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Aggregation Operators Based onIndistinguishability OperatorsJ. Jacas,1,† J. Recasens2,*1Sec Matemàtiques i Informàtica, ETS Arquitecura de Barcelona,Universitat Politècnica de Catalunya, Diagonal 649, 08028 Barcelona, Spain2Secció Matemàtiques i Informàtica, ETS Arquitectura del Vallès,Universitat Politècnica de Catalunya, Pere Serra 1-15, 08190 Sant Cugatdel Vallès, Spain

This article gives a new approach to aggregating assuming that there is an indistinguishabilityoperator or similarity defined on the universe of discourse. The very simple idea is that when wewant to aggregate two values a and b we are looking for a value l that is as similar to a as to b or,in a more logical language, the degrees of equivalence of l with a and b must coincide. Inter-esting aggregation operators on the unit interval are obtained from natural indistinguishabilityoperators associated to t-norms that are ordinal sums. © 2006 Wiley Periodicals, Inc.

1. INTRODUCTION

When we aggregate two values a and b we may want to get a number l that isas similar to a as to b or, in other words, l should be equivalent to both values. Soif we have defined some kind of similarity E on our universe, the aggregation l ofa and b should satisfy E~a,l!� E~b,l!. We will develop this idea when the uni-verse is the unit interval @0,1# and the similarity is the natural T-indistinguishabilityoperator ET associated to a ~continuous! t-norm T. In particular we will show thatfor an Archimedean t-norm T with additive generator t the aggregation operatorassociated with ET is the quasi-arithmetic mean m generated by t ~m~x, y! �t�1 ~@t~x! � t~ y!#/2!!. This can give a justification for using a concrete quasi-arithmetic mean in a real problem, because it will be related to a logical systemhaving T as conjunction ~and ET as bi-implication!. If T is an ordinal sum, inter-esting aggregation operators are obtained because the way they aggregate two val-ues varies locally: For points in a piece @ai , bi #

2 where we have a copy of anArchimedean t-norm with additive generator ti , their aggregation is related to the

*Author to whom all correspondence should be addressed: e-mail: [email protected].†e-mail: [email protected].

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 21, 857–873 ~2006!© 2006 Wiley Periodicals, Inc. Published online in Wiley InterScience~www.interscience.wiley.com!. • DOI 10.1002/int.20165

quasi-arithmetic mean generated by ti . Points outside these pieces with the smallestcoordinate c in some @ai , bi # have

ai � ~bi � ai !ti�1 � t� c � ai

bi � ai�

2�

as aggregation whereas the aggregation of the rest of the points is their smallestcoordinate.

This idea can be easily generalized to weighted aggregations and aggrega-tions of more than two objects. Roughly speaking, if the weights are p and q �1 � p, then we can replace E~a,l! and E~b,l! by T p~E~a,l!! and T q~E~b,l!!,respectively. If we want to aggregate n numbers a1 � a2 � {{{� an , we can findl satisfying T ~E~a1,l!, . . . , E~ai ,l!!� T ~E~ai�1,l!, . . . , E~an ,l!! for some i . Inthis case we can have some problems if the t-norm has nilpotent elements. Then itwould be possible that both sides of the previous equality were 0. This problem issolved in this article by softening the indistinguishability operator E, replacing itby E 1/n .

2. PRELIMINARIES

This section contains some results on t-norms and indistinguishability opera-tors that will be needed later on in the article. Besides well-known definitions andtheorems, the power T n of a t-norm is generalized to irrational exponents in Def-inition 2 and given explicitly for continuous Archimedean t-norms in Propo-sition 1. Theorem 4 and Corollary 1 give new representations of the naturalT-indistinguishability operator ET associated to a t-norm T.

Though many results remain valid for arbitrary t-norms and especially forleft continuous ones, for the sake of simplicity we will assume continuity for thet-norms throughout the article.

Definition 1. A continuous t-norm is a continuous map T : @0,1#� @0,1#r @0,1#satisfying for all x, y, z, x ', y ' � @0,1#

(1) T ~x, T ~ y, z!!� T ~T ~x, y!, z! (Associativity)(2) T ~x, y!� T ~ y, x! (Commutativity)(3) If x � x ' and y � y ', then T ~x, y!� T ~x ', y ' ! (Monotonicity)(4) T ~1, x!� x

Because a t-norm T is associative, we can extend it to an n-ary operation in thestandard way:

T ~x! � x

T ~x1, x2, . . . , xn ! � T ~x1, T ~x2, . . . , xn !!

858 JACAS AND RECASENS

International Journal of Intelligent Systems DOI 10.1002/int

In particular,

T ~

n times

AssDssGx, x, . . . , x !

will be denoted by xT~n! .

If T is continuous, the nth root xT~1/n! of x wrt T is defined by

xT~1/n! � sup$z � @0,1# 6 zT

~n!� x%

and for m, n � N, xT~m/n!� ~xT

~1/n! !T~m! .

Lemma 1.1 If k, m, n � N, k, n � 0, then xT~km/kn! � xT

~m/n! .

Lemma 2. Let x1, . . . , xn � ~0,1# and n � N. T ~x1T

~1/n! , . . . , xnT

~1/n! !� 0.

Proof. Let xi � Min~x1, . . . , xn !. Then

T ~x1T

~1/n! , . . . , xnT

~1/n! ! � T ~

n times

AsssssDsssssGxiT~1/n! , . . . , xiT

~1/n!!� ~xiT

~1/n! !T~n!� xiT

~n/n!� xi � 0 �

Assuming continuity for the t-norm T, the powers xT~m/n! can be extended to

irrational exponents in a straightforward way.

Definition 2. If r � R� is a positive real number, let $an %n�N be a sequence ofrational numbers with limnr` an � r. For any x � @0,1# , the power xT

~r! is

xT~r! � limnr` xT

~an !

Continuity assures the existence of last limit and independence of the sequence$an %n�N .

Proposition 1. Let T be an Archimedean t-norm with additive generator t, x �@0,1# , and r � R�. Then

xT~r! � t @�1# ~rt~x!!

Proof. Due to continuity of t we need to prove it only for rational r.If r is a natural number m, then trivially xT

~m!� t @�1#~mt~x!!.If r � 1/n with n � N, then xT

~1/n! � z with zT~n! � x or t @�1#~nt~z!! � x and

xT~1/n!� t @�1#~t~x!/n!.

AGGREGATION OPERATORS 859


For a rational number m/n,

xT~m/n! � ~xT

~1/n! !T~m!� t @�1# ~mt~xT

~1/n! !!

� t @�1#�mt�t @�1#� t~x!

n�� t @�1#�m

nt~x!� �

Let E~T !� $x � @0,1# 6 xT~2!� x% be the set of idempotent elements of T and

NIL~T !� $x � @0,1# 6 xT~n!� 0 for some n � N % the set of nilpotent elements of T.

Definition 3. A continuous t-norm T is Archimedean if and only if E~T !� $0,1%.T is called strict when NIL~T ! � @0,1!. Otherwise it is called nonstrict andNIL~T !� $0%.

Theorem 1.2 A continuous t-norm T is Archimedean if and only if there exists acontinuous decreasing map t : @0,1#r @0,`# with t~1!� 0 such that

T ~x, y! � t @�1# ~t~x!� t~ y!!

where t @�1# stands for the pseudo-inverse of t defined by

t @�1# ~x! � �1 if x � 0

t�1~x! if x � @0, t~0!#

0 otherwise

T is strict if t~0!�` and nonstrict otherwise.

t is called an additive generator of T, and two additive generators of the samet-norm differ only by a multiplicative constant.

The next theorem states that all continuous t-norms can be built from Archi-medean ones.

Theorem 2.1,3 T is a continuous t-norm if and only of there exists a familyof continuous Archimedean t-norms ~Ti !i�I ~I finite or countably infinite) and~#ai , bi @!i�I a family of nonempty pairwise disjoint open subintervals of @0,1#such that

T ~x, y! � �ai � ~bi � ai !Ti� x � ai

bi � ai

,y � ai

bi � ai� if ~x, y! � @ai , bi #

2

Min~x, y! otherwise

T is called the ordinal sum of the summands ^@ai , bi # , Ti &, i � I and will bedenoted T � ~^@ai , bi # , Ti &!i�I .



Proposition 2.4 Let T be a continuous t-norm. There exists a family of strictlydecreasing and continuous maps ti : @ai , bi # r @0,`# with ti ~bi ! � 0, i � I suchthat

T ~x, y! � �ti@@�1## ~ti ~x!� ti ~ y!! if ~x, y! � @ai , bi #

2

Min~x, y! otherwise

where here ti@@�1## are defined by

ti@@�1## ~x! � �

bi if x � 0

ti�1 ~x! if x � @0, t~ai !#

ai otherwise

Proposition 3.1 Let T � ~^ @ai , bi # , Ti &!i�I be the ordinal sum of the summands^@ai , bi # , Ti &, i � I and let ti be an additive generator of Ti for all i � I. Then

T ~x, y! � �hi�1 ~hi ~x!� hi ~ y!! if ~x, y! � @ai , bi #

2

Min~x, y! otherwise

for each i � I hi : @ai , bi #r @0,`# is given by

hi ~x! � ti� x � ai

bi � ai�

Definition 4. The residuation <T of a t-norm T is defined by

<T ~x 6y! � sup$a � @0,1# 6 T ~x,a!� y!

In Ref. 4 we can find the following theorem characterizing the residuations ofcontinuous t-norms.

Theorem 3. <T : @0,1#� @0,1#r @0,1# is the residuation of a continuous t-norm Tif and only if there exists a family ~#ai , bi @!i�I of pairwise disjoint open subinter-vals of @0,1# and a family of strictly decreasing and continuous maps ti : @ai , bi #r@0, ti ~ai !# with ti ~bi !� 0 such that

<T ~x 6y! � �1 if x � y

ti@@�1## ~ti ~ y!� ti ~x!! if ~x, y! � @ai , bi @

2

y otherwise

Definition 5. The natural T-indistinguishability ET associated to a given t-normT is the fuzzy relation on @0,1# defined by

ET ~x, y! � T ~ <T ~x 6y!, <T ~ y 6x!!



ET is indeed a special kind of ~one-dimensional! T-indistinguishability oper-ator ~Definition 6!,5 and in a logical context where T plays the role of the conjunc-tion, ET is interpreted as the bi-implication associated to T.6

For readers interested in a deeper study of indistinguishability operators, werecall its definition below and refer them to Ref. 5.

Definition 6. Given a t-norm T, a T-indistinguishability operator E on a set Xis a fuzzy relation E : X � Xr @0,1# satisfying for all x, y, z � X

(1) E~x, x!� 1 (Reflexivity)(2) E~x, y!� E~ y, x! (Symmetry)(3) T ~E~x, y!, E~ y, z!!� E~x, z! ~T-transitivity)

Example 1.

~1! If T is the Lukasiewicz t-norm, then ET ~x, y!� 1 � 6x � y 6 for all x, y � @0,1# .~2! If T is the Product t-norm, then ET ~x, y! � Min~x/y, y/x! for all x, y � @0,1# where

z/0 � 1.~3! If T is the Minimum t-norm, then

ET ~x, y! � �Min~x, y! if x � y

1 otherwise

It is important to note that even if the t-norm T is continuous, ET need not beso. It is easy to see, for instance, that for the Minimum t-norm, ET is not continu-ous in the diagonal D of @0,1# ~D� $~x, x! 6 x � @0,1#%! and for a strict Archime-dean t-norm T ET is not continuous in ~0,0!.

More generally, if T is an ordinal sum of ~Ti !i�I , then the points of disconti-nuity of ET lie on the diagonal of @0,1# as follows directly from the next theorem,Theorem 4, which is a straightforward modification of Theorem 3 and character-izes natural T-indistinguishability operators for continuous t-norms T.

Theorem 4. ET is the natural T-indistinguishability operator associated to acontinuous t-norm T if and only if there exists a family ~ #ai , bi @ !i�I of pairwisedisjoint open subintervals of @0,1# and a family of strictly decreasing and contin-uous maps ti : @ai , bi #r @0, ti ~ai !# with ti ~bi !� 0 such that

ET ~x, y! � �1 if x � y

ti@@�1## ~ti ~Min~x, y!!� ti ~Max~x, y!!! if ~x, y! � @ai , bi @

2 x � y

Min~x, y! otherwise

Proof. Theorem 3 and Definition 5. �

Corollary 1. Let T � ~^ @ai , bi # , Ti &!i�I be the ordinal sum of the summands^@ai , bi # , Ti &, i � I and let ti be an additive generator of Ti for all i � I. The naturalT-indistinguishability operator ET associated to T is given by



ET ~x, y! � �1 if x � y

hi�1 ~hi ~Min~x, y!!� hi ~Max~x, y!!! if ~x, y! � @ai , bi @

2

x � y

Min~x, y! otherwise

where for each i � I hi : @ai , bi #r @0,`# is given by

hi ~x! � ti� x � ai

bi � ai�

Finally, let us recall in this preliminary section the definition of aggregationoperator.

Definition 7.7 An aggregation operator is a map h : �n�N @0,1# n r @0,1#satisfying

(1) h~0, . . . ,0!� 0 and h~1, . . . ,1!� 1(2) h~x!� x ∀x � @0,1#(3) h~x1, . . . , xn !� h~ y1, . . . , yn ! if x1 � y1, . . . , xn � yn ~monotonicity!

3. AGGREGATING TWO VARIABLES

To make it clearer, let us first consider the aggregation of two variables.When two numbers x and y need to be aggregated, it is expected to obtain a

value that in some sense is equivalent to both of them. More precisely, we may askfor a value l with the same degree of similarity to both x and y or in a more logicallanguage, we may want to obtain l with the same degree of equivalence to x as toy. Hence the following definition seems reasonable, though it has a small short-coming that will be overcome with Definition 9.

Definition 8 ~First attempt!. The aggregation of x, y � @0,1# , ~x � y! with respectto the natural T-indistinguishability operator ET is the number l, ~x � l � y!such that

ET ~x,l! � ET ~ y,l!

Example 2.

~1! If T is the Lukasiewicz t-norm, then the aggregation l of x and y with respect to thenatural indistinguishability operator ET is their arithmetic mean l� ~x � y!/2.

~2! If T is the Product t-norm, then the aggregation l of x and y ~x{y � 0! with respect tothe natural indistinguishability operator ET is their geometric mean l�Mxy.

Nevertheless, Definition 8 has the problem that if the natural indistinguish-ability operator is not continuous, the aggregation l may not exist, for example, ifT is the minimum t-norm and x � y, or for strict Archimedean t-norms when exactly



one of the two values is 0. To overcome this, it should be modified in the followingway.

Definition 9. The aggregation of x, y � @0,1# , ~x � y! with respect to the natu-ral T-indistinguishability operator ET is the number l, ~x � l� y! such that

limzrl ET ~a, z! � limzrl ET ~b, z!

It is worth noticing that if l of Definition 8 exists, then it coincides with theaggregation of Definition 9, and hence this last definition is more general thanDefinition 8.

Example 3.

~1! If T is the Minimum t-norm, then the aggregation l of x and y with respect to thenatural indistinguishability operator ET is the Minimum of x and y.

~2! If T is the Lukasiewicz t-norm, then the aggregation l of x and y with respect to thenatural indistinguishability operator ET is their arithmetic mean l� ~x � y!/2.

~3! If T is the Product t-norm, then the aggregation l of x and y with respect to the naturalindistinguishability operator ET is their geometric mean l�Mxy.

These last two examples belong to a more general framework in the sensethat the aggregation with respect to a natural indistinguishability operator associ-ated to a continuous Archimedean t-norm is a quasi-arithmetic mean.

The next proposition is well known.

Proposition 4.1,8 m is a quasi-arithmetic mean in @0,1# if and only if there existsa continuous monotonic map t : @0,1#r @�`,`# such that for all x, y � @0,1#

m~x, y! � t�1� t~x!� t~ y!

2�

m is continuous if and only if Ran t � @�`,`# .

Lemma 3. Let t, t ' : @0,1# r @�`,`# be two continuous strict monotonic mapswith Ran t, Ran t '� @�`,`# differing only by a nonzero multiplicative constant a~t ' � at) and mt , mt ' the quasi-arithmetic means generated by them respectively.Then mt � mt ' .

Proof. If t '� at, then

mt ' ~x, y! � t '�1� t '~x!� t '~ y!

2�� t�1�at~x!� at~ y!

2a�

� t�1� t~x!� t~ y!

2�� mt ~x, y! �



Lemma 4. Let t, t ' : @0,1# r @�`,`# be two continuous strict monotonic mapswith Ran t, Ran t '� @�`,`# differing only by an additive constant and mt , mt ' thequasi-arithmetic means generated by them respectively. Then mt � mt ' .

Proof. If t '� t � a, then

mt ' ~x, y! � t '�1� t '~x!� t '~ y!

2�� t�1� t~x!� a � t~ y!� a

2� a�

� t�1� t~x!� t~ y!

2�� mt ~x, y! �

Lemma 5. Let t : @0,1# r @�`,`# be a continuous strict monotonic map. Thenmt � m�t .

Proof.

m�t ~x, y! � ~�t !�1� ~�t !~x!� ~�t !~ y!

2�� t�1��~�t~x!� t~ y!!

2�

� t�1� t~x!� t~ y!

2�� mt ~x, y! �

Proposition 5. The map assigning to every continuous Archimedean t-norm Twith generator t the mean mt generated by t is a bijection between the set of con-tinuous Archimedean t-norms and the set of continuous quasi-arithmetic means.

Proof. It is a consequence of Lemmas 3, 4, and 5. �

Proposition 6. Let T be a continuous Archimedean t-norm with additive gener-ator t and ET the natural T-indistinguishability operator. The aggregation l of x, ywith respect to ET is the quasi-arithmetic mean mt ~x, y! .

Proof. We may assume that x � y. In this case, since x � l � y, ET ~x,l! �<T ~l 6x! and ET ~ y,l!� <T ~ y 6l!. Therefore

<T ~l 6x! � <T ~ y 6l!

t�1~t~x!� t~l!! � t�1~t~l!� t~ y!!

t~x!� t~l! � t~l!� t~ y!

2 t~l! � t~x!� t~ y!

and

l � t�1� t~x!� t~ y!

2�



If x � 0, y � 0 and T is strict Archimedean, then limzr0 ET ~z, x! �limzr0 ET ~z, y!. �

If we use natural indistinguishability operators associated to a t-norm that isan ordinal sum, we obtain interesting new aggregation operators.

Example 4. Let T be the ordinal sum with a single copy of Lukasiewicz t-normin @0, 1

2_ # 2 . Then

ET ~x, y! � �1 if x � y

1

2� 6x � y 6 if ~x, y! � �0,

1

2�x � y

Min~x, y! otherwise

and the aggregation operator is

l � �x � y

2if ~x, y! � �0,

1

2� 2

Min~x, y! if ~x, y! � � 1

2, 1�2

Min~x, y!

2�

1

4otherwise

~See Figure 1.!More generally, we have Example 5.

Example 5. If T is the ordinal sum with a single copy of an Archimedean t-normT with additive generator t in @0, 1

2_ # 2 . Then the aggregation operator is

l � �1

2t�1� t~2x!� t~2y!

2� if ~x, y! � �0,

1

2� 2

Min~x, y! if ~x, y! � � 1

2, 1�2

1

2t�1� t~2 Min~x, y!!

2� otherwise

In fact, the general result for aggregating with respect to the naturalT-indistinguishability operator ET associated to a continuous t-norm is next theorem.



Theorem 5. Let T � ~^ @ai , bi # , Ti &!i�I be an ordinal sum, ~ti !i�I a family of addi-tive generators of ~Ti !i�I , and M � @0,1#� �i�I #ai , bi @. The aggregation opera-tor associated to the natural T-indistinguishability operator ET is

l � �Min~x, y! if Min~x, y! � M

ai � ~bi � ai ! ti�1 � ti� x � ai

bi � ai�� ti� y � ai

bi � ai�

2� if ~x, y! � @ai , bi #

2

ai � ~bi � ai ! ti�1 � ti�Min~x, y!� ai

bi � ai�

2� if Min~x, y! � @ai , bi # and

Max~x, y! � @ai , bi #

Proof. If x, y � @ai , bi # , then it can be proved in the same way as in Proposi-tion 6 that l� h�1~~h~x!� h~ y!!/2!where h~x!� t~~x � ai !/~bi � ai !!. Therefore

l � ai � ~bi � ai ! ti�1 � ti� x � ai

bi � ai�� ti� y � ai

bi � ai�

2�

Figure 1. l in Example 4.



If Min~x, y! � @ai , bi # and Max~x, y! � @ai , bi # , then ET ~Min~x, y!,l! �ET ~Max~x, y!,l! is h�1~h~Min~x, y! � h~l!! � l, and therefore

l � h�1�h~Min~x, y!!

2�� ai � ~bi � ai ! ti

�1 � ti�Min~x, y!� ai

bi � ai�

2� �

Example 6. Figure 2 shows the aggregation operator obtained from the ordinalsum T � ~^ @0.2, 0.4# , T1&^ @0.6, 0.9# , T2 &! with T1 the product t-norm and T2 thet-norm with additive generator t~x!� ~1/x!� 1 ~t generates the harmonic mean!.

Theorem 5 gives a new way to aggregate two values that may vary locally.Indeed, for the points in a piece @ai , bi #

2 their aggregation is related to the quasi-arithmetic mean generated by ti . Points outside these pieces ~points in M ! with thesmallest coordinate in M have this coordinate as aggregation, and points in M withthe smallest coordinate c in some @ai , bi # have

ai � ~bi � ai !ti�1 � ti� c � ai

bi � ai�

2�

as aggregation.

Figure 2. l in Example 6.



4. AGGREGATING USING INDISTINGUISHABILITY OPERATORS

The previous ideas can be generalized in order to aggregate more than twovariables or using weights. If we want to aggregate x1 � x2 � {{{ � xn , we canlook for a l having the same degree of equivalence with all the values on its left-and on its right-hand side. So a first attempt would be the number l such that thereexists i � $1,2, . . . , n% with

T ~ET ~x1,l!, ET ~x2,l!, . . . , ET ~xi ,l!! � T ~ET ~xi�1,l!, . . . , ET ~xn ,l!!

or, better,

limzrl T ~ET ~x1, z!, ET ~x2, z!, . . . , ET ~xi , z!!

� limzrl T ~ET ~xi�1, z!, . . . , ET ~xn , z!!

The problem could be if the t-norm had nilpotent elements. In this case itcould be likely that the sides of the previous equality were 0.

For example, let us try to aggregate the values 0.1, 0.15, 0.2, 0.8, 0.85, and0.9 using the natural indistinguishability operator ET associated to the Lukasiewiczt-norm. l should be between 0.2 and 0.8 and the solutions of the following equation,

T ~E~0.1,l!, E~0.15,l!, E~0.2,l!! � T ~E~0.8,l!, E~0.85,l!, E~0.9,l!!

range between 0.483 and 0.517. Fortunately, this interval of solutions can bereduced to a single point in the following way.

Proposition 7. Let E be a T indistinguishability operator on a set X. The fuzzyrelation E n defined by

E n~x, y! � T ~

n times

AssssssDssssssGE~x, y!, . . . E~x, y! ! ∀x, y � X

is a T-indistinguishability operator.

Corollary 2.9 Let ET be the natural T-indistinguishability operator on @0,1#associated to T. ET

n is a T-indistinguishability operator.

The powers ETn of the natural T-indistinguishability operators have been stud-

ied in relation to antonymy and fuzzy partitions in Ref. 9.

Proposition 8. Let E be a T-indistinguishability operator on a set X. E 1/n is aT-indistinguishability operator on X.

Proof. Reflexivity and symmetry are trivial.



Transitivity: If E 1/n � F, then F n � E. Because E is a T-indistinguishabilityoperator, ∀x, y, z � X

F n~x, z! � T ~F n~x, y!, F n~ y, z!!� ~T ~F~x, y!, F~ y, z!!!T~n!

~F n~x, z!!T1/n � ~~T ~F~x, y!, F~ y, z!!!T

~n! !T1/n

and from Lemma 1

F~x, z! � T ~F~x, y!, F~ y, z!! �

Corollary 3. Let ET be the natural T-indistinguishability operator on @0,1#associated with T. ET

1/n is a T-indistinguishability operator.

Corollary 4. Let E be a T-indistinguishability operator on a set X. E m/n is aT-indistinguishability operator on X.

Proof. Propositions 7 and 8. �

Corollary 5. Let ET be the natural T-indistinguishability operator on @0,1#associated with T. ET

m/n is a T-indistinguishability operator.

Example 7.

~1! If T is the Lukasiewicz t-norm, then ETm/n ~x, y! � Max~0,1 � ~m/n!6x � y 6! for all

x, y � @0,1# .~2! If T is the Product t-norm, then ET ~x, y!m/n � ~Min~x/y, y/x!!m/n for all x, y � @0,1#

where z/0 � 1.~3! If T is the Minimum t-norm, then ET

m/n ~x, y!� ET ~x, y!.

Continuity of the t-norm T allows us to extend the powers of aT-indistinguishability operator to irrational numbers in the same way as in Defini-tion 2.

With the previous results we can relax or strengthen the equivalence rela-tions. Indeed, ET

p� ET

q iff p � q.

Lemma 6. Let a, b � ~0,1# , n � N, and T be an Archimedean t-norm. If aT~n! �

bT~n! , then a � b.

Proposition 9. Let ET be the natural T-indistinguishability operator with respectto an Archimedean t-norm T. If l is a solution of

T ~ETn ~x1,l!, ET

n ~x2,l!, . . . , ETn ~xi ,l!! � T ~ET

n ~xi�1,l!, . . . , ETn ~xn ,l!!



and both sides of the equality are different from 0, then l is a solution of

T ~ET ~x1,l!, ET ~x2,l!, . . . , ET ~xi ,l!! � T ~ET ~xi�1,l!, . . . , ET ~xn ,l!!

as well.

Proof.

T ~ETn ~x1,l!, ET

n ~x2,l!, . . . , ETn ~xi ,l!!� ~T ~E~x1,l!, ET ~x2,l!, . . . , ET ~xi ,l!!!T

~n!

and

T ~ETn ~xi�1,l!, . . . , ET

n ~xn ,l!! � ~T ~ET ~xi�1,l!, . . . , ET ~xn ,l!!!T~n!

Then the result follows from Lemma 6. �

Lemma 7. T ~ET1/n ~x1,l!, ET

1/n ~x2,l!, . . . , ET1/n ~xn ,l!! � 0 for all x1, x2, . . . ,

xn ,l � ~0,1# .

Proof. Noting that if a{b � 0, then ET ~a, b!� 0, the result follows directly fromLemma 2. �

Thanks to this lemma we can extend the aggregation with respect to ET tomore than two variables.

Definition 10. Let x1 � x2 � {{{� xn � @0,1# l ~x1 � l� xn ! be the aggrega-tion of x1, x2, . . . , xn with respect to ET if and only if

limzrl T ~ET1/n ~x1, z!, . . . , ET

1/n ~xi , z!! � limzrl T ~ET1/n ~xi�1, z!, . . . , ET

1/n ~xn , z!!

Example 8. The aggregation of the values 0.1, 0.15, 0.2, 0.8, 0.85, and 0.9 of theexample at the beginning of this section is l� 0.5.

More generally, we have the following propositions.

Proposition 10. Let x1 � x2 � {{{ � xn � @0,1# and T be an Archimedeant-norm with additive generator t. The aggregation l of x1, x2, . . . , xn with respectto ET is its quasi-arithmetic mean mt generated by t:

l � mt ~x1, . . . , xn !� t�1� t~x1!� {{{� t~xn !

n�

Proposition 11. Let x1 � x2 � {{{ � xn � @0,1# . The aggregation l ofx1, x2, . . . , xn with respect to EMin is the minimum of x1, x2, . . . , xn

l � Min~x1, x2, . . . , xn !

In a similar way we can aggregate weighting the objects.



Definition 11. Let x1 � x2 � {{{ � xn � @0,1# and p1, p2, . . . , pn � @0,1# with(n

i�1 pi �1. l is the weighted aggregation of x1, x2, . . . , xn with respect to ET andp1, p2, . . . , pn if and only if

limzrl T ~ETp1 ~x1, z!, . . . , ET

pi ~xi , z!! � limzrl T ~ETpi�1 ~xi�1, z!, . . . , ET

pn ~xn , z!!

Example 9. Let T be a continuous Archimedean t-norm with additive generator tand p1, p2, . . . , pn � @0,1# with (n

i�1 pi � 1.The weighted aggregation l of x1, x2, . . . , xn � @0,1# with respect to ET and

p1, p2, . . . , pn is their weighted quasi-arithmetic mean generated by t:

l � t�1 �(i�1

n

pi t~xi !

2�

5. CONCLUDING REMARKS

In this article we have developed a new semantics for aggregating based onthe idea that when we want to aggregate some values we try to find a number assimilar to all of them as possible. We have studied the case of the unit intervalprovided with the natural indistinguishability operator ET associated to a t-normT. This may provide a justification to the use of a concrete aggregation operator,because it can be related to a specific logic having T as conjunction and ET asbi-implication.

If T is an Archimedean t-norm with additive generator t, we have obtained the~weighted! arithmetic mean mt generated by t ~mt ~x1, . . . , xn ! � t�1~ p1 t~x1! �{{{� pn t~xn !!!.

For ordinal sums the obtained aggregation operators vary locally.It is worth noticing that we would have obtained the same results ~i.e., the

same aggregation operators! defining the aggregation operator mT associated to at-norm T by mT ~x1, . . . , xn !� T ~x1T

1/n , . . . , xnT

1/n !. The relation between the seman-tics of this formula and our similarity-based approach will be studied by the authorsin a forthcoming paper.

Acknowledgments

Research partially supported by DGICYT project number TIC2003-04564.

References

1. Klement EP, Mesiar R, Pap E. Triangular norms. Dordrecht: Kluwer; 2000.2. Ling CM. Representation of associative functions. Publ Math Debrecen 1965;12:189–212.3. Schweizer B, Sklar A. Probabilistic metric spaces. Amsterdam: North-Holland; 1983.4. Boixader D. Some properties concerning the quasi-inverse of a t-norm. Mathware Soft Com-

put 1998;5:5–12.



5. Boixader D, Jacas J, Recasens J. Fuzzy equivalence relations: Advanced material. In: DuboisD, Prade H, editors. Fundamentals of fuzzy sets. Dordrecht: Kluwer; 2000. pp 261–290.

6. Hájek P. Metamathematics of fuzzy logic. Dordrecht: Kluwer; 1998.7. Calvo T, Kolesárova A, Komorníková M, Mesiar R. Aggregation operators: Properties,

classes and construction methods. In: Calvo T, Mayor G, Mesiar R, editors. Aggregationoperators: New trends and applications. Studies in fuzziness and soft computing. Berlin:Springer; 2002. pp 3–104.

8. Aczél J. Lectures on functional equations and their applications. New York/London: Aca-demic Press; 1966.

9. De Soto AR, Recasens J. Modeling a linguistic variable as a hierarchical family of parti-tions induced by an indistinguishability operator. Fuzzy Set Syst 2000;121:427– 437.

