Generalized qualitative Sugeno integrals

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Dubois, Didier and Prade, Henri and Rico,

Agnés and Teheux, Bruno Generalized qualitative Sugeno integrals.

(2017) Information Sciences, 415 - 416. 429-445. ISSN 0020-0255

Official URL

DOI :

https://doi.org/10.1016/j.ins.2017.05.037

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Generalized

qualitative

Sugeno

integrals

Didier

Dubois

a,∗

_,

_Henri

_Prade

a

_,

_Agnès

_Rico

b

_,

_Bruno

_Teheux

c a IRIT, Université Paul Sabatier 118 route de Narbonne, 31062 Toulouse cedex 9 (France)

b ERIC, Université Claude Bernard Lyon 1, 43 bld du 11 novembre, 69100 Villeurbanne (France) c Mathematics Research Unit, FSTC University of Luxembourg, L-1359 Luxembourg (Luxembourg)

Keywords:

Sugeno integrals Conjunctions Implications

a b s t r a c t

Sugeno integralsareaggregationoperationsinvolvingacriterionweightingschemebased on theuseofset functionscalledcapacities or fuzzymeasures. Inthis paper, wedefine generalizedversionsofSugeno integralsontotallyorderedboundedchains, byextending theoperation thatcombinesthevalueofthecapacityoneachsubsetofcriteriaandthe value oftheutility function overelements ofthesubset.We showthat thegeneralized conceptofSugeno integral splitsintotwo functionals,one based onageneral multiple-valuedconjunction(wecallintegral)andonebasedonageneralmultiple-valued implica-tion(wecallcointegral).These fuzzyconjunctionandimplicationconnectivesarerelated viaaso-calledsemidualityproperty,involvinganinvolutivenegation.Sugenointegrals cor-respondtothecasewhenthefuzzyconjunctionistheminimumandthefuzzyimplication isKleene-Dienesimplication,inwhichcaseintegralsand cointegralscoincide.Inthis pa-per, weconsider averygeneral classoffuzzy conjunctionoperationsonafinite setting, thatreducetoBooleanconjunctionsonextremevaluesoftheboundedchain,andare non-decreasingineachplace,andthecorrespondinggeneralclassofimplications(their semid-uals).The meritofthese newaggregationoperatorsistogobeyondpurelattice polyno-mials,thusenhancingtheexpressivepowerofqualitativeaggregationfunctions,especially astothewayanimportanceweightcanaffectalocalratingofanobjecttobechosen.

1. Introduction

In thesetting of ArtificialIntelligence (for instance, recommendersystems, cognitiverobotics, but other fieldsas well),

theuseofdecisionrulesbasedonnumericalaggregationfunctionsisnotalwaysnatural.Forinstance,probabilities,utilities,

importanceweightscannotalwaysbeeasilyelicitedfromtheuser,bylackoftimeorlackofprecision.Informationsystems

advisingpersonscannotasktoomanyquestionstousersformodelingtheirpreferences,norcollectfrom themmeaningful

numbers representing probabilities or criteria importance levels, or yet utilityvalues. Even if they get them, making

nu-mericaloperationson themlooksdebatable.Anotherexampleiswhenarefereehasto fillaformtoassessthemeritsofa

paperforajournal, and numericalratingsarerequired. What istheprecisemeaningofthese ratings?Does itmakesense

tocomputeaveragesfromthem?

In suchsituations it ismore natural to resortto aqualitative approach to multicriteria evaluation. The rationaleis to

refrain from using numbers that look arbitrary or hard to collect, if we can evaluate decisions in a reasoned approach,

∗ _{Corresponding author.}

E-mail addresses: dubois@irit.fr (D. Dubois), prade@irit.fr (H. Prade), rico@eric.fr (A. Rico), bruno.teheux@uni.lu (B. Teheux).

http://dx.doi.org/10.1016/j.ins.2017.05.037

without numericalcalculations and address decisionproblems inthe ordinalsetting. There are twoadvantages: (i)a gain

inrobustnessand theneedforless data;(ii)qualitativemethodslend themselvestoalogicalrepresentation (whichmakes

proposed choices moreeasily explainable). There are two possible choices of qualitativesettings for representing notions

suchasutilityratingsbyseveralagents,importancelevelsand likelihooddegrees:

• Use distinct non-commensuratescales. This makes theframework very restrictiveas impossibility theorems regarding

rationalaggregationprocessesareoften obtained(e.g.,invotingtheory).

• Usefinitecommensuratescales(takingadvantageofnotionsfacilitatingcommensuratenesssuchascertaintyequivalents)

thenoneworkswithafiniteorderedset ofvalueclasses.

In multi-criteria decision making, Sugeno integrals [39,40] are commonly used as qualitative aggregation functions

[25] using finite scales and a commensurateness assumption between them. The definitions of these integrals are based

onamonotonicset-function namedcapacityorfuzzymeasurethat aimsto qualitativelyrepresent thelikelihoodofsets of

possiblestatesofnature,theimportanceofsetsofcriteria,etc.Thesesetfunctionsarecurrentlyusedinmanyareassuchas

uncertaintymodeling[13,14],multiplecriteriaaggregation[3,23,24] oringametheory[37].Seealsoarecentbookdevoted

tocapacitiesinsuchareas [28].Moreover,Sugenointegralsnaturallylendthemselvestoarepresentationintermsofif-then

rulesinvolvingthresholds[7,15,23],whichmakesthemeasytointerpret.

Capacitiescanbeexploited indifferentwayswhenaggregatinglocalratingsofobjectsaccordingtovariouscriteria, and

different qualitativeintegrals (q-integrals, for short) canbe obtained. Inthe caseof Sugenointegrals, the capacityis used

as abound that restricts the globalevaluation from below or from above. In other cases, the capacityis considered as a

tolerancethresholdsuchthat overcomingitleads toimproving theglobalevaluationoftheobject understudy.When this

thresholdisnotreachedtherearetwo possibilities.Eitherthelocal ratingremainsasitstandsorit ismodified:improved

ifthecriterionislittleimportant,downgradedifitisimportant.

Theseconsiderationsgiverisetonewaggregationoperationsinthequalitativesetting, suchassoftand drasticintegrals,

studiedinarecentpaper[16].Namely, weintroducedvariantsofSugenointegralsbasedonGödelimplicationand its

con-trapositiveversion,usinganinvolutivenegation.Itmodelsqualitativeaggregationmethodsthatextendminandmax,based

on the idea oftolerance threshold beyond which acriterion is consideredsatisfied. These variants of integrals arevalued

ina scaleequipped with aresiduated implicationand an involutivenegation. Moreprecisely, in[16], theevaluation scale

isbothatotallyorderedHeytingalgebraand aKleenealgebra.Thesenewaggregationoperationshavebeenaxiomatizedin

[17]in the setting of a complete bounded chain with an involutive negation.

In the present paper, which extends a conference paper [18], we try to cast this approach in a more general totally

ordered algebraic setting, usingmultivalued conjunctionoperations that are notnecessarily commutative,and implication

operationsinducedfromthembymeansofaninvolutivenegation,withaviewtopreservecharacterizationtheoremsproved

in[17]forGödel implicationanditscontrapositiveversion,andnon-commutative fuzzyconjunctionsobtainedviaan

invo-lutivenegation.Weshowthat,oncegeneralizedinthisway,theconceptofSugenointegral splitsintotwofunctionals,one

basedonageneralizedconjunction(wecallq-integral)andonebasedonanimplication(wecallq-cointegral).Thesefuzzy

conjunctionandimplicationconnectivesarerelatedviaasemidualityproperty,involvinganinvolutivenegation.Sugeno

in-tegralscorrespondtothecasewhenthefuzzyconjunctionistheminimumandtheimplicationisKleene-Dienesimplication,

inwhichcaseintegrals andcointegralscoincide.

Thepaperisstructuredasfollows.Section2presentsthemainmotivations forthestudyofnewaggregationoperations,

namelythenotionofweightedqualitativeaggregationfunction.Itstudiesthewayimportanceweightsofcriteriaaffectlocal

ratingsof objectswith respectto such criteria. Section3 providesinsights on the algebraicsetting useful for generalizing

Sugenointegrals,namelythepossibledependencebetweenfuzzyconjunctionsand implicationsviaresiduationanda

prop-erty calledsemiduality. Section 4presents theproposed generalizations ofSugeno integrals, emphasizingthe existenceof

distinct integralsand cointegrals respectivelydefined bymeansoffuzzy conjunctionsand implications.Section 5presents

representationtheoremsforq-integralsand q-cointegrals.

2. Motivations

After recalling the weighted min and max aggregation functions, this section extends these aggregation functions to

otherweightingschemes,focusingonthepossibleeffects ofweightingcriteriaonthecorrespondingratings.Thisextension

requiresmoregeneral conjunctionsand implications.

Weadopttheterminologyand notationsusual inmulti-criteriadecisionmaking, wheresomealternatives areevaluated

accordingto acommon set C=

{

1,...,n

}

=[n]ofcriteria. Acommon evaluation scaleL isassumed to provideratings

ac-cordingto the criteria:each alternativeis thus identified with a function f ∈LC _{which maps every criterioni of}

C to the

localratingfiofthealternativewithregardtothiscriterion.

Weassume that L isa finitetotally ordered set with 1and 0astop and bottom, respectively(L maybe asubset

{

0=

a0<a1<· · · <ak=1

}

oftherealunitinterval[0, 1]for instance).For anya ∈L, wedenotebyaC theconstantalternative

whoseratingsequalaforallcriteriainC.Inaddition,weassumethatLisequipped withaunaryorderreversinginvolutive

operationa7→1−a, thatwecallnegation.Inthefinitesetting, thereisauniquesuchnegationoperation.1 _{Werespectively}

denote by∧and ∨ theminimum and maximumoperations inducedby≥on L. Suchafinitetotally ordered scaledefined

by

(

L,≥,1−,0,1

)

iscalledqualitativeinthispaper.

2.1. Weightedminimum andmaximum

There aretwo elementarysymmetric qualitativeaggregationschemesonafinitetotallyordered chain:

• Thepessimisticone Vn_i=1f_iwhichisverydemandingsinceinordertoobtainagoodevaluationanobjectneedstosatisfy

allthecriteria.

• Theoptimisticone, Wni=1fiwhich isvery loosesinceonefulfilledcriterionisenoughtoobtainagood evaluation.

These two aggregation schemescan be slightly generalizedby meansof importance levels or priorities

π

i ∈ L,on the

criteriai ∈[n],thus yieldingweightedminimum andmaximum [11]. Suppose

π

i isincreasing withtheimportance ofi. A

fullyimportant criterionhasimportance weight

π

i=1. Auseless criterionhasimportance weight

π

i=0. Ingeneral these

criteria can be eliminated. We also assume

π

i=1, for at least one criterion i (the most important ones). It is a kind of

normalizationassumptionthat ensuresthatthewholepositivepartofthescaleLisuseful.Itistypicalofpossibilitytheory

[12].Indeed,thesetofweightscanbeviewedasapossibilitydistributionovercriteria,justlikethesetofnumericalweights

inaweighted averagecanbeviewedasaprobabilitydistribution.

These importancelevels caninteract with eachlocalevaluation f_i indifferent manners.Usually,aqualitative weight

π

i

actsasasaturationthresholdthatblockstheglobalscoreunderoraboveacertainvaluedependentontheimportancelevel

ofcriterioni.Suchweightstruncatetheevaluationscalefrom aboveorfrombelow.Usually,theratingf_i ismodifiedinthe

form of either

(

1−

π

i

)

∨fi∈[1−

π

i,1], or

π

i∧fi ∈ [0,

π

i].In the first expression, low ratingsof unimportant criteria are

upgraded;inthesecondone, highratingsofunimportantcriteriaaredowngraded.The weightedminimumand maximum

operationsthentakethefollowingforms[11]:

MINπ

(

f

)

= n ^ i=1

¡

(

1−

π

i

)

∨fi

¢

; MAXπ

(

f

)

= n _ i=1

(

π

i∧fi

)

. (1)

In these aggregation operations,a fully important criterioncan alone bring thewhole global scoreto 0(MINπ(f))or to 1

(MAXπ(f)),accordingtothechosenattitude(resp.pessimisticoroptimistic).

Itiswell-knownthatiftherangeofthelocalratingsf_i isreducedto{0,1}(Booleancriteria)thenlettingA_f=

{

i:f_i=1

}

bethesetofcriteriasatisfiedbyalternativef,thefunction

5

:A_f 7→MAXπ

(

f

)

=

_

{

π

i:i∈Af

}

isapossibilitymeasureonC [42](i.e,asetfunction

5

thatsatisfies

5(

A∪B

)

=

5(

A

)

∨

5(

B

)

foreveryA,B⊆ C),and,

N:Af 7→MINπ

(

f

)

=

^

{

1−

π

i:i6∈Af

}

is a necessity measure [12] (i.e., a set function N that satisfies N

(

A∩B

)

=N

(

A

)

∧N

(

B

)

for every A,B⊆ C). Note that the

well-knownduality property

5(

A

)

=1−N

(

A

)

, (2)

whereAdenotestheset complementofAinC,immediatelygeneralizestothescaleLinthefollowingway:

MAXπ

(

f

)

=1−MINπ

(

1−f

)

. (3)

Inorderto getamaximalglobalrating withtheweighted minimum,we forceallcriteria withaweightnot equalto0to

besatisfied;and withtheweightedmaximumweforceonlyonefullyimportantcriteriatobesatisfied.

Example 1. Let L=

{

0,0.4,0.6,1

}

⊂[0,1] and considerthreecriteria withtheimportance profile

π

and theBoolean

alter-nativesfandg:

1 2 3 MINπ MAXπ

π 1 1 0.6

f 1 1 0 0.4 1

g 1 0 0 0 1

We canseethat fviolatestheleastimportantcriterion 3only,and itmakestheglobalevaluation positiveaccordingto

thepessimistic criterion,whilegisruled outbecause itviolatesone essentialcriterion.Buttheoptimistic evaluationputs

thetwo decisionson aparbecausetheybothsatisfyoneimportantcriterion.

Note that the form (1) of the criteria is not arbitrary. It we consider weighted aggregations of the form ∧n_i=1

π

i∨fi,

and ∨n_i=1

(

1−

π

i

)

∧ fi for instance, the effect of weights will beradically different. In the case of the former, you get the

maximalvalue 1 forthe globalratingas soonasall criteriaare important, whateverthevalue of thelocal ratings,which

overall rating, again counterintuitive. So, the mathematical form of the combination between a criterion weight and the

local rating is affected by the aggregation scheme used and cannot be chosen arbitrarilyif intuitively satisfactory results

mustbeobtained.

Inthefollowingweexplorevariousalternativeweightingschemesintheordinalsetting.Theweightedminandmaxcan

begeneralizedbyconjunctiveand disjunctiveexpressionsoftheform:

MIN⊲ π

(

f

)

= n ^ i=1

¡π

i⊲fi

¢

; MAX π

(

f

)

= n _ i=1

(

π

ifi

)

. (4)

forsuitablechoicesofratingmodificationoperations⊲and.Inthesequelwestudyconditionsthattheseoperationsmust

satisfyinorderforthemtomakesense.Weconsiderthecasesofconjunctiveanddisjunctiveaggregationsrespectively.

2.2. Ageneralizedrating modificationschemefortheconjunctiveaggregation

Let

π

i⊲ fibetheresultofapplyingtheweight

π

itotheratingfi.Usingaconjunctiveaggregation,analternativeshould

notberejectedduetoapoorratingonanunimportantcriterion,especiallytheglobalratingshouldnotbeaffectedby

use-lesscriteria,whateverthecorrespondingrating.Theimportanceweight0shouldturnthelocalratingonauselesscriterion

into1,sincea∧1=a,i.e.,thisextremevalueisnottakenintoaccountbytheconjunctiveaggregation.Sowemustassume

that 0⊲f_i=1. Moreover,it isclearthat if thecriterionhasmaximalimportance,a verypoor localratingon thiscriterion

shouldbe enoughto bring theglobal evaluationdown to 0,which implies 1⊲0=0. Besides, itis natural that thebetter

thelocalratingfi,thegreaterthemodified localrating.However, thelessimportantthecriterion,themorelenient should

bethemodifiedlocalrating.

According to the above rationale, the operation ⊲ should formally be a multiple-valued implication (as defined in

[1,21]forL=[0,1]):

Definition1. Afuzzyimplicationisatwo-place operation→onLsuchthat:

1. 0→1=1;1→0=0;1→1=1;0→0=1.

2. a→ bisincreasinginthewidesensewithbforalla∈L.

3. a→ bisdecreasing inthewidesensewithaforallb ∈L.

As weusea finitescalewithan involutivenegation, itisnatural to consider thespecial casewhen theset ofpossible

values for

π

i⊲fi isrestricted to one of thearguments, their negations,and extreme values, i.e.,

{

0,1−

π

i,1−fi,

π

i,fi,1

}

accordingtotherelativepositionoff_iwithrespectto

π

ior1−

π

i.Inordertorespectthepropertiesofthefuzzyimplication,

wecanmakethefollowingobservations:

• Wecannot have

π

i⊲fi=

π

i nor

π

i⊲fi=1−fi, forconsecutivevaluesoffi,

π

i 6=0,1. Indeed ifaj⊲ fi=aj andaj+1⊲

fi=aj+1, the modified score would be increasing with

π

i, and if

π

i⊲aj=1−aj and

π

i⊲aj+1=1−aj+1, then the

modifiedscorewouldbedecreasing withf_i,whichcontradictstheclaimthat⊲shouldbeafuzzy implicationfunction.

• Wecannothave

π

i⊲1=0, otherwisewewouldget1⊲1=0,as

π

i⊲ fi decreaseswith

π

i, whichmeansthat ⊲would

notgeneralizeimplication.

Underthese restrictions on the range of

π

i⊲fi, some well-knownfuzzy implications[1], used in [16]or [17], are

re-trieved: • Rescher-Gainesimplication:

π

i⊲ fi=

π

i⇒RG fi=

½

_{1 if}

π

i≤ fi, 0 otherwise; • Gödelimplication:

π

i⊲fi=

π

i→G fi=

½

_{1 if}

π

i≤ fi, f_i otherwise;

• ThecontrapositivesymmetricofGödelimplication:

π

i⊲ fi=

π

i→GC fi=

½

1 if

π

i≤ fi,

1−

π

i otherwise;

• Kleene-Dienesimplication:

π

i⊲fi=

π

i→KD fi=

(

1−

π

i

)

∨ fi.

• Theresiduum ofthenilpotentminimum:

π

i→fi=

½

_{1 if}

π

i≤fi

(

1−

π

i

)

∨fi otherwise.

Otherfuzzyimplications,likeŁukasiewicz’s

π

i→L fi=min

(

1−

π

i+fi,1

)

arepossibleifL=

{

0,1/k,2/k,...,

(

k−1

)

/k,1

}

,

buttheresultsmaystandoutside

{

0,1−

π

i,1,fi

}

.Notethat ourframeworkexcludesZadeh’simplication[41]

(

1−a

)

∨

(

a∧

b

)

asthelatterisnotdecreasingwitha.

Thecorrespondingconjunctiveaggregationschemes,alternativetoMINπ arenatural.Using→RG,theaggregationselects

allobjectsthatpass theprescribedthresholdsforallcriteriaand totallyrejectstheotherones. Using→G,theaggregation

selectsallobjectsthatpass theprescribedthresholdsforallcriteriaandranks theremainingonesaccordingtotheirworst

localratings(formingawaitinglist). Using→GC,theaggregationselects allobjectsthat passtheprescribedthresholdsfor

allcriteria and ranks theremaining onesaccording to theimportance of violatedcriteria, putting the objectsthat violate

Example2. ConsiderL=

{

j/10:j=0,1,...,10

}

⊂[0,1], threecriteriawiththeimportanceprofile

π

and thealternatives f andg: 1 2 3 ∧iπi → RG f i ∧iπi → G f i ∧iπi → GC f i ∧i(1 −πi) ∨ f i π 1 0.5 0.2 f 1 0.4 0.3 0 0.4 0.5 0.5 g 1 0.5 0.3 1 1 1 0.5

The excellent global ratingof g for all the threenew aggregation operations is justified bythe fact that it passes the

importancethresholdsforallcriteria whileffailsforone ofthem.Using→RG,fisthencompletelyeliminated.Incontrast,

theusual weightedminimumcannottellffromgalthoughthelatterPareto-dominatestheformer.

2.3. Ageneralizedrating modificationschemeforthedisjunctiveaggregation

Let

π

i fibetheresultofapplyingtheweight

π

i totheratingfi.Usingadisjunctiveaggregation,analternativeshould

berejected only dueto a poor ratingon allimportant criteria. Again, the globalrating should notbe affected byuseless

criteria,whateverthecorrespondingrating.Astheaggregationoperationisdisjunctive,thelocalratingonauselesscriterion

shouldalways be turnedinto 0, sincea∨0=a, i.e.,0 does notalter thedisjunctive aggregation. So 0f_i=0. Moreover,

it is clear that if thecriterion has maximal importance, a very good local rating on this criterion is enough to bring the

global evaluation up to 1. So 11=1. But if fi=0, criterion i should not participate to theimprovement of the global

rating,even ifthecriterionisimportant.So

π

i0=0.Besides,itisnaturalthat thebetterthelocal ratingf_i,thebetterthe

modifiedlocal rating

π

i fi. However,themoreimportantthecriterion,thehighershouldbe

π

i fi sinceagiven rating

on animportant criterionshould havea morepositive influenceon the globalevaluation than thesamerating on alittle

importantcriterion.

According totheabove rationale,theoperationshouldformallybeamultiple-valuedconjunction:

Definition2. Afuzzyconjunctionisatwo-placeoperation on Lsuchthat:

1. 01=0;10=0;11=1;00=0.

2. abisincreasinginthewidesense witha,forallb∈L.

3. abisincreasinginthewidesense withbforalla∈L.

Note that weassume neither associativity norcommutativity (e.g., localrating and weight donot play the samerole).

Moreoverthetopelementcanbetheidentityontheleft(1a=a)orontheright(a1=a),inwhichcaseweshallspeak

of left- or right-conjunctions, respectively; or an identity on both sides (1a=a1=a, namely semicopulas [19] when

L=[0,1]).Besides,duetomonotonicityandlimitconditions,a0=01=0.

Weshallagainstudythecasewhenthesetofpossiblevaluesfor

π

if_iislimitedto

{

0,1−

π

i,

π

i,fi,1−fi,1

}

.Itisclear

thatone cannotassume

π

ifi∈

{

1−

π

i,1−fi

}

for

π

i,fi 6=0,1,asafuzzyconjunctionisincreasinginbothplaces.So one

islefttohave

π

i fi∈{0,

π

i,fi,1},whichasweshallseeleavesthepossibilitytohave

π

ifi >

π

i∧fi,includingexamples

where

π

ifi=1when

π

i <1, fi<1, evenif

π

i fimustextendtheBooleanconjunction.Inthenextsubsection,weuse

asystematicwayofproducingfuzzyconjunctionsfromfuzzyimplications.

2.4. Semiduality

In order togenerate interesting fuzzy conjunctions

π

i fi, wecan use thejoint extension (3) of theDe Morgan

rela-tionship∨n_i=1f_i=1− ∧n_i=1

(

1− f_i

)

existingbetweenconjunctiveand disjunctiveaggregations,and ofthewell-knownduality

propertybetweenpossibilityandnecessitymeasures(2).ApplyingthisDeMorgandualitytogeneralizedweighted

conjunc-tiveaggregationMIN⊲

π

(

f

)

=∧ni=1

π

i⊲fi yieldsafamilyofgeneralizedweighteddisjunctiveaggregations:

MAX π

(

f

)

=1−MIN ⊲ π

(

1−f

)

=1− n ^ i=1

π

i⊲

(

1−fi

)

= n _ i=1

π

ifi. (5)

In other words, to each connective ⊲ (fuzzy implication) that modifies local ratings in a conjunctive aggregation,

corre-sponds a connective =S

(

⊲

)

(fuzzy conjunction) that modifies local ratings in a disjunctive aggregation, and is of the

formaS

(

⊲

)

b:=1−

(

a⊲

(

1−b

))

.Weshallsaythatsuchapairofoperations

(

⊲,

)

aresemidualoperations.Itisobviousto

seethat⊲ isafuzzyfuzzy implicationinthesenseofDefinition 1if and onlyifitssemidualisafuzzy conjunctioninthe

sense ofDefinition 2.It isalsoobvious that theconnectivescan beexchanged,that is, a⊲b=aS

(



)

b=1−

(

a

(

1−b

))

,

i.e.,semidualityisaninvolutivetransformation.Notethata⊲b=1ifandonlyifaS

(

⊲

)(

1−b

)

=0thatis,aand1−b,when

positive,aredivisorsof0forthefuzzyconjunction= S

(

⊲

)

.

Under semiduality, some fuzzy conjunctionssuch that

π

i fi ∈{0,

π

i,fi, 1}are recoveredfrom previouslymentioned

fuzzyimplications2_:

• Rescher-Gainesconjunction[17]:

π

i⋆RGfi=1−

(π

i⇒RG

(

1−fi

))

=

½

0 if

π

i≤1−fi,

1 otherwise. Inthiscase,wemayhavethat

π

i⋆RG fi=1evenif

π

i<1, fi<1.

• Gödelconjunction[9]:

π

iG fi=1−

(π

i→G

(

1− fi

))

=

½

_{0 if}

π

i≤1−fi,

fi otherwise.

It isanon-commutative left-conjunction notupper-boundedbytheminimum (sinceforinstance,

π

iG1=1when

π

i

>0) .

• ThereciprocalGödelconjunction[9]:

π

iGC fi=1−

(π

i→GC

(

1−fi

))

=

½

_{0 if}

π

i≤1−fi

π

i otherwise.

It isanon-commutative right-conjunctionnotupper-boundedby

theminimum(sinceforinstance, 1_G f_i=1whenf_i >0).

• Kleene-Dienesconjunction:

π

iKD fi=1−

(π

i→KD

(

1− fi

))

=

π

i∧fi, whichistheminimum.

• Thenilpotent minimum[22]:

π

i∧fi=1−

(π

i→

(

1−fi

))

=

½

_{0 if}

π

i≤1−fi

π

i∧ fi otherwise.

Moregeneralexamples offuzzy conjunctionsare

• T-norms, i.e.,increasingsemi-groups with identity 1and annihilator0. Inthe finitesetting itincludes ∧, thenilpotent

minimum.TheŁukasiewiczt-normmax

(

a+b−1,0

)

canalsobeused ifL=

{

0,1/k,2/k,...,

(

k−1

)

/k,1

}

.

• Weakt-norms[21],i.e.,left-conjunctionssuchthata1≤a.

• Thenon-commutative left-conjunction_CRGdefinedbya_CRGb=0ifa=0,and a_CRGb=b ifa6=0(see[17]).

• Theright-conjunction a_RGb=b_CRGaassociatedwiththepreviousfuzzyconjunction.

• Fuzzyconjunctionshavingidentity 1.On [0,+∞

)

theyarecalledpseudo-multiplications afterKlementetal.[33] inthe

definitionofuniversalintegrals,andsemicopulas[19]ontheunitinterval.

Thesefuzzy conjunctionsyielddisjunctiveweightedaggregationswithvariousbehaviors:

• Using ⋆RG theaggregationrejectsallobjectsthat havealocalratingless than 1−

π

i forallcriteriaand fullyselects the

other ones.Thelocalratingf_i isturnedinto0foranonimportantcriterion,even ifthisratingishigh.

• Using _G theaggregation rejectsallobjectsthat havealocal ratingless than 1−

π

i forallcriteria and rankstheother

onesaccordingtotheirbestlocalratings,formingawaiting list.

• Using _GC theaggregationrejectsallobjectsthathavealocal ratinglessthan 1−

π

ifor allcriteria andrankstheother

satisfiedonesaccordingtotheleastimportance degreeofviolatedcriteria,formingawaitinglist.Notethat_G and _GC

arenotcommutative,anda_GCb=b_Ga.

• Usingthenilpotentminimum,theaggregationdowngradesto0alllocalratingslessthan1−

π

i,andperformsastandard

weighted minimumoflocalratingsontherestofthecriteria.

Example3. LetusconsiderL=

{

j/10:j=0,1,...,10

}

⊂[0,1],threecriteriawithimportancevector

π

andthealternatives

f,gandh: 1 2 3 ∨iπi ⋆ RG f i ∨iπiG f i ∨iπiGC f i ∨iπi ∧ f i π 1 0.5 0.2 f 0 0.4 0.8 0 0 0 0.4 g 0.1 0.6 0.8 1 0.6 1 0.5 h 0 0.6 0.8 1 0.6 0.5 0.5

Optionf iseliminated bythreeoperations because itdoes notpass the requested thresholds.In particular,a grade0.8

isinsufficient to make itfor satisfyinga poorly importantcriterion with weight0.2. Option g isbetter evaluated because

it passes the threshold for the most important criteria 1 and 2. Option h is Pareto-dominated by g and only passes the

thresholdforcriterion2.handgreceivedistinctglobalratingswiththethirdaggregationbecausegpassesthethresholdon

criterion1thatismoreimportant thancriterion2.The usualweighted maximum,incontrast yieldsamorelenient global

evaluationforf,butcannotdistinguishgfromh.

Remark1. ThetransformationofanaggregationoperationAon theunitinterval intoafuzzyimplicationusinganegation

isthe topic ofan recentextensive review[34]. Howeverthese authors focuson the transformationA(a, b)7→A(n(a),b) for

a general negation n, in thespirit of material implication induced bya disjunction. The reader can find manyproperties

inducedonfuzzy implicationsfromthepropertiesoftheaggregationfunctionbythistransformation,whichisthe

counter-partfor fuzzy disjunctionsofsemidualityfor fuzzy conjunctions.Namelytheir resultscould beequivalentlyspelled outin

termsofsemiduality.

3. Algebraicconsiderations

Let us consider the following transformations that can be applied to some fuzzy conjunction ⋆ on a qualitative scale

Fig. 1. Connectives induced by the minimum on a finite chain.

Fig. 2. Residuation and semiduality on a finite chain.

• Residuation:aRes

(

⋆

)

b:= W

{

x:a⋆x≤b

}

,and wedenote →=Res

(

⋆

)

.

• Semiduality:aS

(

⋆

)

b:=1−

(

a⋆

(

1−b

))

, andwedenote⇒=S

(

⋆

)

.

• Contraposition:aC

(

⋆

)

b:=

(

1−b

)

⋆

(

1−a

)

, andwedenote⋆C=C

(

⋆

)

.

• Argumentexchange:aE

(

⋆

)

b:=b⋆a.

• DeMorganduality:aD

(

⋆

)

b:=1−

(

1−a

)

⋆

(

1−b

)

.

Note that semiduality, contraposition, and argument exchange are involutive transformations that are always defined.

However, residuationis notdefinedfor any operation. Itis well-definedonlyif theset {x:a ⋆ x ≤b} isnot empty, which

meansthat

∀

a,b∈L,

∃

xs.t.a⋆x ≤b.Thiscondition holdsforfuzzyconjunctionsasa⋆0=0≤b.

3.1. Relationshipsbetweenfuzzyimplicationsandconjunctions

It isknown thatsome fuzzyimplicationsand conjunctionspresentedinSection 2.2areinterdefinableviasuch

transfor-mations[9].As anexample, thediagraminFig.1,where⋆=∧, commutesand involves Gödeland Kleene-Dienes

implica-tions.

Note that thediagramalsocommutes ifwestartwiththenilpotentminimuminsteadoftheminimum,but itcollapses

toasingletransformationsinceitiswell-knownthat Res

(

∧

)

=S

(

∧

)

=→andC

(

→

)

=→.

Thisdiagramcanbegeneralizedtomoregeneralfuzzy conjunctions[21].Weconsideraqualitativescale

(

L,≥,0,1,1−

)

,

equippedwithafuzzyconjunction⋆(seeDefinition2).ThediagramonFig.2presentsthismoregeneralsettingwhere

oper-ations⋆,=S

(

Res

(

⋆

))

arefuzzyconjunctions,andoperations→=Res

(

⋆

)

,⇒=S

(

⋆

)

arefuzzyimplications(seeDefinition1).

Fodor [21] has proved that this diagram commutes if ⋆ is commutative. More precisely in the finite setting (the

left-continuityof⋆isnotneeded) Fodorprovedthat:

• Res◦ S◦Res

(

⋆

)

=S

(

⋆

)

if andonlyif {a:a⋆b≤c}isneverempty.Thisconditionisverifiedforallfuzzyconjunctionsin

thesenseofDefinition2.

• Res◦ E◦ S◦Res

(

⋆

)

=C◦Res

(

⋆

)

(thatis,onthediagram,Res◦ E

(



)

=→C),if andonlyif ⋆iscommutative.

There are some usefulconnections between the properties of the commutative fuzzy conjunction ⋆ at the top left of

Fig.2andthepropertiesofotherconnectivesinthediagram.Thereadercancheckthefollowingeasyresults:

Proposition1. Let⋆be acommutativefuzzyconjunctionwithidentity1:

• ⇒=S

(

⋆

)

isafuzzyimplicationsuchthat1⇒b=b,anda⇒0=1−a.

• →=Res

(

⋆

)

isafuzzyimplicationsuchthata→b=1whenevera≤b,1→b=b,andifa>0,a→0=a⋆wherewedenote a⋆=W

{

x:a⋆x=0

}

.

•  =S

(

→

)

isaleft-conjunctionsuchthatab=0wheneverb≤1−a,anda1=1−a⋆.

• →C=C

(

→

)

isafuzzyimplicationsuchthata→b=1whenevera≤b,1→Cb=

(

1−b

)

⋆,anda→C0=1−a.

Note thattheabovepropositionapplieswhen⋆isanytriangularnormonafinitechain(andany left-continuoust-norm

ontheunitinterval, moregenerallyany left-continuoussemi-copula).

The properties, neutrality of truth 1⇒b=b and strong negation principle a⇒0=1−a [34], can be related to the

propertiesofitssemidual.Namelythefollowingclaimsareeasilyverified:

Proposition2. Letbeafuzzyconjunctionand→be itssemidualfuzzyimplication:

•  isaleft-conjunctionifandonlyifitssemidualverifies1→b=b.

Fig. 3. Connectives associated to the Rescher-Gaines implication on a finite chain.

Fig. 4. Connectives induced by De Morgan law.

NotethatRescher-Gainesconjunction⋆RG (definedinSection2.3)isneitherarightnoraleft-conjunction, butitis

com-mutative,so we cangetanother special caseofthe abovecommutative diagram.Actually, ⋆RG is acommutativeoperation

that has no identity. In a numerical setting, it considers that obtaining a ratingb for a criterionof importance a isfully

acceptableiftheaverageofaandbisstrictlygreaterthan1/2,andfullyrejectedotherwise.

The (largely ignored, except in thethree-valued case, see [6]) three pairsof semidual fuzzy implication functions and

conjunctions related to Rescher-Gaines implication, namely (⇒RG, ⋆RG), (→ RG, RG), and (→ CRG, CRG), are pictured on

Fig.3where(⇒RGand ⋆RG)aredefinedinSections2.2 and2.4respectively, andtheotherconnectivesaresuchthat

a→RGb=

½

1 ifb=1, 1−a ifb6=1; aRGb=

½

0 ifb=0, a ifb6=0; (6) a→CRGb=

(

1−b

)

→RG

(

1−a

)

; a_CRGb=b_RGa. (7)

Wecancheckthat

• Thesemidualof⋆RG(Rescher-Gainesimplication)isa{0,1}-valuedimplicationfunction,hencesuchthat1⇒RGb6=b,and

a⇒RG06=1−a,ifa,b6= 0,1.

• Itsresiduum →RG isafuzzy implicationfunctionsuch that1→RGb6= bfor b6= 0,1, but a→RG0=1−a. Asexpected,

thesemidualfuzzyconjunction_RG of→RGisaright-conjunction:aRG1=a.

• Similarresultsholdforthelastpairofresidualconnectives,butnow,1→CRGb=b, asCRGisaleft-conjunction.

3.2. Relationshipsbetweenfuzzydisjunctions anddifferences

Interestinglyenough,thediagraminFig.2canbechangedintothediagraminFig.4byDeMorgandualitywithD

(

⋆

)

=◦

(adisjunction),D

(

→

)

=(aset-difference),D

(



)

=(anon-commutativedisjunction).

Itcanbecheckedthat thedisjunctionsand differences inFig.4arestillexchanged byS , C andE asthecorresponding

fuzzy conjunctionand implicationfunctions inFig.2.Moreover, thetransformationRes thatlinksother verticesinFig.4is

definedby

aRes

(

•

)

b=^

{

x

|

a•x≥b

}

Toseeit:

ba =1−

(

1−a

)

→

(

1−b

)

=1−_

{

x:

(

1−a

)

⋆x≤

(

1−b

)

}

=^

{

1−x:1−

(

1−a

)

⋆x≥b

}

=^

{

1−x:a◦

(

1−x

)

≥b

}

.

This transformation has been considered in the fuzzy set literature for a long time [10]. Res may be called an

anti-residuation.

Proposition 3. Considering a fuzzy conjunction ⋆ and the associated disjunction D

(

⋆

)

=•, then Res

(

Res

(

⋆

))

=⋆ and Res

(

Res

(

•

))

=•,i.e.,ResandResaretheinverseofeachother.

Proof. ConsideringFigs. 2and 4and assuming the existenceof theinfimum and ofthesupremum,let usfirstprove that V

{

y,a→y≥b

}

=V

{

y:W

{

x,a⋆x≤y

}

≥b

}

=a⋆b.3 LetE

(

a,b

)

=

{

y:W

{

x,a⋆x≤y

}

≥b

}

.Note that:

Fig. _{5. Connectives induced by the maximum on a finite chain.}

1. a⋆b∈ E(a,b):indeed,takey=a⋆b andx=b.Thenitistruethat W

{

x,a⋆x≤a⋆b

}

≥b.

2. If y < a ⋆ b, then y6∈E

(

a,b

)

. Indeed, it would mean

∃

x ≥ b: a ⋆ x ≤ y < a ⋆ b, which is impossible since the fuzzy

conjunction⋆isorder-preserving.

So V_E

(

a,b

)

=a⋆b. ¤

Example4. Weconsider◦ =∨(seeFig.5),then,

• aRes

(

∨

)

b=b_Ga=1−

(

1−a

)

→G

(

1−b

)

=

½

0 ifb≤a

b ifb>a.

• b_Ga=1ifb≥1−aandb_Ga=b ifb<1−a.

Operation _G is a non-commutative disjunction (it coincides with classical disjunction for Boolean values), and b_Ga

expressesadifference(itcoincideswithb∧¬aforBooleanvalues).Ascanbeseen,b_Gaisaninterestingmodifiersuchthat

thevalue bispreservedifbisstrictlylessthan threshold1−aandisputto1otherwise.

The study ofdifferences and disjunctionsinrelationwith conjunctiveand disjunctive aggregations,and moregenerally

withqualitativeaggregation,isleftforfurtherstudies.

4. Sugeno-likeq-integralsandq-cointegrals

The weighted minimumand maximum can begeneralizedbyassigning relative weightsto subsets ofcriteria via a

ca-pacity, which is aninclusion-monotonic map

γ

:2C

→L (if A ⊆ B, then

γ

(A) ≤

γ

(B)),that also satisfies limit conditions

γ (

∅

₎

₌0and

γ (

C

)

=1.Thisgeneral formofweightingallowsustoexpress dependenciesbetweencriteriaordimensions.

The Sugeno integral [39,40] of analternative f canbe definedby meansof several expressions,among which the two

followingnormalforms[32]:

Z γ f =_ A⊆C

Ã

γ

(

A

)

∧^ i∈A fi

!

=^ A⊆C

Ã

γ

(

A

)

∨_ i∈A fi

!

. (8)

These expressions,whichgeneralizetheconjunctive and disjunctivenormal formsinlogic,can besimplified asfollows

[29,32,Th.4.1]: Z γ f =_ a∈L

γ

(

{

i: fi≥a

}

)

∧a= ^ a∈L

γ

(

{

i: fi>a

}

)

∨a. (9)

The first expression in (8) is the original formulation of Sugeno integral [39], where it is proven to be equivalent to

thefirstequalityin(9). Moreover,for thenecessitymeasureNassociated withapossibilitydistribution

π

,wehave R

Nf=

MINπ

(

f

)

;and forthepossibility measure

5

associated with

π

,wehaveR₅ f=MAXπ

(

f

)

[16,27].

4.1. Motivationsanddefinitions

Let the conjugate capacity

γ

c of

γ

be defined by

γ

c

(

A

)

=1−

γ (

A

)

for every A⊆ C. This duality relation extends to

Sugenointegral withrespecttotheconjugatecapacity,also extendingtheDeMorgandualitybetween weightedminimum

andmaximumgiven in(3):

Z γ f =1− Z γc

(

1−f

)

. (10)

Thisequality,provedin[27],stemsfromEq.(8).Itisinterestingtonoticethatthisequalitybetweenthetwonormalforms

canbeexpressed usingtheconjugatecapacityand theKleene-Dienesimplication →KD=S

(

∧

)

(definedin Section2.2), the

semidualof∧: Z γ f =_ A⊆C

Ã

γ

(

A

)

∧^ i∈A fi

!

=^ A⊆C

Ã

γ

c

(

A

)

→KD _ i∈A fi

!

. (11)

So,Sugenointegralcanbegeneralizedwithotherinnerfuzzyconjunctionorimplicationfunctionslinkedbysemiduality,

using expressions we call q-integrals and q-cointegrals, denoted respectively by R

γ f and

R→

γ f, for every capacity

γ

and

every f ∈LC_.

Definition 3. Let be a fuzzy conjunction in the sense of Definition 2and

γ

:2C

→L be a capacity. A q-integral is the

mappingR γ :L C →L definedby Z  γ f =_ A⊆C

Ã

γ

(

A

)

^ i∈A fi

!

, forall f ∈LC .

When  isthe product and L=[0,1], thisis Shilkret integral [38] proposed in 1971(and laterreintroduced by

Kauf-mann[30]in1978, underthenameof“admissibility”).Grabischetal.[27]introduceso-called Sugeno-likeintegrals,which

are similar to q-integrals, where  is a triangular norm. Borzová-Molnárová et al. [2] study this type of integrals in the

continuous case as well when  is a semicopula and L=[0,1]. This kind of definition is also proposed by Dvoˇrák and

Holˇcapek [20]assuming (L, ,1) isa commutative monoidand considering thecomplete residuated lattice generated by

thismonoidaloperation.Infact, whattheystudy isanextension ofDefinition3:Namely, theystudyfuzzy integralsofthis

typeextendedtoalgebrasoffuzzysets,thatis,where

γ

isnow afuzzysetfunctionthat assignsanimportancevalue

γ (

A˜

)

to anyfuzzysubsetA˜ ofC.

Definition4. Let →beafuzzy implicationinthesenseofDefinition 1and let

γ

:2C

→L beacapacity.Aq-cointegralisa mappingR→ γ :L C →Ldefinedby Z → γ f =^ A⊆C

Ã

γ

c

(

A

)

→_ i∈A fi

!

, for all f ∈LC .

There are not many works considering q-cointegrals in the above sense. Grabisch et al.[27, p. 302] notice the failure

of the duality relation (10) for Sugeno-like integrals that uset-norms other than min, which hints at co-integrals using

implicationsthat aresemi-dualsoft-norms.

Withthisterminology, Eq.(8)actuallystatesthat

Z γ f = Z ∧ γ f = Z →KD γ f

forevery capacity

γ

and every f ∈LC_{.ItmeansthatSugeno integralsand cointegralsdefinedbymeansoftheoperation}

∧

and itssemidualKleene-Dienes implication,respectively, coincide. Asalready seenusingGödel implicationfor→G in [16],

thisisnotgenerallythecase.Sowestudybothq-integralsandq-cointegralsseparatelyinthesequel.

Inthefollowingwhenweconsider f∈LC

,themapping i7→(i)denotesapermutationonthesetofcriteria suchthatf₍₁₎

≤ ÅÅÅ ≤ f(n)and weletF(i)=

{

(

i

)

,. . .,

(

n

)

}

withtheconventionF(n+1)=∅.ForallfinLC wehave f=Wni=11F(i)∧f(i),where

1AisthecharacteristicfunctionofA.

4.2. Elementarypropertiesofq-integrals

WefirstprovethatthecounterpartoftheSugenointegralexpressionontheleft-handsideof(9),intermsofthenested

familyofsubsetsF_(i)inducedbyf,isstillvalidforfuzzyconjunction-based q-integrals:

Proposition4. R γ f = Wn i=1

γ (

F(i)

)

f(i). Proof. R γ f= W

A⊆C

γ (

A

)

Vi∈Afi.Letindex iA ∈ Abesuchthat Vi∈Afi= fiA.ItfollowsthatA⊆F(iA) which entails

γ (

A

)

≤

γ (

F_(i

A)

)

and

γ (

A

)

fiA ≤

γ (

F(iA)

)

fiA.So,if themaximum isattainedforset A,itisalsoattainedbyset F(iA). And

R

γ f =

Wn

i=1

γ (

F(i)

)

f(i),noticingthat f(i)= f(iF_(i)). ¤

Theconjunction-basedq-integralismonotonic:

Lemma1. For everycapacity

γ

,themapR

γ :L

C

→Lisorder-preserving.

Proof. Directlyfromtheassumptionthat themapx7→axisorder-preserving. ¤

Anothersimplified formofSugenoq-integral isvalidforconjunction-basedq-integrals:

Lemma2. R

γ f =Wa∈L

γ ({

f ≥a

}

)

a.

Proof. WeuseProposition4.Let a∈L

Ifa>f_(n) then

γ ({

f ≥a

}

)

a=0a=0.

If f_(i−1)<a≤ f_(i) then

γ ({

f≥a

}

)

a=

γ ({

f ≥f_(i)

}

)

a≤

γ ({

f ≥f_(i)

}

)

f_(i). ¤

Borzová-Molnárová et al. [2]also prove this result when  is asemicopula.We show here it is validfor a larger class of

fuzzyconjunctions.

Finally, if

γ

isapossibilitymeasureweretrievethe-weightedmaximum:

Proposition5. If

γ

isapossibility measure

5

basedonpossibilitydistribution

π

thenR

5 f =MAX



π

(

f

)

.

Proof. W

A⊆C

(5(

A

)

Vi∈Afi

)

=WA⊆C

(

∨i∈A

π

i

)

Vi∈Afi=∨ni=1

π

ifi since letting

π

iA=∨i∈A

π

i, we do have the inequality

5(

A

)

V

i∈Afi≤

π

iAfiA,asisorder-preserving. ¤

The proof of this proposition is the same as for Sugeno integral proper [8, p. 138–139] and [27]. However, if

γ

is a

necessitymeasure,theexpression oftheq-integral will notsimplify when 6=min.In other wordswe do nothave that

R

N f=MINπ→

(

f

)

for→=S

(



)

.

4.3. Elementarypropertiesofq-cointegrals

Using semiduality,q-cointegralscanbeexpressedintermsofq-integralsbyextendingthedualityformula(10):

Proposition6. If=S

(

→

)

,thenR→ γ f =1− R γc

(

1−f

)

. Proof. 1−W A⊆C

(γ

c

(

A

)

Vi∈A

(

1−fi

))

=VA⊆C1−

(γ

c

(

A

)



(

1−Wi∈Afi

)

=VA⊆C

γ

c

(

A

)

→Wi∈Afi=Rγ→f . ¤

As in [17] (but now in a more general setting), using semiduality, we derive the following results from the ones on

(conjunction-based)q-integrals: Z → γ f = n ^ i=1

γ

c

₍

_F (i+1)

)

→ f(i) = ^ a∈L

γ

c

₍

_{

_{f ≤a}

_}

₎

_{→a. (12)}

We also get that when

γ

is a necessity measure N based on possibility measure

π

, the q-cointegral reduces to the

→-based weightedminimum:

Z →

N

f =MIN→

π

(

f

)

=∧ni=1

π

i→ fi

Toseeit,justusethesemidualfuzzy conjunction=S

(

→

)

:

Z → N f =1− Z  5

(

1−f

)

=1−MAX π

(

1−f

)

=MIN S() π

(

f

)

.

Howevertheq-cointegralwithrespecttoapossibilitymeasureR→

5 f doesnotreducetoaweightedmaximum.

4.4. Somehintsonthecomparisonbetweenq-cointegralsandq-integrals

The equality(8)betweenq-cointegralsand q-integralsusingfuzzy conjunction∧and itssemidual→KDdoesnotextend

toconjunction-basedq-integrals.Forexample,usingGödelimplication-relatedconnectives,onlytheinequality:R

γ f≥

R→

γ f

holdsfor(,→)∈ {(_G,→G),(GC,→GC)}[16].Thisinequalitycannoteven begeneralizedtootherfuzzy conjunctions.

To getabetter insight on thecomparison betweenq-integrals and q-cointegrals,consider thecase ofaprofile f =aAb

suchthat f_i=a>b if i∈A and f_i=b otherwise. Thenitis easytocheck usingLemma 2, andEq. (12)that theq-integral

andq-cointegralofaAbreduceto

Z  γ

(

aAb

)

=

(

1b

)

∨

(

γ

(

A

)

a

)

; Z → γ

(

aAb

)

=

(

γ

c

(

A

)

→b

)

∧

(

1→a

)

.

Supposeforsimplicitythatisaleft-conjunction and→isitssemidual.Thenthefirstexpressionsimplifiesasfollows:

Z 

γ

(

aAb

)

=b∨

(

γ

(

A

)

a

)

.

Besides

(γ

c

(

A

)

→b

)

∧

(

1→a

)

=a∧

((

1−

γ (

A

))

→b

)

=a∧

(

1−

(

1−

γ (

A

))



(

1−b

))

. So, usingthe De Morgandual =

D

(



)

of,wecanwrite theq-cointegralas

Z →

γ

(

aAb

)

=a∧

(

γ

(

A

)

b

)

.

• Sincewhen=∧theq-integralandq-cointegralscoincide,wenoticethat ifisatriangularnorm,orasemicopula,in

factanyfuzzyconjunctiondominatedbytheminimum,thenR→

γ

(

aAb

)

≥

R

γ

(

aAb

)

,withoutequalityingeneralwhenever

0<

γ

(A)<1.

For example, consider the nilpotent minimum ∧ [22] and its De Morgan dual a∨b=

½

a∨b ifa≤1−b

1 otherwise called the

nilpotent maximum. Let us show a case where the inequality R∧

γ f <

R→

γ f holds, where the fuzzy implication

func-tion →=S

(

∧

)

=Res

(

∧

)

. Suppose b<a<1−

γ (

A

)

and b < a <

γ

(A). Then R∧

γ aAb=b∨

(γ (

A

)

∧a

)

=b∨0=b while

R→

γ aAb=a∧

(γ (

A

)

b

)

=a∧

(γ (

A

)

∨b

)

=a.

• However, if  is not upper-bounded by the minimum, then the other inequality R_γ→

(

aAb

)

≤R_γ

(

aAb

)

may hold. For

instance, thisis the casewith the semidual of the contrapositivesymmetric of Gödel implication, without equalityin

general [16,p.773].

5. Characterizationresults

Sugenointegrals havebeen characterized bymeansofafew properties pertainingtoco-monotonic minitivity or

maxi-tivityand thecorrespondingformsofhomogeneity.Letusrecallthemainresults.

Twoalternatives f,g∈LC _{are said tobe comonotonicif for every i,j}

∈ [n], if f(i) < f(j) theng(i) ≤g(j).Sugeno integral

canthenbecharacterizedasfollows:

Theorem1. LetI:LC

→L.Thereisacapacity

γ

suchthatI

(

f

)

=R

γ f forevery f∈L

C _{ifandonlyifthefollowingpropertiesare} satisfied

1. I

(

f∨g

)

=I

(

f

)

∨I

(

g

)

,foranycomonotonic f,g∈LC_.

2. I

(

a∧f

)

=a∧I

(

f

)

, foreverya∈L and f ∈LC ₍

∧-homogeneity).

3. I

(

1C

)

=1.

Mostolder formulationsofthistheorem[4,35]addanassumptionofincreasingmonotonicityofthefunctionalI(iff≥g

thenI(f)≥I(g))tothethreeconditions (1–3).However, theproof thatconditions (1–3) arenecessaryand sufficientseems

tofirstappearonly inthethesisofRico[36](laterreproduced inthebook[26],and alsoin[28]).

Conditions(1–3) inTheorem1canbeequivalentlyreplacedbyconditions(1’–3’)below.

1’. I

(

f∧g

)

=I

(

f

)

∧I

(

g

)

,foranycomonotonic f,g∈LC_.

2’. I

(

b∨f

)

=b∨I

(

f

)

,foreverya∈L and f∈LC ₍

∨-homogeneity).

3’. I

(

0C

)

=0.

TheexistenceofthesetwoequivalentcharacterizationsisduetothepossibilityofwritingSugenointegralinconjunctive

and disjunctive forms(8) equivalently. As this identity isno longer valid for moregeneral forms ofthisintegral, weshall

characterizeq-integralsw.r.t.afuzzyconjunctionandq-cointegralsw.r.t.afuzzyimplicationseparatelyinthefollowing.The

aimisto lookfor minimalproperties offuzzy conjunctionsand implicationsthat ensurethevalidityof such

characteriza-tionsinthestyle ofTheorem1.

5.1. Characterizationofq-integralsassociatedtoafuzzyconjunction

Wefirstverifythatq-integralssatisfycomonotonicmaxitivityand ageneralizedformof∧-homogeneity.

Lemma3. For anyfuzzyconjunctionandanytwocomonotonicfunctions f,g∈LC

,wehaveR γ

(

f∨g

)

= R γ f∨ R γ g.

Proof. TheinequalityR

γ

(

f∨g

)

≥

R

γ f∨

R

γ g follows from Lemma1. Letusprove theother inequality.Let a∈ L. Forany

twocomonotonicfunctions f,g∈LC _{wehaveeither{f}

≥a}⊆{g≥a} or{g≥a}⊆{f≥a}.

If {f ≥a}⊆{g≥ a} then

{

f∨g≥a

}

=

{

f≥a

}

∪

{

g≥a

}

=

{

g≥a

}

and

γ ({

f∨g≥a

}

)

a=

γ ({

g≥a

}

)

a≤

(γ ({

g≥a

}

)



a

)

∨

(γ ({

f≥a

}

)

a

)

.Bysymmetry,theinequalityisalsotruewhen{g≥a}⊆{f≥a};henceR

γ

(

f∨g

)

≤ ∨a∈L

((γ ({

g≥a

}

)



a

)

∨

(γ ({

f≥a

}

)

a

))

=R

γ f∨

R

γ g. ¤

Lemma4. For every f ∈LC _andevery

ℓ∈

{

1,...,n−1

}

,themaps1F_(ℓ)∧ f(ℓ) and Wni=ℓ+11F_(i)∧f(i) arecomonotonic.

Proof. We represent both maps as vectors of components ordered according to

(

1

)

,. . .,

(

n

)

, so that F_(ℓ)=

{

(

ℓ

)

,. . .,

(

n

)

}

. In consequence, 1F_(ℓ)∧f(ℓ)

(

i

)

=0 if i ≤ ℓ while 1F_(ℓ)∧f(ℓ)=

(

0,...,0,f(ℓ),...,f(ℓ)

)

; and Wni=ℓ+11F_(i)∧f(i)=

(

0,· · · ,0,0,f_(ℓ+1),...,f_(n)

)

.Henceitiseasytocheckthat thetwomapsarecomonotonic. ¤

Lemma5. For anycapacity

γ

,anyB⊆ C andanya∈LwehaveR

γ

(

a∧1B

)

=

γ (

B

)

a.InparticularR_γ1C=1.

Proof. UsingLemma2,wealreadyestablishedthattheq-integralofaAbreducestoR

γ

(

aBb

)

=

(

1b

)

∨

(γ (

B

)

a

)

.Ifb=0,

weimmediatelyconcludesinceR

Note that contrary to when =∧, we do not have R γ

(

a∧f

)

= R γ

(

f

)

a; namely if

{

i: f(i)≥a

}

=B, R γ

(

a∧f

)

=

(γ (

B

)

a

)

W ∨_i6∈B

(γ (

F_(i)

)

f_(i)

)

clearlydiffersfromR

γ

(

f

)

a=∨ni=1

(γ (

F(i)

)

f(i)a

)

,ingeneral.

Wecannowproveourfirstcharacterizationresult.

Theorem 2. Let I:LC

→L be a mapping. There are a capacity

γ

and a fuzzy conjunction  such that I

(

f

)

=R

γ f for every

f∈LC _ifandonlyif

1. I

(

f∨g

)

=I

(

f

)

∨I

(

g

)

,foranycomonotonic f,g∈LC_.

2. Thereareacapacity

λ

: 2C

→Landabinaryoperation⋆onLsuchthatI

(

a∧1A

)

=

λ(

A

)

⋆aforeverya∈LandeveryA⊆ C.

3. I

(

1C

)

=1andI

(

0C

)

=0.

In thatcase, wehave

γ

=

λ

and=⋆.

Proof. Necessity isobtained bypreviousLemmas 3and5. Forsufficiency,assume that Iisa mappingthat satisfies

condi-tions1and 2andlet f∈LC_.WehaveI

(

f

)

=I

(

Wn i=1f(i)∧1F(i)

)

.UsingLemma 4, I

(

f

)

= Wn i=1I

(

f(i)∧1F(i)

)

= Wn i=1

λ(

F(i)

)

⋆f(i).

Nowwemustprovethat⋆isafuzzyconjunction.Namely

• ⋆ is increasing in the wide sense in both places. If a ≥ b, then a∧1_A ≥ b∧1_A pointwise, and the two profiles are

comonotonic.Thus bycondition 1, wefindthat I

((

a∧1A

)

∨

(

b∧1A

))

=I

(

a∧1A

)

∨I

(

b∧1A

)

=I

(

a∧1A

)

, since

(

a∧1A

)

∨

(

b∧1A

)

=a∧1A.Using condition 2, itreads

λ

(A)⋆a ≥

λ

(A)⋆b. Likewiseusing A⊂B, wecan provewith thesame

argu-mentthat

λ

(B)⋆a≥

λ

(A)⋆a.Indeed,1A∧aand1B∧aarecomonotonicand

(

1A∧a

)

∨

(

1B∧a

)

=1B∧a.Condition1implies

I

((

1A∧a

)

∨

(

1B∧a

))

=I

(

1B∧a

)

=I

(

1A∧a

)

∨I

(

1B∧a

)

.SoI(1B∧a)≤I(1A∧a),whichreads

λ

(B)⋆a≥

λ

(A)⋆a.So,since

λ

is

acapacity,⋆isorder-preservingontheleftplaceaswell.

• I

(

1∧1C

)

=1⋆1=I

(

1C

)

=1

• I

(

0∧1A

)

=

λ(

A

)

⋆0=I

(

0C

)

=0,

• I

(

a∧1∅

)

=0⋆a=I

(

0C

)

=0.

So wecanlet

γ

=

λ

and =⋆andgetI

(

f

)

=R_λ f. ¤

This result is not a generalization of Theorem 1, since Theorem 2 assumes

λ

is a capacity and characterizes

operation ⋆. A counterpart of Theorem 1 would assume a fuzzy conjunction , i.e., respecting the three conditions of

Definition 2. It is easy to prove, following the same steps as in [26,28,36] for the proof of Theorem 1, that due to the

monotonicity and extremal properties of , the set-function A7→I

(

1A

)

=

γ (

A

)

1 is a capacity

γ

ˆ that generally differs

from

γ

, and contrary to universalintegrals [33],R

γ 1A=

γ (

ˆ A

)

6=

γ (

A

)

, generally, since1is not necessarilyanidentity on

theright. Thefollowingcounter-example showsthat wemayhave

γ (

ˆ A

)

6=

γ (

A

)

and R

γ 6=

R

ˆ

γ .

Example5. C=

{

1,2

}

, L=

{

0,0.2,0.8,1

}

and _G istheGödelconjunction.Wehave

γ (

A

)

_G1=1if

γ

(A)>0and0

other-wiseso

γ

(A)_G16=

γ

(A)if0<

γ

(A)< 1.Letusconsider

γ

suchthat

γ ({

1

}

)

=0and

γ ({

2

}

)

=0.3.Iffisdefinedby f₁=0

and f₂=0.8weobtain RG

γ f=

(

1G0

)

∨

(

0.3G0.8

)

=0.8and R_γ_ˆ G f=1since

γ ({

ˆ 2

}

)

=1.

Corollary 1. Ifthe fuzzy conjunctionis aright-conjunction, thenunder theassumptionsof Theorem2,the functional Iis of

theformI

(

f

)

=R

γ f where

γ (

A

)

=I

(

1A

)

.

Proof. Due toTheorem2,weknowthat I

(

1A

)

=

γ (

A

)

1foracapacity

γ

.As isaright-conjunction,

γ (

A

)

1=

γ (

A

)

=

I

(

1A

)

. ¤

Due to Proposition1,operation canbebuilt from exchangingargumentsto afuzzy conjunctionthat isthesemidual

oftheresiduum ofanotherfuzzyconjunctionthat hasatwo-sidedidentity1,suchasasemicopula.

InthecasewhenthefunctionalIisfullymaxitive,wecanextendtherepresentationTheorem2ifwe slighlyrestrict the

notionofafuzzyconjunction:

Definition5. Abinaryoperation J is said tobex-right-cancellative ifand only if for every a,b ∈ L, if aJ_{x =b}J_{x then}

a=b.

Theorem3. LetI:LC

→Lbeamapping.Thereareapossibilitymeasure

5

andafuzzyconjunctionsuchthatI

(

f

)

=R

5 f for

every f∈LC _{ifandonlyifIsatisfiesthefollowingproperties:}

1. I

(

f∨g

)

=I

(

f

)

∨I

(

g

)

,forany f,g∈LC_.

2. Thereareacapacity

λ

: 2C

→Landabinaryoperation⋆onLsuchthatI

(

a∧1_A

)

=

λ(

A

)

⋆aforeverya∈LandeveryA⊆ C.

3. I

(

1C

)

=1andI

(

0C

)

=0.

In thatcase, wehave⋆=and,if⋆is1-right-cancellative,wehave

5

=

λ

.

Proof. A q-integral with respect to a possibility measure satisfies the requested properties. Let us prove the converse.

According to the previous theorem, there exists a capacity

λ

and a fuzzy conjunction ⋆ such that I

(

f

)

=R⋆

λ f . For all