Aggregation Operators and Commuting

1032 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 15, NO. 6, DECEMBER 2007

Aggregation Operators and CommutingSusanne Saminger-Platz, Radko Mesiar, and Didier Dubois

Abstract—Commuting is an important property in any two-stepinformation merging procedure where the results should notdepend on the order in which the single steps are performed. Weinvestigate the property of commuting for aggregation operatorsin connection with their relationship to bisymmetry. In case ofbisymmetric aggregation operators we show a sufficient conditionensuring that two operators commute, while for bisymmetricaggregation operators with neutral element we even provide afull characterization of commuting -ary operators by means ofunary distributive functions. The case of associative operations,especially uninorms, is considered in detail.

Index Terms—Aggregation operators, bisymmetry, commutingoperators, consensus.

I. INTRODUCTION

I N VARIOUS applications where information fusion or mul-tifactorial evaluation is needed, an aggregation process is

carried out as a two-stepped procedure whereby several localfusion operations are performed in parallel and then the resultsare merged into a global result. It may happen that in practicethe two steps can be exchanged because there is no reason toperform either of the steps first. For instance, in a multi-personmulti-aspect decision problem, each alternative is evaluated bya matrix of ratings where the rows represent evaluations bypersons and the columns represent evaluations by criteria. Onemay, for each row, merge the ratings according to each columnwith some aggregation operation and form as such the globalrating of each person, and then merge the persons’ opinionsusing another aggregation operation . On the other hand, onemay decide first to merge the ratings in each column using theaggregation operation , thus forming the global ratings ac-cording to each criterion, and then merge these social evalu-ations across the criteria with aggregation operation . Theproblem is that it is not guaranteed that the results of the twoprocedures will be the same, while one would expect them to beso in any sensible approach. When the two procedures yield thesame results operations and are said to commute.

This paper is devoted to a mathematical investigation of com-muting aggregation operators which are used, e.g., in utility

Manuscript received May 4, 2006; revised July 15, 2006. Parts of this workwere supported by Grants VEGA 1/3006/06, VEGA 1/2032/05, and APVT20-003204. Their support is gratefully acknowledged.

S. Saminger-Platz is with the Department of Knowledge-Based MathematicalSystems, Johannes Kepler University, A-4040 Linz, Austria (e-mail: [email protected]).

R. Mesiar is with the Department of Mathematics and Descriptive Geometry,the Faculty of Civil Engineering at Slovak University of Technology, SK-81368 Bratislava, and also with the Institute of the Theory of Information and Au-tomation of the Czech Academy of Sciences, Prague, Czech Republic (e-mail:[email protected]).

D. Dubois is with the I.R.I.T-CNRS, Université de Toulouse, 31062 ToulouseCedex 4, France (e-mail: [email protected]).

Digital Object Identifier 10.1109/TFUZZ.2006.890687

theory [15], but also in extension theorems for functional equa-tions [33]. Very often, the commuting property is instrumentalin the preservation of some property during an aggregationprocess, like transitivity when aggregating preference matricesor fuzzy relations (see, e.g., [13] and [34]), or some form ofadditivity when aggregating set functions (see, e.g., [15]). Infact, early examples of commuting appear in probability theoryfor the merging of probability distributions. Suppose two jointprobability distributions are merged by combining degreesof probability point-wisely. It is natural that the marginals ofthe resulting joint probability function are the aggregates ofthe marginals of the original joint probabilities. To fulfill thisrequirement the aggregation operation must commute with theaddition operation involved in the derivation of the marginals.It enforces a weighted arithmetic mean as the only possible ag-gregation operation for probability functions [31]. This result isclosely related to the theory of probabilistic mixtures that playsa key-role in the axiomatic derivation of expected utility theory[22]. In [15], the same question is solved for more general setfunctions, where the addition is replaced by a t-conorm and theconsequences for generalized utility theory are pointed out.

In this paper, the problem of commuting operators is con-sidered with more generality. After a section presenting neces-sary definitions and background, Section III considers the caseof commuting unary operations, called distributive functions,that play a key role in the representation of commuting oper-ators. Section IV provides characterization results concerningbisymmetric operations, i.e., aggregation operations that com-mute with themselves. Sections V and VI focus on functions dis-tributive over continuous t-(co)norms and particular uninorms,respectively.

II. PRELIMINARIES

A. Aggregation Operators

Aggregation by itself is an important task in any disciplinewhere the fusion of information is of vital interest. It compre-hends the transformation of several items of input data into asingle output value which is characteristic for the input data it-self or some of its aspects. In case of aggregation operators itis assumed that a finite number of inputs from the same (nu-merical) scale, most often the unit interval, are being aggre-gated. Moreover, interpreting the inputs as evaluation resultsof objects according to some criterion, the monotonicity andboundary conditions of its formal definition look very natural:

Definition 1: A function is calledan aggregation operator if it fulfills the following proper-ties [10]:

(AO1) wheneverfor all ;

(AO2) for all ;(AO3) and .

1063-6706/$25.00 © 2007 IEEE

SAMINGER-PLATZ et al.: AGGREGATION OPERATORS AND COMMUTING 1033

Each aggregation operator can be represented bya family of -ary operations, i.e., functions

given by

In that case, and, for , each is nonde-creasing and satisfies and

. Usually, the aggregation operator and the correspondingfamily of -ary operations are identified with eachother. Note that, -ary operations

, which fulfill properties (AO1) and (AO3) are referred to as-ary aggregation operators.Depending on the requirements applied to the aggregation

process several properties for aggregation operators have beenintroduced. We only mention those few which are relevant forour further investigations. For more elaborated details on aggre-gation operators we refer to, e.g., [10].

Definition 2: Consider some aggregation operator.

i) is called symmetric if for all and for all

for all permutations of .ii) is called bisymmetric if for all and all

with and

iii) is called associative if for all and alland all with and

iv) An element is called neutral element of if forall and for all it holdsthat if for some then

v) An element is called an idempotent element ofif for all . We will abbreviate

the set of idempotent elements by

In case that , the aggregation operator is calledidempotent.

Associative aggregation operators are completely charac-terized by their binary operators since all -ary, ,aggregation operators can be constructed by the recursiveapplication of the binary operator .

Depending on the additional properties, several subclassesof aggregation operators can and have been distinguished, like,e.g., symmetric and associative operators with some neutral el-ement : For , they are referred to as triangular norms

(t-norm for short), for , they are called t-conorms, forwe will refer to them as uninorms (see also [6], [17],

and [24]).Note that associative and symmetric aggregation operators

are also bisymmetric. On the other hand, bisymmetric aggre-gation operators with some neutral element are associative.Therefore, as just mentioned, the class of all associative andsymmetric and, therefore, bisymmetric, aggregation operatorswith neutral element consists of all t-norms, t-conorms anduninorms.

Note that not all aggregation processes are carried out oninput data from the unit interval, therefore, aggregation opera-tors on other intervals as well as methods for transforming inputdata are needed to model the required aggregation process. Ag-gregation operators can be defined as acting on any closed in-terval . We will then speak of an aggre-gation operator acting on . While (AO1) and (AO2) basicallyremain the same, only (AO3), expressing the preservation of theboundaries, has to be modified accordingly

(AO3’) and .

Such aggregation operators can also be achieved from standardaggregation operators by means of isomorphic transformations.By such transformations many of the before mentioned proper-ties are being preserved.

For an isomorphic transformation , i.e., amonotone bijection, the isomorphic transformation of anaggregation operator is given by

and is an aggregation operator on . If for two aggregationoperators on (possibly) different intervals, there exists amonotone bijection such that or we referto and as isomorphic aggregation operators.

By means of increasing bijections, we can introduce t-normsand t-conorms on arbitrary interval preserving the

boundary elements as the corresponding neutral elements. Wewill denote such t-norms, respectively, t-conorms as t-(co)normson the corresponding interval .

B. Commuting and Dominance

Definition 3: Consider two aggregation operators and .We say that dominates if for all andfor all , with and ,the following property holds:

(1)

Definition 4: Consider an -ary aggregation operatorand an -ary aggregation operator . Then, we say that

commutes with if for all withand , the following property holds:

(2)


Two aggregation operators and commute with each other ifcommutes with for all . We will also refer

to and as commuting aggregation operators.Observe that the property of commuting as expressed by (2) is

a special case of the so called generalized bisymmetry equationas introduced and discussed in [4] and [5] and plays a key rolein consistent aggregation.

It is an immediate consequence of the definition of com-muting that two aggregation operators commute if and only ifthey dominate each other; further, that any aggregation operatorcommuting with itself is bisymmetric and vice versa. Note thatin case of two associative aggregation operators, commutingbetween the binary operators is a necessary and sufficientcondition for their commuting in general.

Because of the preservation properties of dominance duringisomorphic transformations (see also [34]) we immediately canstate the following result:

Corollary 5: Let and be two aggregation operators.Then, the following are equivalent:

i) commutes with ;ii) commutes with for some isomorphic transforma-

tion ;iii) commutes with for all isomorphic transforma-

tions .Example 6: The projections to the first coordinate resp. to the

last coordinate, i.e.,

commute with arbitrary aggregation operator .

III. DISTRIBUTIVE FUNCTIONS

A. Basic Property

There is a close relationship between commuting aggrega-tion operators and unary functions being distributive over one ofthe two aggregation operators involved. On the one hand, suchfunctions can be constructed from commuting aggregation op-erators, on the other hand — as we will show in the next section— they can be used for constructing commuting operators. Notethat such distributive functions are in fact commuting with theinvolved aggregation operator.

Proposition 7: For any -ary aggregation operatorand any -ary aggregation operator , itholds that if commutes with , then the function

defined by

(3)

with and some idempotent element of ,is distributive over , i.e., it fulfills for alland all with

Moreover, is nondecreasing.

Proof: Consider some -ary aggregation operator ,one of its idempotent elements , e.g., 0 or 1, and some -aryaggregation operator such that commutes with

. Then, it holds for defined by(3) with arbitrary that

The nondecreasingness of follows immediately fromthe monotonicity of .

Analogously, we can define nondecreasing functionswhich are distributive over with some

idempotent element of .

B. Distributive Functions and Lattice Polynomials

We will denote by the set of all nondecreasing func-tions that are distributive over the -aryaggregation operator , i.e.,

is nondecreasing

Observe that is the identity function and thus con-tains all nondecreasing functions . For thereaders’ convenience we will abbreviate this set simply by ,i.e.,

is nondecreasing

Evidently, is the set of all functionsthat are distributive over the aggregation operator . Note that

as well as contain at least the following functions:

and are, therefore, not empty for arbitrary aggregation operator. The following proposition shows that is maximal in case

of lattice polynomials only, i.e., can be expressed by andits arguments only [8], compare also, e.g., [29] and [30].

Proposition 8: Consider an aggregation operator . Then thefollowing holds:

is a lattice polynomial

Proof: If all with are lattice polynomials, itfollows immediately from the nondecreasingness of alland the definition of that .


Before showing the sufficiency, note that any -variable lat-tice polynomial can be put in the followingdisjunctive normal form [8]:

(4)

where and is a nonde-creasing set function fulfilling and .Therefore, in order to show that some -ary aggregation oper-ator is a lattice polynomial, we have to show that a setfunction fulfilling the above conditions ex-ists and that can be written in the form of (4). For betterreadability, we will use in the sequel of this proof insteadof , as well as the additional notationswhere if and otherwise, and .Now assume that .

• First, we show that for all. In case

, depending on the value of , one of thefollowing functions

ifotherwise

ifotherwise

ifotherwise,

with and ,contradicts .Therefore, in particular is idempotent, i.e.,

and for all .• Since for all the functions

resp. fulfillwe can conclude the following for all

since , and.

• Due to the monotonicity of we can further conclude thatfor arbitrary

by replacing each either by 0, if , or by , if, for arbitrary choice of . Therefore, also

We abbreviate by such thatthe previous inequality can be written as

Since it is clear that the setis not empty. Moreover, the following

holds for its complement

so that necessarily .If we replace each in either by incase that or by 1 in case that , we can alsoconclude, due to the monotonicity of and the propertiesshown before, that

showing that

Finally, we define a set function by, then it is immediate to show that it is non-

decreasing and fulfills , and thatis indeed a lattice polynomial.

Let us now focus on additional properties of in case ofparticular properties of the aggregation operator involved.

C. Distributive Functions for Bisymmetric and AssociativeAggregation Operators

Proposition 9: Let be a bisymmetric aggregation oper-ator and fix some . If we choose some

, not necessarily different, then alsodefined by

(5)

belongs to , i.e., is closed under .Proof: Consider some bisymmetric aggregation operator

and fix some arbitrary , , for some. Define a function by (5) then the fol-

lowing holds for arbitrary due to the bisym-metry of and the distributivity of all over :

Corollary 10: If is a bisymmetric aggregation operatorand additionally fulfills for all and all ,

,

then defined by


also belongs to for arbitrary and arbitrary, i.e., is closed under any

.Moreover, in case of an associative aggregation operator

the relationship can be generalized, expressing that it is suffi-cient (and necessary) to characterize all functions distributiveover the binary aggregation operator only in order to char-acterize the set of all unary mappings distributive overwith arbitrary arity.

Proposition 11: Let be an associative aggregation oper-ator, then the following holds:

Proof: Consider an associative aggregation operator .If some nondecreasing function ful-fills , it is distributive over all -ary aggregationoperators , , in particular over the binary ag-gregation operator . On the other hand ifthe property follows directly from the associativity of ,i.e., the fact that for all with it holds that

.Note that the associativity of an aggregation operator is a suf-

ficient condition for . However, as the followingexample will demonstrate, it is not necessary.

Example 12: Consider the arithmetic mean, .

Then (compare also [2] and [3])

although clearly the arithmetic mean is not associative.Example 13: Examples of associative and symmetric and

therefore bisymmetric aggregation operators are -medians

with [18]. The set of distributive functions ischaracterized in the following way: Some nondecreasing func-tion is distributive over , i.e.,

if and only if either oror .

Besides associativity and bisymmetry, the possibility ofbuilding isomorphic aggregation operators leads to furtherinsight to relationships between sets of distributive functions.

D. Distributive Functions and Isomorphisms

Proposition 14: Consider an aggregation operator andsome bijection . Then for all it holdsthat where

is nondecreasing and

distributive over

Proof: Consider the isomorphic aggregation operatorsand with some bijection. Further assume

, then the following are equivalent since for all

, there exists a uniquewith

showing that .Example 15: Following Aczél [1], [3], the class of all contin-

uous, strictly monotone, bisymmetric, and idempotent aggrega-tion operators on the unit interval are just weighted quasi-arith-metic means

with some monotone nondecreasing bijec-tion and weights with for all and

. It is immediate that weighted quasi-arithmeticmeans are isomorphic transformations of weighted arithmeticmeans with corresponding weights. Due to Proposition 14,the set of distributive functions is, therefore, given by

and

since

and

such that

in case that for all and .Example 16: For invariant aggregation operators , i.e.,

aggregation operators fulfilling for all bijections, it immediately holds that all nondecreasing

bijections are included in (see also, e.g., [29], [30] forcharacterizations of aggregation operators invariant undernondecreasing bijections). This is, e.g., the case for the drasticproduct and the weakest aggregation operator beingdefined by

ifotherwise

ifotherwise

However, their set of distributive functions does not only containall nondecreasing bijections, but is even much richer, namely

and


Similarly, lattice polynomials are invariant aggregation opera-tors and we know already their sets of distributive functionsequal the set of all nondecreasing functions.

However, for arbitrary aggregation operators at least thefollowing relationship between a bijective distributive functionand its inverse can be stated.

Lemma 17: Consider an aggregation operator . Ifis bijective then also .

IV. OPERATORS COMMUTING WITH BISYMMETRIC

AGGREGATION OPERATORS

After discussing unary operators being distributive over someaggregation operator and as such commuting, let us now turn tomore general commuting operators.

Proposition 18: Let be a bisymmetric aggregation oper-ator. Then any -ary operator , on defined by

(6)

with for commutes with .Proof: Consider some bisymmetric aggregation oper-

ator , choose some and arbitrary. Then, the following holds for arbitrary

with and

Note that the involved operator need not be an aggregationoperator, e.g., choose for all ,then

for arbitrary , and therefore theboundary conditions (AO2) and (AO3) are not fulfilled.

Remark 19: Note that the previous proposition provides asufficient, but not a necessary condition for an operation tocommute with . As mentioned above, any aggregation oper-ator commutes with the projection to the first coordinatewhich is a bisymmetric aggregation operator. However, using

, only aggregation operatorsdepending just on the first coordinate can be obtained althoughwe have that , since is a lattice polynomial.

A. Commuting Aggregation Operators

Let us briefly focus on the restrictions which additionallyhave to be applied to the selected functions such thatthe constructed operator also fulfills the requirements of anaggregation operator. If , the correspondingmust be the identity function in order to guarantee .

For , the functions , must bechosen accordingly to such that

are both fulfilled at the same time. This is for sure guaranteed iffor all it holds that and , but it need notbe the case as the following example shows.

Example 20: The class of all aggregation operators com-muting with the minimum

with for all

for at least one

is also the class of all aggregation operators dominating the min-imum in the sense of Definition 3 (see also [34]).

B. The Role of Neutral Elements

Let us now consider for which bisymmetric aggregation op-erators , operators defined by (6) are the only commutingoperators, i.e., if (6) does provide a sufficient as well as a neces-sary condition. For better readability, we will briefly restrict our-selves to binary operators only. Since the projections commutewith any aggregation operator , they particularly commutealso with such operators for which (6) indeed is necessaryand sufficient. In this case, there necessarily exist

, such that for all

If there exists some such thatand it follows from the

monotonicity of that

Therefore, independently of , we have that

i.e., such an element is unique. A typical candidate fulfilling thelast property is a neutral element . In such a case, it suffices tochoose and for all .

Indeed, we obtain a necessary and sufficient condition if theinvolved aggregation operator is bisymmetric and possessesa neutral element .

Proposition 21: Let be a bisymmetric aggregation operatorwith neutral element . An -ary operator , commuteswith if and only if there exist , suchthat

(7)


Proof: Consider some bisymmetric aggregation operatorwith neutral element . If is defined by (7) for some

then it commutes with due to Proposition 18. In orderto show the necessity assume that commutes with , i.e.,especially for all it holds that

with defined by (3), thus fulfilling andproving that can be expressed as in (7).

Recall once again that any bisymmetric aggregation oper-ator with neutral element is also associative and symmetric andtherefore is either a t-norm, a t-conorm or a uninorm. However,note that it is impossible that commuting operators having neu-tral elements are different operators.

Proposition 22: Consider two aggregation operators andwith neutral elements , respectively, . If commutes

with , then . Moreover, also .Proof: Assume that and are commuting aggregation

operators with neutral elements , respectively, . Therefore

and

for all and arbitrary .As a consequence commuting does not work between

t-norms, t-conorms, or uninorms respectively. The only opera-tors commuting with such bisymmetric operators with neutralelement are, besides the operator itself, aggregation operatorswith no neutral element.

Example 23: As mentioned before the projection to the firstcoordinate commutes with any aggregation operator andtherefore also, e.g., with the product t-norm . Observe that

is bisymmetric but has no neutral element, while is abisymmetric aggregation operator with neutral element 1. Ac-cording to Proposition 21, corresponding functions

, can be chosen such that

namely and all other for .However, for any the operator

can never represent theproduct .

C. Consequences

Since Proposition 21 provides a full characterization of com-muting operators in case that one of them is bisymmetric withsome neutral element and further shows that these operators canbe attained through functions distributive over the bisymmetricaggregation operator with neutral element involved, we will nowfocus on the set of such functions.

Note that a full characterization of all bisymmetric aggrega-tion operators with neutral element, in particular if the neutralelement is from the open interval, is still missing. Since the char-acterization of the set of unary functions distributing with suchoperators is heavily influenced by the structure of the underlyingoperator, we will later on focus on special subclasses of bisym-metric aggregation operators with neutral element only, namelyon

• continuous t-norms;• continuous t-conorms;• particular classes of uninorms.

Therefore, consider to be some continuous t-norm , somecontinuous t-conorm , or some uninorm . Note thatis equivalent to the fact that fulfills a Cauchy like equation,i.e., for all

(8)

Observe that besides and alsothe constant function is included in .

Lemma 24: If , thenfor all fulfills .

V. CHARACTERIZATION OF FOR CONTINUOUS T-(CO)NORMS

For the case of continuous t-conorms (8) has been solved byBenvenuti et al. in [7] and as such by duality also for continuoust-norms. Continuous t-(co)norms are particularly important sub-classes of t-(co)norms. We briefly recall a few basic facts andproperties, but refer the interested reader for more details to themonographs [6], [24] and the articles [25]–[27].

The class of continuous t-(co)norms exactly consists of allso called continuous Archimedean t-(co)norms and of ordinalsums of such continuous Archimedean t-(co)norms. Let us firstturn to continuous Archimedean t-(co)norms resp. . Theyare in turn characterized as being generated by some continuousadditive generator resp. , i.e., they can be written as

In case of (continuous) t-norms, the additive generatoris a strictly decreasing (continuous) function which fulfills

and for which . In caseof (continuous) t-conorms, the additive generatoris a strictly increasing (continuous) function which fulfills

and for which Notethat in both cases additive generators are unique up to apositive multiplicative constant. For continuous Archimedeant-(co)norms two subclasses can be further distinguished, namelynilpotent t-(co)norms for which resp. , andstrict t-(co)norms with resp. .

Let us now turn to ordinal sum t-(co)norms, a concept appli-cable to all kinds of t-(co)norms. The main properties are basedon results in the framework of semigroups, however, the basicidea of ordinal sums can be described the following way: De-fine a t-(co)norm , respectively, by t-(co)norms on pairwisenonoverlapping subsquares along the diagonal of the unit squareand choose for all other cases in case of t-norms andin case of t-conorms. Formally, consider a familyof pairwise disjoint open subintervals of the unit-interval and acorresponding family of t-(co)norms resp. ,


then the ordinal sums ,respectively, are givenby (9), resp., (10), shown at the bottom of the page, and are in-deed a t-norm resp. a t-conorm. The ordinal sum t-norm aswell as the ordinal sum t-conorm are continuous if and only ifall resp. are continuous. Based on these facts let us nowbriefly recall the main results of [7] which will be further rele-vant for the investigation of particular classes of uninorms.

A. Continuous T-Conorms

Theorem 25 ([7]): Consider a continuous t-conorm . Thenfor some index set and there

exists a family of continuous strictly increasing mappingswith such that (11), shown at

the bottom of the page, holds. Let and denote by itsrestriction to the interval .

i) If , then one of the following holds:(ssi) with and

;(ssg) for some

and some such thatand .

ii) If , then one of the following holds:(sni) ;(sng) for some

so that isfinite and .

Note that in case of (ssi) and (sni), is constant on the wholecorresponding interval , respectively, attainingits value at an idempotent element of . In case of (ssg) and(sng), there exists at least one such that

so that necessarily there exists some fulfilling.

The previous theorem already indicates how all distributivefunctions for some continuous t-conorm can beobtained:

Theorem 26 ([7]): Consider some continuous t-conormand use the notations as introduced in Theorem 25. Anyis obtained from a generic function whichis monotone nondecreasing and from its restrictions for everyinterval whereas each restriction is chosen either byexpression (ssi), respectively, (ssg) in case that orby expression (sni), respectively, (sng) in case that .

Example 27: Consider the basic t-conorm. It is continuous with and

. Its set of distributive functions is given by

where is defined by

ifotherwise

Example 28: Consider the basic t-conorm. It is continuous with

, and

B. Continuous T-Norms

Since t-norms are dual to t-conorms we can get analogousresults for functions being distributive over some continuoust-norm .

Corollary 29: Consider a continuous t-norm . Thenfor some index set and there

exists a family of continuous strictly decreasing mappingswith such that (12), shown at

the bottom of the page, holds. Let and denote by itsrestriction to the interval .

i) If , then one of the following holds:(tsi) with and

;(tsg) for some

and some such thatand .

ii) If , then one of the following holds:(tni) ;(tng) for some

such that isfinite and .

Analogous to the case of continuous t-conorms all functionsbeing distributive over some continuous t-norm can be found.

if

otherwise(9)

if

otherwise(10)

ifotherwise

(11)

ifotherwise

(12)


Corollary 30: Consider some continuous t-norm and usethe notations as introduced in Corollary 29. Any isobtained from a generic function which ismonotone nondecreasing and from its restrictions for everyinterval whereas each restriction is chosen either byexpression (tsi), respectively, (tsg) in case that or byexpression (tni), respectively, (tng) in case that .

Example 31: Consider the two basic t-normsand . For both we have that

and their additive generators aregiven by and , respectively.Further, we get that

with

with

VI. CHARACTERIZATION OF FOR (PARTICULAR

CLASSES OF) UNINORMS

Let us now turn to the last class of bisymmetric aggregationoperators with some neutral element, namely uninorms whoseneutral elements fulfill (see also [11], [17]). Note thatuninorms can be interpreted as combination of some t-normand some t-conorm, i.e.,

with some t-norm acting on and some t-conorm actingon . To express explicitly that some uninorm is related tosome t-norm and some t-conorm , we will use the notation

.Such created uninorms cover a quite large class of aggrega-

tion operators since on the remainder of their domains they canbe chosen such that the monotonicity and associativity condi-tion are not violated but otherwise arbitrarily. However, due toits properties any uninorm fulfills

whenever and for all, giving rise to the particular classes of

uninorms. Note further, there exists no uninorm which is con-tinuous on the whole domain [17]. Generated uninorms, whichwe will discuss later in more detail, therefore, form another im-portant subclass of uninorms, since they are continuous on thewhole domain up to the case where .

As the next section will show, functions distributing withsome uninorm heavily depend on the structure of the uni-norm. Therefore, since a full characterization of all uninorms isstill missing, we restrict the discussion of to two particularsubclasses of uninorms—namely to uninorms which are eitheracting as the minimum or as the maximum on their remaindersand to generated uninorms.

A. Distributive Functions on Uninorms

First of all let us investigate necessary and sufficient condi-tions for some nondecreasing function beingdistributive over some uninorm , i.e., for all

If we choose we see that forall , expressing that acts as a neutral element of

on the range of . Moreover, so thatnecessarily .

From this, we see already, that the set of idempotent elementsas well as the range of will play a crucial role in char-acterizing .

Lemma 32: Consider some . Then, the followingholds:

i) if , then ;ii) if , then also .

Proof: Consider some . If then thereexists some , such that and

Moreover, if then also

i.e., .Let us now briefly focus on particular cases where

.Proposition 33: Consider some uninorm with neutral

element and some with either or. Then, the following holds:

In case if and only ifi) ;

ii) ;iii) is distributive over ;iv) .In case if and only ifi) ;

ii) ;iii) is distributive overiv) .

Proof: Consider some uninorm with neutral el-ement , some with . Assume that .

Since is an idempotent element of and , itimmediately follows that , i.e., is anidempotent element of the t-norm involved.

Further, since acts as a neutral element on weknow that for all it holds that

Moreover, due to the nondecreasingness of forall such that indeed for all .


Fig. 1. Uninorm U and some f 2 F as discussed in Examples 34 and 40.

The fact that is distributive over follows immedi-ately from . Finally, choose arbitrary , thendue to property ii)

To prove the sufficiency, assume that andthat conditions i)-iv) are fulfilled. If both then also

, such that distributes over due to conditioniii). In case that both , also such that

due to condition ii) and the fact that is an idempotent el-ement of . Finally, let us consider w.l.o.g. some .Due to condition iv) and the nondecreasingness of and wecan conclude that

Moreover, since commutes with resp., and condi-tion ii), we also know that

such that . Analogously, the remaining case and thecharacterization of in case of can beshown.

Let us illustrate the previous results by some examples.Example 34: Consider the following uninorm

with neutral element

ifotherwise

Note that with isan isomorphic transformation of the product and

(see also Fig. 1). Its set of idem-potent elements is given by since the contin-uous t-norm has its boundaries as its only trivial idempotentelements.

• Therefore, there is only one function with, namely the constant function , sinceand has to be nondecreasing.

• On the other hand, there exist several functionswith : We can choose

arbitrarily and fix as suchfor all . Because is a lattice poly-nomial, has just to be nondecreasing on todistribute over such that condition iii) of Proposition33 is fulfilled. Finally, condition iv) trivially holds since

in case of andfor all .

Therefore, e.g., all functions withgiven by

ifotherwise

distribute over (see also Fig. 1).Example 35: Consider the following uninorm

with neutral element

ifififotherwise


Fig. 2. Uninorm U and some f 2 F as discussed in Examples 35 and 41.

Again with on ordinal sum t-norm onwith twice the product as its summands and a basict-conorm on (see also Fig. 2). The set of idempotentelements is given by .

Let us now focus just on those with ,i.e., .

• : Necessarily due to the nondecreasingnessof and the necessary properties given in Proposition 33.Therefore, is the only element of for which

and .• : Necessarily, we fix for all

and as such fulfill conditions i), ii), and iv) of Proposition33 immediately, i.e.,

ifotherwise.

The function and as such alsodistributes over the ordinal sum t-norm if it is one of thefollowing functions (see also Fig. 2):

ifotherwise

with

ifotherwise

ifotherwise

ifotherwise

withifotherwise

So far, we have investigated nondecreasing functions withparticular domains being distributive over some uninorm .However, in case that the characterization of those

heavily depends on the structure of the uninorminvolved. Therefore, we will now turn to special subclasses ofuninorms.

B. Special Case: Uninorms

We now assume that the uninorm is such that issome t-norm on some t-conorm on andon the remainder acts either as the minimum or as the max-imum. We will denote such uninorms by , respectively,

. In case that the t-norm as well as the t-conorminvolved are continuous, we refer to the uninorm as weaklycontinuous t-norm.

We will focus on functions based on a composition of func-tions distributive over , respectively, , i.e., on functions

defined by

ififif

(13)

with some and . We will use the abbreviation.

Note that not all are of the type as thefollowing example shows.


Example 36: Consider some weakly continuous uninorm. Then, defined by

ifif

fulfills and , but .However, since uninorms can be interpreted as operators

acting on a bipolar scale with neutral element , it is natural toinvestigate distributive functions preserving that neutralitylevel, i.e., fulfilling .

By the construction provided by (13), it isguaranteed that the restrictions of some to resp.

are distributive over the corresponding , respectively, .Note that this construction also ensures that, due to the nonde-creasingness of , that for all andfor all . Depending on whether or

has to fulfill additional properties for .Proposition 37: Consider some weakly continuous uninorm

, further some and and defineby .

i) , if and only if or.

ii) if and only if or.

Proof: Consider some weakly continuous uninorm ,further some and and define

as by (13).Assume that . Further assume that there exists

some with and some with, then the following holds:

leading to a contradiction. Vice versa, sinceit distributes over for all and for all

due to its construction. Therefore, it suffices toprove that distributes over for all

.Assume that additionally fulfills either for all

or for all and choose anarbitrary and an arbitrary . Therefore, either

or , in any case , such that

In case that and , it immediately holds that

Analogously, the distributivity of over for somecan be shown as well as the characteri-

zation of all .Based on this result, we can immediately state which func-

tions are distributive over both as well.

Lemma 38: Consider some weakly continuous uninorm, further some and and define

by . if andonly if either

• and ;

• .Moreover, due to Proposition 37 and the nondecreasingness

of we can further draw the following conclusions.Corollary 39: Consider some weakly continuous uninorm

, further some and and defineby .

i) If and there exists some suchthat , then for all .

ii) If and there exists some suchthat , then for all .

Example 40: (Continuation of Example 34) Let us once againconsider the uninorm as introduced in Example 34, i.e.,

ifotherwise

It is of the type withan isomorphic transformation of the product

and the maximum. Now, we are looking forthose which are constructed by . Thesets and of nondecreasing functions distributing with

, respectively, are given by

with or

or

is nondecreasing

In accordance with Proposition 37, we now have to choose eitherfor all or for all such

that fulfills , so, e.g.,, (see also Fig. 1)

ifotherwise

Example 41: (Continuation of Example 35) Note that the uni-norm defined by

if

if

ifotherwise

is a uninorm of the type . Since is an ordinal sumt-norm on , its set of distributive functions is ratherlarge. Some of its elements are already listed in Example 35.Similarly, also contains many, namely all nondecreasingfunctions on . In accordance with Proposition 37 we haveto choose functions such that either forall or that for all , such that

, e.g., ,(see also Fig. 2)

ifotherwise

ififotherwise


C. Special Case: Generated Uninorms

An important subclass of uninorms are those generated bysome additive generator. They are continuous on the whole do-main up to the case where .

Definition 42: An operator is anArchimedean uninorm continuous in all points up to

, if and only if thereexists a monotone bijection such that

with convention . The uninorm is thencalled a generated uninorm with additive generator .

Note that the neutral element of such a generated uninormis given by . The increasingness of the additive gen-erator is equivalent to its conjunctive form. Moreover, generateduninorms are related to strict t-norms and strict t-conorms, since

and are additive genera-tors of a strict t-norm, respectively, t-conorm, associated with .

In case of some with generated by the additivegenerator , we get

Since is a bijection this is equivalent to

with and both elements from suchthat for and arbitrary it holdsthat . In case that is continuousthe solutions of this equation (see also [3]) are given by

with . As a consequence

leading to the following lemma.Lemma 43: Consider some uninorm generated by some

additive generator . If and continuous, but not con-stant, then there exists some such that

for all .Example 44: Consider some uninorm generated by some

additive generator and choose for alland for at least one . Then, the operator

defined by

commutes with .

Example 45: Consider the additive generator. The generated uninorm

is then given by

with neutral element . Note that is also known as-operator and has already been discussed by several authors

[14], [17], [23], [35], [37]. It is worth remarking that it plays animportant role as combining functions of uncertainty factors inexpert systems like MYCIN and PROSPECTOR (see also [9],[12], and [21]).

In accordance with the previous example, aggregation opera-tors defined by

with for all and for at least onecommute with .

VII. FINAL REMARKS

The issue of commuting aggregations has been consideredin the general case and in some important particular cases,especially the one of uninorms, where new nontrivial resultsare obtained. Finding commuting operations can be a difficultexercise sometimes leading to impossibility results. So, e.g.,in the class of OWA operators [36], the set of all aggregationoperators commuting with an -ary OWA operator differentfrom , or the arithmetic mean, respectively, is trivial,namely, consisting just of the projections [32]. However, forbisymmetric operations such as the weighted arithmetic mean,results on commuting exist for some 25 years in connectionwith the problem of consensus functions for probabilities [28],more recently for t-norms and conorms in connection withgeneralized utility theory [15] or transitivity preservation in theaggregation of fuzzy relations [34]. Commuting operators foruninorms can be relevant in multicriteria decision-making withbipolar scales where bipolar set-functions are used to evaluatethe importance of criteria [19], [20]. Indeed the neutral elementof the uninorm separates a bipolar evaluation scale in its posi-tive and negative parts [16]. Our results can be instrumental inlaying bare consensus functions for multiperson multicriteriadecision-making problems on bipolar scales, a topic to beinvestigated at a further stage.

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[3] J. Aczél, Lectures on Functional Equations and their Applications.New York: Academic, 1966.

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[6] C. Alsina, M. Frank, and B. Schweizer, Associative Functions: Trian-gular Norms and Copulas. Singapore: World Scientific, 2006.

[7] P. Benvenuti, D. Vivona, and M. Divari, “The Cauchy equation onI-semigroups,” Aequationes Math., vol. 63, no. 3, pp. 220–230, 2002.

[8] G. Birkhoff, Lattice Theory. Providence: AMS, 1973.[9] B. Buchanan and E. Shortliffe, Rule-Based Expert Systems, The

MYCIN Experiments of the Standford Heuristic ProgrammingProject. Reading, MA: Addison-Wesley, 1984.

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[11] T. Calvo, A. Kolesárová, M. Komorníková, and R. Mesiar, , T. Calvo,G. Mayor, and R. Mesiar, Eds., “Aggregation operators: Properties,classes and construction methods,” in Ser. Studies in Fuzziness and SoftComputing. Heidelberg, Germany: Physica-Verlag, 2002, vol. 97, pp.3–104.

[12] B. De Baets and J. Fodor, “Van Melle’s combining function in MYCINis a representable uninorm: An alternative proof.,” Fuzzy Sets Syst., vol.104, pp. 133–136, 1999.

[13] S. Díaz, S. Montes, and B. De Baets, “Transitivity bounds in additivefuzzy preference structures,” IEEE Trans. Fuzzy Syst., vol. 15, no. 2,pp. 275–286, Apr. 2007.

[14] J. Dombi, “Basic concepts for a theory of evaluation: The aggregativeoperator,” Eur. J. Oper. Res., vol. 10, pp. 282–293, 1982.

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[29] J.-L. Marichal, “On order invariant synthesizing functions,” J. Math.Psychol., vol. 46, no. 6, pp. 661–676, 2002.

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[35] W. Silvert, “Symmetric summation: A class of operations on fuzzysets,” IEEE Trans. Syst., Man, Cybern., vol. SMC-9, pp. 657–659,1979.

[36] R. R. Yager, “On ordered weighted averaging aggregation operators inmulticriteria decisionmaking,” IEEE Trans. Syst., Man, Cybern., vol.18, no. 1, pp. 183–190, Feb. 1988.

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Susanne Saminger-Platz received the Ph.D. degree in mathematics withhighest distinction from the Johannes Kepler University, Linz, Austria, in 2003.

She is an Assistant Professor at the Department of Knowledge-Based Mathe-matical Systems, Johannes Kepler University, Linz, Austria. Her major researchinterests focus on the preservation of properties during uni- and bipolar aggre-gation processes and as such have significant connections to fuzzy rule-basedmodeling, preference modeling, and intelligent data analysis, in particular dom-inance of aggregation operators, construction principles for aggregation oper-ators, fuzzy measures, and fuzzy relations. She is author/coauthor of severaljournal papers and chapters in edited volumes.

Dr. Saminger-Platz is a member of the European Association for Fuzzy Logicand Technology (EUSFLAT), of the EURO Working Group on Fuzzy Sets (EU-ROFUSE), and of the Austrian Mathematical Society.

Radko Mesiar received the Ph.D. degree from the Comenius UniversityBratislava and the D.Sc. degree from the Czech Academy of Sciences, Prague,in 1979 and 1996, respectively.

He is a Professor of Mathematics at the Slovak University of Technology,Bratislava, Slovakia. His major research interests are in the areas of uncertaintymodeling, fuzzy logic, and several types of aggregation techniques, nonadditivemeasures, and integral theory. He is a coauthor of a monograph on triangularnorms and an author/coauthor of more than 100 journal papers and chapters inedited volumes.

Dr. Mesiar is an Associate Editor of four international journals. He is amember of the European Association for Fuzzy Logic and Technology board.He is a Fellow Researcher at UTIA AV CR Prague (since 1995) and at IRAFMOstrava (since 2005).

Didier Dubois received the Ph.D. degree in engineering from ENSAE,Toulouse, France, in 1977, the Doctorat d’Etat from Grenoble University,Grenoble, France, in 1983, and an Honorary Doctorate from the FacultéPolytechnique de Mons, Belgium, in 1997.

He is a Research Advisor at IRIT, the Computer Science Department, PaulSabatier University, Toulouse, France, and belongs to the French NationalCentre for Scientific Research (CNRS). He is the coauthor of two books onfuzzy sets and possibility theory, and 15 edited volumes on uncertain reasoningand fuzzy sets. He coordinated the Handbook of Fuzzy Sets series published byKluwer (7 volumes, 1998–2000), including the book Fundamentals of FuzzySets. He has contributed about 200 technical journal papers on uncertaintytheories and applications. His topics of interest range from artificial intelligenceto operations research and decision sciences, with emphasis on the modeling,representation and processing of imprecise and uncertain information inreasoning and problem-solving tasks.

Dr. Dubois is the Co-Editor-in-Chief of the journal Fuzzy Sets and Systems,an Advisory Editor of the IEEE TRANSACTIONS ON FUZZY SYSTEMS (TFS), anda member of the Editorial Board of several technical journals, such as the Inter-national Journal on Approximate Reasoning, General Systems, Applied Logic,and Information Sciences, among others. He is a former President of the Inter-national Fuzzy Systems Association (1995–1997). He received the 2002 Pio-neer Award of the IEEE Neural Network Society, and the 2005 IEEE TFS Out-standing Paper Award.

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Aggregation Operators and Commuting