ [c , d] if and only if a ≤ c, b ≤ d. In [18] we have obtained a family of negations on (L, ⊑) that transform singletons into singletons and are closed on the set of characteristic functions of closed intervals. Theorem 2 ([18]). Let n be a strong negation in ([0, 1], ≤) and let α be an order automorphism in ([0, 1], ≤). And the operation Nn,α : L → L defined as Nn,α (f ) = (α ◦ f R ◦ n) ∧ (α −1 ◦ f L ◦ n). Then (a) Nn,α is a strong negation in L. (b) Nn,α (a¯ ) = n(a) and Nn,α ([a, b]) = [n(b), n(a)]. (c) Nn,α (p) = p and Nn,α (1) = 1, being p the fixed point of the strong negation, n, in [0, 1] and 1 the constant function 1.
α strong negation and automorphism, respectively, on ([0, 1], ≤), the operations Nn,α are strong negations on (L, ⊑). Moreover, for all a, b ∈ [0, 1], with a ≤ b, they transform the function [a, b] on the function [n(b), n(a)], and, in particular, they transform the function 1 = [0, 1] into the same function 1 = [n(1), n(0)] = [0, 1]. That is, those strong negations leave the function 1 fixed. In order to characterize the strong negations on (L, ⊑) satisfying this condition, the present section is devoted to study the properties of these negations. In particular, we will present some previous results that, step by step, will allow us to get a final theorem. A first lemma, will characterize increasing and decreasing functions in L. Let us denote Lc = {f ∈ L : f increasing } and Ld = {f ∈ L : f decreasing }. Lemma 1. Given f ∈ L, we have: f ∈ Ld ⇔ f ⊑ 1, f ∈ Lc ⇔ 1 ⊑ f , f L = f R ⇔ f = 1. Proof. f ∈ Ld ⇔ f R = f ≤ 1 = 1R and f L = 1 = 1L (by Proposition 1) ⇔ f ⊑ 1 (by Theorem 1). In a similar way, we have the second equivalence, and the third one is trivial. □ Proposition 3. Let N : L → L be a strong negation on L, with N(1) = 1. Then f ∈ Ld ⇔ N(f ) ∈ Lc , f ∈ Lc ⇔ N(f ) ∈ Ld . Proof. By the previous Lemma, f ∈ Ld ⇔ f ⊑ 1. Then, as N is a strong negation with N(1) = 1, it is f ⊑ 1 ⇔ N(1) = 1 ⊑ N(f ) ⇔ N(f ) ∈ Lc . The proof of the second equivalence is analogous. □ Remark 1. Now, it is easy to proof that, if N : L → L is a strong negation with N(1) = 1, then N(Lc ) = Ld and N(Ld ) = Lc . Before obtaining new properties for the strong negations with N(1) = 1, we need the following lemmas and corollary on (L, ⊑). Lemma 2. If f , g ∈ L then f ⊑ g if and only if f L ⊑ g L and f R ⊑ g R . Proof. Straight from Theorem 1 and Proposition 1. □ Lemma 3. If f ∈ L, then f ⊑ f L and f R ⊑ f . Proof. By Lemma 2, it is f ⊑ f L ⇔ f L ⊑ (f L )L = f L and f R ⊑ (f L )R = 1 that is always true, as f R is decreasing (see Fig. 4). f R ⊑ f ⇔ (f R )L = 1 ⊑ f L , that is always true as f L is increasing, and (f R )R = f R ⊑ f R (see Fig. 4). □ Corollary 1. R
If f ∈ L, then,
f = max⊑ {g ∈ Ld : g ⊑ f }
and
f L = min⊑ {g ∈ Lc : f ⊑ g }.
Proof. As f R is decreasing, by Lemma 3, we have f R ∈ {g ∈ Ld : g ⊑ f }. Moreover, for all g ∈ Ld such that g ⊑ f , it is g = g R ⊑ f R , and so f R is an upper bound of the set {g ∈ Ld : g ⊑ f } and belongs to it. Then f R is the maximum of that set. The proof of the second equality is similar. □
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Moreover, N(f R )
N(g R ), and so
L
a = (N(f ))L = N(f R )
N(g R ) = (N(g))L
L
R
that is, (N(g))L ≨ a . Then a ⊑ N(g) with (N(g))R = a . Also, as a is a singleton, according to Lemma 4 it is N(g) = a, and N(a) = g, attaining a contradiction. In a similar way, that (ii) is not possible can be proved. Therefore, N(a) have to be a singleton. □ Fig. 4. Example of f ∈ L, f L and f R showing the result of Lemma 3.
Remark 2. Let (A, ≤A ) be a partially ordered set and D : A → A a decreasing function (respectively C : A → A increasing). Given a subset F ⊂ A, if there exists the maximum of F then D(max F ) = min D(F ) (respectively C (max F ) = max C (F )). And if there exists the minimum of F then D(min F ) = max D(F ) (respectively C (min F ) = min C (F )). Proposition 4. Let N : L → L be a strong negation with N(1) = 1. Then, for all f ∈ L it is (N(f ))L = N(f R ) and (N(f ))R = N(f L ). Proof. Direct from Corollary 1, and Remarks 1 and 2. □ This Proposition allows us, given a strong negation N on L with N(1) = 1, to obtain the negation of any f ∈ L as the minimum of the negation of a decreasing function and the negation of an increasing function. In fact, by Proposition 1, N(f ) = (N(f ))L ∧ (N(f ))R = N(f R ) ∧ N(f L )
Now, the following results are aimed at proving that the image of characteristic functions of intervals are also characteristic functions of intervals. To do this, firstly we will define a strong negation n on [0, 1] from the strong negation N on L. Theorem 3. If N : L → L is a negation such that the image of any singleton is also a singleton, then the function n : [0, 1] → [0, 1], ¯ is a negation on [0, 1]. Moreover, if given by n(a) = b if N(a¯ ) = b, N is strong, then n is also a strong negation. Proof. Trivial according to definition of n. □ From now on, nN will designate the negation in [0, 1] defined ¯ in this Theorem 3. That is, nN (a) = b if N(a¯ ) = b. Proposition 6. ( ) If N : L → L is a strong negation with N(1) = 1, then N [a, b] = [nN (b), nN (a)]. Proof.
( ) ( ( ))R ( ) L L [a, b] = a¯ L ⇒ N [a, b] = N [a, b] = N a¯ L = (N(a¯ ))R , ( ) ( ( ))L ( ) R R ¯ L. [a, b] = b¯ R ⇒ N [a, b] = N [a, b] = N b¯ R = (N(b))
Therefore, the negation N is determined by the images of Ld and Lc . Besides, as N is involutive, it also is determined by N(Ld ) or N(Lc ). The following results are aimed at proving that the image of a singleton is also a singleton.
By Proposition 5, N transforms singletons into singletons. Let ¯ = d. ¯ As a ≤ b, it is a¯ ⊑ b¯ us suppose that N(a¯ ) = c¯ and N(b) ¯ ⊑ N(a¯ ) = c¯ . So, d ≤ c. Moreover, because of and d¯ = N(b) ( ( ))R ( ( ))L N [a, b] = c¯ R and N [a, b] = d¯ L , it is that
Lemma 4. Let f ∈ L and let a be a singleton. Then,
N [a, b] = d¯ L ∧ c¯ R = [d, c ] = [nN (b), nN (a)]. □
(
L
1. If f ⊑ a such that f L = a ⇒ f = a R 2. If a ⊑ f such that f R = a ⇒ f = a L
)
Lemma 5. Let N be a strong negation in (L, ⊑) such that N(1) = 1. Then, R
Proof. Let f ⊑ a such that f L = a . f ⊑ a implies f R ≤ a , and R L so f (x) ≤ f R (x) ≤ a (x) = 0, ∀ x > a. On the other hand, f L = a L L implies f (x) ≤ f (x) = a (x) = 0, ∀ x < a. Then f (x) = 0, ∀ x ̸ = a, and since f is normal, f (a) = 1. Therefore f = a. The proof of the second property is analogous. □ Remark 3. If f ∈ L is not a singleton, there exists g ∈ L such that g ⊑ f with g L = f L and g R ≨ f R (see Fig. 5) or there exists g ∈ L such that f ⊑ g with g R = f R and g L ≨ f L (see Fig. 6). Proposition 5. Let N : L → L be a strong negation such that N(1) = 1. Then the image of a singleton by N is also a singleton.
N [0, a) = (nN (a), 1], ∀a ∈ (0, 1],
(
)
N (a, 1] = [0, nN (a)), ∀a ∈ [0, 1).
(
)
Proof. By the previous Proposition, N [0, a] = [nN (a), 1] and ( ) N [a, 1] = [0, nN (a)].
(
)
Let H = {f ∈ Ld : [0, b] ⊑ f ⊑ [0, a], ∀b < a}
= {f ∈ Ld : f (x) = 1, ∀x < a, f (x) = 0, ∀x > a and f (a) ∈ [0, 1]} (see Fig. 7a). Therefore, N(H) = {N(f ) ∈ Lc : f ∈ H }
Proof. Let a be a singleton. Let us suppose that N(a) = f ∈ L is not a singleton. By the previous Remark, there are only two possible cases: (i) There exists g ∈ L such that g ⊑ f with g L = f L and g R ≨ f R , that is, g L = f L and g R f R (g R ⊑ f R and g R ̸ = f R ). See Fig. 5. (ii) There exists g ∈ L such that f ⊑ g with g R = f R and g L ≨ f L (that is, g R = f R and f L g L ). See Fig. 6. In the case (i), N(g L ) = N(f L ) and by Proposition 4, (N(g))R = N(g L ) = N(f L ) = (N(f ))R = a
R
= {N(f ) ∈ Lc : [nN (a), 1] ⊑ N(f ) ⊑ [nN (b), 1], ∀nN (b) > nN (a)} = {g ∈ Lc : [nN (a), 1] ⊑ g ⊑ [nN (b), 1], ∀nN (b) > nN (a)} = {g ∈ Lc : g(x) = 0, ∀x < nN (a), g(x) = 1, ∀x > nN (a) and g(nN (a)) ∈ [0, 1]} (see Fig. 7b). Moreover, given f1 , f2 ∈ H, f1 ⊑ f2 if and only if f1 (a) ≤ f2 (a), as f1 , f2 are decreasing (see Fig. 7a). Therefore, min⊑ (H) = [0, a).
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Fig. 5. Function f is not a singleton, g
f with g L = f L and g R ≨ f R .
Fig. 6. Function f is not a singleton, f
g with f R = g R and g L ≨ f L .
5
Fig. 9. Example of N(fa,1 ) = f b,0 . Fig. 7. (a) Functions of H in the proof of Lemma 5, (b) Functions of N(H).
that [0, 1) ⊑ f ⊑ 1, then f = fa,1 for some a ∈ [0, 1] as [0, 1) ⊑ f ⊑ 1 ⇔ [0, 1) ≤ f ≤ 1, because they are decreasing. In a similar way, given f ∈ L such that 1 ⊑ f ⊑ (0, 1], then f = f a,0 for some a ∈ [0, 1]. Proposition 7. Let N : L → L be a strong negation with N(1) = 1. Then for each a ∈ [0, 1] there exists a unique b ∈ [0, 1] such that N(fa,1 ) = f b,0 (see Fig. 9), and for each c ∈ [0, 1] there exists a unique d ∈ [0, 1] such that N(f c ,0 ) = fd,1 . Fig. 8. (a) Function fa,1 , (b) function f a,0 .
Proof. Any function fa,1 satisfies that [0, 1) ⊑ fa,1 ⊑ 1. Then 1 = N(1) ⊑ N(fa,1 ) ⊑ N([0, 1)) = (0, 1] by Lemma 5.
And given g1 , g2 ∈ N(H), g1 ⊑ g2 if and only if g2 (nN (a)) ≤ g1 (nN (a)), as g1 , g2 increasing (see Fig. 7b). Therefore, max⊑ (N(H)) = (nN (a), 1]. ( ) Finally, by Remark 2, N [0, a) = N(min⊑ (H)) = max⊑ (N(H)) = (nN (a), 1]. ( ) In a similar way, we can prove that N (a, 1] = [0, nN (a)). □ The following functions will allow us to construct an order automorphism on ([0, 1], ≤), that, together with the negations obtained in Theorem 3, will lead us to obtain all the strong negations leaving the function 1 fixed. These functions are constantly 1, except in the 0 or in the 1 (see Fig. 8), and will be denoted as: 1, a,
{ fa,1 (x) =
if x ̸ = 1, if x = 1,
f
a,0
{ (x) =
a, 1,
if x = 0, if x ̸ = 0.
Remark 4. Let us observe that, [0, 1) ⊑ fa,1 ⊑ 1 and 1 ⊑ f a,0 ⊑ (0, 1] for any a ∈ [0, 1]. Moreover, given f ∈ L such
So, according to Remark 4, it is N(fa,1 ) = f b,0 for some b ∈ [0, 1]. ′ If there exists b′ ∈ [0, 1] such that N(fa,1 ) = f b ,0 , we have that ′ ′ f b,0 = f b ,0 and b = f b,0 (0) = f b ,0 (0) = b′ . Then b is unique. In a similar way, we can prove that N(f c ,0 ) = fd,1 for a unique d ∈ [0, 1]. □ The following remark characterizes the order on the set of the functions fa,1 and f a,0 throughout the order ≤ of the real numbers. Remark 5. Given a, a′ ∈ [0, 1], it is:
• fa,1 ⊑ fa′ ,1 ⇐⇒ a ≤ a′ . Consequently, min⊑ {fa,1 : a ∈ [0, 1]} = f0,1 = [0, 1), max⊑ {fa,1 : a ∈ [0, 1]} = f1,1 = 1. ′ • f a ,0 ⊑ f a,0 ⇐⇒ a ≤ a′ . Consequently, min⊑ {f a,0 : a ∈ [0, 1]} = f 1,0 = 1, max⊑ {f a,0 : a ∈ [0, 1]} = f 0,0 = (0, 1]. Theorem 4. If N : L → L is a strong negation with N(1) = 1, the function α : [0, 1] → [0, 1], given by α (a) = b if N(fa,1 ) = f b,0
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Fig. 10. Functions f , g ∈ Ld with f (a) = g(a) and (N(f ))(nN (a)) ̸ = (N(g))(nN (a)).
is an order automorphism on [0, 1]. Moreover, if N(f c ,0 ) = fd,1 we have α −1 (c) = d. Proof. (i) By Proposition 7, the function α is well defined. (ii) α is injective. Given a, a′ ∈ [0, 1] such that α (a) = α (a′ ), it is N(fa,1 ) = N(fa′ ,1 ) and since N is involutive, fa,1 = N(N(fa,1 )) = N(N(fa′ ,1 )) = fa′ ,1 . Consequently, a = fa,1 (1) = fa′ ,1 (1) = a′ , and α es injective. (iii) α is surjective. Let b ∈ [0, 1], according to Proposition 7, N(f b,0 ) = fb′ ,1 for some b′ ∈ [0, 1]. Then, N(fb′ ,1 ) = N(N(f b,0 )) = f b,0 and α (b′ ) = b. Therefore, α is surjective. (iv) α is increasing. If a ≤ a′ , it is fa,1 ≤ fa′ ,1 and fa,1 ⊑ fa′ ,1 . ′ ′ Then f b ,0 = N(fa′ ,1 ) ⊑ N(fa,1 ) = f b,0 and f b,0 ≤ f b ,0 . So, ′ α (a) = b = f b,0 (0) ≤ f b ,0 (0) = b′ = α (a′ ). (v) N(f1,1 ) = N(1) = 1 = f 1,0 , and then α (1) = 1. (vi) According to Lemma 5 and Remark 2, N(f0,1 ) = N(min{fa,1 : a ∈ [0, 1]})
Fig. 11. Infimum and supremum of two decreasing functions and their negations.
According to Proposition 2, N(I) = N(inf{f , g }) = sup{N(f ), N(g)}, ⊑
⊑
N(S) = N(sup{f , g }) = inf{N(f ), N(g)}. ⊑
⊑
Also, as N(f ) and N(g) are increasing, (N(I))(x) = min{(N(f ))(x), (N(g))(x)} and (N(S))(x) = max{(N(f ))(x), (N(g))(x)}, ∀x ∈ [0, 1] (see Fig. 11). 2. Now, let us consider M = inf⊑ {S , [0, a]} and P = sup⊑ {I , [0, a)}. As they are decreasing, we have M(x) = min{S(x), [0, a](x)}
{ =
S(x) = max{f (x), g(x)}, 0,
if x ≤ a, if x > a.
P(x) = max{I(x), [0, a)(x)}
{
⊑
=
= max{N(fa,1 ) : a ∈ [0, 1]}
1, I(x) = min{f (x), g(x)},
if x < a, if x ≥ a,
⊑
= max{f a ,0 : a′ ∈ [0, 1]} = f 0,0 , ′
⊑
and so α (0) = 0. (vii) Finally, if N(f c ,0 ) = fd,1 then f c ,0 = N(N(f c ,0 )) = N(fd,1 ). It is α (d) = c and, as α is bijective, α −1 (c) = d. □ This automorphism on [0, 1] will be denoted, in the rest of the paper, by αN .
see Fig. 12. Therefore N(M) = N(inf{S , [0, a]}) = sup{N(S), N [0, a] }
(
⊑
)
⊑
= sup{N(S), [nN (a), 1]}, ⊑
N(P) = N(sup{I , [0, a)}) = inf{N(I), N [0, a) }
(
⊑
)
⊑
= inf{N(I), (nN (a), 1]}. ⊑
4. Characterization of strong negations on L that leave 1 fixed
As these functions are increasing, The following theorem is a significant result that will allow us to build a method to pointwise obtaining the image of a function f ∈ L throughout a strong negation N in L that leave the constant function 1 fixed. Specifically, we will be able to know the image of the functions that take the value 1 in the interval [0, m) or (m, 1], and a constant value in the rest of the interval [0, 1]. The image of these functions together with the strong negation nN and the order automorphism αN in [0, 1] associated to N, will allow us to obtain N(f ). Finally, this result will lead us to characterize the strong negations in L with fixed point the function 1. All this will be obtained from Theorem 5. Theorem 5. Let N be a strong negation in (L, ⊑) such that N(1) = 1. Let f , g ∈ Ld and a ∈ [0, 1]. If f (a) = g(a), then (N(f ))(nN (a)) = (N(g))(nN (a)). Moreover, if f , g ∈ Lc with f (a) = g(a) then, (N(f ))(nN (a)) = (N(g))(nN (a)).
(N(M))(x) = min{(N(S))(x), [nN (a), 1](x)}
{ =
0, (N(S))(x) = max{(N(f ))(x), (N(g))(x)},
if x < nN (a), if x ≥ nN (a).
(N(P))(x) = max{(N(I))(x), (nN (a), 1](x)} (N(I))(x) = min{(N(f ))(x), (N(g))(x)}, 1,
{ =
if x ≤ nN (a), if x > nN (a),
see Fig. 12. 3. Let us denote m = min{(N(f ))(nN (a)), (N(g))(nN (a))}, p = max{(N(f ))(nN (a)), (N(g))(nN (a))}. By hypothesis, m < p. Now, let us consider the following function µ (see Fig. 13a)).
Proof. The proof will be done by reduction to the absurd. Let us suppose that there exists a ∈ [0, 1] and f , g ∈ Ld such that f (a) = g(a), but (N(f ))(nN (a)) ̸ = (N(g))(nN (a)) (see Fig. 10).
⎧ ⎨max{f (x), g(x)}, f (a) = g(a), µ(x) = ⎩ min{f (x), g(x)},
si x < a,
1. Let us consider I = inf⊑ {f , g } and S = sup⊑ {f , g }. Taking into account that f and g are decreasing I(x) = min{f (x), g(x)} and S(x) = max{f (x), g(x)}, ∀x ∈ [0, 1] (see Fig. 11).
it is easy to note that M ⊑ µ ⊑ P (see Fig. 13a)). Then N(P) ⊑ N(µ) ⊑ N(M), that is equivalent to (N(M))(x) ≤ (N(µ))(x) ≤ (N(P))(x) (see Fig. 13b)), as the functions are increasing.
si x = a, si x > a,
Please cite this article as: S. Cubillo, C. Torres-Blanc and P. Hernández-Varela, A characterization for some type-2 fuzzy strong negations, Knowledge-Based Systems (2019) 105281, https://doi.org/10.1016/j.knosys.2019.105281.
S. Cubillo, C. Torres-Blanc and P. Hernández-Varela / Knowledge-Based Systems xxx (xxxx) xxx
Fig. 12. Functions infimum and supremum of decreasing and increasing functions.
Fig. 13. (a) Function µ in the proof of Theorem 5, (b) Impossible construction of N(µ) in the proof of Theorem 5.
≤
(N(µ))(nN (a))
≤
(N(P))
(N(M))(nN (a)) = max{(N(f ))(nN (a)), (N(g))(nN (a))} = p, (N(P))(nN (a)) = min{(N(f ))(nN (a)), (N(g))(nN (a))} = m, so p ≤ (N(µ))(nN (a)) ≤ m, that is impossible, as p > m by hypothesis. Therefore, it have to be (N(f ))(n(a)) = (N(g))(n(a)). In a similar way, we can prove that if f , g ∈ Lc with f (a) = g(a) then, (N(f ))(nN (a)) = (N(g))(nN (a)). □ In the following, we will build some new functions, including as a particular case the functions given in Fig. 8. They will lead us to give a method to pointwise obtain the image of the functions in Lc or in Ld throughout the negation N in (L, ⊑). For any a ∈ [0, 1] and for any m ∈ (0, 1], fa,m will denote the following function (see Fig. 14a)), 1, a,
{ fa,m (x) =
if x < m, if x ≥ m,
and for any a ∈ [0, 1] and any m ∈ [0, 1), f a,m will denote the function (see Fig. 14b)), f a,m (x) =
{
a, 1,
Fig. 15. Decreasing function f with f (1) = a and (N(f ))(0) = αN (a).
Fig. 16. Example of N(fa,m ) = f αN (a),nN (m) .
Fig. 17. (a) Function f ∈ Ld with f (1) = a and [0, m) ⊑ f , (b) function g ∈ Lc with g(0) = αN (a) and g ⊑ (nN (m), 1].
Fig. 14. (a) Function fa,m , (b) function f a,m .
In particular, (N(M))(nN (a)) (nN (a)). But,
7
if x ≤ m, if x > m.
Let us remember that, according to Proposition 7, for each a ∈ [0, 1] there is a unique b ∈ [0, 1] such that N(fa,1 ) = f b,0 .
Moreover, we have built nN : [0, 1] → [0, 1] the strong negation associated with N (nN (x) = y if and only if N(x¯ ) = y¯ ) and αN : [0, 1] → [0, 1] the order automorphism associated with N (αN (a) = b if and only if N(fa,1 ) = f b,0 ). Lemma 6. Let N be a strong negation in (L, ⊑) with N(1) = 1. If f ∈ Ld such that f (1) = a then (N(f ))(0) = αN (a) (see Fig. 15). Proof. As f (1) = fa,1 (1), then, by Theorem 5, (N(f ))(nN (1)) = (N(fa,1 ))(nN (1)) = (N(fa,1 ))(0)
= (f αN (a),0 )(0) = αN (a). Therefore, (N(f ))(0) = αN (a).
□
Lemma 7. Let N be a strong negation in (L, ⊑) with N(1) = 1. Then, ∀a ∈ [0, 1] and ∀m ∈ (0, 1] it holds that, N(fa,m ) = f αN (a),nN (m) (see Fig. 16). Proof. N(fa,m ) = N(min⊑ {f ∈ Ld : f (1) = a and [0, m) ⊑ f }) = max⊑ {g ∈ Lc : g(0) = αN (a) and g ⊑ (nN (m), 1]} = f αN (a),nN (m) (see Fig. 17). □ Theorem 6. Let N be a strong negation in (L, ⊑) with N(1) = 1. Then, there are an order automorphism, α , in [0, 1] and a strong negation, n, in [0, 1] such that ∀x ∈ [0, 1] it holds (N(f ))(x) = α (f (n(x))), ∀f ∈ Ld , and
Please cite this article as: S. Cubillo, C. Torres-Blanc and P. Hernández-Varela, A characterization for some type-2 fuzzy strong negations, Knowledge-Based Systems (2019) 105281, https://doi.org/10.1016/j.knosys.2019.105281.
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S. Cubillo, C. Torres-Blanc and P. Hernández-Varela / Knowledge-Based Systems xxx (xxxx) xxx
Fig. 18. Example f (x) = ff (x),x (x) and so (N(f ))(nN (x)) = (f αN (f (x)),nN (x) )(nN (x)).
(N(f ))(x) = α −1 (f (n(x))), ∀f ∈ Lc . Proof. Let f ∈ Ld and an x ∈ [0, 1]. Let us consider the function ff (x),x that satisfies ff (x),x (x) = f (x) (see Fig. 18). Therefore, by Theorem 5 and Lemma 7 it holds (N(f ))(nN (x)) = (N(ff (x),x ))(nN (x)) = (f αN (f (x)),nN (x) )(nN (x))
= αN (f (x)). Then, (N(f ))(nN (x)) = αN (f (x)), ∀x ∈ [0, 1], and so, (N(f ))(x) = αN (f (nN (x))), ∀x ∈ [0, 1]. Now, if g ∈ Lc then N(g) ∈ Ld and so, g = N(N(g)) = αN ◦ N(g) ◦ nN . Then, N(g) = αN−1 ◦ g ◦ nN . □ Theorem 7. N : L → L is a strong negation in (L, ⊑) such that N(1) = 1 if and only if there exists a strong negation n and an order automorphism α in ([0, 1], ≤), such that ∀x ∈ [0, 1], (N(f ))(x) = α (f R (n(x))) ∧ α −1 (f L (n(x))). Proof. ⇒) Let f ∈ L, by Proposition 4, N(f )L = N(f R ), N(f )R = N(f L ), and N(f ) = N(f )L ∧ N(f )R . Therefore, N(f ) = N(f R ) ∧ N(f L ) = (αN ◦ f R ◦ nN ) ∧ (αN−1 ◦ f L ◦ nN ).
⇐) This result is included in Theorem 2. □ Remark 6.
• When α = Id and n = 1 − Id we obtain (N(f ))(x) = f (1 − x), operation in M proposed by Walker and Walker [5]. Nevertheless, while they did not studied if it was a negation, we proved in [15] that it is not a negation in M, but a strong negation in L. • When α = Id and n is any strong negation in [0, 1], it results (N(f ))(x) = f (n(x)), strong negations in L presented by the authors in [15]. • Finally, if α is any automorphism in [0, 1] and n is any strong negation in [0,1], it turns out (Nn,α (f ))(x) = α (f R (n(x))) ∧ α −1 (f L (n(x))), strong negations in L presented by the authors in [18]. However, until now, the problem of whether there were more strong negations in L, apart from those determined by the previous formula, remained open. And, in particular, strong negations that leave the label 1 fixed. This work answers this question by characterizing the negations that satisfy these conditions and by proving that those obtained by said formula are all that exist, there is no other. 5. Conclusions Through literature about fuzzy sets, and in particular, regarding negations, a lot of results have been obtained. For example,
the characterization of strong negations in [0, 1] by Trillas in [19] is well known. Also, the negations in the membership degrees of Atanassov’s intuitionistic fuzzy sets, have been characterized by Deschrijver in [21]. However, in the case of the type-2 fuzzy sets and their membership degrees, the situation is not the same, there are still many gaps to be filled. This is because in this case the study entails much greater complexity. In this regard, this paper has showed a lot of new results. All of them have been directed, in an orderly manner and step by step, to obtain a characterization of a particular type of negations in the membership degrees of type-2 fuzzy sets: those negations in the set of convex and normal membership degrees that furthermore, leave the constant function 1 fixed. Of course, many questions remain open, since there are many other negations that do not verify these conditions, and that will have to be studied in future work. CRediT authorship contribution statement S. Cubillo: Conceptualization, Methodology, Writing - original draft, Investigation, Writing - review & editing. C. Torres-Blanc: Conceptualization, Methodology, Writing - original draft, Investigation, Writing - review & editing. P. Hernández-Varela: Conceptualization, Methodology, Writing - original draft, Investigation, Writing - review & editing. Acknowledgments This paper was partially supported by MCIU (Spain) project PGC2018-096509-B-100, Universidad Politécnica de Madrid (Spain) and Universidad Mayor (Chile). References [1] L. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-I, Inform. Sci. 8 (1975) 199–249. [2] L. Zadeh, Fuzzy sets, Inf. Control 20 (1965) 301–312. [3] J. Mendel, R. Jhon, Type-2 fuzzy sets made simple, IEEE Trans. Fuzzy Syst. 10 (2) (2002) 117–127. [4] M. Mizumoto, K. Tanaka, Some properties of fuzzy sets of type-2, Inf. Control 31 (1976) 312–340. [5] C. Walker, E. Walker, The algebra of fuzzy truth values, Fuzzy Sets and Systems 149 (2005) 309–347. [6] O. Castillo, M.A. Sánchez, C.I. González, G.E. Martínez, Review of recent type-2 fuzzy image processing applications, Information 8 (3) (2017) 97. [7] O. Castillo, L. Amador-Angulo, A generalized type-2 fuzzy logic approach for dynamic parameter adaptation in bee colony optimization applied to fuzzy controller design, Inf. Sci. 460–461 (2018) 476–496. [8] D.S. Comas, J.I. Pastore, A. Bouchet, V.L. Ballarin, G.J. Meschino, Interpretable interval type-2 fuzzy predicates for data clustering: A new automatic generation method based on self-organizing maps, Knowl.-Based Syst. 133 (2017) 234–254. [9] T.E. Dalkili, S. Sairkaya, Parameter prediction based on type-2 fuzzy clustering, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 26 (6) (2018) 877–892. [10] A. Doostparast Torshizi, M.H. Fazel Zarandi, A new cluster validity measure based on general type-2 fuzzy sets: Application in gene expression data clustering, Knowl-Based Syst. 64 (2014) 81–93. [11] S. Greenfield, F. Chiclana, Type-Reduced Set structure and the truncated type-2 fuzzy set, Fuzzy Sets and Systems 352 (2018) 119–141. [12] B.Q. Hu, C.K. Kwong, On type-2 fuzzy sets and their t-norm operations, Inform. Sci. 255 (2014) 58–81. [13] M. Najariyan, M. Mazandarani, R. John, Type-2 fuzzy linear systems, Granul. Comput. 2 (2017) 175–186. [14] T. Wu, X.W. Liu, An interval type-2 fuzzy clustering solution for large-scale multiple-criteria group decision-making problems, Knowl.-Based Syst. 114 (2016) 118–127. [15] P. Hernández, S. Cubillo, C. Torres-Blanc, Negations on type-2 fuzzy sets, Fuzzy Sets and Systems 252 (2014) 111–124. [16] E.E. Castiñeira, C. Torres-Blanc, S. Cubillo, Measuring contradiction on A-IFS defined in finite universes, Knowl.-Based Syst. 24 (2011) 1297–1309.
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Please cite this article as: S. Cubillo, C. Torres-Blanc and P. Hernández-Varela, A characterization for some type-2 fuzzy strong negations, Knowledge-Based Systems (2019) 105281, https://doi.org/10.1016/j.knosys.2019.105281.