Ordinal sums of the main classes of fuzzy negations and the natural negations of t-norms, t-conorms and fuzzy implications

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Ordinal sums of the main classes of fuzzy negations and the natural negations of t-norms, t-conorms and fuzzy implications Annaxsuel A. de Lima a,b , Benjamin Bedregal b , Ivan Mezzomo c Instituto Federal de Educação, Ciência e Tecnologia do Rio Grande do Norte – IFRN, Campus São Paulo do Potengi, Rio Grande do Norte, Brazil Programa de Pós-Graduação em Sistemas e Computação, Departamento de Informática e Matemática Aplicada – DIMAp, Universidade Federal do Rio Grande do Norte – UFRN, Natal, Rio Grande do Norte, Brazil c Departamento de Ciências Naturais, Matemática e Estatística – DCME, Universidade Federal Rural de Semi-Árido – UFERSA, Mossoró, Rio Grande do Norte, Brazil b

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Article history: Received 15 June 2019 Received in revised form 21 August 2019 Accepted 8 October 2019 Available online xxxx Keywords: Fuzzy connectives and aggregation operators Fuzzy negations Ordinal sums Classes of fuzzy negations Fuzzy implications

a b s t r a c t

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In the context of fuzzy logic, ordinal sums provide a method for constructing new functions from existing functions, which can be triangular norms, triangular conorms, fuzzy negations, copulas, overlaps, uninorms, fuzzy implications, among others. As our main contribution, we establish conditions for the ordinal sum of a family of fuzzy negations to be a fuzzy negation of a specific class, such as strong, strict, continuous, invertible and frontier. Also, we relate the natural negation of the ordinal sum on families of tnorms, t-conorms and fuzzy implications with the ordinal sum of the natural negations of the respective families of t-norms, t-conorms and fuzzy implications. This motivates us to introduce a new kind of ordinal sum for families of fuzzy implications. © 2019 Elsevier Inc. All rights reserved.

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1. Introduction

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The concept of the fuzzy set was introduced by Zadeh (1965) and, since then, several mathematical concepts such as number, group, topology, differential equation, and so on, have been fuzzified. There are several ways to extend the propositional connectives for a set [0, 1], but in general these extensions do not preserve all the properties of the classical logical connectives. Triangular norms (t-norms) and triangular conorms (t-conorms) were first studied by Menger [31] and also by Schweizer and Sklar [39] in probabilistic metric spaces and they are used to represent the logical conjunction in fuzzy logic and the intersection of fuzzy sets, whereas t-conorms are used to represent the logical disjunction in fuzzy logic and the union in fuzzy set theory. In 1965, L. A. Zadeh introduced the notion of fuzzy negation in [46], known as standard negation, in order to represent the logical negation and the complement of fuzzy sets. Since then, several important classes of fuzzy negations have been proposed with different motivations, as we can see in [22,23,26,29,37,41]. Fuzzy negations have applications in several areas, such as decision making, stock investment, computing with words, mathematical morphology and associative memory, as the presented in [6,10,23,25,43,47].

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E-mail addresses: [email protected] (A.A. de Lima), [email protected] (B. Bedregal), [email protected] (I. Mezzomo). https://doi.org/10.1016/j.ijar.2019.10.004 0888-613X/© 2019 Elsevier Inc. All rights reserved.

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The ordinal sums construction was first introduced, in the context of semigroups, by Climescu in [14] and Clifford in [15]. In the context of fuzzy logic, the ordinal sums were first studied for triangular norms and triangular conorms in [40] in order to provide a method to construct new t-norms and t-conorms from other t-norms and t-conorms preserving the most of common properties of the summands (for more details see [27]). However, the ordinal sums of several other important fuzzy connectives also has been studied, such as, for example, the ordinal sums of copulas [35], overlap functions [17], uninorms [32,33], fuzzy implications [18,42] and fuzzy negations [7]. The ordinal sums of fuzzy negations were introduced in [7] with the goal of study the ordinal sums of families of t-norms, t-conorms and fuzzy negations when they form De Morgan’s triples and in this paper, we prove some new results on ordinal sums of families of fuzzy negations. In particular, we establish conditions for the ordinal sum of a family of fuzzy negations resulting in a fuzzy negation belonging to a given class of fuzzy negations, such as strict, strong, frontier, continuous and invertible. In addition, we also investigate the relation between the ordinal sum of the natural negations of families of t-norms, t-conorms and fuzzy implications and the natural negations of the ordinal sums of the same families of t-norms, t-conorms and fuzzy implications. This paper is organized as follows: Section 2 provides a review of concepts such as t-norms, t-conorms, fuzzy implications, fuzzy negations, natural fuzzy negations, ordinal sums of a family of t-norms, t-conorms and fuzzy implications. In Section 3, we prove that the ordinal sum of a family of fuzzy negations is a fuzzy negation and we prove results involving concepts of ordinal sums of a family of fuzzy negations and equilibrium point. In Section 4, we establish conditions for the ordinal sum of a family of fuzzy negations of a given class to result in a fuzzy negation belonging to the same class. In particular, we consider the classes of fuzzy negations strict, strong, frontier, continuous and invertible. In Section 5, we define the left ordinal sum of a family of fuzzy implications and prove results involving ordinal sums of a family of t-norms, t-conorms and fuzzy implications. Also, we prove that the natural negation of the left ordinal sum of a family of fuzzy implications is the same as the ordinal sum of a family of fuzzy negations. Finally, Section 6 contains the final considerations and future works. 2. Preliminaries

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Symmetry: T (x, y ) = T ( y , x); Associativity: T (x, T ( y , z)) = T ( T (x, y ), z); Monotonicity: If x ≤ y, then T (x, z) ≤ T ( y , z); One identity: T (x, 1) = x.

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Gödel t-norm: T G (x, y ) = min(x, y ); Product t-norm: T P (x, y ) = x · y; Łukasiewicz t-norm: T L (x, y ) = max(0, x + y − 1); Drastic t-norm:



T D (x, y ) =

0, min(x, y ),

if (x, y ) ∈ [0, 1[ ; otherwise.

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1. 2. 3. 4.

Symmetry: S (x, y ) = S ( y , x); Associativity: S (x, S ( y , z)) = S ( S (x, y ), z); Monotonicity: If x ≤ y, then S ( z, x) ≤ S ( z, y ); Zero identity: S (x, 0) = x.

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A t-conorm S is called positive if it satisfies the condition: S (x, y ) = 1 iff x = 1 or y = 1.

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Example 2.2. Some examples of t-conorms:

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Definition 2.2. A function S : [0, 1]2 → [0, 1] is a t-conorm if, for all x, y , z ∈ [0, 1], the following axioms are satisfied:

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1. 2. 3. 4.

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Example 2.1. Some examples of t-norms:

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A t-norm T is called positive if it satisfies the condition: T (x, y ) = 0 iff x = 0 or y = 0.

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1. 2. 3. 4.

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Definition 2.1. A function T : [0, 1]2 → [0, 1] is a t-norm if, for all x, y , z ∈ [0, 1], the following axioms are satisfied:

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2.1. t-Norms, t-conorms, fuzzy implications and fuzzy negations

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In this section, we will briefly review some basic concepts which are necessary for the development of this paper. The definitions and additional results can be found in [1,2,5,6,12,13,24,27].

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1. Gödel t-conorm: S G (x, y ) = max(x, y );

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2. Probabilistic sum: S P (x, y ) = x + y − x · y; 3. Łukasiewicz t-conorm: S L (x, y ) = min(x + y , 1); 4. Drastic sum:



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S D (x, y ) =

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Definition 2.3 ([4, Definition 1.1.1]). A function J : [0, 1]2 → [0, 1] is called a fuzzy implication if it satisfies the following conditions:

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J1: J2: J3: J4: J5:

J is non-increasing with respect to the first variable; J is non-decreasing with respect to the second variable; J (0, 0) = 1; J (1, 1) = 1; J (1, 0) = 0.

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Example 2.3. Some examples of fuzzy implications:

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1. Gödel implication:



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J G (x, y ) =

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2. Rescher implication:



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J R S (x, y ) =

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if x ≤ y ; otherwise.

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3. Kleene-Dienes implication: J K D (x, y ) = max(1 − x, y ).



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Definition 2.4. A function N : [0, 1] → [0, 1] is a fuzzy negation if

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N1: N (0) = 1 and N (1) = 0; N2: If x ≤ y, then N (x) ≥ N ( y ), for all x, y ∈ [0, 1].

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Fuzzy negations N is strict if it is continuous and strictly decreasing, i.e., N (x) < N ( y ) when y < x. A fuzzy negation N satisfying the condition N3 is called strong.

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N3: N ( N (x)) = x for each x ∈ [0, 1].

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A fuzzy negation is called crisp if it satisfies N4

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N4: For all x ∈ [0, 1], N (x) ∈ {0, 1}.

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A fuzzy negation is invertible if there is a function N −1 x ∈ [0, 1].

: [0, 1] → [0, 1]

such that N −1 ( N (x))

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= x for each

Clearly, if a fuzzy negation N is invertible then N −1 , their inverse, is unique and is also an invertible fuzzy negation.

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Example 2.4. Some examples of fuzzy negations: 1. Standard negation: N S (x) = 1 − x; 2 2. A strict non-strong negation: N S (x) = 1 −√x2 ; 3. A strong non-standard negation: N 2S (x) = 1 − x2 .

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Note that:

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1. If N is strong then it has an inverse N −1 which is also a strong fuzzy negation; 2. If N is strong then N is strict.

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An equilibrium point of a fuzzy negation N is a value e ∈ [0, 1] such that N (e ) = e.

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Definition 2.5 ([4, Definition 1.4.2]). A fuzzy negation N is said to be non-vanishing if N (x) = 0 if and only if x = 1 and it is said to be non-filling if N (x) = 1 if and only if x = 0. A fuzzy negation N that is simultaneously non-vanishing and non-filling, it is called frontier.

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N T (x) = sup{ y ∈ [0, 1] : T (x, y ) = 0}

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for each x ∈ [0, 1], is called the natural fuzzy negation of T or the negation induced by T . In addition, let S be a t-conorm. The function N S : [0, 1] → [0, 1] defined as

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for each x ∈ [0, 1], is called the natural fuzzy negation of S or the negation induced by S.

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Remark 2.1 ([4, Remark 2.3.2 (i)]). Clearly N T and N S are, in fact, fuzzy negations.

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Lemma 2.1 ([4, Lemma 1.4.14]). If a function J : [0, 1] → [0, 1] satisfies J 1, J 3 and J 5, then the function N J : [0, 1] → [0, 1] defined by 2

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N J (x) = J (x, 0)

(1)

for all x ∈ [0, 1], is a fuzzy negation.

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Definition 2.8 ([7, Definition 2.5]). Let T be a t-norm, S be a t-conorm and N be a strict fuzzy negation. T N is the N-dual of T if, for all x, y ∈ [0, 1], T N (x, y ) = N −1 ( T ( N (x), N ( y ))). Similarly, S N is the N-dual of S if, for all x, y ∈ [0, 1], S N (x, y ) = N −1 ( S ( N (x), N ( y ))). Proposition 2.1 ([45, Theorem 3.2]). Let T be a t-norm, S be a t-conorm and N be a strict fuzzy negation. Then, T N is a t-conorm and S N is a t-norm.

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T I (x, y ) =

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ai + (b i − ai ) T i

x − ai

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, if (x, y ) ∈ [ai , bi ] ;

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b i − ai b i − ai min(x, y ), otherwise,

(2)

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Proposition 2.3 ([27]). Let ( S i )i ∈ I be a family of t-conorms and (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1]. Then the function S I : [0, 1]2 → [0, 1] defined by

S I (x, y ) =

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ai + (b i − ai ) S i

x − ai

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y − ai

b i − ai b i − ai ⎩ max(x, y ), otherwise,



, if (x, y ) ∈ [ai , bi ]2 ;

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(3)

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is a t-conorm which is called the ordinal sum of the summands (ai , b i , S i )i ∈ I .

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is a t-norm which is called the ordinal sum of the summands (ai , b i , T i )i ∈ I .

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Proposition 2.2 ([27]). Let ( T i )i ∈ I be a family of t-norms and (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1]. Then the function T I : [0, 1]2 → [0, 1] defined by

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In this subsection, we will recall the notion of ordinal sums of a family of t-norms and t-conorms, and some important results that will be used in the course of this work. For more information, see [7,27,45].

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2.2. Ordinal sums of t-norms, t-conorms and fuzzy implications

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If the negation is standard, then T N is called dual t-conorm of T and S N is called dual t-norm of S.

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Definition 2.7 ([4, Definition 1.4.15]). Let J be a fuzzy implication. The function N J defined by Eq. (1) is called the natural negation of J or the negation induced by J .

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N S (x) = inf{ y ∈ [0, 1] : S (x, y ) = 1}

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Definition 2.6 ([4, Definition 2.3.1]). Let T be a t-norm. The function N T : [0, 1] → [0, 1] defined as

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Nevertheless, for fuzzy implications there are several proposals of ordinal sums. For example,

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Proposition 2.4 ([19, Theorem 7]). Let ( J i )i ∈ I be a family of implications and (]ai , b i [)i ∈ I be a family of nonempty pairwise disjoint open subintervals of [0, 1] such that ai > 0 for each i ∈ I . Then the function J I : [0, 1]2 → [0, 1] given by

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J I (x, y ) =

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⎧ ⎨



ai + (b i − ai ) J i

x − ai

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y − ai



b i − ai b i − ai ⎩ J R S (x, y ), otherwise

, if x, y ∈ [ai , bi ];

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(4)

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is an implication which is called the ordinal sum of the summands (ai , b i , J i )i ∈ I .

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Other proposals of ordinal sums for fuzzy implications can be found, for example, in [3,18–20,42].

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3. Ordinal sums of fuzzy negations

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In this section, we will use the definition of ordinal sums of fuzzy negations introduced in [7], to show some results involving equilibrium point, relations between some classes of fuzzy negations and that ordinal sum of a fuzzy negation family is a fuzzy negation.

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Definition 3.1 ([7, Definition 3.1]). Let ( N i )i ∈ I be a family of fuzzy negations and (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1]. Then the function N I : [0, 1] → [0, 1] defined by

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N I (x) =

⎧ ⎨ ⎩



(1 − bi ) + (bi − ai ) N i N S (x), otherwise

x − ai b i − ai



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, if x ∈ [ai , bi ];

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Lemma 3.1 ([7, Lemma 3.1]). Let (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1], ( N i )i ∈ I be a family of fuzzy negations and N I the ordinal sum of the summands (ai , b i , N i )i ∈ I . Then,

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If ( N i )i ∈ I is a family of fuzzy negations such that (] N i (b i ), N i (ai )[)i ∈ I is also a family of nonempty, pairwise disjoint open subintervals of [0, 1], then the ordinal sum of ( T i )i ∈ I and ( S i )i ∈ I with respect to (] N i (b i ), N i (ai )[)i ∈ I will be denoted by T IN and S N I , respectively.

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Proposition 3.2. Let (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1], ( N i )i ∈ I be a family of fuzzy negations and N I be the ordinal sum of the summands (ai , b i , N i )i ∈ I . If, for some i ∈ I , N i has an equilibrium point e i and b i = 1 − ai , then ai + (b i − ai )e i is the equilibrium point of N I .

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Proof.Supposethat N i (e i ) = e i and b i = 1 − ai , for some i ∈ I . As xi = ai + (b i − ai )e i ∈ [ai , b i ], then N I (xi ) = (1 − b i ) + (b i − xi − a i xi − a i ai ) N i . Since, = e i and ai = 1 − bi , we have that b i − ai b i − ai

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= 1 − bi + (bi − ai )e i

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= ai + (bi − ai )e i . Therefore, ai + (b i − ai )e i is the equilibrium point of N I .

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N I (ai + (b i − ai )e i ) = 1 − b i + (b i − ai ) N i (e i )

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Proposition 3.1 ([7, Proposition 3.1]). Let (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1] and ( N i )i ∈ I be a family of fuzzy negations. Then the ordinal sum N I of the summands (ai , b i , N i )i ∈ I is a fuzzy negation.

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[ai , bi ] for some i ∈ I , then N I (x) ∈ [1 − b i , 1 − ai ]; 1) If x ∈  2) If x ∈ / i ∈ I [ai , bi ], then N I (x) ∈ / i ∈ I [1 − b i , 1 − a i ] .

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is called of the ordinal sum of the summands (ai , b i , N i )i ∈ I .

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Proposition 3.3. Let (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1], ( N i )i ∈ I be a family of fuzzy negations and N I be the ordinal sum of the summands (ai , b i , N i )i ∈ I . Then N I ≤ N S if and only if N i ≤ N S for all i ∈ I .

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Proof. (⇒) Let i ∈ I and x ∈ [0, 1]. Then xi = ai + (b i − ai )x ∈ [ai , b i ]. So,

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N i (x) = N i

xi − a i b i − ai



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N I (xi ) − (1 − b i )

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b i − ai

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b i − ai

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b i − xi

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b i − ai

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S

= 1 − x = N (x).

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(⇐) Suppose that N i ≤ N

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N I (x) =

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⎧ ⎨

S

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for all i ∈ I . Then

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x − ai

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, if x ∈ [ai , bi ];

b i − ai ⎩ S N (x), otherwise. ⎧   x − ai ⎨ , if x ∈ [ai , bi ]; (1 − bi ) + (bi − ai ) N S ≤ b i − ai ⎩ S N (x), otherwise.

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= N S (x). 2

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Proposition 3.4. Let (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1], ( N i )i ∈ I be a family of fuzzy negations and N I be the ordinal sum of the summands (ai , b i , N i )i ∈ I . Then, N I ≥ N S if and only if N i ≥ N S for all i ∈ I . Proof. Analogous from Proposition 3.3.

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Proposition 3.5. Let (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1], ( N i )i ∈ I and ( N i )i ∈ I be two families of fuzzy negations and N I and N I the ordinal sum of the summands (ai , b i , N i )i ∈ I and (ai , b i , N i )i ∈ I , respectively. Then N I = N if and only if N = N for each i ∈ I . i

I

i

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Proof. (⇒) If N i = N i , for some i ∈ I , then there exists x ∈ [0, 1] such that N i (x) = N i (x) and therefore, N i N i



y −ai b i −ai

ai ) N i





for y = b i x + ai (1 − x). Hence, y ∈ [ai , b i ] and so N I ( y ) = (1 − b i ) + (b i − ai ) N i

y −ai b i −ai





y −ai b i −ai





y −ai b i −ai



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= (1 − bi ) + (bi −

= N I ( y ). So by contrapositive, if N I = N I then N i = N i for each i ∈ I . (⇐) Straightforward. 2

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Lemma 4.1. Let (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1] and ( N i )i ∈ I be a family of functions and N I the function obtained as in Eq. (5). Then N I (ai ) = 1 − ai and N I (b i ) = 1 − b i , for each i ∈ I . Proof. For each i ∈ I , N I (ai ) = 1 − b i + (b i − ai ) N i



ai ) N i

b i −ai b i −ai





ai −ai b i −ai



= 1 − bi + (bi − ai ) N i (0) = 1 − ai and N I (bi ) = 1 − bi + (bi −

= 1 − bi + (bi − ai ) N i (1) = 1 − bi . 2

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In this section, we will prove some propositions and theorems using definitions and results seen in previous sections. We will establish conditions for the ordinal sum of a family of fuzzy negations resulting in a fuzzy negation belonging to a class of fuzzy negations such as strict, strong, frontier, continuous and invertible.

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4. Ordinal sums of fuzzy negations and classes of fuzzy negations

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28

38

The Proposition 3.5 shows that the definition by ordinal sums of a fuzzy negation is unique for a fixed family of nonempty, pairwise disjoint open subintervals of [0, 1]. But, if we consider different families of subintervals, then in general the representation is not unique, in fact, any fuzzy negation N is equal to N I for I = {1}, a1 = 0, b1 = 1 and N 1 = N, called the trivial ordinal sum representation of N. It is worthing to note that there are fuzzy negations with a unique ordinal sum representation, the trivial one, and for that are called of ordinally irreducible, for example, the fuzzy negation N (x) = 1 − x2 .

44 45

27

31

=

38 39

23 24

24 25

22

Proposition 4.1. Let (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1] and ( N i )i ∈ I be a family of functions and N I the function obtained as in Eq. (5). All the N i ’s are (continuous, strictly decreasing) fuzzy negations if and only if N I is a (continuous, strictly decreasing) fuzzy negation.

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1 2 3 4

Proof. (⇒) If all the N i ’s are fuzzy negations, then, by Proposition 3.1, N I is a fuzzy negation.   x − ai Now, suppose that for each i ∈ I , N i is continuous. Then, N I |[a ,b ] (x) = (1−b i ) + (b i −ai ) N i is clearly continuous. i i b i − ai S S S Since, N is continuous then it is sufficient to prove that for each i ∈ I , lim N (x) = N I (ai ) and lim N (x) = N I (b i ). In fact, xai

5 6 7

11 12 13 14

17 18

21 22 23 24 25 26 27 28 29

= 1 − ai

8 9 10 11 12 13 14 15

x b i

Now we will prove that N I is strictly decreasing when all N i are strictly decreasing. If x < y then we have the following cases: Case 1: If x, y ∈ [ai , b i ] for some i ∈ I , then

x − ai

y − ai



y − ai



x − ai

34 35

19 21

1 − bi ≤ N I (x) and by Eq. (5) N I ( y ) = 1 − y, then follows that N I ( y ) < N I (x). / j ∈ I [a j , b j ] and y ∈ [ai , bi ] for some i ∈ I , then x < ai and, by Eq. (5) N I (x) = 1 − x. Therefore, by Lemma 3.1, Case 4: If x ∈ N I ( y) ≤  1 − ai < 1 − x = N I (x). / [ai , bi ] then by Eq. (5), N I ( y ) = 1 − y < 1 − x = N I (x). Case 5: If x, y ∈

26

b i − ai

b i − ai

b i − ai

j∈ I

i∈ I



N i (0) = N i

=

39 40

=

41 42

=

43 44

b i − ai 1 − ai − (1 − b i )

58 59 60 61

27 28 29

33 34



ai −ai b i −ai



36 37 38

b i − ai

39 40 41 42

by Lemma 4.1

43 44 45 46

N i (1) = N i

= =

55 57

b i − ai



53

56

25

35

N I (ai ) − (1 − b i )

Analogously,

50

54

ai − ai

−(1 − bi ) + (1 − bi ) + (bi − ai ) N i

47

52

24

32



b i − ai

= 1.

45

51

23

31

38

49

22

30

Therefore, N I is strictly decreasing. (⇐) If N I is a fuzzy negation then, for each i ∈ I ,

37

48

18

N I ( y ) < N I (x). Case 2: If x ∈ [ai , b i ] and y ∈ [a j , b j ] for some i , j ∈ I such that i = j then ai < b i ≤ a j < b j . So, by Lemma 3.1, N I ( y ) ∈ [1 − b j , 1 − a j ] and N I (x) ∈ [1 − bi , 1 − ai ]. Thus, since 1 − b j < 1 − ai , then N I ( y ) < N I (x). / [a j , b j ], then bi < y and therefore 1 − y < 1 − bi . Since, by Lemma 3.1, Case 3: If x ∈ [ai , b i ] for some i ∈ I and y ∈

36

46

17

20

b i − ai

< Ni



16

. So, by Eq. (5),

<

and therefore N i



31 33

4

7

Analogously, we prove that lim N S (x) = N I (b i ).

30 32

3

6

19 20

2

xai

= (1 − bi ) + (bi − ai ) N i (0)   ai − ai = (1 − bi ) + (bi − ai ) N i b i − ai = N I (ai ).

10

1

5

lim N (x) = lim 1 − x

xai

9

16

x b i

S

8

15

7

=

b i − ai



47 48

b i − ai

−(1 − bi ) + (1 − bi ) + (bi − ai ) N i N I (b i ) − (1 − b i ) b i − ai 1 − b i − (1 − b i )

= 0.

b i − ai



b i −ai b i −ai

49



50 51

b i − ai

52 53 54 55 56

by Lemma 4.1

57 58

Let i ∈ I , x, y ∈ [0, 1] such that x ≤ y, xi = ai + (b i − ai )x and y i = ai + (b i − ai ) y. So, ai ≤ xi ≤ y i ≤

b i which implies N I ( y i ) ≤ y i −ai xi −ai N I (xi ). Therefrom, 1 − b i + (b i − ai ) N i ( y ) = 1 − b i + (b i − ai ) N i b −a ≤ 1 − b i + (b i − ai ) N i b −a = 1 − b i + (b i − ai ) N i (x). i

i

i

i

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8

1 2

Hence, N i ( y ) ≤ N i (x) and therefore N i is a fuzzy negation for each i ∈ I . In addition, from Eq. (5) clearly for each i ∈ I , if N I is continuous (strictly), then N i also will be. 2

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

Proposition 4.2. Let (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1] and ( N i )i ∈ I be a family of fuzzy negations such that N i is non-filling when ai = 0 and non-vanishing when b i = 1. Then, the ordinal sum N I of the summands (ai , bi , N i )i ∈ I is frontier. Proof. By Proposition 3.1, N I is a fuzzy negation. Let x ∈]0, 1[. If x ∈ /

 i∈ I

[ai , bi ], then from Eq. (5) it follows that N I (x) =

N S (x) ∈]0, 1[. Suppose x ∈ [ai , b i ] for some i ∈ I , then by Lemma 3.1 we have that N I (x) ∈ [1 − b i , 1 − ai ]. If ai = 0 and x−a b i = 1, then 0 < 1 − b i ≤ N I (x) ≤ 1 − ai < 1, i.e., N I (x) ∈]0, 1[. If ai = 0, also by Eq. (5) we obtain b −ai = bx ∈]0, 1[ and



i

i

i

35 36 37 38 39 40 41 42

x−ai 1−ai

Proposition 4.3. Let (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1] and ( N i )i ∈ I be a family of fuzzy negations. If the ordinal sum N I of the summands (ai , b i , N i )i ∈ I satisfies the following two properties

11 13 14 15 16 18 19 20 21 22

26 27

i

i

1 − b i + (b i − ai ) N i (x) = 1 − b i . So, by the second property, xi = b i and therefore, x = 1 which is a contradiction. The second case is analogous. Therefore, for each i ∈ I , the fuzzy negation N i is frontier. 2

28 29 30 31 32 33

Theorem 4.1. Let (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1] and ( N i )i ∈ I be a family of fuzzy negations. If all the N i ’s are strict fuzzy negations and, for each i ∈ I , there exists j ∈ I such that [a j , b j ] = [1 − b i , 1 − ai ] and N j = N i−1 , then the ordinal sum N I of the summands (ai , b i , N i )i ∈ I , is a strong fuzzy negation. Proof. From Proposition 3.1, N I is a fuzzy negation. Besides this, for any x ∈ [0, 1] if x ∈ [ai , b i ] for some i ∈I , then for x − ai ∈ hypothesis there exists j ∈ I such that [a j , b j ] = [1 − b i , 1 − ai ] and by Eq. (5), N I (x) = (1 − b i ) + (b i − ai ) N i b i − ai [1 − bi , 1 − ai ] = [a j , b j ]. Therefore,

⎛ ⎜

N I ( N I (x)) = (1−b j )+(b j −a j ) N j ⎜ ⎝

⎛ ⎜ = ai +(bi −ai ) N i−1 ⎜ ⎝

49 50 51 52 53

If x ∈ /

 i∈ I



(1−bi )+(bi −ai ) N i

x−ai b i −ai

bj −aj

 (1−bi )+(bi −ai ) N i

x−ai b i −ai

b i − ai



−a j

⎞ ⎟ ⎟ ⎠

 −(1−bi )

36 37 38 39 40 41 42 44 45 46 47

⎟ ⎟ ⎠

49

[ai , bi ], then N I (x) = 1 − x and, by Lemma 3.1 and hypothesis, N I (x) ∈ / 2

35

⎞

= x.

1 − (1 − x) = x.

34

43

48 50 51 52

 i∈ I

[1 − b i , 1 − a i ] =

 i∈ I

53

[ai , bi ]. So, N I ( N I (x)) =

54 55 56 57

57

61

10

25

Proof. Suppose that for some i ∈ I , N i is not frontier. Then there exists x = 1 such that N i (x) = 0 or there exists x = 0 such

x −a that N i (x) = 1. Let xi = ai + (b i − ai )x, thus xi ∈ [ai , b i ]. In the first case, we have that N I (xi ) = 1 − b i + (b i − ai ) N i bi −ai =

48

60

9

24

then N i is frontier for each i ∈ I .

47

59

8

23

1. N I (x) = 1 − ai for some i ∈ I only when x = ai ; and 2. N I (x) = 1 − b i for some i ∈ I only when x = b i

46

58

7

17

45

56

6

Corollary 4.1. Let (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1] and ( N i )i ∈ I be a family of fuzzy negations. If ai = 0 and b i = 1, for each i ∈ I , then N I is frontier.

44

55

5

12

43

54

4

= ∈]0, 1[ and < 1, since N i is non-filling in this case. Analogously, if bi = 1, then

x−ai hence 0 < N I (x) = (1 − ai ) N i 1−a < 1, because N i is non-vanishing. Therefore, for each x ∈]0, 1[, N I (x) ∈]0, 1[, i.e., N I is i frontier. 2

0 < N I (x) = 1 − b i + b i N i

x−ai b i −ai

x bi

33 34

2 3

3 4

1

Theorem 4.2. Let (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1] and ( N i )i ∈ I be a family of fuzzy negations. If the ordinal sum N I of the summands (ai , b i , N i )i ∈ I , is a strong fuzzy negation then all the N i ’s are strict fuzzy negations and, for each i ∈ I , there exists j ∈ I such that [a j , b j ] = [1 − b i , 1 − ai ] and N j = N i−1 . In addition, if for each i , j ∈ I , ai = b j and N i = N S , then for each i ∈ I there exists j ∈ I such that [a j , b j ] = [1 − b i , 1 − ai ] and N j = N i−1 .

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1 2 3 4 5 6 7 8 9 10

Proof. xi , y i ∈ [ai , b i ], N i (x) =  If x <  y and i ∈ I , thentaking xi = ai + (bi − ai )x and y i = ai + (bi − ai ) y, we have that xi < y i ,  xi − a i y i − ai y i − ai < (1 − bi ) + Ni and N i ( y ) = N i . Since, N I ( y i ) < N I (xi ), then by Eq. (5), (1 − b i ) + (b i − ai ) N i b i − ai  b − a b i i i − ai     xi − a i y i − ai xi − a i (bi − ai ) N i < Ni = N i (x) and therefore, each N i is strictly decreasing. On the . So, N i ( y ) = N i b i − ai b i − ai b i − ai other hand, by Lemma 3.1 is clear that case some N i is non continuous, then by Eq. (5) N I also would be non continuous. Hence, each N i is strict.  Now, suppose that for each i , j ∈ I , ai = b j when i = j and N i = N S . If 1 − ai ∈ [a j , b j ], then 1 − ai ∈ [a j , b j ] for some j ∈ I . If 1 − ai < b j , then there exist

 > 0 such that 1 − ai +  < b j , but ai −  ∈/

11 12 13 14

19 20 21 22

28

N I ( N I (x j )) = N I

34 35 36 37 38 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61



1 − b j + (b j − a j ) N j

xj − aj



10

19 20 21 22 24 x j −a j b j −a j

.

25 26 27 28

bj −aj

29 30 31

= N I (ai + (bi − ai ) N i−1 (x))   ai + (b i − ai ) N i−1 (x) − ai = 1 − bi + (bi − ai ) N i b i − ai

32

= 1 − bi + (bi − ai )x

37

33 34 35 36 38 39

which is a contradiction once N I is strong. Therefore, N j = N i−1 .

2

Corollary 4.2. Let (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1] and ( N i )i ∈ I be a family of fuzzy negations such that for each i , j ∈ I , ai = b j and N i = N S . Then, the ordinal sum N I of the summands (ai , b i , N i )i ∈ I , is a strong fuzzy negation if and only if all the N i ’s are strict fuzzy negations and, for each i ∈ I , there exists j ∈ I such that [a j , b j ] = [1 − b i , 1 − ai ] −1

and N j = N i .

40 41 42 43 44 45 46

Proof. Straightforward from Theorems 4.1 and 4.2.

2

Proposition 4.4. Let (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1], ( N i )i ∈ I be a family of invertible 1 fuzzy negations, N I be the ordinal sum of the summands (ai , b i , N i )i ∈ I and N − be the ordinal sum of the summands (1 − b i , 1 − I −1

9

18

= x j,

39

8

17

= N I (ai + (bi − ai ) N j (x))

33

7

16

= N I (1 − b j + (b j − a j ) N j (x))

32

6

23



31

5

15



Suppose N j = N i−1 , so there exists x ∈ [0, 1] such that N j (x) = N i−1 (x). Let x j = a j + (b j − a j )x, then x =

30

4

14

which is a contradiction, and therefore 1 − ai = b j . Analogously is possible to prove that 1 − b i = a j .

29

3

13

= ai −  ,

23

2

11

 (1 − ai +  ) − a j = 1 − b j + (b j − a j ) N j by Eq. (5) bj −aj   (1 − ai +  ) − a j

= 1 − b j + (b j − a j ) 1 − since N J = N S bj −aj = 1 − b j + (b j − a j ) − ((1 − ai +  ) − a j )

18

27

[a j , b j ]. So, N I (ai −  ) = 1 − ai +  ∈]a j , b j [

1

12

= N I (1 − ai +  )

17

26

j∈ I

j∈ I

ai −  = N I ( N I (ai −  ))

16

25



and therefore,

15

24

9

−1

ai , N i )i ∈ I . Then, N I

is the inverse of N I .

1 −1 Proof. It is sufficient to prove that N I ◦ N − I (x) = N I ◦ N I (x) = x, for each x ∈ [0, 1]. Then

  ⎞ ⎛⎧ x−(1−b i ) ⎨ −1 , 1 −( 1 − a )+( 1 − a −( 1 − b )) N if x ∈ [ 1 − b , 1 − a ]; i i i i i 1 i ⎠ ⎝ NI ◦ N− 1−ai −(1−b i ) I (x) = N I ⎩ S N (x), otherwise.   ⎞ ⎛⎧ ⎨ −1 x−1+b i , a +( b − a ) N if x ∈ [ 1 − b , 1 − a ]; i i i i i i ⎠ = NI ⎝ b i −ai ⎩ S N (x), otherwise.

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10

1 2 3

1 Let y = N − I (x), then y ∈ [ai , b i ] if and only if x ∈ [1 − b i , 1 − ai ]. Therefore,

5 6

=

⎧ ⎪ ⎪ ⎪ ⎪ ⎨

⎛

ai +(b i −ai ) N i−1

⎜ (1−bi )+(bi −ai ) N i ⎜ ⎝



x−1+b i b i −ai

3

 −ai

⎞

4 5

⎟ ⎟ , if y ∈ [ai , bi ]; ⎠

6

b i −ai ⎪ ⎪ ⎪ ⎪ ⎩ S S N ( N (x)), otherwise. ⎧    x − 1 + bi ⎨ , if x ∈ [ai , bi ]; (1 − bi ) + (bi − ai ) N i N i−1 = b i − ai ⎩ x, otherwise. ⎧   x − 1 + bi ⎨ , if x ∈ [ai , bi ]; (1 − bi ) + (bi − ai ) = b i − ai ⎩ x, otherwise. = x.

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

2

1 NI ◦ N− I (x) = N I ( y )

4

1

1 Analogously, we prove that N − I ◦ N I (x) = x.

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

2

19 20

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

Observe that, if (]ai , b i [)i ∈ I is a family of nonempty, pairwise disjoint open subintervals of [0, 1] and ( N i )i ∈ I is a family of fuzzy negations, then there exists a family (]c j , d j [)i ∈ J of nonempty, pairwise disjoint open subintervals of [0, 1] and a family ( N j ) j ∈ J of fuzzy negations where for each l, j ∈ J , cl = d j and N j = N S , such that N I = N J .

41 42

Proposition 5.1. Let ( T i )i ∈ I be a family of t-norms and (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1]. Then





= sup y ∈ [0, 1] : ai +(bi −ai ) T i

47

= 0.

48

(6)

56 57 58 59 60 61

31 32 33

36 37 38

If x ∈ /



x − ai

,

y − ai

b i − ai b i − ai



42



43

=0

44 45 46 47 48

i ∈ I [ai , b i ] ∪ {0}, then

49 50

50

55

30

41

= sup ∅

54

29

40

N T I (x) = sup { y ∈ [0, 1] : T I (x, y ) = 0}

46

53

28

39

45

52

27

35

Proof. If x ∈ [ai , b i ] and ai > 0, then

44

51

24

34

bi

43

49

23

26

As seen in Section 2, it is possible to generate fuzzy negations (called of natural negations) from t-norms, t-conorms and fuzzy implications, for more details see [4]. In this section, we will establish how these natural negations are preserved by ordinal sums.

39 40

22

25

5. Ordinal sums of fuzzy negations and the ordinal sums of t-norms, t-conorms and fuzzy implications

⎧  0, if x ∈ [ai , b i ] and ai > 0 or x ∈ / i ∈ I [ai , bi ] ∪ {0}; ⎪ ⎪ ⎨ 1, if  x = 0; N T I (x) = x ⎪ ⎪ , if x ∈]ai , bi ] and ai = 0. ⎩ NTi

21

N T I (x) = sup { y ∈ [0, 1] : T I (x, y ) = 0}

51

= sup { y ∈ [0, 1] : min(x, y ) = 0}

52

= 0.

54

53 55

If x = 0 then, trivially by Remark 2.1 and Proposition 2.2, N T I (x) = 1. If x ∈]ai , b i ] and ai = 0, then

N T I (x) = sup { y ∈ [0, 1] : T I (x, y ) = 0}



= sup



y ∈ [0, b i ] : b i T i

x

,

y

bi bi

56 57 58



 = 0 ∪ { y ∈]bi , 1] : min(x, y ) = 0}



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 1



y

x

y



11

 1

=0 ∈ [0, 1] : T i , bi bi     x = sup z ∈ [0, 1] : T i ,z =0 bi   x = NTi . 2

= sup

2 3 4 5 6

bi

2 3 4 5 6

bi

7

7 8

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

Proposition 5.2. Let ( S i )i ∈ I be a family of t-conorms and (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1]. Then

⎧  1, if x ∈ [ai , b i ], b i < 1 or x ∈ / i ∈ I [ai , bi ] ∪ {1}; ⎪ ⎪ ⎨ 0, if x = 1;   N S I (x) = x − ai ⎪ ⎪ + ai , if x ∈ [ai , bi [ and bi = 1. ⎩ (1 − ai ) N S i 1 − ai



= inf

21 22 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

(7)

If x ∈ /



x−ai

,

y −ai



 =1

b i −ai b i −ai

y ∈ [ai , 1] : ai + (1 − ai ) S i

58

28 29 30

61

31 32 33 34



x−ai y −ai

,

1−ai 1−ai



 =1

∪ { y ∈ [0, ai [: max(x, y ) = 1}}      x − ai y − ai =1 ∪∅ , = inf y ∈ [ai , 1] : S i 1 − ai 1 − ai     y − ai x − ai y − ai = (1−ai ) inf =1 + ai ∈ [0, 1] : S i , 1 − ai 1 − ai 1 − ai     x − ai = (1 − ai ) inf z ∈ [0, 1] : S i , z = 1 + ai 1 − ai   x − ai + ai . 2 = (1 − ai ) N S i 1 − ai In the case of fuzzy implications, we have that the natural negation of the ordinal sums of a family of fuzzy implications, according to the established in Proposition 2.4 (and also for the proposal in [3,18–20,42]) always results in bottom negation, i.e.,

0, if x > 0; 1, if x = 0.

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56

So, in order to obtain, more interesting natural negations we propose the following notion of ordinal sums for fuzzy implications:

57 58 59

59 60

22

27

If x = 1 then, trivially by Remark 2.1 and Proposition 2.3, N S I (x) = 0. If x ∈ [ai , b i [ and b i = 1, then



21

26

then

N S I (x) = inf { y ∈ [0, 1] : S I (x, y ) = 1}

20

25

56 57

16

24

= inf { y ∈ [0, 1] : max(x, y ) = 1} = 1.

N ⊥ (x) =

15

23

N S I (x) = inf { y ∈ [0, 1] : S I (x, y ) = 1}



14

19



= inf(∅ ∪ {1}) = 1.

= inf

13

18

y ∈ [ai , b i ] : ai +(b i −ai ) S i

i ∈ I [ai , b i ] ∪ {1},

11

17

∪ { y ∈ [0, ai [∪]bi , 1] : max(x, y ) = 1}}

23

10 12

Proof. If x ∈ [ai , b i ] and b i < 1, then

N S I (x) = inf { y ∈ [0, 1] : S I (x, y ) = 1}

9

Definition 5.1. Let ( J i )i ∈ I be a family of fuzzy implications and (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1] such that b i < 1 for each i ∈ I . Then the function  J I : [0, 1]2 → [0, 1] defined by

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J I (x, y ) =

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12

⎧ ⎨ ⎩

 (1 − bi ) + (bi − ai ) J i J K D (x, y ), otherwise,

x − ai b i − ai

 , y , if x ∈ [ai , bi ];

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is called of the left ordinal sum of the summands (ai , b i , J i )i ∈ I . Proposition 5.3. Let ( J i )i ∈ I be a family of fuzzy implications and (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1] such that b i < 1 for each i ∈ I . Then the function  J I : [0, 1]2 → [0, 1] defined in Equation (8) is a fuzzy implication.

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Proof. J1) Let x, y , z ∈ [0, 1] such that x ≤ y. We have several cases:   x − ai y − ai y − ai 1. If x, y ∈ [ai , b i ] for some i ∈ I , then because ≤ and J i satisfy J1, we have that J i ,z ≤ b i − ai b i − ai     b i − ai  x − ai y − ai x − ai Ji , z and therefore J I ( y , z) = (1 − bi ) + (bi − ai ) J i , z ≤ (1 − bi ) + (bi − ai ) J i ,z = b i − ai b i − ai b i − ai J I (x, z). 2. If x ∈ [ai , b i ] and y ∈ [a j , b j ] for some i = j ∈ I , then  J I (x, z) ∈ [1 − b i , 1 − ai ] and  J I ( y , z) ∈ [1 − b j , 1 − a j ]. But, because x ≤ y and [ai , b i ] be disjoint of [a j , b j ], we have that b i < a j and so 1 − a j < 1 − b i . Therefore,  J I ( y , z) < J I (x, z).  3. If x ∈ [ai , b i ] for some i ∈ I and y ∈ / j ∈ I [a j , b j ], then J I (x, z) ∈ [1 − bi , 1 − ai ] where J I ( y , z) = J K D ( y , z) = max{1 −

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J4) J5)

y , z}. But, because x < y then 1 − y < 1 − b i and therefore,  J I ( y , z) <  J I (x, z).  4. The casex ∈ / i ∈ I [ai , bi ] and y ∈ [a j , b j ] for some i = j ∈ I is analogous to previous one. 5. If x, y ∈ / i ∈ I [ai , bi ], then J I ( y , z) = J K D ( y , z) = max{1 − y , z} ≤ max{1 − x, z} = J K D (x, z) = J I (x, z). Let x, y , z ∈ [0, 1] such that y ≤ z. We have two cases:   x − ai 1. If x ∈ [ai , b i ] for some i ∈ I , then because J i satisfy J2,  J I (x, y ) = (1 − b i ) + (b i − ai ) J i , y ≤ (1 − bi ) + (bi − b i − ai   x − ai ai ) J i , z = J I (x, z). bi − ai  2. If x ∈ / i ∈ I [ai , bi ], then J I (x, y ) = J K D (x, y ) = max{1  − x, y} ≤ max{1 − x, z} = J K D (x, z) = J I (x, z). 0 If ai = 0 for some i ∈ I , then  J I (0, 0) = (1 − b i ) + b i J i , 0 = 1 − bi + bi = 1. If ai = 0 for each i ∈ I , then J I (0, 0) = bi J K D (0, 0) = 1. Since b i = 1 for each i ∈ I , then  J I (1, 1) = J K D (1, 1) = 1. Since b i = 1 for each i ∈ I , then  J I (1, 0) = J K D (1, 0) = 0. 2

 Remark 5.1. By   Equation (8), for each x ∈ [ai , bi ], some i ∈ I and y ∈ [0, 1], clearly J I (x, y ) = (1 − bi ) + (bi − ai ) x − ai Ji , y ∈ [1 − b i , 1 − a i ] . b i − ai

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Theorem 5.1. Let ( J i )i ∈ I be a family of fuzzy implications and (]ai , b i [)i ∈ I be a family of nonempty, pairwise disjoint open subintervals of [0, 1] such that b i < 1 for each i ∈ I . For each i ∈ I let N i the natural negation of the implication J i and N I their ordinal sums. Then, the natural negation of the left ordinal sum of the summands (ai , b i , J i )i ∈ I , denoted by NJ I , is such that NJ I = N I .

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Proof. Since, by Proposition 5.3,  J I is a fuzzy implication then their natural negation NJ I is in fact a fuzzy negation. On the other hand, by Proposition 3.1, N I is also a fuzzy negation. Then NJ I (0) = N I (0) and NJ I (1) = N I (1). Let x ∈ (0, 1). If     x − ai x − ai = N I (x). x ∈ [ai , b i ] for some i ∈ I then NJ I (x) =  J I (x, 0) = (1 − b i ) +(b i − ai ) J i , 0 = (1 − bi ) +(bi − ai ) N i b i − ai b i − ai Now, if x ∈ / [ai , bi ] for each i ∈ I then NJ I (x) = J I (x, 0) = J K D (x, 0) = 1 − x = N I (x). 2 6. Final remarks

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Ordinal sum is an important method that allows us to construct, from a family of operators of a certain class, a new operator of the same class. For instance, in [40], it was proved that the ordinal sums of a family of triangular norms and triangular conorms result in triangular norms and triangular conorms, respectively. This same idea was used for copulas [35], overlaps functions [17], uninorms [32,33], fuzzy implications [18,42] and fuzzy negations [7]. In [7], it was proved that ordinal sums of a family of fuzzy negations result in a fuzzy negation. In this paper, we studied the ordinal sums of some known classes of fuzzy negations and some properties of fuzzy negations, as well as, we studied the ordinal sums of a family of t-norms, t-conorms and fuzzy implications. In particular, we introduce a new way to make an ordinal sum of fuzzy implications which only consider the first variable to decide the case to be considered. The advantage of this new ordinal sums of fuzzy implication, when compared with the several proposes existent in the literature, is that

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the natural negation is not trivial and commute with the ordinal sums of the natural negations of the summands in the following sense

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J I Lemma 2.1

P roposition 3.1

NJ I

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In [28] was proved that continuous t-norms (and therefore of continuous t-conorms) can be obtained as ordinal sums of continuous Archimedean t-norms (t-conorms), i.e., ordinal sums where the summands are conjugated of the product or Łukasiewicz t-norm (for more details see [27]). This characterization theorem had important consequences in t-norm based fuzzy logics, which are axiomatic extensions of some classes of algebraic logics, as can be seen in [21,36,44]. In addition, this characterization also had been useful in a neural net approach of multi-adjoint logic programming, which are unified frameworks where uncertainty, imprecise data and incomplete information can coexist [30]. Future developments will be devoted to the study and potential application of ordinal sums of fuzzy negations and fuzzy implications in some classes of algebraic logics and in multi-adjoint logic programming, relating it with the ordinal sums of t-norms and t-conorms in these contexts. Several extensions or types of fuzzy set theory had been proposed in order to solve the problem of constructing the membership degree functions of fuzzy sets or/and to represent the uncertainty associated to the considered problem in a way different from fuzzy set theory [11]. In other future works, we intend to extend the study of ordinal sums of fuzzy negations to some of the most important extensions of fuzzy set theory, such as, ordinal sums of n-dimensional fuzzy negations [8], ordinal sums of intuitionistic fuzzy negations [16], ordinal sums of interval-valued fuzzy negations [5], ordinal sums of interval-valued Atanassov’s intuitionistic fuzzy negations [38] and ordinal sums of typical hesitant fuzzy negations [9]. Declaration of competing interest

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement

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This work is supported by Brazilian National Council of Technological and Scientific Development CNPq (Proc. 307781/2016-0 and 404382/2016-9) and UFERSA.

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