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Aggregation Operators Based onIndistinguishability OperatorsJ. Jacas,1, J. Recasens2,*1Sec Matemtiques i Informtica, ETS Arquitecura de Barcelona,Universitat Politcnica de Catalunya, Diagonal 649, 08028 Barcelona, Spain2Secci Matemtiques i Informtica, ETS Arquitectura del Valls,Universitat Politcnica de Catalunya, Pere Serra 1-15, 08190 Sant Cugatdel Valls, 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! t1 ~@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: j.recasens@upc.edu.e-mail: joan.jacas@upc.edu.

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 21, 857873 ~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 !ti1 tc 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~ai1,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 !!

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In particular,

T ~

n timesAssDssG

x, 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 xiMin~x1, . . . , xn !. Then

T ~x1T~1/n!

, . . . , xnT~1/n! ! T ~

n timesAsssssDsssssGxiT~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 toirrational exponents in a straightforward way.

Definition 2. If r R is a positive real number, let $an %nN 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 %nN .

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!.

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For a rational number m/n,

xT~m/n! ~xT

~1/n! !T~m! t @1# ~mt~xT

~1/n! !!

t @1#mtt @1# t~x!n

t @1#mn

t~x! Let E~T ! $x @0,1# 6 xT

~2! x% be the set of idempotent elements of T andNIL~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 0t1~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 !iI ~I finite or countably infinite) and~#ai , bi @!iI a family of nonempty pairwise disjoint open subintervals of @0,1#such that

T ~x, y! ai ~bi ai !Tix ai

bi ai,

y aibi ai

if ~x, y! @ai , bi # 2Min~x, y! otherwise

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

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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 # 2Min~x, y! otherwisewhere here ti

@@1##are defined by

ti@@1## ~x!

bi if x 0ti1 ~x! if x @0, t~ai !#

ai otherwise

Proposition 3.1 Let T ~^ @ai , bi # , Ti &!iI 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! hi1 ~hi ~x! hi ~ y!! if ~x, y! @ai , bi # 2Min~x, y! otherwisefor each i I hi : @ai , bi #r @0,`# is given by

hi ~x! ti x aibi aiDefinition 4. The residuation

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/01.~3! If T is the Minimum t-norm, then

ET ~x, y! Min~x, y! if x y1 otherwiseIt is important to note that even if the t-norm T is continuous, ET need not be

so. 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 !iI , 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 @ !iI 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 yti@@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 &!iI 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

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ET ~x, y! 1 if x yhi1 ~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 aibi aiFinally, let us recall in this preliminary section the definition of aggregation

operator.

Definition 7.7 An aggregation operator is a map h :nN @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 VARIABLESTo 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 lMxy.

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

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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 lMxy.

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! t1 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 mtmt ' .

Proof. If t ' at, then

mt ' ~x, y! t '1 t '~x! t '~ y!2 t1at~x! at~ y!2a t1 t~x! t~ y!2 mt ~x, y!

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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 mtmt ' .

Proof. If t ' t a, then

mt ' ~x, y! t '1 t '~x! t '~ y!2 t1 t~x! a t~ y! a2 a t1 t~x! t~ y!2 mt ~x, y!

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

Proof.

mt ~x, y! ~t !1 ~t !~x! ~t !~ y!2 t1~t~x! t~ y!!2 t1 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...