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Information Sciences 180 (2010) 1373–1389
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Information Sciences
journal homepage: www.elsevier .com/locate / ins
On interval fuzzy S-implications
B.C. Bedregal a,*, G.P. Dimuro b, R.H.N. Santiago a, R.H.S. Reiser c
a Universidade Federal do Rio Grande do Norte, Departamento de Informática e Matemática Aplicada, Campus Universitário, 59072-970 Natal, Brazilb Universidade Federal do Rio Grande, Programa de Pós-Graduação em Modelagem Computacional, Campus Carreiros, 96201-090 Rio Grande, Brazilc Universidade Católica de Pelotas, Programa de Pós-Graduação em Informática, Rua Felix da Cunha 412, 96010-000 Pelotas, Brazil
a r t i c l e i n f o
Article history:Received 6 March 2008Received in revised form 21 August 2009Accepted 21 November 2009
Keywords:Fuzzy logicInterval mathematicsInterval representationInterval fuzzy implicationInterval S-implicationInterval automorphism
0020-0255/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.ins.2009.11.035
* Corresponding author. Tel.: +55 84 32153814; fE-mail addresses: [email protected] (B.C
ucpel.tche.br (R.H.S. Reiser).
a b s t r a c t
This paper presents an analysis of interval-valued S-implications and interval-valued auto-morphisms, showing a way to obtain an interval-valued S-implication from two S-implica-tions, such that the resulting interval-valued S-implication is said to be obtainable. Someconsequences of that are: (1) the resulting interval-valued S-implication satisfies the cor-rectness property, and (2) some important properties of usual S-implications are preservedby such interval representations. A relation between S-implications and interval-valued S-implications is outlined, showing that the action of an interval-valued automorphism on aninterval-valued S-implication produces another interval-valued S-implication.
� 2009 Elsevier Inc. All rights reserved.
1. Introduction
Fuzzy set theory was introduced by Zadeh [76], allowing the development of soft computing techniques centered on theidea that computation, reasoning and decision making should exploit, whenever possible, the tolerance for imprecision anduncertainty [78].
Like classical set theory, the corresponding fuzzy logic has been developed as formal deductive systems, but with a com-parative notion of truth that formalizes deduction under vagueness. It provides tools for approximate reasoning and decisionmaking together with a framework to deal with imprecision, uncertainty, incompleteness of information, conflicting infor-mation, partiality of truth and partiality of possibility [79], improving the design of flexible information processing systems[51]. It has been applied in several areas, such as control systems [18], decision making [17], expert systems [67], patternrecognition [19,50], etc. On the other hand, fuzzy logic may be viewed as an attempt to formalize/mechanize the humancapability to perform a wide variety of physical and mental tasks without any measurements or computations [79].
Fuzzy sets were originally defined by membership functions of the form lA : X ! ½0;1�, where any membership degreelAðxÞwas a precise number. However, in some situations, we do not have precise knowledge about the membership function(or the membership degree) that should be taken into account. This consideration has led to some extensions of fuzzy sets,giving rise to type-n fuzzy sets [77], which incorporated uncertainty about membership functions and membership degreesinto fuzzy set theory, where the ‘‘precise number” representing a membership degree was generalized to a value carrying itsuncertainty.
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ax: +55 84 32153813.. Bedregal), [email protected] (G.P. Dimuro), [email protected] (R.H.N. Santiago), reiser@
1374 B.C. Bedregal et al. / Information Sciences 180 (2010) 1373–1389
Type-2 fuzzy sets have been largely applied since the works of Jerry Mendel [48] in the 90s, with the increase of the the-oretical research on their properties [49,71]. Interval-valued fuzzy sets are a particular case of type-2 fuzzy sets with a richstructure provided by Interval Mathematics [52].
Interval Mathematics is a mathematical theory that aims at the representation of uncertain input data and parameters,originally interested in the automatic and rigorous control of the errors that arise in numerical computations [37]. It has beenapplied to deal with the uncertainties in the results of numerical algorithms in Engineering and Scientific Computing, withcontributions1http://www.cs.utep.edu/interval-comp/. in several areas [40,41], such as electrical power systems [8], mechan-ical engineering [54], chemical engineering [68], artificial intelligence [38], multiagent systems [26] and geophysics [2].
Interval mathematics is another form of information theory which is related to, but independent from, fuzzy logic. In onehand, intervals can be considered to be a particular type of fuzzy set. On the other hand, interval membership degrees can beused to represent the uncertainty and the difficulty of an expert to precisely determine the fairest membership degree of anelement with respect to a linguistic term, as considered in interval-valued fuzzy sets. In this case, the radius of an interval isused as an error measure [57], providing an estimation of the uncertainty during membership assignment. Interval degreescan also be viewed as summarizing the opinions of several experts about the exact membership degree for an element withrespect to a linguistic term. In this case, the left and right interval endpoints are, respectively, the least and the greatest de-grees provided by a group of experts [29,57,70]. In both cases, the richness of interval structures provides tools to deal withsuch notions of uncertainty.
Interval-valued fuzzy sets were introduced independently by Zadeh [77] and other authors in the 70s (e.g., [36,39,63]),allowing to deal not only with vagueness (lack of sharp class boundaries), but also with uncertainty (lack of information)[29,45]. Since then, the integration of fuzzy theory with interval mathematics has been studied from different viewpoints,as properly pointed out by Lodwick [45], generating several different approaches (as in [21,27–29,33,44,45,53,56,57,71,75]).
In this paper, we follow the approach first introduced in Bedregal and Takahashi’s works [12,13], which has already beenapplied in our previous papers, where we provided interval extensions for some fuzzy connectives (see [9,15,25,59]) by con-sidering both correctness (accuracy) and optimality aspects of interval methods [37,64]. In particular, we are interested inthe investigations of interval extensions of the various types of fuzzy implications and their related properties.
Fuzzy implications [4–7,11,31,46,47,58,61,62,66,65,74] play an important role in fuzzy logic. In a broad sense, fuzzyimplications are important not only because they are used to formalize ‘‘If. . .then” rules in fuzzy systems, but also becausethey have different meanings (e.g., S-implications, R-implications, QL-implications, D-implications etc.) to be used in per-forming inferences in approximate reasoning and fuzzy control [46]. The role of fuzzy implications on the development ofapplications also motivates the research in the narrow sense through the investigation of related logical aspects [47].
The aim of this work is to introduce an interval generalization for a particular meaning of fuzzy implications, namely, S-implications [5,6,16,47,61,30,31]. This generalization, which we call interval S-implication, satisfies the correctness propertymentioned above. We present an analysis of interval S-implications and interval automorphisms, showing a way to obtain aninterval S-implication from two S-implications, so that the resulting interval S-implication is said to be obtainable. We provethat interval S-implications are closed under the action of the interval-valued automorphisms introduced in [33,34]. We alsoprove that several analogous important properties of S-implications are also valid for interval S-implications, showing theirapplicability on interval-based fuzzy systems. Thus, this work is an important step towards the fundamentals for the devel-opment of such interval-based fuzzy systems.
The paper is organized as follows: Section 2 discusses the notions of interval representations of real functions providingthe related definitions and results. The main results related to the interval extensions of fuzzy t-conorms introduced in pre-vious works [12–14] are presented in Section 3. In Section 4, we discuss the interval extensions of fuzzy negations. A briefreview about fuzzy implications, and, in particular, S-implications, is presented in Section 5, where the main properties of S-implications are presented. Section 6 introduces interval fuzzy implications and the definition of interval S-implications,showing that several analogous properties of S-implications also hold for interval S-implications. The action of an intervalautomorphism on an interval S-implication is analyzed in Section 7. Section 8 concludes this paper, summarizing its mainresults, presenting some final remarks and pointing out future works.
2. Interval representations
Consider the real unit interval U ¼ ½0;1�# R and the set U ¼ f½a; b�j0 6 a 6 b 6 1g of subintervals of U. The left and rightprojections of an interval ½a; b� 2 U are given by the functions l; r : U! U, defined, respectively, by
1 For
lð½a; b�Þ ¼ a and rð½a; b�Þ ¼ b: ð1Þ
For a given interval X 2 U; lðXÞ and rðXÞ are also denoted, respectively, by X and X.The following partial orders play important roles in this paper:
(i) The product order (also called component-wise order or Kulisch-Miranker order), defined, for all X;Y 2 U, by:
X 6 Y () X 6 Y ^ X 6 Y; ð2Þ
a survey on applications of Interval Mathematics, see, e.g., http://www.cs.utep.edu/interval-comp/.
B.C. Bedregal et al. / Information Sciences 180 (2010) 1373–1389 1375
(ii) The inclusion order, defined, for all X;Y 2 U, by:
X # Y () X P Y ^ X 6 Y : ð3Þ
These partial orders can be naturally extended to Un. For example, considering the product order defined in Eq. (2), forany ~X ¼ ðX1; . . . ;XnÞ; ~Y ¼ ðY1; . . . ;YnÞ 2 Un, one has that
~X 6 ~Y () X1 6 Y1 ^ � � � ^ Xn 6 Yn: ð4Þ
An interval function F : Un ! U is said to be strictly increasing if, for each~X;~Y 2 Un, whenever~X < ~Y (that is,~X 6 ~Y and~X – ~Y)it holds that Fð~XÞ < Fð~YÞ.
The notion of interval correctness plays a very important role in numerical computations [64]. A correct interval methodcan always guarantee that if x 2 X then f ðxÞ 2 FðXÞ, where F is the interval method that evaluates a real function f. In [64], thenotion of correctness is formalized by the so-called Interval Representation, considering that interval methods are represen-tations of punctual methods. In what follows, we reproduce such definition, but, instead of considering the set of real num-bers R, we consider the set U ¼ ½0;1�# R.
Definition 1 [64]. An interval X 2 U is said to be a representation for a real number a if a 2 X. Considering two intervalrepresentations X and Y for a real number a; X is said to be a better interval representation of a than Y, denoted by Y v X, ifX # Y .
The notion of better interval representation can also be easily extended for n-tuples of intervals.
Definition 2 [64]. A function F : Un ! U is said to be an interval representation of a real function f : Un ! U if, for each~X 2 Un and ~x 2 ~X; f ð~xÞ 2 Fð~XÞ. F is also said to be correct with respect to f.
An interval function F : Un ! U is said to be a better interval representation of a real function f : Un ! U than an intervalfunction G : Un ! U, denoted by G v F, if Fð~XÞ# Gð~XÞ, for each ~X 2 Un [64].
In [64], the notion of optimality of interval methods was formalized by the so-called canonical interval representations ofreal functions, also known by the best interval representations [12] of real functions:
Definition 3 [64]. The best interval representation of a real function f : Un ! U is the interval function bf : Un ! U, defined by
bf ð~XÞ ¼ ½infff ð~xÞj~x 2 ~Xg; supff ð~xÞj~x 2 ~Xg�: ð5Þ
Notice that the interval function bf is well defined and it is clearly an interval representation of f. Moreover, for any otherinterval representation F of f, F v bf . This means that bf always returns a narrower interval than the intervals produced by anyother interval representation of f. Thus, bf has the optimality property of interval algorithms mentioned by Hickey et al. [37],when it is seen as an algorithm to compute a real function f.
Observe that if the real function f is continuous in the usual sense then, for each ~X 2 Un, one has that
bf ð~XÞ ¼ ff ð~xÞj~x 2 ~Xg ¼ f ð~XÞ; ð6Þ
that is, the best interval representation bf of a real function f coincides with its range [64].
Definition 4. An interval function F : Un ! U is obtainable if there exist projections P1; . . . ; P2n : U! U, where Pi 2 fl; rg, fori ¼ 1; . . . ;2n, and functions f1; f2 : Un ! U such that, for each X1; . . . ;Xn 2 U, it holds that
FðX1; . . . ;XnÞ ¼ ½f1ðP1ðX1Þ; . . . ; PnðXnÞÞ; f2ðPnþ1ðX1Þ; . . . ; P2nðXnÞÞ�: ð7Þ
The concept of obtainable function generalizes the notion of representable function, as proposed by Deschrijver et al. [22–24,32] in the context of interval t-norms. On the other hand, observe that every obtainable interval function F is an intervalrepresentation of some real function f (at least f1 and f2). However, the converse is not true. For example, the interval functionF : Un ! U, defined by FðXÞ ¼ ½maxð0;X � X
10Þ;minð1;X þ X10Þ�, is an interval representation of the identity on U, idUðxÞ ¼ x, but
F is not obtainable.
An interval function F : Un ! U preserves degenerate intervals, if it maps degenerate intervals into degenerate intervals,that is, if, for each x1; . . . ; xn 2 U, there exists y 2 U such that Fð½x1; x1�; . . . ; ½xn; xn�Þ ¼ ½y; y�.
Notice that the best interval representation of any real function is # -monotonic (inclusion-monotonic), obtainable andpreserves degenerate intervals.
In this paper, we adopt the following notions of continuity defined on the set U of subintervals of ½0;1�:
(i) Moore continuity [52]: is defined as an extension of the continuity on the set of the real numbers by considering themetric given by the distance between two intervals X;Y 2 U, which is defined by: dðX;YÞ ¼maxfjX � Yj; jX � Y jg.
(ii) Scott continuity: is defined as an extension of the continuity on the set of the real numbers, considering the quasi-met-ric qðX; YÞ ¼maxfY � X;X � Y; 0g defined over U, introduced in [1,64]. An alternative way to define the Scott continu-ity on U is to consider the set U with the reverse inclusion order as a continuous domain [35], and a function
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f : ðU;�Þ ! ðU;�Þ is said to be Scott-continuous if it is monotonic and preserves the least upper bound of directed sets[35].2
The main result in [64] can be adapted to our context, considering the set U instead of R, as shown in the following:
Theorem 5. Let f : Un ! U be a real function. The following statements are equivalent:
(i) f is continuous;(ii) bf is Scott-continuous;
(iii) bf is Moore-continuous.
3. Interval t-conorms
A triangular conorm (t-conorm for short) is a function S : U2 ! U that is commutative, associative, monotonic and has anidentity ‘‘ 0”, generalizing the classical disjunction. Among several t-conorms, in this paper, we consider the maximum t-con-orm SM : U ! U, defined as
2 A d3 sup
SMðx; yÞ ¼maxfx; yg: ð8Þ
An interval generalization of t-conorms was introduced in [13], applying the principles discussed in Section 2. The so-calledinterval t-conorm is defined as an interval representation of a t-conorm. This generalization fits the idea of interval member-ship degrees as approximations of exact degrees.
Definition 6 [13]. A function S : U2 ! U is an interval t-conorm, whenever it is commutative, associative, monotonic withrespect to the product and inclusion orders, and ½0; 0� is the identity element.
In the following, the main results related to interval t-conorms are presented.
Proposition 7 [13, Theorems 5.1 and 5.2]. If S is a t-conorm, then its best interval representation bS : U2 ! U is an interval t-conorm.
For example, the supremum interval t-conorm SM : U2 ! U, defined by
SMðX;YÞ ¼ supfX;Yg; ð9Þ
is the best interval representation of the maximum t-conorm SMðx; yÞ, given in Eq. (8), that is, SM ¼ cSM .3
Proposition 8 [14, Corollary 5.3]. The function S : U2 ! U is an interval t-conorm if, and only if, the real functionsS;S : U2 ! U, defined by
Sðx; yÞ ¼ lðSð½x; x�; ½y; y�ÞÞ and Sðx; yÞ ¼ rðSð½x; x�; ½y; y�ÞÞ; ð10Þ
are t-conorms and
SðX; YÞ ¼ ½SðX; YÞ;SðX;YÞ�; ð11Þ
where l and r are, respectively, the left and right projections defined in Eq. (1).
Therefore, one has that interval t-conorms are obtainable. The following result is immediate:
Corollary 9. Let S : U2 ! U be an interval t-conorm and S : U2 ! U be a t-conorm. If S represents S then S 6 S 6 S.
Given a t-conorm S, the interval t-conorm bS can be expressed by:
bSðX; YÞ ¼ ½SðX;YÞ; SðX; YÞ�: ð12Þ
4. Interval fuzzy negations
Like t-conorms, fuzzy negations generalize the classical negations. A function N : U ! U is a fuzzy negation if
N1: Nð0Þ ¼ 1 and Nð1Þ ¼ 0;N2: If x P y then NðxÞ 6 NðyÞ; 8x; y 2 U.Fuzzy negations satisfying the involutive property N3 are called strong fuzzy nega-
tions [16,43]:N3: NðNðxÞÞ ¼ x; 8x 2 U.In addition, a continuous fuzzy negation is strict whenN4: If x > y then NðxÞ < NðyÞ; 8x; y 2 U.
irected set of ðU;�Þ is a non-empty subset S # U such that every pair of intervals in S has an upper bound in S.denotes the supremum related to the Kulisch-Miranker or product order.
B.C. Bedregal et al. / Information Sciences 180 (2010) 1373–1389 1377
As is well known, all strong fuzzy negations are strict.An element e 2 U is said to be an equilibrium point of a fuzzy negation N whenever NðeÞ ¼ e. If N is a strict fuzzy negation,
then there exists a unique equilibrium point eN 2 U and it holds that NðxÞP eN , for all x 6 eN . Conversely, one has thatNðxÞ 6 eN , for all x P eN .
Definition 10. An interval function N : U! U is an interval fuzzy negation if, for all X;Y in U, the following properties hold:
N1: Nð½0;0�Þ ¼ ½1;1� and Nð½1;1�Þ ¼ ½0; 0�;N2a If X P Y then NðXÞ 6 NðYÞ;N2b If X # Y then NðXÞ# NðYÞ.If N also satisfies the involutive property N3, then it is said to be a strong interval fuzzy negation:N3: NðNðXÞÞ ¼ X; 8X 2 U.
A Moore and Scott-continuous interval fuzzy negation N is strict if it also satisfies the following properties:
N4a If X < Y then NðYÞ < NðXÞ;N4b If X � Y then NðXÞ � NðYÞ.
The concepts of interval representation and obtainability show their strength on the context of fuzzy negations in the fol-lowing results. We show that those concepts guarantee that punctual properties are preserved by the interval generalizationof fuzzy negations.
Let N : U ! U be a fuzzy negation. The interval function bN can be expressed as:
bNðXÞ ¼ ½NðXÞ;NðXÞ�: ð13Þ
The proofs of the next propositions in this section can be found in [10].
Proposition 11. A function N : U! U is a (strict) interval fuzzy negation if, and only if, the functions N;N : U ! U, defined,respectively, by
NðxÞ ¼ lðNð½x; x�ÞÞ and NðxÞ ¼ rðNð½x; x�ÞÞ; ð14Þ
are (strict) fuzzy negations and
NðXÞ ¼ ½NðXÞ;NðXÞ�; ð15Þ
where l and r are, respectively, the left and right projections defined in Eq. (1).
Remark 12. If N ¼ N, then NðXÞ ¼ bNðXÞ ¼ bNðXÞ.Therefore, one has that (strict) interval fuzzy negations are obtainable.
Proposition 13. A function N : U! U is a strong interval fuzzy negation if, and only if, there exists a strong fuzzy negation N suchthat N ¼ bN.
From Eq. (13) and Remark 12, if the conditions of Proposition 13 hold, then it follows that N ¼ N ¼ N.From Propositions 11 and 13, it is immediate that:
Corollary 14. Let N : U ! U be a fuzzy negation. Then bN is an interval fuzzy negation. In addition, if N is a strong (strict) fuzzynegation then bN is a strong (strict) interval fuzzy negation.
An interval E 2 U is an equilibrium point of an interval fuzzy negation N if NðEÞ ¼ E. Trivially, ½0;1� is an equilibrium pointof any interval fuzzy negation. Thus, if an equilibrium interval E is such that E – ½0;1� then E is said to be a non-trivial equi-librium point.
Proposition 15. If N is a strong interval fuzzy negation, then N has a degenerate equilibrium. Moreover, it is the unique non-trivial equilibrium point.
5. Fuzzy implications and S-implications
Several definitions for fuzzy implications together with related properties have been given (see, e.g., [4–7,16,30,31,46,47,58,61,60,62,66,65,72–74]). However, there is just one consensus on what a fuzzy implication should be,namely: ‘‘a fuzzy implication should present the behavior of the classical implication when the crisp case is considered”[47]. In other words, a function I : U2 ! U is a fuzzy implication whenever it satisfies the minimal boundary conditions:
Ið1;1Þ ¼ Ið0;1Þ ¼ Ið0;0Þ ¼ 1 and Ið1; 0Þ ¼ 0: ð16Þ
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Several reasonable properties may be required for fuzzy implications, among them we consider the following:
I1: If x 6 z then Iðx; yÞP Iðz; yÞ (first place antitonicity);I2: If y 6 z then Iðx; yÞ 6 Iðx; zÞ (second place isotonicity);I3: Ið1; xÞ ¼ x (left neutrality principle);I4: Iðx; Iðy; zÞÞ ¼ Iðy; Iðx; zÞÞ (exchange principle);I5: Iðx; yÞ ¼ Iðx; Iðx; yÞÞ (iterative boolean-like law);I6: Iðx;NðxÞÞ ¼ NðxÞ, where N is a strong fuzzy negation;I7: NðxÞ ¼ Iðx;0Þ is a strong fuzzy negation;I8: Iðx;1Þ ¼ 1 (dominance of truth of consequent);I9: Iðx; yÞP y;
I10: Iðx; yÞ ¼ IðNðyÞ;NðxÞÞ, where N is a strong fuzzy negation (contra-positive);I11: Ið0; xÞ ¼ 1 (dominance falsity).
Some relations between classical implications and negations can be recovered for the fuzzy case. For example, ifI : U2 ! U is a fuzzy implication satisfying the Property I1, then there is a fuzzy negation NI : U ! U that can be definedby [6, Lemma 2.1]:
NIðxÞ ¼ Iðx;0Þ: ð17Þ
Another relation between negation and implication follows the opposite direction, showing that it is possible to define a fuz-zy implication from a fuzzy negation. Let S be a t-conorm and N be a fuzzy negation. An S-implication [5,6,16,30,31,47,61] is afuzzy implication IS;N : U2 ! U defined by
IS;Nðx; yÞ ¼ SðNðxÞ; yÞ: ð18Þ
In some texts (e.g., [16,30,31]), the definition of an S-implication requires a strong fuzzy negation. Such S-implications arecalled here strong S-implications. Similar definitions can be introduced for continuous S-implications and strict S-implications.
Trillas and Valverde [69, Theorem 3.2] (see also [30, Theorem 1.13] and [6, Theorem 1.6]) provided the following char-acterization for strong S-implications: a function I : U2 ! U is a strong S-implication if, and only if, it satisfies the PropertiesI1–I4, and I10. Lately, Baczynsky and Jayaram [6, Theorem 2.6]) introduced a new characterization of strong S-implications,considering properties I1, I4 and I7. Strong S-implications also satisfy the properties I8–I11 and the following two extraproperties below:
I12: Iðx; yÞP NIðxÞ;I13: Iðx; yÞ ¼ 0 if, and only if, x ¼ 1 and y ¼ 0.
Notice that any S-implication IS;N satisfies the properties I1–I3,I8,I9, and I11.If a fuzzy implication I is an S-implication then NI , as given in Eq. (17), is the underlying negation of I, that is:
NIS;N ðxÞ ¼ IS;Nðx;0Þ ¼ SðNðxÞ;0Þ ¼ NðxÞ: ð19Þ
Therefore, NI is a strict fuzzy negation if, and only if, I is a strict S-implication.Baczynsky and Jayaram [6, Theorem 5.2] provided a characterization for strict S-implications, where an S-implication IS;N
is strict if and only if NIS;N is strict and the properties I1 and I10 also hold.If a fuzzy implication I : U2 ! U is a strong S-implication, then I satisfies I6 if, and only if, the underlying t-conorm of I is
the maximum t-conorm SM , given in Eq. (8), and, therefore, one has I ¼ ISM ;N , where N is a strong fuzzy negation. The strong S-implication ISM ;N also satisfies the properties I1–I11. Moreover, it is the only S-implication satisfying I6. Given an equilibriumpoint eN , if x 2 U and x P eN then one has that ISM ;Nðx; xÞ ¼ x.
6. Interval fuzzy implications
Since real numbers may be identified with degenerate intervals in the context of interval mathematics, the boundary con-ditions that must be satisfied by the classical fuzzy implications can be naturally extended to interval fuzzy degrees, when-ever degenerate intervals are considered. Then, a function I : U2 ! U is said to be an interval fuzzy implication if the followinginterval-based boundary conditions hold:
(i) Ið½1;1�; ½1;1�Þ ¼ Ið½0; 0�; ½0;0�Þ ¼ Ið½0;0�; ½1;1�Þ ¼ ½1;1�;(ii) Ið½1;1�; ½0; 0�Þ ¼ ½0;0�.
The properties presented in Section 5 can then also be naturally extended to an interval-based approach:
I1: If X 6 Z then IðX;YÞP IðZ;YÞ (first place antitonicity);I2: If Y 6 Z then IðX;YÞ 6 IðX; ZÞ (second place isotonicity);
B.C. Bedregal et al. / Information Sciences 180 (2010) 1373–1389 1379
I3: Ið½1;1�;XÞ ¼ X (left neutrality principle);I4: IðX; IðY ; ZÞÞ ¼ IðY; IðX; ZÞÞ (exchange principle);I5: IðX;YÞ ¼ IðX; IðX;YÞÞ (iterative boolean-like law);I6: NðXÞ ¼ IðX;NðXÞÞ, where N is a strong interval fuzzy negation;I7: NðXÞ ¼ IðX; ½0;0�Þ is a strong interval fuzzy negation;I8: IðX; ½1;1�Þ ¼ ½1;1� (dominance of truth of consequent);I9: IðX;YÞP Y;
I10: IðX;YÞ ¼ IðNðYÞ;NðXÞÞ, where N is a strong interval fuzzy negation (contra-positive);I11: Ið½0;0�;XÞ ¼ ½1;1� (dominance falsity).
It is always possible to canonically obtain an interval fuzzy implication from any fuzzy implication. The interval fuzzyimplication satisfies the optimality property and preserves the properties satisfied by the corresponding fuzzy implication.
Proposition 16. If I is a fuzzy implication then bI is an interval fuzzy implication.
Proof. It is straightforward. h
In the next results, we adopt a canonical way to construct, under some conditions, interval fuzzy implication from fuzzyimplication and vice-versa. The properties of fuzzy implications presented in Section 5 are related with the respective prop-erties of interval fuzzy implications enrolled above.
Theorem 17. Let I1 and I2 be fuzzy implications satisfying the Properties I1 and I2 and such that I1 6 I2. If I1 and I2 satisfy aProperty Ik, for k ¼ 1; . . . ;6;8; . . . ;11, then I : U2 ! U, defined by
IðX; YÞ ¼ ½I1ðX;YÞ; I2ðX; YÞ�; ð20Þ
satisfies the Property Ik.
Proof. Since I1 6 I2 and by the Properties I1 and I2, one has that I1ðX;YÞ 6 I1ðX;YÞ 6 I2ðX;YÞ 6 I2ðX;YÞ, and, therefore, I iswell defined. It follows that:
I1: Let X;Y ; Z 2 U such that X 6 Z. Since X 6 Z;X 6 Z, and I1 and I2 satisfy Property I1, then it holds that I1ðZ;YÞ 6 I1ðX;YÞand I2ðZ;YÞ 6 I2ðX;YÞ. So, by Eq. (20), it follows that IðZ;YÞ 6 IðX;YÞ.
I2: Let X;Y ; Z 2 U such that Y 6 Z. Since Y 6 Z;Y 6 Z, and I1 and I2 satisfy Property I2, then it holds that I1ðX;YÞ 6 I1ðX; ZÞand I2ðX;YÞ 6 I2ðX; ZÞ. Then, by Eq. (20), it follows that IðX;YÞ 6 IðX; ZÞ.
I3: It holds that Ið½1;1�;XÞ ¼ ½I1ð1;XÞ; I2ð1;XÞ� ¼ ½X;X� ¼ X.I4: By Property I4, it follows that:
IðX; IðY; ZÞÞ ¼ IðX; ½I1ðY; ZÞ; I2ðY ; ZÞ�Þ ¼ ½I1ðX; I1ðY; ZÞÞ; I2ðX; I2ðY ; ZÞÞ� ¼ ½I1ðY; I1ðX; ZÞÞ; I2ðY; I2ðX; ZÞÞ� ¼ IðY; IðX; ZÞÞ:
I5: By Property I5, it follows that:
IðX; YÞ ¼ ½I1ðX;YÞ; I2ðX; YÞ� ¼ ½I1ðX; I1ðX;YÞÞ; I2ðX; I2ðX; YÞÞ� ¼ IðX; ½I1ðX;YÞ; I2ðX;YÞ�Þ ¼ IðX; IðX; YÞÞ:
I6: Let N be a strong interval fuzzy negation. By Proposition 13, there exists a strong fuzzy negation N such thatNðXÞ ¼ ½NðXÞ;NðXÞ�. It follows that:
IðX;NðXÞÞ ¼ IðX; ½NðXÞ;NðXÞ�Þ ¼ ½I1ðX;NðXÞÞ; I2ðX;NðXÞÞ� ¼ ½NðXÞ;NðXÞ� ðby Property I6Þ ¼ NðXÞ:
I8: One has that IðX; ½1;1�Þ ¼ ½I1ðX;1Þ; I2ðX;1Þ� ¼ ½1;1�.I9: By Property I9, it holds that I1ðX;YÞP Y and I2ðX;YÞP Y . Then, it follows that IðX;YÞ ¼ ½I1ðX;YÞ; I2ðX;YÞ�P Y .
I10: Let N be a strong interval fuzzy negation. By Proposition 13, there exists a strong fuzzy negation N such thatNðXÞ ¼ ½NðXÞ;NðXÞ�. So, by Property I10, it follows that:
IðX; YÞ ¼ ½I1ðX;YÞ; I2ðX; YÞ� ¼ ½I1ðNðYÞ;NðXÞÞ; I2ðNðYÞ;NðXÞÞ� ¼ Ið½NðYÞ;NðYÞ�; ½NðXÞ;NðXÞ�Þ ¼ IðNðYÞ;NðXÞÞ:
I11: One has that Ið½0;0�;XÞ ¼ ½I1ð0;XÞ; I2ð0;XÞ� ¼ ½1;1�. h
Remark 18. According to the conditions stated by Theorem 17, the Property I7 does not hold even if both I1 and I2 satisfy theProperty I7. For example, considering I1ðx; yÞ ¼minf1� xþ y;1g and I2ðx; yÞ ¼minf
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� x2 þ y
p;1g, it is immediate that I1
and I2 are fuzzy implications, as they satisfy Eq. (16), and it holds that I1 6 I2. Moreover, consideringNI1 ðxÞ ¼ I1ðx;0Þ ¼ 1� x and NI2 ðxÞ ¼ I2ðx;0Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� x2p
, it is immediate that NI1 is a strong fuzzy negation, and, since
1380 B.C. Bedregal et al. / Information Sciences 180 (2010) 1373–1389
NI2 ðNI2 ðxÞÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ð
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� x2p
Þ2q
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ð1� x2Þ
p¼ x, it holds that NI2 is also a strong fuzzy negation. Thus, I1 and I2 satisfy the
Property I7. However, IðX;YÞ, defined as in Eq. (20), does not satisfy I7. Observe that
NðXÞ ¼ IðX; ½0; 0�Þ ¼ ½I1ðX;0Þ; I2ðX;0Þ� ¼ ½NI1 ðXÞ;NI2 ðXÞ� ¼ ½1� X;ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� X2
p�:
Then, for example, one has that NðNð½0:4;0:5�ÞÞ ¼ Nð½1� 0:5;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 0:42
p�Þ ¼ ½1�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 0:42
p;
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ð1� 0:5Þ2
q�– ½0:4;0:5�.
The next theorem establishes a sufficient condition for interval fuzzy implications to be obtainable.
Theorem 19. An interval fuzzy implication I : U2 ! U is obtainable if it is inclusion-monotonic and satisfies the Properties I1 andI2.
Proof. Consider X;Y 2 U. Since ½X;X�# X and ½Y ;Y�# Y , then, by the inclusion monotonicity of I, we have thatIð½X;X�; ½Y ;Y�Þ# IðX;YÞ, and, therefore, lðIð½X;X�; ½Y ;Y�ÞÞP lðIðX;YÞÞ. On the other hand, since X 6 ½X;X� and ½Y ;Y� 6 Y , then,by Properties I1 and I2 of I, it holds that Ið½X;X�; ½Y ;Y�Þ 6 IðX;YÞ, and, therefore, lðIð½X;X�; ½Y ;Y�ÞÞ 6 lðIðX;YÞÞ. Then, it followsthat lðIð½X;X�; ½Y;Y �ÞÞ ¼ lðIðX;YÞÞ.
Analogously, since ½X;X�# X and ½Y; Y�# Y , then, by the inclusion monotonicity of I, we have that Ið½X;X�; ½Y;Y �Þ# IðX;YÞ,and, therefore, it holds that rðIð½X;X�; ½Y ;Y�ÞÞ 6 rðIðX;YÞÞ. On the other hand, since ½X;X� 6 X and Y 6 ½Y;Y �, then, by PropertiesI1 and I2 of I, it holds that IðX;YÞ 6 Ið½X;X�; ½Y;Y �Þ, and, therefore, rðIð½X;X�; ½Y;Y �ÞÞP rðIðX;YÞÞ. Thus, one has thatrðIð½X;X�; ½Y ;Y�ÞÞ ¼ rðIðX;YÞÞ. So, considering the left and right projections defined, respectively, by
Iðx; yÞ ¼ lðIð½x; x�; ½y; y�ÞÞ and Iðx; yÞ ¼ rðIð½x; x�; ½y; y�ÞÞ; ð21Þ
we have that IðX;YÞ ¼ ½IðX;YÞ; IðX;YÞ�. h
The next results establish conditions in order to relate the properties of fuzzy implications with the respective propertiesof interval fuzzy implications.
Theorem 20. Let I : U2 ! U be an interval fuzzy implication and I; I : U2 ! U be the left and right projections, defined as in Eq.(21). If I satisfies a Property Ik, for some k ¼ 1; . . . ;11, then I and I satisfy the Property Ik. In the case of Properties I4 and I5, theinterval fuzzy implication I must also preserve degenerate intervals.
Proof. In the following, we present the proof for the left projection I. The proof for the right projection I is analogous.
I1: Suppose that x 6 z. By Property I1, it holds that Ið½x; x�; ½y; y�ÞP Ið½z; z�; ½y; y�Þ, and then Iðx; yÞ ¼ lðIð½x; x�; ½y; y�ÞÞPlðIð½z; z�; ½y; y�ÞÞ ¼ Iðz; yÞ.
I2: Suppose that y 6 z. By Property I2, it holds that Ið½x; x�; ½y; y�Þ 6 Ið½x; x�; ½z; z�Þ, and thus Iðx; yÞ ¼ lðIð½x; x�; ½y; y�ÞÞ 6lðIð½x; x�; ½z; z�ÞÞ ¼ Iðx; zÞ.
I3: By Property I3, it holds that Ið½1;1�; ½x; x�Þ ¼ ½x; x�. So, it follows that Ið1; xÞ ¼ lðIð½1;1�; ½x; x�ÞÞ ¼ x.I4: By Property I4, one has that Ið½x; x�; Ið½y; y�; ½z; z�ÞÞ ¼ Ið½y; y�; Ið½x; x�; ½z; z�ÞÞ. Considering that the interval fuzzy implication
I preserves degenerate intervals, it follows that:
Iðx; Iðy; zÞÞ ¼ Iðx; lðIð½y; y�; ½z; z�ÞÞÞ ¼ lðIð½x; x; �; ½lðIð½y; y�; ½z; z�ÞÞ; lðIð½y; y�; ½z; z�ÞÞ�ÞÞ¼ lðIð½x; x; �; ½lðIð½y; y�; ½z; z�ÞÞ; rðIð½y; y�; ½z; z�ÞÞ�ÞÞ ¼ lðIð½x; x�; Ið½y; y�; ½z; z�ÞÞÞ ¼ lðIð½y; y�; Ið½x; x�; ½z; z�ÞÞÞ¼ lðIð½y; y�; ½lðIð½x; x�; ½z; z�ÞÞ; rðIð½x; x�; ½z; z�ÞÞ�ÞÞ ¼ lðIð½y; y�; ½lðIð½x; x�; ½z; z�ÞÞ; lðIð½x; x�; ½z; z�ÞÞ�ÞÞ¼ Iðy; lðIð½x; x�; ½z; z�ÞÞÞ ¼ Iðy; Iðx; zÞÞ:
I5: By Property I5, it holds that Ið½x; x�; ½y; y�Þ ¼ Ið½x; x�; Ið½x; x�; ½y; y�ÞÞ. Considering that the interval fuzzy implication I pre-serves degenerate intervals, it follows that:
Iðx; yÞ ¼ lðIð½x; x�; ½y; y�ÞÞ ¼ lðIð½x; x�; Ið½x; x�; ½y; y�ÞÞÞ ¼ lðIð½x; x�; ½lðIð½x; x�; ½y; y�ÞÞ; rðIð½x; x�; ½y; y�ÞÞ�ÞÞ¼ lðIð½x; x�; ½lðIð½x; x�; ½y; y�ÞÞ; lðIð½x; x�; ½y; y�ÞÞ�ÞÞ ¼ Iðx; lðIð½x; x�; ½y; y�ÞÞÞ ¼ Iðx; Iðx; yÞÞ:
I6: By Property I6, it holds that Ið½x; x�;Nð½x; x�ÞÞ ¼ Nð½x; x�Þ, where N is a strong interval fuzzy negation. Considering Prop-osition 13, we have that Nð½x; x�Þ ¼ ½NðxÞ;NðxÞ�, for some strong fuzzy negation N. Then, it follows that:
Iðx;NðxÞÞ ¼ lðIð½x; x�; ½NðxÞ;NðxÞ�ÞÞ ¼ lðIð½x; x�;Nð½x; x�ÞÞÞ ¼ lðNð½x; x�ÞÞ ¼ NðxÞ:
I7: By Property I7, it holds that NðXÞ ¼ IðX; ½0;0�Þ is a strong interval fuzzy negation. Then, it follows that:
N1: Nð0Þ ¼ Ið0;0Þ ¼ lðIð½0;0�; ½0;0�ÞÞ ¼ lð½1;1�Þ ¼ 1 and Nð1Þ ¼ Ið1;0Þ ¼ lðIð½1;1�; ½0;0�ÞÞ ¼ lð½0;0�Þ ¼ 0.N2: If x P y then ½x; x�P ½y; y�. Therefore, by Property N2a, it holds that Nð½x; x�Þ 6 Nð½y; y�Þ. It follows thatNðxÞ ¼ Iðx;0Þ ¼ lðIð½x; x�; ½0;0�ÞÞ ¼ lðNð½x; x�ÞÞ 6 lðNð½y; y�ÞÞ ¼ NðyÞ.
B.C. Bedregal et al. / Information Sciences 180 (2010) 1373–1389 1381
N3: By Property N3, it holds that NðNð½x; x�ÞÞ ¼ ½x; x�. By Proposition 13, we have that the strong interval fuzzy nega-tion N preserves degenerate interval. Then, it follows that:
NðNðxÞÞ ¼ NðIðx;0ÞÞ ¼ IðIðx;0Þ;0Þ ¼ IðlðIð½x; x�; ½0;0�ÞÞ; 0Þ ¼ lðIð½lðIð½x; x�; ½0;0�ÞÞ; lðIð½x; x�; ½0;0�ÞÞ�; ½0;0�ÞÞ¼ lðIð½lðNð½x; x�ÞÞ; lðNð½x; x�ÞÞ�; ½0;0�ÞÞ ¼ lðNð½lðNð½x; x�ÞÞ; lðNð½x; x�ÞÞ�ÞÞ ¼ lðNðNð½x; x�ÞÞÞ ¼ lð½x; x�Þ ¼ x:
I8: By Property I8, it holds that Ið½x; x�; ½1;1�Þ ¼ ½1;1�. Then, it follows that Iðx;1Þ ¼ lðIð½x; x�; ½1;1�ÞÞ ¼ lð½1;1�Þ ¼ 1.I9: By Property I9, it holds that Ið½x; x�; ½y; y�ÞP ½y; y�. Thus, it follows that Iðx; yÞ ¼ lðIð½x; x�; ½y; y�ÞÞP lð½y; y�Þ ¼ y.
I10: By Property I10, it holds that Ið½x; x�; ½y; y�Þ ¼ IðNð½y; y�Þ;Nð½x; x�ÞÞ, where N is a strong interval fuzzy negation. In addi-tion, based on Proposition 13, there exists a strong fuzzy negation N such that Nð½x; x�Þ ¼ ½NðxÞ;NðxÞ�. Then, it followsthat:
Iðx; yÞ ¼ lðIð½x; x�; ½y; y�ÞÞ ¼ lðIðNð½y; y�Þ;Nð½x; x�ÞÞÞ ¼ lðIð½NðyÞ;NðyÞ�; ½NðxÞ;NðxÞ�ÞÞ ¼ IðNðyÞ;NðxÞÞ:
I11: By Property I11, it holds that Ið½0;0�; ½x; x�Þ ¼ ½1;1�. Thus, it follows that Ið0; xÞ ¼ lðIð½0;0�; ½x; x�ÞÞ ¼ lð½1;1�Þ ¼ 1. h
Proposition 21. A fuzzy implication I : U2 ! U satisfies the Properties I1 and I2 if, and only if, the interval fuzzy implication bI canbe expressed as
bIðX;YÞ ¼ ½IðX;YÞ; IðX;YÞ�: ð22Þ
Proof. ()) By definition of bI , we have that I ¼ fIðx; yÞjx 2 X; y 2 Yg#bIðX;YÞ, and so IðX;YÞ and IðX;YÞ are both in I. IfX 6 x 6 X;Y 6 y 6 Y , then, by the Properties I1 and I2, it follows that IðX;YÞ 6 Iðx; yÞ 6 IðX;YÞ, and then IðX;YÞ and IðX;YÞare the infimum and the supremum of I, respectively.
(() Let x; z; y 2 U be such that x 6 z. By Eq. (22), one has that bIð½x; z�; ½y; y�Þ ¼ ½Iðz; yÞ; Iðx; yÞ�, and so Iðz; yÞ 6 Iðx; yÞ.Therefore, I satisfies the Property I1. Analogously, let x; z; y 2 U be such that y 6 z. Then, it holds thatbIð½x; x�; ½y; z�Þ ¼ ½Iðx; yÞ; Iðx; zÞ�, and so Iðx; yÞ 6 Iðx; zÞ. Therefore, I satisfies the Property I2. h
Corollary 22. Let I be a fuzzy implication satisfying I1 and I2. I satisfies a Property Ik, for some k ¼ 1; . . . ;11 if and only if bI sat-isfies the Property Ik.
Proof. The proofs for k ¼ 1; . . . ;6;8; . . . ;11 are immediate, following from Theorems 17, 20 and Proposition 21. On the otherhand, for the case in which k=7, from Theorem 20 it is immediate that if bI satisfies the Property I7, then I satisfies a PropertyI7. Conversely, by the Property I7, NðxÞ ¼ Iðx;0Þ is a strong fuzzy negation, and, therefore, it follows that:
NðXÞ ¼ bIðX; ½0;0�Þ ¼ ½inffIðx;0Þjx 2 Xg; supfIðx;0Þjx 2 Xg� ¼ ½inffNðxÞjx 2 Xg; supfNðxÞjx 2 Xg� ¼ ½NðXÞ;NðXÞ� ¼ bNðXÞ:
Thus, by Proposition 13, one has that N is a strong interval fuzzy negation. hWith little effort, it is possible to substitute the constraints above by the continuity of the fuzzy implication, wheneverproperties I5 and I6 are not considered:
Proposition 23. A continuous fuzzy implication I satisfies a Property Ik, for k ¼ 1; . . . ;4;7; . . . ;11, if and only if bI satisfies theProperty Ik.
Proof. ()) Since I is continuous, then by Eq. (6), it holds that bIðX;YÞ ¼ fIðx; yÞjx 2 X ^ y 2 Yg. It follows that:
I1: If u 2 bIðX;YÞ, then there exist x 2 X and y 2 Y such that Iðx; yÞ ¼ u. Thus, if X 6 Z then there exists z 2 Z such that x 6 z.So, by the Property I1, it follows that u ¼ Iðx; yÞP Iðz; yÞ. Therefore, for each u 2 bIðX;YÞ, there exists v 2 bIðZ;YÞ suchthat u 6 v . On the other hand, if v 2 bIðZ;YÞ, then there exist z 2 Z and y 2 Y such that Iðz; yÞ ¼ v . If X 6 Z thenx 6 z, for some x 2 X. Then, by the Property I1, it holds that Iðx; yÞP Iðz; yÞ ¼ v . Therefore, for each v 2 bIðZ;YÞ, thereis u 2 bIðX;YÞ such that u P v . Hence, we conclude that bIðX; YÞP bIðZ; YÞ.
I2: If u 2 bIðX;YÞ, then there exist x 2 X and y 2 Y such that Iðx; yÞ ¼ u. If Y 6 Z then there exists z 2 Z such that y 6 z. Thus,by the Property I2, it holds that u ¼ Iðx; yÞ 6 Iðx; zÞ. Therefore, for each u 2 bIðX;YÞ there exists v 2 bðX; ZÞ such thatu 6 v . On the other hand, if v 2 bIðX; ZÞ, then there exist z 2 Z and x 2 X such that Iðx; zÞ ¼ v . If Y 6 Z then y 6 z, forsome y 2 Y . Therefore, by the Property I2, it follows that Iðx; yÞP Iðx; zÞ ¼ v . Thus, for each v 2 bIðX; ZÞ, there isu 2 bIðX;YÞ such that u 6 v . Hence, one concludes that bIðX;YÞ 6 bIðX; ZÞ.
I3: Trivially, by the Property I3, for each x 2 X, it holds that Ið1; xÞ ¼ x, and so fIð1; xÞjx 2 Xg ¼ X. Therefore, it follows thatbIð½1;1�;XÞ ¼ X.I4: If u 2 bIðX;bIðY ; ZÞÞ, then there exist x 2 X; y 2 Y and z 2 Z such that Iðx; Iðy; zÞÞ ¼ u. But, by the Property I4, it holds that
u ¼ Iðy; Iðx; zÞÞ. Then, one has that u 2 bIðY ;bIðX; ZÞÞ, and, thus, bIðX;bIðY; ZÞÞ#bIðY ;bIðX; ZÞÞ. Analogously, if u 2 bIðY ;bIðX; ZÞÞ,
1382 B.C. Bedregal et al. / Information Sciences 180 (2010) 1373–1389
then there exist x 2 X; y 2 Y and z 2 Z such that Iðy; Iðyx; zÞÞ ¼ u. However, by the Property I4, it holds thatu ¼ Iðx; Iðy; zÞÞ. Thus, one has that u 2 bIðX;bIðY ; ZÞÞ, and then bIðY ;bIðX; ZÞÞ#bIðX;bIðY; ZÞÞ. Hence, it follows thatbIðX;bIðY ; ZÞÞ ¼ bIðY;bIðX; ZÞÞ.
I7: By the Property I7, NðxÞ ¼ Iðx;0Þ is a strong fuzzy negation and so N is continuous. Therefore, by the continuity of I andN, it follows that:
NðXÞ ¼ bIðX; ½0;0�Þ ¼ fIðx; 0Þjx 2 Xg ¼ fNðxÞjx 2 Xg ¼ bNðXÞ:
Thus, by Proposition 13, it follows that N is a strong interval fuzzy negation.I8: By the Property I8, it follows that bIðX; ½1;1�Þ ¼ fIðx;1Þjx 2 Xg ¼ f1g ¼ ½1;1�.I9: By the Property I9, for each x 2 X and y 2 Y , it holds that Iðx; yÞP y. So, for each y 2 Y there exists u 2 bIðX;YÞ such that
u P y. On the other hand, if u 2 bIðX;YÞ there exist x 2 X and y 2 Y such that u ¼ Iðx; yÞ. Thus, by the Property I9, it holdsthat u P y. Therefore, for each u 2 bIðX;YÞ, there exists y 2 Y such that u P y. Hence, it follows that bIðX;YÞP Y .
I10: By Proposition 13, N is a strong interval fuzzy negation if and only if there exists a strong fuzzy negation N, such thatN ¼ bN . By the continuity of I;u 2 bIðX;YÞ if and only if there exist x 2 X and y 2 Y such that u ¼ Iðx; yÞ. By the PropertyI10, u ¼ Iðx; yÞ if and only if u ¼ IðNðxÞ;NðyÞÞ, that is, u 2 bIðNðXÞ;NðYÞÞ. Hence, one concludes thatbIðX;YÞ ¼ bIðNðXÞ;NðYÞÞ.
I11: By the Property I11, it follows that bIð½0;0�;XÞ ¼ fIð0; xÞjx 2 Xg ¼ f1g ¼ ½1;1�.
(() It is straightforward, since bIð½x; x�; ½y; y�Þ ¼ fIðx; yÞg ¼ ½Iðx; yÞ; Iðx; yÞ�. h
6.1. Interval S-implications
An interval fuzzy implication IS;N is said to be an interval S-implication, if there exists an interval t-conorm S and an inter-val fuzzy negation N such that
IS;NðX; YÞ ¼ SðNðXÞ;YÞ: ð23Þ
If an interval fuzzy negation is strong (strict) then the related interval S-implication is called strong (strict) interval S-implication.
The next theorem provides a characterization for the best interval representation of an S-implication and proves that suchinterval fuzzy implication is an interval S-implication.
Theorem 24. Let S be a t-conorm and N be a fuzzy negation. Then,
IbS;bN ¼dIS;N : ð24Þ
Proof. Consider X;Y 2 U. Then, it follows that:
IbS;bN ðX;YÞ ¼ bSðbNðXÞ;YÞ ¼ bSð½NðXÞ;NðXÞ�;YÞ ¼ ½SðNðXÞ;YÞ; SðNðXÞ;YÞ� ¼ ½IS;NðX;YÞ; IS;NðX; YÞ� ¼dIS;N ðX;YÞ: �
The next corollary is straightforward.
Corollary 25. If I is an S-implication then bI is an interval S-implication.
These results together with Theorem 24 state the commutativity of the diagram in Fig. 1, where CðSÞ (CðSÞ) denotes theclass of (interval) t-conorms, CðNÞ (CðNÞ) indicates the class of (interval) fuzzy negations and CðIÞ (CðIÞ) is the class of (inter-val) S-implications.
The next theorem shows that interval S-implications are obtainable by S-implications.
Theorem 26. I : U2 ! U is a (strong, strict) interval S-implication if, and only if, there exist (strong, strict) S-implications I1 and I2
such that I1 6 I2 and, for all X;Y 2 U, it holds that
IðX; YÞ ¼ ½I1ðX; YÞ; I2ðX;YÞ�: ð25Þ
Fig. 1. Commutative classes of interval S-implications.
B.C. Bedregal et al. / Information Sciences 180 (2010) 1373–1389 1383
Proof. ()) Let S and N be the underlying interval t-conorm and interval fuzzy negation of I, respectively. So, it follows that:
IðX; YÞ ¼ IS;NðX;YÞ¼ SðNðXÞ;YÞ by Eq: ð23Þ¼ Sð½NðXÞ;NðXÞ�;YÞ by Proposition 11
¼ ½SðNðXÞ;YÞ;SðNðXÞ;YÞ� by Proposition 8
¼ ½IS;NðX; YÞ; IS;NðX; YÞ� by Eq: ð18Þ
¼ ½I1ðX;YÞ; I2ðX;YÞ�
If N is a strong interval fuzzy negation, instead of considering Proposition 11, we must then consider Proposition 13 andN ¼ N.
(() The proof of the converse follows the reverse structure of the above proof. h
The next proposition guarantees that I1 and I2 in Eq. (25) are, respectively, equal to I and I.
Proposition 27. Let S be an interval t-conorm and N an interval fuzzy negation. Then it holds that
IS;N ¼ IS;N and IS;N ¼ IS;N: ð26Þ
Proof. We will only show the proof for the first case, as the proof of the second case is analogous. Since S and N areinclusion-monotonic, then IS;N is also inclusion-monotonic. One has that:
IS;Nðx; yÞ ¼ lðIS;Nð½x; x�; ½y; y�ÞÞ by Eq: ð21Þ
¼ lðSðNð½x; x�Þ; ½y; y�ÞÞ by Eq: ð23Þ¼ lðSð½NðxÞ;NðxÞ�Þ; ½y; y�Þ by Eq: ð15Þ¼ lð½SðNðxÞ; yÞ;SðNðxÞ; yÞ�Þ by Eq: ð11Þ¼ SðNðxÞ; yÞ¼ IS;Nðx; yÞ by Eq: ð18Þ �
Lemma 28. If I is an interval S-implication, then I satisfies I1 and I2.
Proof. If X 6 Z then, by the Property N2, it holds that NðZÞ 6 NðXÞ. So, by the 6-monotonicity of S, we have thatIS;NðZ;YÞ ¼ SðNðZÞ;YÞ 6 SðNðXÞ;YÞ ¼ IS;NðX;YÞ. Thus, IS;N satisfies the Property I1. Analogously, if Y 6 Z then, by the 6-monotonicity of S, we have that IS;NðX;YÞ ¼ SðNðXÞ;YÞ 6 SðNðXÞ; ZÞ ¼ IS;NðX; ZÞ. Thus, IS;N satisfies I2. h
The results of the following theorem and propositions are analogous to the results for classical S-implications presentedin [6, Theorem 1.7,Lemma 2.1, Theorem 5.2], in [16, Lemma 1(x), Corollary 2, Theorem 5, Theorem 6] and also in [31].
Theorem 29. An interval fuzzy implication I is a strong interval S-implication if, and only if, the Properties I1–I4 and I10 hold.
Proof. Let I and I the left and right projection of I defined as in Eq. (21).()) By Propositions 27 and 13, I and I are strong S-implications. Therefore, by [69, Theorem 3.2] (or [6, Theorem 1.7]), I
and I satisfy the properties I1–I4 and I10. So, by Theorem 17, I satisfies the Properties I1; I2, I3; I4 and I10.(() If I satisfies I1 and I2 then by Theorem 17, I and I satisfy I1–I4 and I10. Therefore, by [69, Theorem 3.2] (or [6,
Theorem 1.7]), I and I are strong S-implications. Thus, by Theorem 26, I is an interval strong S-implication. h
Proposition 30. Let I : U2 ! U be an interval fuzzy implication satisfying the Property I1. The interval function NI : U! U
defined by NIðXÞ ¼ IðX; ½0;0�Þ is an interval fuzzy negation.
Proof. Since I is an interval fuzzy implication, then it is immediate that NIð½0;0�Þ ¼ Ið½0;0�; ½0;0�Þ ¼ ½1;1� andNIð½1;1�Þ ¼ Ið½1;1�; ½0;0�Þ ¼ ½0;0�. If X P Y then, by Property I1;NIðXÞ ¼ IðX; ½0;0�Þ 6 IðY; ½0;0�Þ ¼ NIðYÞ. Analogously, ifX # Y then, by the inclusion monotonicity of I, it holds that NIðXÞ ¼ IðX; ½0;0�Þ# IðY ; ½0;0�Þ ¼ NIðYÞ. So, one concludes thatNI is an interval fuzzy negation. h
Notice that, analogously to the punctual case, if I is an interval S-implication then NI is the underlying interval fuzzynegation of I. In fact, it holds that
NIS;N ðXÞ ¼ IS;NðX; ½0;0�Þ ¼ SðNðXÞ; ½0; 0�Þ ¼ NðXÞ: ð27Þ
1384 B.C. Bedregal et al. / Information Sciences 180 (2010) 1373–1389
Proposition 31. An interval fuzzy implication I : U2 ! U is a strict interval S-implication if, and only if, the properties I1 and I10hold whenever NI is a strict interval fuzzy negation.
Proof. ()) It is straightforward, following from the definition of strict interval S-implication, from [6, Theorem 5.2] and alsofrom Proposition 11, Theorem 26, Proposition 27 and Theorem 17.
(() It is straightforward, following from Proposition 11, Theorem 20, [6, Theorem 5.2] and Theorem 26. h
Proposition 32. Let I be an interval S-implication such that NI is a strong interval fuzzy negation. Then I satisfies the Properties I2and I7–I11.
Proof. It is straightforward, following from [31], [6, Theorem 1.7], [16, Lemma 1(x)] and also from Theorems 17, 20 and26. h
In the following, we prove two properties of interval S-implications which are analogous to properties I12 and I13 of clas-sical S-implications:
Proposition 33. Let I : U2 ! U be an interval fuzzy implication. If I is an interval S-implication such that NI is a strong intervalfuzzy negation, then I satisfies the properties I7 and
I12: IðX;YÞP NIðXÞ;I13: IðX;YÞ ¼ ½0;0� if, and only, if X ¼ ½1;1� and Y ¼ ½0;0�.
Proof. It is straightforward, following from Proposition 13, Theorem 26 and Properties I12 and I13. h
Proposition 34. If an interval fuzzy implication I : U2 ! U is a strong interval S-implication then I satisfies I6 if, and only if, thesupremum interval t-conorm SM : U2 ! U, defined in Eq. (9), is the underlying interval t-conorm of I.
Proof. It is straightforward, following from [16, Theorem 5] and also from Proposition 13, Theorems 17, 20, 26 and Propo-sition 27. h
Proposition 35. Let N be a strong interval fuzzy negation. An inclusion-monotonic function I : U2 ! U satisfies the PropertiesI1; I2, I4; I6 and I7 if, and only if, I ¼ ISM ;N, where SM is the supremum interval t-conorm defined in Eq. (9).
Proof. It is straightforward, following from [16, Theorem 6] and also from Proposition 13, Theorems 17, 20 and 26. h
Corollary 36. Let N be a strong interval fuzzy negation. The interval S-implication ISM ;N, where SM is the supremum interval t-conorm defined in Eq. (9), is such that
(i) The Properties I1-I4 and I7–I11 are satisfied;(ii) It is the only interval S-implication satisfying I6;
(iii) Whenever EN is an equilibrium point, if X 2 U and X P EN then IðX;XÞ ¼ X.
Proof. It is straightforward, following from [16, Corollary 2], and also from Proposition 15, Theorems 17 and 26. h
Proposition 37. A strong interval S-implication I satisfies the Property I5 if and only if I ¼ ISM ;N, where the interval fuzzy negationN is strong and SM is the supremum interval t-conorm defined in Eq. (9).
Proof. It is straightforward, following from [16, Theorem 5], and also from Theorems 17, 20 and 26. h
7. Interval automorphisms
In this section, the concept of interval automorphism is considered. The related properties are also studied in order toanalyze the effects of the action of an interval automorphism on an interval S-implication.
Definition 38 ([42,55]). A mapping q : U ! U is said to be an automorphism if it is bijective and monotonic (that is, x 6 yimplies that qðxÞ 6 qðyÞ). AutðUÞ denotes the set of all automorphisms.
An equivalent definition is given in [16], where q : U ! U is said to be an automorphism if it is a continuous and strictlyincreasing function such that qð0Þ ¼ 0 and qð1Þ ¼ 1.
The inverse of an automorphism is also an automorphism and automorphisms are closed under composition, that is, if qand q0 are automorphisms then q � q0ðxÞ ¼ qðq0ðxÞÞ is also an automorphism.
B.C. Bedregal et al. / Information Sciences 180 (2010) 1373–1389 1385
The action of q on a function f : Un ! U, denoted by f q, is defined as follows
f qðx1; . . . ; xnÞ ¼ q�1ðf ðqðx1Þ; . . . ;qðxnÞÞÞ: ð28Þ
In particular, whenever f is a t-conorm or a (strong) fuzzy negation then f q is also a t-conorm or a (strong) fuzzy negation,respectively. Moreover, if I is an S-implication then Iq is also an S-implication.
Proposition 39 [4, Proposition 21]. Let S be a t-conorm and N be a strong fuzzy negation. Then it holds that IqS;N ¼ ISq;Nq .
7.1. Canonical construction of an interval automorphism
A mapping . : U! U is an interval automorphism if it is bijective and monotonic w.r.t. the product order (that is, X 6 Yimplies that .ðXÞ 6 .ðYÞ) [33,34]. The set of all interval automorphisms is denoted by AutðUÞ.
Theorem 40 [33, Theorem 2]. Let . : U! U be an interval automorphism. Then there exists an automorphism q : U ! U suchthat
.ðXÞ ¼ ½qðXÞ;qðXÞ�: ð29Þ
Eq. (29) also provides a canonical construction of interval automorphisms from automorphisms and therefore a bijectionbetween the sets AutðUÞ and AutðUÞ [33, Theorem 3].
7.2. The best interval representation of an automorphism
In the following, interval automorphisms are discussed from the point of view of representation of automorphisms.
Theorem 41 (Automorphism representation theorem [12, Theorem 5.2]). Let q : U ! U be an automorphism. Then bq is aninterval automorphism given by:
bqðXÞ ¼ ½qðXÞ;qðXÞ�: ð30Þ
Therefore, interval automorphisms are the best interval representations of classical automorphisms.
Corollary 42. . : U! U is an interval automorphism if, and only if, . ¼ b., where . : U ! U is defined by .ðxÞ ¼ lð.ð½x; x�ÞÞ.
Proof. It is straightforward, following from Theorems 40 and 41. h
Notice that t-conorms are required, by definition, to be inclusion-monotonic. Nevertheless, this property is not required,by definition, for interval automorphisms. In the following, we show that interval automorphisms are also inclusion-mono-tonic [12].
Corollary 43 [12, Corollary 5.1]. If . is an interval automorphism then . is inclusion-monotonic, that is, if X # Y then.ðXÞ# .ðYÞ.
Considering the alternative definition of automorphism introduced in [16], we can provide alternative characterizationsfor interval automorphisms based on the Moore and Scott continuity notions.
Proposition 44 [12, Proposition 5.1]. . : U! U is an interval automorphism if and only if . is Moore-continuous, strictlyincreasing, .ð½0;0�Þ ¼ ½0;0� and .ð½1;1�Þ ¼ ½1;1�.
Corollary 45 [12, Corollary 5.2]. Let . : U! U be a Moore-continuous and strictly increasing function such that .ð½0;0�Þ ¼ ½0;0�and .ð½1;1�Þ ¼ ½1;1�. Then there exists an automorphism q such that . ¼ bq.
The case of Scott continuity is analogous.
Proposition 46. Let q be an automorphism. Then, it holds that dq�1 ¼ bq�1.
Proof. It follows that:
bqðdq�1ðXÞÞ ¼ bqð½q�1ðXÞ;q�1ðXÞ�Þ by Eq: ð30Þ¼ ½qðq�1ðXÞÞ;qðq�1ðXÞÞ�Þ by Eq: ð30Þ¼ X
Then, dq�1 is the inverse of bq, that is, dq�1 ¼ bq�1. h
1386 B.C. Bedregal et al. / Information Sciences 180 (2010) 1373–1389
7.3. Interval automorphisms acting on interval fuzzy negations and interval t-conorms
Analogously to the punctual case, an interval automorphism . acts on an interval-valued fuzzy function F : Un ! U, asfollows:
F.ðX1; . . . ;XnÞ ¼ .�1ðFð.ðX1Þ; . . . ;.ðXnÞÞÞ: ð31Þ
Theorem 47 [10]. Let . : U! U be an interval automorphism and N : U! U be an interval fuzzy negation. Then the mappingN. : U! U, defined by
N.ðXÞ ¼ .�1ðNð.ðXÞÞÞ; ð32Þ
is an interval fuzzy negation.
Theorem 48 [14, Proposition 7.1]. Let . : U! U be an interval automorphism and S : U2 ! U be an interval t-conorm. Thenthe mapping S. : U2 ! U, defined by
S.ðX; YÞ ¼ .�1ðSð.ðXÞ;.ðYÞÞÞ; ð33Þ
is an interval t-conorm.
Proposition 49. Let N be a fuzzy negation, S be a t-conorm and q be an automorphism. Then, it holds that
(i) bNbq ¼ cNq ; and(ii) bSbq ¼ cSq .
Proof. It follows that:
bNbqðXÞ ¼ bq�1ðbNðbqðXÞÞÞ by Eq: ð32Þ¼dq�1ðbNðbqðXÞÞÞ by Prop: ð46Þ¼ ½q�1ðNðqðXÞÞÞ;q�1ðqðNðXÞÞÞ� by Eq: ð5Þ¼ ½NqðXÞ;NqðXÞ� by Eq: ð28Þ
¼ cNqðXÞ by Eq: ð13Þ
The proof of the second case is analogous. h
7.4. Interval automorphisms acting on interval S-implications
In the following theorem, we show how interval automorphisms act on interval S-implications, generating new intervalS-implications.
Theorem 50. Let. : U! U be an interval automorphism and IS;N : U2 ! U be an interval S-implication. Then I.S;N : U2 ! U is
an interval S-implication defined by
I.S;N ¼ IS. ;N. : ð34Þ
Proof. Considering X; Y 2 U, then it follows that:
I.S;NðX; YÞ ¼ .�1ðIS;Nð.ðXÞ;.ðYÞÞÞ by Eq: ð31Þ
¼ .�1ðSðNð.ðXÞÞ;.ðYÞÞÞ by Eq: ð23Þ¼ .�1ðSð. � .�1ðNð.ðXÞÞÞ;.ðYÞÞÞ¼ .�1ðSð.ðN.ðXÞÞ;.ðX;YÞÞ by Eq: ð32Þ¼ S.ðN.ðXÞ;YÞ by Eq: ð33Þ¼ IS. ;N. ðX; YÞ by Eq: ð23Þ: �
Corollary 51. Let I be an interval S-implication and . be an interval automorphism. Then it holds thatI.ðX;YÞ ¼ ½IqðX;YÞ; IqðX;YÞ�, where I and I are, respectively, the left and right projections defined in Eq. (21), and q is the auto-morphism defined in Eq. (30).
Proof. It is straightforward, following from Theorem 50. h
Fig. 2. Commutative classes of interval S-implications and automorphisms.
B.C. Bedregal et al. / Information Sciences 180 (2010) 1373–1389 1387
According to Theorem 50, the commutative diagram pictured in Fig. 2 holds.Based on Theorem 50, (interval) S-implications and (interval) automorphisms can be seen as objects and morphisms,
respectively, of the category CðCðIÞ;AutðIÞÞ (CðCðIÞ;AutðIÞÞ). In a categorical approach, the action of an interval automorphismon an interval S-implication can be conceived as a covariant functor whose application over S-implications and automor-phisms in CðCðIÞ;AutðIÞÞ returns the related best interval representations in CðCðIÞ;AutðIÞÞ.
Proposition 52. Let IS;N be an S-implication and q be an automorphism. Then it holds that dIqS;N ¼dIS;Nbq :
Proof. It follows that:
4 The
dIqS;N ðX;YÞ ¼ ½inffIqS;Nðx; yÞjx 2 X; y 2 Yg;supfIqS;Nðx; yÞjx 2 X; y 2 Yg� by Eq: ð5Þ
¼ ½IqS;NðX;YÞ; IqS;NðX;YÞ� by Properties I1 and I2
¼ ½q�1ðIS;NðqðXÞ;qðYÞÞÞ;q�1ðIS;NðqðXÞ;qðYÞÞÞ� by Eq: ð28Þ¼dq�1ð½IS;NðqðXÞ;qðYÞÞ; IS;NðqðXÞ;qðYÞÞ�Þ by Eq: ð30Þ¼ bq�1ð½IS;NðqðXÞ;qðYÞÞ; IS;NðqðXÞ;qðYÞÞ�Þ by Prop: 46
¼ bq�1ðdIS;N ðbqðXÞ;qðYÞÞÞ by Properties I1 and I2 and Eq: ð30Þ
¼dIS;NbqðX;YÞ �
Corollary 53. Let IS;N be an S-implication and q be an automorphism. Then it holds that dIqS;N ¼ IbSq ; bNq:
Proof. It is straightforward, following from Theorem 24, Proposition 49, Theorem 50 and Proposition 52. h
8. Conclusion and final remarks
This work considers the particular case of type-2 fuzzy set theory that integrates Interval Mathematics and Fuzzy Set The-ory as a tool to deal with both vagueness and the uncertainty of membership functions in Fuzzy Set Theory. The paper fol-lows the line of our previous work [9,11–13,15,25,59], where interval extensions of some fuzzy connectives were provided astheir interval representations.
Considering the importance of the different kinds of fuzzy implications in the development of practical applications infuzzy systems and fuzzy control [47], the main contribution of this paper was the introduction of the concept of intervalS-implication, based on the notions of interval t-conorm and interval fuzzy negation, together with the analysis of the relatedproperties.
This paper also shows under which conditions interval S-implications preserve the main properties of S-implications. Itdemonstrates that several properties that are satisfied by interval S-implications are analogous to the ones presented by S-implications, showing, in this way, the applicability of our fuzzy interval approach for S-implications.
In particular, we proved that interval S-implications are obtainable. In addition, we showed that the action of an intervalautomorphism on an interval S-implication generates another interval S-implication.
Although the notion of interval-valued S-implication is not new (see, e.g., [20,22]), this paper brings contributions intotwo directions, in comparison with related works. On one hand, it provides a general way to treat the subject, throughthe interval representation approach, providing more general results. For example, the result in [20, Theorem 4]4 is weakerthan our Theorem 29.
On the other hand, the resulting class of interval S-implications proposed here is a subclass of the S-implications proposedin [22], since we require the inclusion monotonicity property. However, we provided some new non-trivial results for inclu-sion-monotonic interval S-implications.
paper [20] investigates Intuitionistic Fuzzy Sets (IFS) and it is well known that IFS is equivalent to Interval-Valued Fuzzy Sets [3,20].
1388 B.C. Bedregal et al. / Information Sciences 180 (2010) 1373–1389
Further works are concerned with the study of interval additive generators of interval t-conorms and of interval fuzzynegations (extending the work on interval additive generators of interval t-norms presented in [25]), and, therefore, of inter-val S-implications, with the analysis of the related properties based on their additive generators. Future works will also con-sider the investigation of the properties of interval extensions for other kinds of fuzzy implications, such as R-implications,QL-implications [65] and their contrapositions D-implications.5
Finally, we are also interested in the study of interval generalizations of S-implications and R-implications derived fromuninorms6 [47,61].
Acknowledgments
This work was supported by the Brazilian Funding Agency CNPq (under the process numbers 473201/07-0, 307185/07-9,307879/06-2). We are very grateful to the anonymous referees for their valuable comments and suggestions that helped usto improve the paper.
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