Information Sciences 271 (2014) 1–13

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Information Sciences

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Aggregation of fuzzy truth values

http://dx.doi.org/10.1016/j.ins.2014.02.1160020-0255/� 2014 Elsevier Inc. All rights reserved.

⇑ Tel.: +421 905857197; fax: +421 252495177.E-mail address: zdenko.takac@stuba.sk

Zdenko Takác ⇑Institute of Information Engineering, Automation and Mathematics, Faculty of Chemical and Food Technology, Slovak University of Technology in Bratislava,Radlinského 9, 812 37 Bratislava, Slovak Republic

a r t i c l e i n f o a b s t r a c t

Article history:Received 2 February 2013Received in revised form 3 February 2014Accepted 15 February 2014Available online 27 February 2014

Keywords:Aggregation operatorFuzzy truth valuesType-2 fuzzy setsType-2 aggregation operatorType-1 aggregation operatorFTV aggregation operatorn-Dimensional fuzzy setFuzzy interval analysis

The main aim of this paper is to propose an FTV aggregation operator, which is a tool foraggregation of fuzzy truth values (fuzzy sets in ½0;1�), and to provide a theoretical basisfor the concept of FTV aggregation operator. We extend (ordinary) aggregation operatorvia convolution and show that the extension satisfies stated axioms, i.e. this approach leadsto constructing of FTV aggregation operator. Furthermore, we show that our definition ofFTV aggregation operator is an extension of usual definition of aggregation operator as wellas of definition of aggregation operators for intervals and for n-dimensional intervals.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

The purpose of aggregation operators is reducing a set of values into a unique representative value. In all kinds of knowl-edge based systems the aggregation is of basic concerns, at some point it plays a fundamental role [4,12]. The theory ofaggregation of real numbers, which is well established (see e.g. [3,15,20–22,34,45,49]), is useful in fuzzy logic systems basedon fuzzy sets (we will refer to as type-1 fuzzy sets).

Recently, the aggregation operators for intervals were proposed and studied [11]. These operators are applicable in sys-tems based on the interval-valued fuzzy sets [51], Atanassov’s intuitionistic fuzzy sets [1] and interval type-2 fuzzy sets [26].Note that the three kinds of extensions of type-1 fuzzy sets are equivalent from syntactical point of view (e.g. [9,10]). Shanget al. [39] generalized the concept of interval-valued fuzzy sets to n-dimensional fuzzy sets (also called fuzzy multisets) andBedregal et al. [2] proposed aggregation operator for n-dimensional intervals (membership grades of n-dimensional fuzzysets).

The concept of type-2 fuzzy sets was introduced by Zadeh [50] as an extension of type-1 fuzzy sets. The membershipgrades of type-2 fuzzy sets are type-1 fuzzy sets in ½0;1�, we will refer to as fuzzy truth values. The algebra of fuzzy truthvalues was closely discussed in [35,36] and recently in [47,41]. We focus on this algebra, particularly on the aggregationof its elements. Our motivation rose from the need of aggregation of a family of fuzzy truth values into a unique value intype-2 fuzzy logic systems [25]. So the paper is placed under the scope of type-2 fuzzy sets, however, our results come alsounder the scope of fuzzy interval analysis (see [13,14,31]), where the focus is on arithmetic of fuzzy intervals (fuzzy subsets

2 Z. Takác / Information Sciences 271 (2014) 1–13

of real line whose a-cuts are closed intervals) [13]. Some of the results in Section 3 are, with different hypotheses, alreadyknown in fuzzy interval analysis – we will discuss this fact later, in Section 3.

According to [17], there are two different views of type-2 fuzzy sets (or any extension of fuzzy sets). 1. They are fuzzy setswith some new algebra of truth values which is a matter of adopting a new representation convection. 2. They are fuzzy setswith ill-known truth values from ½0;1� which means that the truth values are supposed to be precise, even if out of ourknowledge. The foundations of this paper follows from the first point of view.

The type-2 fuzzy sets are very useful in circumstances when there is need to handle more uncertainty than it is possibleusing type-1 or interval-valued fuzzy sets. This makes them to be very attractive tool in many real problems. However, thetheory for aggregating of fuzzy truth values is not sufficiently developed. Some particular generalized aggregation operatorswere studied: type-2 t-norms and type-2 t-conorms in [18,23,24,40,41]; type-2 implications in [19]; a-level approach totype-1 OWA operator (for discussion on terminology: type-1 versus type-2 aggregation operators, see Remark 2 of this arti-cle – Section 4) is developed in [52], and it is applied in [7]; an overview of linguistic aggregation operators is given in [48]summarizing results from [5,6,28,29]. Theoretical aspects of aggregation operators for fuzzy truth values are presented in[32,44]. The authors of [32] focus on multi-dimensional aggregation of fuzzy numbers, especially with trapezoidal shape.They applied the extension principle to multi-dimensional functions (with certain conditions) and obtained multi-dimen-sional aggregation functions on the lattice of fuzzy numbers. The author of [44] proves that the generalization of classicalaggregation operators (by extension principle) fulfils axioms stated for general aggregation operators (Theorem 3). But, thisproof is not correct, because some necessary condition is missing in Theorem 3 – we give a counterexample in Example 1.Our approach is similar to that in [32,44], but, we propose results for fuzzy truth values; formulate correct version of The-orem 3 of [44]; moreover, we study basic properties of generalized aggregation operators like symmetry, idempotency, neu-tral element and annihilator based on the properties of original classical aggregation operator. Applying of convolution fordistribution functions is widely described in [37,38].

Despite the previous list of results on aggregation of functions, the theory for aggregating of fuzzy truth values is not suf-ficiently developed; this is one of the several obstacles for applicability of systems based on type-2 fuzzy sets. Our goal is toovercome this lack of knowledge.

The main aim of this paper is to propose an aggregation operator for fuzzy truth values (FTV aggregation operator). Weprovide a theoretical basis for the concept of FTV aggregation operator. Then we extend (ordinary) aggregation operator viaconvolution (extension principle) and demonstrate that this extension (on convex normal fuzzy truth values) satisfies statedaxioms. Moreover, we show that our definition of FTV aggregation operator is an extension: of usual definition of (ordinary)aggregation operator; of the definition of aggregation operator for intervals [11]; and also of the definition of aggregationoperator for n-dimensional intervals [2].

Our approach is appropriate in the situations where membership grades are fuzzy without representing an ill-knownmembership function. For example, when evaluating a car’s design, linguistic labels like ’good’, ’very good’, ’excellent’ areusually used; and these labels can be expressed by functions on ½0;1�, i.e. by fuzzy truth values.

The paper is organized as follows. Section 2 contains basic definitions and notations that are used in the remaining partsof the paper. Section 3 presents the extension of (ordinary) aggregation operator via convolution. In Section 4, we provide theaxiomatic basis for FTV aggregation operator and prove that the proposed extension satisfies these axioms. Some propertiesof FTV aggregation operators are studied in Section 5. The conclusions are discussed in Section 6.

2. Preliminaries

In this section we present some basic concepts and terminology that will be used throughout the paper.A mapping f : X ! ½0;1� is called a fuzzy set (or type-1 fuzzy set) in a set X, the value f ðxÞ is called a membership grade of

x. A fuzzy set f in X is normal if there exists x 2 X such that f ðxÞ ¼ 1. The crisp set Kerðf Þ ¼ fx 2 X j f ðxÞ ¼ 1g is called a kernelof f. The crisp set fa ¼ fx 2 X j f ðxÞP ag, where a 2�0;1� is called an a-cut of f. Let X be a linear space, a fuzzy set f in X is con-vex if it is satisfied f ðkx1 þ ð1� kÞx2ÞP minðf ðx1Þ; f ðx2ÞÞ for all k 2 ½0;1�, where x1; x2 are arbitrary elements of X.

Definition 1. A function A : ½0;1�n ! ½0;1� is called an n-ary aggregation operator on ½0;1� if and only if it satisfies theconditions:

ðA1Þ Að0; . . . ;0Þ ¼ 0;ðA2Þ Að1; . . . ;1Þ ¼ 1;ðA3Þ x1 6 y1; . . . ; xn 6 yn implies Aðx1; . . . ; xnÞ 6 Aðy1; . . . ; ynÞ.for all x1; y1; . . . xn; yn 2 ½0;1�.

An n-ary aggregation operator A is called: symmetric if for each permutation r : f1; . . . ;ng ! f1; . . . ; ng and eachx1; . . . ; xn 2 ½0;1� it holds Aðx1; . . . ; xnÞ ¼ Aðxrð1Þ; . . . ; xrðnÞÞ; idempotent if for each x 2 ½0;1� it holds Aðx; . . . ; xÞ ¼ x. An elementa 2 ½0;1� is called an annihilator of n-ary aggregation operator A if for each x1; . . . ; xn 2 ½0;1� when xk ¼ a for somek ¼ 1; . . . ; n, then Aðx1; . . . ; xnÞ ¼ a. An element e 2 ½0;1� is called a neutral element of n-ary aggregation operator A if foreach k ¼ 1; . . . ; n and each x 2 ½0;1� it holds

Z. Takác / Information Sciences 271 (2014) 1–13 3

A e; . . . ; e|fflfflfflffl{zfflfflfflffl}ðk�1Þ�times

; x; e; . . . ; e|fflfflfflffl{zfflfflfflffl}ðn�kÞ�times

0B@1CA ¼ x:

A type-2 fuzzy set in X is a fuzzy set whose membership grades are type-1 fuzzy sets in ½0;1�. Let F denotes the class of alltype-1 fuzzy sets in ½0;1�. Then type-2 fuzzy set in X is a mapping ef : X ! F and elements of F are called fuzzy truth values.We denote by FN=F C=FNC the class of all normal/convex/normal convex fuzzy truth values, respectively. The algebra of fuzzytruth values F ¼ ðF ;t;u;0;1;v;�Þ is closely described in [35,47], whereas

ðf t gÞðzÞ ¼ supx_y¼z

ðf ðxÞ ^ gðyÞÞ; f v g iff f u g ¼ f ;

ðf u gÞðzÞ ¼ supx^y¼z

ðf ðxÞ ^ gðyÞÞ; f � g iff f t g ¼ g;

0ðxÞ ¼1; if x ¼ 0;0; otherwise;

�1ðxÞ ¼

1; if x ¼ 1;0; otherwise:

�

Let f 2 F . Then unary operations

f LðxÞ ¼ supy6x

f ðyÞ and f RðxÞ ¼ supyPx

f ðyÞ: ð1Þ

enable us to express the operations t; u and relations v; � pointwise (see [8,16,47]):

f t g ¼ ðf ^ gLÞ _ ðf L ^ gÞ ¼ ðf _ gÞ ^ ðf L ^ gLÞ; ð2Þf u g ¼ ðf ^ gRÞ _ ðf R ^ gÞ ¼ ðf _ gÞ ^ ðf R ^ gRÞ; ð3Þf v g iff f R ^ g 6 f 6 gR; f � g iff f ^ gL

6 g 6 f L: ð4Þ

Moreover, a fuzzy truth value f is convex if and only if f ¼ f L ^ f R (Proposition 33 in [47]).

3. Extension of aggregation operators

In this section we extend an aggregation operator and show under which conditions the extended aggregation operatorpreserves normality and convexity.

According to Zadeh’s extension principle [50] n-ary aggregation operator A : ½0;1�n ! ½0;1� can be extended by the con-volution with respect to minimum ^ and maximum _ to n-ary operator eA : F n ! F as follows:

eAðf1; . . . ; fnÞðyÞ ¼ sup

Aðx1 ;...;xnÞ¼yðf1ðx1Þ ^ . . . ^ fnðxnÞÞ; ð5Þ

where y; x1; . . . ; xn 2 ½0;1� and f1; . . . ; f n 2 F .

Remark 1. Note that some extensions of t-norms and t-conorms proposed in [18,47] are special cases of (5). Differentapproach to the subject used Zhou et al. [53] and proposed so-called type-1 OWA operators. For discussion on terminology(type-1 versus type-2 aggregation operators) see Remark 2 of this article – Section 4.

The following two theorems are, with different hypotheses, formulated in terms of fuzzy intervals in [14] (Theorem 1),however, we focus on a special class of functions, namely aggregation operators. First we show that for each aggregationoperator A the extended n-ary aggregation operator eAðf1; . . . ; fnÞ is normal whenever f1; . . . ; f n are normal fuzzy truth values.

Theorem 1. Let A be an n-ary aggregation operator, let f1; . . . ; f n be normal fuzzy truth values. Then eAðf1; . . . ; fnÞ given by (5) isnormal fuzzy truth value.

Proof. Let xpi 2 KerðfiÞ, for all i ¼ 1; . . . ;n. Let Aðxp

1; . . . ; xpnÞ ¼ xp. Then

eAðf1; . . . ; fnÞðxpÞ ¼ sup

Aðx1 ;...;xnÞ¼xpðf1ðx1Þ ^ . . . ^ fnðxnÞÞ ¼ f1ðxp

1Þ ^ . . . ^ fnðxpnÞ ¼ 1:

Thus, xp 2 Ker eAðf1; . . . ; fnÞ� �

, consequently eAðf1; . . . ; fnÞ is normal. h

By the following theorem we show the conditions under which convex fuzzy truth values f1; . . . ; fn give convex fuzzy truthvalue eAðf1; . . . ; fnÞ. Unlike [14], we do not require continuity of involved fuzzy truth values. It is sufficient that their supportsare (uninterrupted) intervals, which follows from the convexity of fuzzy truth values.

Theorem 2. Let A be a continuous n-ary aggregation operator, let f1; . . . ; f n be normal convex fuzzy truth values. TheneAðf1; . . . ; fnÞ given by (5) is normal convex fuzzy truth value.

4 Z. Takác / Information Sciences 271 (2014) 1–13

Proof. We prove the proposition for n ¼ 2, the generalization for arbitrary n is straightforward. From Theorem 1 it follows

that eAðf1; f2Þ is normal. It remains to show that it is also convex, i.e., eAðf1; f2Þ ¼ eAðf1; f2Þ� �L

^ eAðf1; f2Þ� �R

. The inequality 6 fol-

lows from f 6 f L; f 6 f R for all f 2 F (Proposition 3 in [47]), so we are going to show for all y 2 ½0;1�:

eAðf1; f2ÞðyÞP eAðf1; f2Þ� �L

^ eAðf1; f2Þ� �R

� �ðyÞ: ð6Þ

From (1) we get: ! !

eAðf1; f2Þ� �L

^ eAðf1; f2Þ� �R

� �ðyÞ ¼ sup

Aðs1 ;s2Þ¼yðf1ðs1Þ ^ f2ðs2ÞÞ

L

^ supAðt1 ;t2Þ¼y

ðf1ðt1Þ ^ f2ðt2ÞÞR

¼ supu6y

supAðs1 ;s2Þ¼u

ðf1ðs1Þ ^ f2ðs2ÞÞ !

^ supvPy

supAðt1 ;t2Þ¼v

ðf1ðt1Þ ^ f2ðt2ÞÞ !

¼ supAðs1 ;s2Þ6y

ðf1ðs1Þ ^ f2ðs2ÞÞ !

^ supAðt1 ;t2ÞPy

ðf1ðt1Þ ^ f2ðt2ÞÞ !

;

where s1; s2; t1; t2 2 ½0;1�. We denote the very last term by Term. Then (6) can be expressed:

supAðx1 ;x2Þ¼y

ðf1ðx1Þ ^ f2ðx2ÞÞP Term; ð7Þ

which we are going to prove. We will consider the following three cases:

1. Let y 2 Ker eAðf1; f2Þ� �

. Then supAðx1 ;x2Þ¼y ðf1ðx1Þ ^ f2ðx2ÞÞ ¼ 1, hence, (7) holds.

2. Let y 6 inf Ker eAðf1; f2Þ� �� �

. Let Aðs1; s2Þ 6 y. Then (see Fig. 1):

(i) Let s1 6 sup Kerðf1Þð Þ and s2 6 sup Kerðf2Þð Þ. Then there exist x01; x0

2 2 ½0;1� such that Aðx01; x

02Þ ¼ y and

s1 6 x01 6 sup Kerðf1Þð Þ; s2 6 x0

2 6 sup Kerðf2Þð Þ. Recall that f1; f 2 are convex, so they are increasing on intervals½0; sup Kerðf1Þð Þ�; ½0; sup Kerðf2Þð Þ�, respectively. Thus, f1ðx0

1Þ ^ f2ðx02ÞP f1ðs1Þ ^ f2ðs2Þ and consequently (7) holds.

(ii) Let s1 6 sup Kerðf1Þð Þ and s2 > sup Kerðf2Þð Þ. Then there exist x01; x0

2 2 ½0;1� such that Aðx01; x

02Þ ¼ y and

s1 6 x01 6 sup Kerðf1Þð Þ; s2 ¼ x0

2. Recall that f1 is increasing on interval ½0; sup Kerðf1Þð Þ�. Thus,f1ðx0

1Þ ^ f2ðx02ÞP f1ðs1Þ ^ f2ðs2Þ and consequently (7) holds.

(iii) Let s1 > sup Kerðf1Þð Þ and s2 6 sup Kerðf2Þð Þ. Then there exist x01; x0

2 2 ½0;1� such that Aðx01; x

02Þ ¼ y and

s1 ¼ x01 s2 6 x0

2 6 sup Kerðf2Þð Þ. Recall that f2 is increasing on interval ½0; sup Kerðf2Þð Þ�. Thus,

Fig. 1. Figure to the proof of Theorem 2.

Z. Takác / Information Sciences 271 (2014) 1–13 5

f1ðx01Þ ^ f2ðx0

2ÞP f1ðs1Þ ^ f2ðs2Þ and consequently (7) holds.From the previous three cases (i)–(iii) it follows:

00

0.5

1

00

0.5

1

00

0.5

1

Fig. 2.second

supAðx1 ;x2Þ¼y

ðf1ðx1Þ ^ f2ðx2ÞÞP supAðs1 ;s2Þ6y

ðf1ðs1Þ ^ f2ðs2ÞÞ; ð8Þ

for all y 6 inf Ker eAðf1; f2Þ� �� �

.

3. Let y P sup Ker eAðf1; f2Þ� �� �

. The proof of

supAðx1 ;x2Þ¼y

ðf1ðx1Þ ^ f2ðx2ÞÞP supAðt1 ;t2ÞPy

ðf1ðt1Þ ^ f2ðt2ÞÞ ð9Þ

is similar to item 2 with the exception that functions f1; f2 are decreasing on intervals ½inf Kerðf1Þð Þ;1�; ½inf Kerðf2Þð Þ;1�,respectively.

Finally, (8) and (9) implies (7). h

Let f be a normal fuzzy truth value. We define:

f LPðxÞ ¼ f LðxÞ; if x 6 supðKerðf ÞÞ;0; otherwise

�ð10Þ

and

f RPðxÞ ¼ f RðxÞ ; if x P infðKerðf ÞÞ;0 ;otherwise;

�ð11Þ

which means that f LP is f L without the part behind kernel and f RP is f R without the part before kernel. Note that, for convexf ; f LP is just the increasing part of f and f RP is just the decreasing part of f, furthermore f ¼ f LP ^ f RP . See the first two rows ofFig. 2. Then, for aggregation of convex fuzzy truth values the following condition holds.

Lemma 3. Let A be a continuous n-ary aggregation operator, let f1; . . . ; f n be normal convex fuzzy truth values. Then

eAðf1; . . . ; fnÞLP ¼ eA f LP1 ; . . . ; f LP

n

� ; eAðf1; . . . ; fnÞRP ¼ eA f RP

1 ; . . . ; f RPn

� :

Proof. We prove the first equality for n ¼ 2, the generalization for arbitrary n is straightforward. The proof for the secondequality is similar.

1. Let y 6 sup KerðeAðf1; f2ÞÞ� �

. Then from Theorem 2 and (10) it follows:

eAðf1; f2ÞLPðyÞ ¼ eAðf1; f2ÞLðyÞ ¼ eAðf1; f2ÞðyÞ: ð12Þ

Let x1; x2 2 ½0;1� be such that Aðx1; x2Þ ¼ y. We will consider three cases (see Fig. 3):

(i) Let x1 6 supðKerðf1ÞÞ; x2 6 supðKerðf2ÞÞ. Then f LP1 ðx1Þ ¼ f1ðx1Þ and f LP

2 ðx2Þ ¼ f2ðx2Þ.(ii) Let x1 6 supðKerðf1ÞÞ; x2 > supðKerðf2ÞÞ. Then there exist x0

1; x02 2 ½0;1� with Aðx0

1; x02Þ ¼ y such that x0

1 P x1 andx0

2 ¼ supðKerðf2ÞÞ < x2. Because f1 is increasing on interval ½x1; x01� and f2 is decreasing on interval ½x0

2; x2� it follows

f1ðx1Þ 6 f1ðx01Þ and f2ðx2Þ 6 f2ðx0

2Þ, which means that the ordered pair ðx1; x2Þ does not affect the eAðf1; f2ÞðyÞ. See Fig. 3.(iii) Let x1 > supðKerðf1ÞÞ; x2 6 supðKerðf2ÞÞ. Similar to the item (ii) we get that the ordered pair ðx1; x2Þ does not affect theeAðf1; f2ÞðyÞ. See Fig. 3.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

y

f1

f2

Ã(f1,f

2)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

x1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

x2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

y

f2LP

f1LP

Ã(f1,f

2)LP =Ã(f

1LP ,f

2LP )

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

x1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

x2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

y

f2RP

f1RP

Ã(f1,f

2)RP =Ã(f

1RP ,f

2RP )

The first column – aggregation of trapezoidal fuzzy truth values f1; f2 by extended arithmetic mean eAðf1; f2ÞðyÞ ¼ supx1þx22 ¼y ðf1ðx1Þ ^ f2ðx2ÞÞ. The

column – aggregation of f LP1 ; f LP

2 . The third column – aggregation of f RP1 ; f RP

2 .

Fig. 3. Figure to the proof of Lemma 3.

6 Z. Takác / Information Sciences 271 (2014) 1–13

From the previous three cases (i)–(iii) it follows that only the ordered pairs ðx1; x2Þ withx1 6 supðKerðf1ÞÞ; x2 6 supðKerðf2ÞÞ affect the eAðf1; f2ÞðyÞ and due to (12) also eAðf1; f2ÞLPðyÞ. Hence,eAðf1; f2ÞLPðyÞ ¼ eA f LP

1 ; f LP2

� ðyÞ.

2. Let y > sup KerðeAðf1; f2ÞÞ� �

. Then eAðf1; f2ÞLP ¼ 0 ¼ eA f LP1 ; f LP

2

� . h

Based on Lemma 3, the aggregation of normal convex fuzzy truth values f1; . . . ; f n can be done separately for increasingparts of the truth values and separately for decreasing parts. See Fig. 2.

Corollary 4. Let A be a continuous n-ary aggregation operator, let f1; . . . ; fn be normal convex fuzzy truth values. Then

eAðf1; . . . ; fnÞ ¼ eA f LP1 ; . . . ; f LP

n

� ^ eA f RP

1 ; . . . ; f RPn

� :

Proof. Straightforward from Lemma 3. h

4. Aggregation of fuzzy truth values

We generalize the definition of aggregation operator on ½0;1� to aggregation operator on the set of fuzzy truth values(called FTV aggregation operator). Then we discuss relation between this new generalized aggregation operator and someknown types of aggregation operators.

4.1. FTV aggregation operator

Definition 2. Let ðU ;t;u;0;1;v;�Þ be a subalgebra of F ¼ ðF ;t;u;0;1;v;�Þ. A function eA : Un ! U is called an n-ary FTVaggregation operator on U if and only if it satisfies the conditions ðeA1Þ and ðeA2Þ and, for all f1; . . . ; f n; g1; . . . ; gn 2 U , atleast one of the conditions ðeA3Þ and ðeA30Þ:

ðeA1Þ eAð0; . . . ;0Þ ¼ 0;ðeA2Þ eAð1; . . . ;1Þ ¼ 1;ðeA3Þ f1 v g1; . . . ; f n v gn implies eAðf1; . . . ; fnÞ v eAðg1; . . . ; gnÞ,ðeA30Þ f1 � g1; . . . ; f n � gn implies eAðf1; . . . ; fnÞ � eAðg1; . . . ; gnÞ.

Z. Takác / Information Sciences 271 (2014) 1–13 7

Recall that for normal convex fuzzy truth values the two orderings coincide, i.e. v¼� (Proposition 19 and Proposition 37in [47]). However, for fuzzy truth values that are not normal or convex this is not true. For example, let f ðxÞ ¼ 0:9 andgðxÞ ¼ 0:8 for all x 2 ½0;1�, i.e. f ; g are not normal. Then f u g ¼ g and f t g ¼ g, so g v f and f � g. Now, let f ; g are fuzzytruth values given by Fig. 4, i.e. f ; g are not convex. Then f u g ¼ f and f t g – g, so f v g and f�g.

Remark 2. There are two different views of terminology. Our original name of the operator proposed in Definition 2 was ‘type-2 aggregation operator’ (see [43,42]). This is in accordance with names (of some special cases of the operator) ‘type-2 t-norms’,‘type-2 t-conorms’ and ‘type-2 implications’, which appeared in some recent papers (e.g. [19,23,41,47]). In this case classicalaggregation operator should be referred to as ‘type-1 aggregation operator’ (this is similar to situation when classical fuzzy setis referred to as type-1 fuzzy set). The other view was proposed by anonymous reviewer of the paper who argued: the operatorproposed in Definition 2 should be called ‘type-1 aggregation operator’, because the use of name ‘type-2 aggregation operator’leads readers to believe that these are operators with inputs being type-2 fuzzy sets not type-1 fuzzy sets; and similarly for‘type-1 aggregation operator’. This point of view is applied e.g. in [53,54]. In my opinion, both points of view are prettyreasonable, however, it also means that both names ‘type-2’ and ‘type-1’ are more or less inappropriate. So, I decided in favor ofstraightforward solution and used the name ’FTV aggregation operator’ (FTV, because inputs are fuzzy truth values).

The main result of this section shows that, on the set of normal convex fuzzy truth values, the extended continuous n-aryaggregation operator given by (5) is an FTV aggregation operator. The theorem legitimates the use of extension principle onaggregation operators – it says that it is the same to use the extension principle on aggregation operators and to define aggre-gation operators by axioms on the fuzzy valued structure (note that it presupposes that we aggregate independent values).

Theorem 5. Let A be a continuous n-ary aggregation operator, let FNC be the set of normal convex fuzzy truth values. Then theextended aggregation operator eA : Fn

NC ! FNC with

Fig. 4.values.

eAðf1; . . . ; fnÞðyÞ ¼ supAðx1 ;...;xnÞ¼y

ðf1ðx1Þ ^ . . . ^ fnðxnÞÞ; ð13Þ

where y; x1; . . . ; xn 2 ½0;1� and f1; . . . ; f n 2 FNC , is an FTV aggregation operator.

Proof. We show that eAðf1; . . . ; fnÞ given by (13) satisfies ðeA1Þ; ðeA2Þ and ðeA3Þ.

eAð0; . . . ;0Þð0Þ ¼ supAðx1 ;...;xnÞ¼0

ð0ðx1Þ ^ . . . ^ 0ðxnÞÞ ¼ 1; ð14Þ

eAð0; . . . ;0Þðy – 0Þ ¼ supAðx1 ;...;xnÞ–0

ð0ðx1Þ ^ . . . ^ 0ðxnÞÞ ¼ 0; ð15Þ

where the last equality of (15) follows from the fact that Aðx1; . . . ; xnÞ – 0 implies existence of i such that xi – 0. Then0ðxiÞ ¼ 0. From (14) and (15) we have ðeA1Þ. Similarly ðeA2Þ.

Now we prove ðeA3Þ. Let fi v gi, for all i ¼ 1; . . . ; n:(i) Then (4) implies fi 6 gR

i , so supðKerðfiÞÞ 6 supðKerðgiÞÞ and consequently sup KerðeAðf1; . . . ; fnÞÞ� �

6 sup KerðeAðg1; . . . ; gnÞÞ� �

. Due to Theorem 2 we have:

eAðf1; . . . ; fnÞLP6

eAðg1; . . . ; gnÞLP

� �R; ð16Þ

eAðf1; . . . ; fnÞRP6

eAðg1; . . . ; gnÞRP

� �Rð17Þ

0 0.2 0.4 0.6 0.8 10

1fgf g

Fuzzy truth values f ; g with f u g ¼ f and f t g – g, so f v g and f�g. It means that the orderings v and � are not the same on non-convex fuzzy truth

8 Z. Takác / Information Sciences 271 (2014) 1–13

and from Corollary 4 it follows

eAðf1; . . . ; fnÞ 6 eAðg1; . . . ; gnÞR: ð18Þ

(ii) Then (4) implies f Ri ^ gi 6 fi, so for xi 6 supðKerðfiÞÞ we have giðxiÞ 6 fiðxiÞ and consequently:

eAðg1; . . . ; gnÞ

LPðyÞ 6 eAðf1; . . . ; fnÞLPðyÞ ð19Þ

and

eAðg1; . . . ; gnÞRPðyÞ 6 eAðf1; . . . ; fnÞRPðyÞ; ð20Þ

for all y 6 sup Ker eAðf1; . . . ; fnÞ� �� �

.

Moreover, from the convexity of eAðf1; . . . ; fnÞ it follows

eAðf1; . . . ; fnÞLP� �R

ðyÞ ¼ eAðf1; . . . ; fnÞLPðyÞ ¼ 0 ð21Þ

and

eAðf1; . . . ; fnÞRP� �R

ðyÞ ¼ eAðf1; . . . ; fnÞRPðyÞ; ð22Þ

for all y > sup Ker eAðf1; . . . ; fnÞ� �� �

.

From (19) and (21) we get

eAðf1; . . . ; fnÞLP� �R

^ eAðg1; . . . ; gnÞLP6eAðf1; . . . ; fnÞLP

; ð23Þ

from (20) and (22) we get

eAðf1; . . . ; fnÞRP� �R

^ eAðg1; . . . ; gnÞRP6eAðf1; . . . ; fnÞRP ð24Þ

‘ and from Corollary 4 it follows

eAðf1; . . . ; fnÞR ^ eAðg1; . . . ; gnÞ 6 eAðf1; . . . ; fnÞ: ð25Þ

Finally, due to (4), (18) and (25) we have eAðf1; . . . ; fnÞ v eAðf1; . . . ; fnÞ, so ðeA3Þ holds. h

Example 1. Fig. 2 depicts an example of extended arithmetic mean Aðx1; . . . ; xnÞ ¼Pn

i¼1xi=n, for n ¼ 2, which is an FTV aggre-gation operator for each n ¼ 1;2; . . .

Now we give an example of discontinuous aggregation operator whose extension is not an FTV aggregation operator anddoes not preserve the convexity. Note that this is a counterexample of Theorem 3 in [44]. Let A be a binary aggregationoperator with:

Aðx1; x2Þ ¼

0; if ðx1; x2Þ 2 ½0;0:5� � ½0; 0:5�;0:3; if ðx1; x2Þ 2 ½0;0:5���0:5;1�;0:7; if ðx1; x2Þ 2�0:5;1� � ½0;0:5�;1; if ðx1; x2Þ 2�0:5;1���0:5;1�

8>>><>>>:

and f1; f 2; g1; g2 be trapezoidal fuzzy truth values with parameters ð0:3;0:4;0:6; 0:7Þ; ð0:1;0:3;0:4;0:6Þ;ð0:35;0:45;0:65;0:75Þ; ð0:6;0:7;0:8;0:9Þ, respectively. Clearly f1 v g1 and f2 v g2. Values of eAðf1; f2Þ and eAðg1; g2Þ are in the

second and third row of the following table. Values of eAðf1; f2Þ u eAðg1; g2Þ computed via (3) are in the fourth row:

We can see that neither eAðf1; f2Þ nor eAðg1; g2Þ is convex. Moreover, eA is not FTV aggregation operator, becauseeAðf1; f2Þ u eAðg1; g2Þ� �

– eAðf1; f2Þ, i.e. eAðf1; f2Þ v eAðg1; g2Þ does not hold.

Z. Takác / Information Sciences 271 (2014) 1–13 9

4.2. FTV aggregation operators on various subalgebras of F

We discuss relations between our generalized aggregation operator given by Definition 2 and some known aggregationoperators on the sets isomorphic to some subalgebras of F .

4.2.1. Fuzzy grades of (type-1) fuzzy setsLet S be a set of all the singletons from F , i.e.,

Fig. 5.6-dime

f 2 S iff f ðxÞ ¼1; if x ¼ a;

0; otherwise

�; for some a 2 ½0;1�:

Then S ¼ ðS;t;u;0;1;vÞ is a subalgebra of F isomorphic with the algebra ð½0;1�;_;^;0;1;6Þ (see [47]), i.e. isomorphicwith the fuzzy grades of type-1 fuzzy sets. If U ¼ S, Definition 2 coincides with the usual notion of n-ary aggregation operator(e.g. [4,34]).

4.2.2. Fuzzy grades of interval-valued fuzzy setsLet I be a set of all the characteristic functions of closed subintervals of ½0;1�, i.e.,

f 2 I iff f ðxÞ ¼1; if x 2 ½a; b�;0; otherwise

�; for some ½a; b�# ½0;1�:

Then I ¼ ðI ;t;u;0;1;vÞ is a subalgebra of F isomorphic with the algebra ðD2;_;^; ½0;0�; ½1;1�;6Þ, whereD2 ¼ f½a; b� j0 6 a 6 b 6 1g (see [47]), i.e. isomorphic with the fuzzy grades of interval-valued fuzzy sets under standardmaximum _, minimum ^ and ordering 6 for intervals. If U ¼ I , Definition 2 coincides with the definition of n-ary aggrega-tion operator for intervals (e.g. [11]).

4.2.3. Fuzzy grades of n-dimensional fuzzy setsShang et al. [39] introduced an n-dimensional fuzzy set A on Z as a mapping A : Z ! ½0;1�n, where AðzÞ ¼ ðA1ðzÞ; . . . ;AnðzÞÞ

with A1ðzÞ 6 A2ðzÞ 6 . . . 6 AnðzÞ is called an n-dimensional interval. Recall that type-1 fuzzy sets and interval-valued fuzzysets are special cases of n-dimensional fuzzy sets for n ¼ 1 and n ¼ 2, respectively. One of the possible interpretations is:an n-dimensional interval can be seen as a chain of nested intervals of length k ¼ n

2 (for even n) or k ¼ nþ12 (for odd n) repre-

senting different uncertainty levels on the membership degree [2]. Now, let us interpret these nested intervals as a-cuts ofsome fuzzy truth value f (fuzzy grade of type-2 fuzzy set for some z 2 Z) for a ¼ 1

k ;2k ; . . . ; 1:

f1k¼ ½A1ðzÞ;AnðzÞ�; f 2

k¼ ½A2ðzÞ;An�1ðzÞ�; . . . ; f 1 ¼ ½AkðzÞ;An�kþ1ðzÞ�;

for even n, and

f1k¼ ½A1ðzÞ;AnðzÞ�; f 2

k¼ ½A2ðzÞ;An�1ðzÞ�; . . . ; f 1 ¼ AkðzÞ;

for odd n. Now, let V be, for some fixed n, a subset of F given by:

g 2 V iff gðxÞ ¼

1k ; if x 2 f1

k� f2

k;

2k ; if x 2 f2

k� f3

k;

..

.

k�1k ; if x 2 fk�1

k� f1;

1; if x 2 f1:

8>>>>>>>><>>>>>>>>:; for some f 2 F

0 0.2 0.4 0.6 0.8 10

1/3

2/3

1

f

f2/3 f1/3f1

0 0.2 0.4 0.6 0.8 10

1/3

2/3

1

g

On the left: a trapezoidal fuzzy truth value f with parameters ð0:1;0:4;0:6;0:9Þ. Its a-cuts f13¼ ½0:2;0:8�; f 2

3¼ ½0:3;0:7� and f1 ¼ ½0:4;0:6� produce

nsional interval AðzÞ ¼ ð0:2;0:3;0:4;0:6;0:7;0:8Þ. On the right: corresponding fuzzy truth value g 2 V.

10 Z. Takác / Information Sciences 271 (2014) 1–13

For example of 6-dimensional fuzzy set and corresponding g 2 V see Fig. 5. Clearly, V ¼ ðV;t;u;0;1;vÞ is a subalgebra ofF isomorphic with the algebra ðDn;_;^; ð0; . . . ;0Þ; ð1; . . . ;1Þ;6Þ, where Dn ¼ fða1; . . . ; anÞ j0 6 a1 6 . . . 6 an 6 1g, i.e. isomor-phic with the fuzzy grades of n-dimensional fuzzy sets under standard maximum _, minimum ^ and ordering 6for n-dimen-sional intervals. If U ¼ V, Definition 2 coincides with the definition of n-ary aggregation operator for n-dimensional intervals(Definition 3.1 in [2]).

Thus, in applications, the computation with membership grades of type-2 fuzzy sets can be approximated via computa-tion with n-dimensional fuzzy sets, i.e. with classes of a-cuts of fuzzy truth values. For instance, the example of fuzzy mul-ticriteria decision making based on n-dimensional fuzzy sets in Section 5 of [2] can be used as an approximate computationfor multicriteria decision making based on type-2 fuzzy sets.

Note that this approach corresponds to a-plane representation [30] and zSlice representation [46] of type-2 fuzzy sets.

5. Properties of FTV aggregation operators

In this section we investigate some properties of FTV aggregation operators that are important for application of operatorsin type-2 fuzzy logic systems. We show that the extension of aggregation operator preserves symmetry, idempotency, exis-tence of neutral element and of annihilator.

5.1. Symmetry, idempotency, neutral element and annihilator

An extended aggregation operator eA is called symmetric if for each permutation r : f1; . . . ;ng ! f1; . . . ; ng and eachf1; . . . ; f n 2 F the following holds:

eAðf1; . . . ; fnÞ ¼ eAðfrð1Þ; . . . ; frðnÞÞ: ð26Þ

Symmetry means that the order of the arguments has not influence on the result – this is an essential property when theaggregation is made of arguments having the same importance.

Theorem 6. Let A be an n-ary aggregation operator and eA : Fn ! F be an extended aggregation operator on fuzzy truth valuesgiven by (5). Then A is symmetric if and only if eA is symmetric.

Proof. Sufficiency: Straightforward from (5).Necessity: Let A be asymmetric. Then there exist z1; . . . ; zn 2 ½0;1� and a permutation r : f1; . . . ;ng ! f1; . . . ;ng such

that Aðz1; . . . ; znÞ– Aðzrð1Þ; . . . ; zrðnÞÞ. Let fi, for all i ¼ 1;2; . . . ;n, be defined by: fiðziÞ ¼ 1 and fiðxÞ ¼ 0, for x 2 ½0;1� � fzig. Lety ¼ Aðz1; . . . ; znÞ. Then

eAðf1; . . . ; fnÞðyÞ ¼ supAðx1 ;...;xnÞ¼y

ðf1ðx1Þ ^ . . . ^ fnðxnÞÞ ¼ f1ðz1Þ ^ . . . ^ fnðznÞ ¼ 1

and from y – Aðzrð1Þ; . . . ; zrðnÞÞ it follows

eAðfrð1Þ; . . . ; frðnÞÞðyÞ ¼ supAðxrð1Þ ;...;xrðnÞÞ¼y

ðfrð1Þðxrð1ÞÞ ^ . . . ^ frðnÞðxrðnÞÞÞ ¼ 0:

So also eA is asymmetric. h

An extended aggregation operator eA is called idempotent if for each f 2 F the following holds:

eAðf ; . . . ; f Þ ¼ f : ð27Þ

Idempotency is based on the expectation (which is essential in many circumstances): aggregation of n times the samevalue results in the initial value.

Theorem 7. Let A be an idempotent n-ary aggregation operator. Then an extended aggregation operator on convex fuzzy truthvalues eA : Fn

C ! FC given by (5) is idempotent.

Proof. Let f 2 F C . Then for each y 2 ½0;1�:

eAðf ; . . . ; f ÞðyÞ ¼ supAðx1 ;...;xnÞ¼y

ðf ðx1Þ ^ . . . ^ f ðxnÞÞP f ðyÞ; ð28Þ

due to the idempotency of A. Now we need to prove the opposite inequality. Suppose that there exist x1; . . . ; xn withAðx1; . . . ; xnÞ ¼ y such that f ðxiÞ > f ðyÞ for all i 2 f1; . . . ;ng. From the convexity of f it follows xi > y for all i 2 f1; . . . ;ng, orxi < y for all i 2 f1; . . . ;ng. If the former is true (the proof of the latter one is similar): let x0 ¼minfx1; . . . ; xng, thenAðx1; . . . ; xnÞP Aðx0; . . . ; x0Þ ¼ x0 > y and we have a contradiction with Aðx1; . . . ; xnÞ ¼ y. h

Z. Takác / Information Sciences 271 (2014) 1–13 11

A function g : ½0;1� ! ½0;1� is called a neutral element of extended aggregation operator eA if for each k ¼ 1; . . . ; n andeach f 2 F it holds that:

eAð g; . . . ; g|fflfflfflffl{zfflfflfflffl}

ðk�1Þ�times

; f ; g; . . . ; g|fflfflfflffl{zfflfflfflffl}ðn�kÞ�times

Þ ¼ f :

Neutral element can be used to be associated to an arguments that have not influence on the result of the aggregation.

Theorem 8. Let A be an n-ary aggregation operator with neutral element e. Then a function g : ½0;1� ! ½0;1� is a neutral elementof extended aggregation operator on fuzzy truth values eA : Fn ! F given by (5) if and only if

gðxÞ ¼1 ; if x ¼ e;

0 ; otherwise:

�ð29Þ

Proof. Let g be given by (29). Then for all f 2 F ; k 2 f1; . . . ; ng it holds:

eA g; . . . ; g|fflfflfflffl{zfflfflfflffl}ðk�1Þ�times

; f ; g; . . . ; g|fflfflfflffl{zfflfflfflffl}ðn�kÞ�times

0B@1CAðyÞ ¼ sup

Aðx1 ;...;xnÞ¼ygðx1Þ ^ . . . ^ gðxk�1Þ ^ f ðxkÞ ^ gðxkþ1Þ ^ . . . ^ gðxnÞð Þ ¼ f ðyÞ:

The uniqueness follows from the uniqueness of neutral element of aggregation operators in general. h

A function g : ½0;1� ! ½0;1� is called an annihilator of extended aggregation operator eA if for each f1; . . . ; f n 2 F whenfk ¼ g for some k ¼ 1; . . . ; n, then

eAðf1; . . . ; fnÞ ¼ g:

Annihilator can be used like an eliminating score, like a veto or as a qualifying score.

Theorem 9. Let A be an n-ary aggregation operator with annihilator a. Then a function g is an annihilator of extended aggregationoperator on normal fuzzy truth values eA : Fn

N ! FN given by (5) if and only if

gðxÞ ¼1 ; if x ¼ a;

0 ; otherwise:

�ð30Þ

Proof. Let g be given by (30). Then for all f1; . . . ; f n, where fk ¼ g for some k 2 f1; . . . ;ng, it holds:

eAðf1; . . . ; fnÞðaÞ ¼ supAðx1 ;...;xnÞ¼a

f1ðx1Þ ^ . . . ^ fk�1ðxk�1Þ ^ gðxkÞ ^ fkþ1ðxkþ1Þ ^ . . . ^ fnðxnÞð Þ ¼ 1;

because for xk ¼ a and xi 2 KerðfiÞ, for i – k, it holds f1ðx1Þ ¼ . . . ¼ fnðxnÞ ¼ 1. Moreover, for y – a we have:

eAðf1; . . . ; fnÞðyÞ ¼ 0;

because Aðx1; . . . ; xnÞ – a implies xi – a for all i ¼ 1; . . . ; n and consequently gðxk – aÞ ¼ 0.The uniqueness follows from the uniqueness of annihilator of aggregation operators in general. h

5.2. Extension of minimum, maximum and compensation property

In this section we study some properties of extended minimum and maximum. Note that minimum and maximum ex-tended via (5):

u ðf1; . . . ; fnÞðyÞ ¼ supx1^...^xn¼y

f1ðx1Þ ^ . . . ^ fnðxnÞð Þ; ð31Þ

t ðf1; . . . ; fnÞðyÞ ¼ supx1_..._xn¼y

f1ðx1Þ ^ . . . ^ fnðxnÞð Þ ð32Þ

corresponds to definition by

f1 u . . . u fn ¼ ðf1 u . . . u fn�1Þ u fn;

f1 t . . . t fn ¼ ðf1 t . . . t fn�1Þ t fn

for all n ¼ 3; 4; . . ., due to associativity (Corollary 7 in [47]) of binary operations u and t, respectively.Note that Walker and Walker [47] expressed (31) and (32) in terms of pointwise operations:

12 Z. Takác / Information Sciences 271 (2014) 1–13

f1 u . . . u fn ¼ ðf1 _ . . . _ fnÞ ^ ðf R1 ^ . . . ^ f R

n Þ;f1 t . . . t fn ¼ ðf1 _ . . . _ fnÞ ^ ðf L

1 ^ . . . ^ f Ln Þ:

Compensation property of an aggregation operator A means that the result of the aggregation is lower than the highestelement aggregated and higher than the lowest one (e.g. [27,12]):

minðx1; . . . ; xnÞ 6 Aðx1; . . . ; xnÞ 6 maxðx1; . . . ; xnÞ;

for all x1; . . . ; xn 2 ½0;1�. It is well-known that an aggregation operator is idempotent if and only if it satisfies the compen-sation property. A similar relationship holds also for extended aggregation operators.

We say that an FTV aggregation operator eA : F n ! F satisfies the compensation property if for all f1; . . . ; f n 2 F it holds:

uðf1; . . . ; fnÞ v eAðf1; . . . ; fnÞ v tðf1; . . . ; fnÞ ð33Þ

or

uðf1; . . . ; fnÞ � eAðf1; . . . ; fnÞ � tðf1; . . . ; fnÞ:

Theorem 10. Let A be a continuous n-ary aggregation operator, let eA : FnNC ! FNC be an extended aggregation operator on

normal convex fuzzy truth values given by (5). Then eA is idempotent if and only if eA satisfies the compensation property.

Proof. Let (33) holds. Then uðf ; . . . ; f Þ v eAðf ; . . . ; f Þ v tðf ; . . . ; f Þ, for all f 2 FNC , which implies eAðf ; . . . ; f Þ ¼ f , i.e. eA isidempotent.

Now, let eA is idempotent and let f1; . . . ; f n 2 FNC . Let fmin ¼ uðf1; . . . ; fnÞ and fmax ¼ tðf1; . . . ; fnÞ. From the axiom ðeA3Þ wehave:

fmin ¼ eAðfmin; . . . ; fminÞ v eAðf1; . . . ; fnÞ v eAðfmax; . . . ; fmaxÞ ¼ fmax:

Hence, eA satisfies the compensation property. h

Corollary 11. Let A be a continuous n-ary aggregation operator satisfying the compensation property. Let eA : F nNC ! FNC be an

extended aggregation operator on normal convex fuzzy truth values given by (5). Then eA satisfies the compensation property.

Proof. Straightforward from Theorem 7 and Theorem 10. h

6. Conclusions

We have presented a theoretical basis for FTV aggregation operators. With these operators we can aggregate fuzzy truthvalues, so it is a step to developing fuzzy logic systems based on type-2 fuzzy sets. The proposed operator is an extension ofknown aggregation operators for real numbers, intervals and n-dimensional intervals.

We also showed that an n-dimensional fuzzy set can be interpreted as a class of a-cuts of some fuzzy truth value. Thus,the computation with membership grades of type-2 fuzzy sets can be approximated via computation with n-dimensionalfuzzy sets. Consequently, various applications of n-dimensional fuzzy sets can be used as an approximate computationfor applications of type-2 fuzzy sets.

In this paper, our emphasis was on the theoretical side. Further research should deal with the theoretical aspects in moredetails, and also with the practical aspects – mainly to study a techniques for computation of FTV aggregation operators forsome specific kinds of fuzzy truth values, e.g. triangular, trapezoidal, gaussian shapes. This is one step to implementation oftype-2 fuzzy logic systems according to [33].

Acknowledgements

The author would like to thank the anonymous reviewers who helped to improve the quality of the paper. This work wassupported in part by a Grant VEGA 1/0419/13.

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