forweightednilpotentoperators Self-dual operators and a general framework International Journal of Approximate Reasoning

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Contents lists available atScienceDirect

International Journal of Approximate Reasoning

www.elsevier.com/locate/ijar

Self-dual operators and a general framework for weighted nilpotent operators

J. Dombi

^a

, O. Csiszár

^b^,^∗

aUniversityofSzeged,InstituteofInformatics,Hungary

bÓbudaUniversity,InstituteofAppliedMathematics,Hungary

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

Articlehistory:

Received2April2016

Receivedinrevisedform11July2016 Accepted7November2016

Availableonline17November2016 Keywords:

Nilpotentoperators Boundedsystem Aggregativeoperator Weightedaggregation Uninorm

Preferencemodeling

The main purpose of this paper is to consider generated nilpotent operators in an integrative frameand to examinethe nilpotentaggregativeoperator. As a startingpoint, instead of associativity, we focus on the necessary and suﬃcient condition of the self- dualproperty.A parametricformofthegeneratedoperatoro_ν isgivenbyusingashifting transformation of the generator function. The parameter has an important semantical meaning as athreshold ofexpectancy(decision level). Nilpotentconjunctive,disjunctive, aggregativeandnegationoperatorscanbeobtainedbychangingtheparametervalue.The properties (De Morgan property, commutativity, self-duality, fulfillment of theboundary conditions,bisymmetry)oftheweightedgeneraloperatorareexaminedandtheformulaof thecommutativeself-dualgeneratedoperator,theso-calledweightedaggregativeoperator is given. It is proved that the two-variable operator with weights w₁=^w2=¹ ∀ⁱ ^is conjunctive for lowinput values,disjunctive for high ones, andaveraging otherwise; i.e.

ahighinputcancompensateforalowerone.

1. Introduction

Oneofthemostsignificantproblemsoffuzzysettheoryistheproperchoiceofset-theoreticoperations[29,32].Triangu- larnormsand conorms havebeen thoroughlyexaminedintheliterature [14,15,19,22],and areoften used asconjunctions anddisjunctions inlogicalstructures[18,27].

The most well-characterized classof t-normsis the so-called representable t-norms. t-norms generatedby continuous additivegeneratorsweredescribedbyMostertand Shield [26].The twomaintypesofrepresentablet-normsarethestrict andnon-strictornilpotentt-norms.Thenilpotentoperatorshavesomenicepropertieswhichmakethemmoreusefulwhen constructinglogicalstructures.Amongthesepropertiesarethefulfillmentofthelawofcontradictionandtheexcludedmid- dle,and thecoincidence oftheresidual and theS-implication[11,31]. In[8], Dombi and Csiszárshowedthat a consistent connective system generated by nilpotent operators is not necessarily isomorphic to the Łukasiewicz-system. Using more thanonegeneratorfunction,consistentnilpotentconnectivesystems(so-calledboundedsystems)canbeobtainedinasig- nificantlydifferentwaywiththreenaturallyderivednegationoperators.DuetothefactthatallcontinuousArchimedean(i.e.

representable) nilpotent t-normsare isomorphic to theŁukasiewicz t-norm [19], thepreviously studiednilpotent systems wereallisomorphictothewell-knownŁukasiewicz-logic. In[9]and in[10], Dombiand Csiszárexamined theimplications andequivalenceoperatorsinboundedsystems.

*

Correspondingauthor.

E-mailaddresses:dombi@inf.u-szeged.hu(J. Dombi),csiszar.orsolya@nik.uni-obuda.hu(O. Csiszár).

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In human thinking, averaging operators, where a high input can compensate for a lower one, play a significant role.

The aggregativeoperator was first introducedin 1982 byDombi [7], byselectinga set ofminimal conceptsthat must be fulfilledbyanevaluation-likeoperator.Theconceptofuninormswasintroducedin[33],asageneralizationofbotht-norms andt-conorms.Byadjustingitsneutralelement ν,a uninormisat-normif ν₌1 and at-conormif ν₌0.Uninormshave turnedouttobeusefulinmany areaslikeexpertsystems[6],aggregation[3,34]and thefuzzy integral[4,21].

The main difference in the definition of the uninorms and aggregative operators is that the self-duality requirement does not appear in uninorms, and the neutral element property is not in the definition for the aggregative operators.

The representation theorem for strict, continuous on [⁰,1]×[⁰,1]\{(0,1),(1,0)} ^uninorms ^(or representable uninorms) was given by Fodor et al. [16](see also Klement etal. [20]). Suchuninorms are calledrepresentable uninorms and they werepreviously introducedasaggregative operators[7]. Recently,a characterization of theclassofuninorms withastrict underlyingt-normand t-conormwaspresentedin[13].In[24],theauthorsshowthatuninorms withnilpotentunderlying t-norm and t-conorm belong to Umin or Umax. Further results on uninorms with fixed values along their borders can be foundin[5].

Ourmainpurpose hereis toconsider generated nilpotent operators inanintegral frame and to examine thenilpotent self-dual generated operators. A general parametric framework for the nilpotent conjunctive, disjunctive, aggregative and negation operators is given and it is demonstrated how the nilpotent generated operator can be applied for preference modeling.

The articleis organizedas follows.After a preliminary discussion in Section 2, a general parametric operator o_ν(x) of nilpotentsystemsisgiveninSection3.Theparameterhas animportantsemanticalmeaningasthethresholdofexpectancy.

In Section 4, the weighted form ofthis operator, a_ν_,w(x) is examined. In Section 5, the properties (De Morgan property, commutativity, self-duality, fulfillment of theboundary conditions, bisymmetry) of theweighted general operator are examined. Here, the formula for the commutative self-De Morgan operator, the so-called weighted aggregative operator is presented. Then in Section 6 we focus on the two-variablecase, where it is proved that the two-variable operator with weights w1=^w2=^{1 is} conjunctive for low input values, disjunctive for high ones, and averaging otherwise; i.e. ahigh input can compensate for a lower one. In Section 7, the main results are summarized and a possible direction of future workismentioned.

2. Preliminaries

2.1. Negations,t-normsandt-conorms

First,werecallsomebasicnotationsandresultsregardingnegationoperators,t-normsandt-conormsthatwillbeuseful inthesequel.

Definition1.Aunary operation n: [⁰,1]→[⁰,1]îs ^calledâ ^negationîf îtîs non-increasing and compatible with classical logic;i.e.n(0)=^{1 and}ⁿ(1)=^0.

Anegationisstrictifitisalsostrictlydecreasing andcontinuous.

Anegationisstrong,if itisalsoinvolutive;i.e.n(n(x))=^x.

Thewell-knownrepresentationtheoremforstrongnegations wasobtainedbyTrillasin[30]:

Proposition1.n(x): [⁰,1]→[⁰,1]îsâ^strong^negationîfândônlyîf^thereêxistsânîncreasing^bijection ^fn(x): [⁰,1]→[⁰,1]^such that

n(x)= ^fn⁻¹(1−^fn(x)).

Remark1.InProposition 1,thebijectionmayalsobedecreasing(seeDombi andCsiszár[8]).

Definition2.Leto(x,y): [⁰,1]²→[⁰,1]^,ând^letⁿ(x)bethenegationgeneratedby f(x): [⁰,1]→[⁰,1]^.^Theôperatorô(x,y) satisfiestheself-DeMorganpropertyifitsatisfiesthefollowingequationforallx,y∈[⁰,1]^:

n(o(x,y)=^o(n(x),n(y)).

Atriangular norm (t-norm for short) T is a binary operation on the closed unit interval [⁰,1] ^such ^that ([⁰,1],T) is anabelian semigroup withneutral element 1 which is totally ordered;i.e., for allx1, x2, y1, y2∈[⁰,1] ^with ^x¹≤^x² ând y1≤^y2,wehave T(x1,y1)≤^T(x2,y2),where≤îs^the^naturalôrderôn[⁰,1]^.

Atriangularconorm(t-conormforshort) S isabinaryoperationontheclosedunitinterval [⁰,1]^such^that ([⁰,1],S) is anabeliansemigroupwithneutralelement 0 whichistotallyordered.

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A continuous t-norm T is said to be Archimedean if T(x,x)<x holds for all x∈(0,1), strict if T is strictly increasing on (0,1]²^; ^i.e. ^T(x,y)<T(x,z) whenever x∈(0,1] ^and ^y<z, and nilpotent if eacha∈(0,1) is anilpotent element;i.e.

∃ⁿ∈{¹,2,...}^such^that ^T(a,a, ...a

! "# $

n-times

)=^{0 for}^any^a∈(0,1).

Fromthedualitybetweent-normsandt-conormswecanreadilyobtain thesimilarpropertiesfort-conormsaswell.

Proposition2.[25]AfunctionT: [⁰,1]²→[⁰,1]îsâ^continuousArchimedeant-normifandonlyifithasacontinuousadditive generator;i.e.thereexistsacontinuousstrictlydecreasingfunctiont: [⁰,1]→[⁰,∞)witht(1)=^0,^whichîsûniquely^determinedûp toapositivemultiplicativeconstant,suchthat

T(x,y)=^t⁻¹(min(t(x)+^t(y),t(0))), x,y∈[⁰,1]. (1) Proposition3.[25]AfunctionS: [⁰,1]²→[⁰,1]îsâ^continuousArchimedeant-conormifandonlyifithasacontinuousadditive generator;i.e.thereexistsacontinuousstrictlyincreasingfunctions: [⁰,1]→[⁰,∞]^with^s(0)=^0,^whichîsûniquely^determinedûp toapositivemultiplicativeconstant,suchthat

S(x,y)=^s⁻¹(min(s(x)+^s(y),s(1))), x,y∈[⁰,1]. (2) Proposition4.[19]

At-normTisstrictifandonlyift(0)=∞^holds^forêach^continuousâdditive^generator^tôf^T.

At-normTisnilpotentifandonlyift(0)<∞^holds^forêach^continuousâdditive^generator^tôf^T.

At-conormSisstrictifandonlyifs(1)=∞^holds^forêach^continuousâdditive^generator^sôf^S.

At-conormSisnilpotentifandonlyifs(1)<∞^holds^forêach^continuousâdditive^generator^sôf^S.

InbothofPropositions 2 and3abovewecanpermitthegeneratorfunctionstobestrictlyincreasingor strictlydecreasing,whichwillmeanthattheycanbedetermineduptoa(notnecessarilypositive)multiplicativeconstant.Inthiscasewe havet(0)= ±∞ând^s(1)= ±∞^for^strict^normsând,^similarly,^t(0)<∞^{or t}(0)>−∞ând^s(1)<∞^{or s}(1)>−∞^for^the nilpotentones.

Fromthedefinitionsoft-normsandt-conormsitfollowsimmediatelythatt-normsareconjunctive,whilet-conormsare disjunctiveaggregationfunctions. Thisiswhytheyarewidelyusedasconjunctionsanddisjunctions inmultivalued logical structures.

Sincethegeneratorfunctionsofthenilpotentt-normsandt-conormsareboundedanddetermineduptoamultiplicative constant(see Propositions 2 and 3),theycan benormalized (see [8]).Let ususethe followingnotationsfor theuniquely definednormalizedgeneratorfunctions:

f_c(x):=^t(x)

t(0), f_d(x):= ^s(x) s(1).

Next, wedefinetheso-called cuttingfunction inordertosimplifythenotations.

Definition3.(SeeDombiand Csiszár[8],Saboand Strezo[28])Letusdefinethecuttingoperation[ ]^by [^x] =

⎧⎨

⎩

0 ifx<0 x if 0≤^x≤¹ 1 if 1<x

and letthenotation[ ] âlsoâct âs^brackets ^when^writing^theârgumentôfânôperator.^Then ^we^can^write ^f[^x] însteadôf f([^x]).

Definition4.Letusdefinethegeneralizedcuttingoperation[ ]^ba by [^x]^ba:=^max(a,min(b,x)),

wherea,b∈R,a<b.

Remark2.Fora=^{0 and}^b=^1,^we^get^the^cutting^function^definedⁱⁿDefinition 3.

Proposition5.(SeeDombiandCsiszár[8])Withthehelpofthecuttingoperator,wecanwritetheconjunctionanddisjunctionoper- atorsinthefollowingform,where fc(x)and fd(x)aredecreasingandincreasingnormalizedgeneratorfunctions,respectively.

c(x,y)= ^fc⁻¹[^fc(x)+ ^fc(y)], (3)

d(x,y)= ^f_d⁻¹[^fd(x)+ ^fd(y)]. (4)

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Remark3.Note that we usethe notationc(x,y) and d(x,y) for the conjunctionand disjunction toemphasize theuseof thenormalizedgeneratorfunctions.

Toconstructalogicalsystem,weneedtodefinetheappropriatelogicaloperators.Asin[8]and[9],weconsiderconnec- tivesystemswheretheconjunctionand disjunctionoperatorsarespecialtypesoft-normsandt-conorms,respectively.

Definition5.[8]Thetriple(c,d,n),wherecisacontinuousArchimedeant-norm,disacontinuousArchimedeant-conorm andnisastrongnegation,iscalledaconnectivesystem.

Definition6.[8] A connective system is nilpotent if the conjunction c is a nilpotent t-norm, and the disjunction d is a nilpotentt-conorm.

Definition7.Anoperatoro(x,y): [⁰,1]²→[⁰,1]^isbisymmetricifitsatisfiesthefollowingequationforallxi∈[⁰,1]^: o(o(x1,x2),o(x3,x4))=^o(o(x1,x3),o(x2,x4)).

Definition8.Anoperator o(x): [⁰,1]ⁿ→[⁰,1]îsîdempotentîf ^forâll^x=(x,. . .x),x∈[⁰,1]^: o(x)=^x.

Theconceptofaggregativeoperatorsand uninormswillplayanimportantroleinthesequel.

Definition9.(SeeDombi[7].) An aggregativeoperatorisafunction a: [⁰,1]→[⁰,1]^with^the^followingproperties:

1. Continuouson[⁰,1]²\{(0,1),(1,0)}^;

2. a(x,y)<a(x,y^′)if y<y^′,x̸=⁰,x̸=¹,a(x,y)<a(x^′,y) ifx<x^′,y̸=⁰,y̸=^1;

3. a(0,0)=^{0 and}^a(1,1)=1 (boundary conditions);

4. Thereexistsastrongnegationηsuchthata(x,y)=η(a(η(x),η(y)))(theself-DeMorganidentity)if {^x,y}̸= {⁰,1}^; 5. a(1,0)=â(0,1)=^{0 or}â(1,0)=â(0,1)=^1.

Definition10.(SeeYagerandRybalov[33].) AmappingU: [⁰,1]×[⁰,1]→[⁰,1]îsâûninorm,îfîtîs^symmetric,associative, nondecreasingandthereexists ane∈[⁰,1]^such^that Û(e,x)=^x ^forâll^x∈[⁰,1]^.

Thestructureofuninormswasfirstexamined byFodoretal.in[16].

Proposition6.(SeeFodoretal.[16].) LetU: [⁰,1]²→[⁰,1]^beâ^functionândν_∈]0,1[^.^The^following^statementsâreequivalent:

1. U isauninormwithneutralelementνwhichisstrictlymonotonicon]⁰,1[²ând^continuousôn[⁰,1]²\{(0,1),(1,0)}^. 2. Thereexistsanincreasingbijectionu: [⁰,1]→(−∞,∞)withu(ν)=⁰^such^that^forâll(x,y)∈[⁰,1]²^,^we^have

U(x,y)=^u⁻¹(u(x)+^u(y)), (5)

where,inthecaseofaconjunctiveuninormU ,weusetheconvention∞+(−∞)=−∞^,^while,ⁱⁿ^thedisjunctivecase,weuse

∞+(−∞)=∞^.

If(5)holds,thefunctionu isuniquelydeterminedbyU uptoapositivemultiplicativeconstant,anditiscalledanadditivegenerator oftheuninormU .

3. Shiftingtransformationsonthegeneratorfunctions–ageneralparametricformula

Fromnowon, weconsidernilpotentlogical systems.First weshowthatbyshiftingthegeneratorfunction ofadisjunc- tion,wecangetaconjunctionandalsooperatorsthat fulfilltheself-DeMorganproperty.Weprovideageneralparametric formula for these operators, where theconjunction, disjunction and the so-called aggregative operator differ only in one singleparameter.SeeFig. 1.

Definition11.Let f : [⁰,1]→[⁰,1] ^beân încreasing^bijection, ν_∈_[0,1]^,ând ^x=(x1,. . . ,xn), where xi∈[⁰,1]ând ^letûs definethegeneraloperator by

oν(x)= ^f⁻¹ ( _n

)

i=¹

(f(x_i)− ^f(

ν

))+ ^f(

ν

)

*

= ^f⁻¹ ( _n

)

i=¹

f(x_i)−(n−¹)f(

ν

)

*

. (6)

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Fig. 1.Shifting transformation in the linear case, f_ν⁻¹(x)forν₌0,ν₌ν^∗,ν₌1; whereν^∗₌f⁻¹(¹₂).

Remark4.Note that ν isaneutral element ofoν(x) and that oν(x) can begeneratedby g(x)= ^f(x)− ^f(ν),since inthis case g⁻¹(x)= ^f⁻¹(x+^f(ν)).

Aggregationfunctionsgeneratedinasimilarwayas(6) werealsodiscussedbyKolesárováandKomorníková[23].

Proposition7.Thegeneraloperatorin(6) 1. Forν₌1iso1(x)=^c(x),a conjunction.

2. Forν₌0iso0(x)=^d(x),a disjunction.

Proof. Since f(1)=^{1 and} ^f(0)=^0,^the^proof^is^trivial. ✷

Remark5.A conjunction and a disjunction differ only in one parameter of the general operator in (6). The parameter has thesemanticalmeaningofthelevel ofexpectancy.Generalizedconjunctionand disjunctionfunctions(GCD) werealso examinedbyDujmovi´cand Larsenin[12].

Next, a moregeneral,weightedformofthisoperator willbeexamined.

4. Theweightedgeneraloperator

If aweightedoperator ow(x): [⁰,1]ⁿ→[⁰,1]^with^w=(w1,. . . ,wn), wi>0 realparameters isrepresented byow(x)= f⁻¹

+ _n ,

i=¹ w_if(x_i)

-

,where f : [⁰,1]→[⁰,1]îsâ^bijection,^thenît^canâlso^be^writtenâsô^w(x)=^f⁻¹ + _n

,

i=¹ f(x^′_i)

-

,wherex^′_i is gotviaaso-calledweightingtransformation:x^′_i= ^f⁻¹(wif(xi)).

Below, we apply thisweighting transformation to the arguments of the operator in (6) to get the so-called weighted generaloperator.

Definition12.Letw=(w1,. . . ,wn)and wi>0 berealparameters, f: [⁰,1]→[⁰,1]^an^increasing^bijection^withν_∈_[0,1]^. Theweightedgeneraloperator isdefined by

aν,w(x):= ^f⁻¹ ( _n

)

i=¹

w_i(f(x_i)−^f(

ν

))+ ^f(

ν

)

*

. (7)

5. Propertiesofthegeneralandtheweightedgeneraloperator

5.1. DeMorganproperty

Thequestionthat immediatelyarisesis:forwhichparameter valuesdoestheabove-definedgeneraloperator satisfy the generalized De Morgan propertyconcerning the negation generated by f(x) (the generator function of o(x)). Thatis, for whichvaluesofν1,ν2 satisfythefollowingequationforallx∈[⁰,1]ⁿ^:

n(oν1(x))=^oν2(n(x)).

Forν1=⁰,ν2=^{1 or} ν1=¹,ν2=^0,^we^get^theclassical DeMorganlaw.

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Proposition8.Let f : [⁰,1]→[⁰,1]^beânîncreasing^bijection,νi∈[⁰,1]ând^x=(x1,. . . ,xn),wherexi∈[⁰,1]^,ⁿ(x)= ^f⁻¹(1− f(x))ando_ν_i(x)thegeneraloperator.Then

n(oν₁(x))=^o^ν2(n(x))

holdsforallx=(x1,. . . ,xn),wherexi∈[⁰,1]îfândônlyîf ^f(ν₁)+^f(ν₂)=^1.

Proof. Usingthefactthatn(x)= ^f⁻¹(1− ^f(x)),weget f⁻¹

. 1−

( _n )

i=¹

f(xi)−(n−¹)f(

ν

1)

*/

= ^f⁻¹ ( _n

)

i=¹

(1− ^f(xi))−(n−¹)f(

ν

2)

* .

1. First,weshowthat f(ν₁)+ ^f(ν₂)=^{1 is}^necessary.

Usingthenotations A:=,ⁿ

i=¹

f(xi), B:=−(n−¹)f(ν1),weget

[^A−(n−¹)f(

ν

₂)] =¹−[^A+^B]. (8) Letusconsiderthefollowingcases:

(a) First letus assume that ν₁_̸=0;^1. ⁽⁸⁾ ^must^hold ^for ^all ^x∈[⁰,1]ⁿ^, ⁱⁿ ^particular ^for ^x=(ν₁, . . . ,ν₁). In this case 0<A+^B= ^f(ν₁)<1,sothecuttingfunctioncan beomitted,and weget B=(1−ⁿ)+(n−¹)f(ν₂),fromwhich

f(ν1)+ ^f(ν2)=^1.

(b) Next,weshowthatthecuttingfunctioncanalsobeomitted,ifν1=^{0 (i.e.} ^B=^0),ν2̸=^1.^This^means^that ^we^have toshowthat

i. n−Â−(n−¹)f(ν2)≤^{0 and} Â+^B=Â≥^1,ôr

ii. n−Â−(n−¹)f(ν2)≥^{1 and} Â+^B=Â≤^{0 cannot}^hold^forâll^x∈[⁰,1]ⁿ^. Forexample,for x=(x, . . . ,x),wherex= ^f⁻¹0₁

n

1̸=^0,^we^get ^A=^1,^and(n−¹)(1−^f(ν2))>0.

(c) Next,weshowthat thecuttingfunction canalsobeomittedif ν1=^{1 (i.e.} ^B=¹−ⁿ⁾^and ν2̸=^0.^This^means ^that wehavetoshowthat

i. n−Â−(n−¹)f(ν2)≤^{0 and} Â+^B=Â+¹−ⁿ≥^1,ôr

ii. n−Â−(n−¹)f(ν2)≥^{1 and} Â+^B=Â+¹−ⁿ≤^{0 cannot}^hold^forâll^x∈[⁰,1]ⁿ^.

Since A≤^n, ^the^first^condition ⁱⁿ^1(c)i ^holdsônly^for ^x=^1,^not^for âll^x∈[⁰,1]ⁿ^.^The ^conditionⁱⁿ ^1(c)ii^does^not holdforx=¹ând Â=^n,^say.

(d) Forν₁₌0,ν₂₌1 orν₁₌1,ν₂₌0,theself-De Morganpropertytriviallyholds.

2. Next,weprovethat f(ν₁)+^f(ν₂)=^{1 is} ^also^suﬃcient.

If f(ν1)+^f(ν2)=^{1 holds,}^than ^f(ν1)=¹−^f(ν2),sowehavetoprovethefollowingequation:

f⁻¹(1−^[^A−ⁿ+¹+^C])= ^f⁻¹^[n−^A−^C^], where A:=,ⁿ

i=¹

f(xi)and C:=(n−¹)f(ν2).Since1−^[A−ⁿ+¹+^C]=^[1−^A+ⁿ−¹−^C],^the^statement^is^trivial. ✷

Remark6.For ν1=ν2, the only solution is ν1=ν2= ^f⁻¹0₁

2

1; i.e. the self-DeMorgan property holdsif and only if the parameterν isthefixpointofthenegation;i.e.ν₌ f⁻¹0

12

1=ν^∗.

Remark7.For ν1=ν2, we find that the operator oν(x,y) fulfills the self-De Morgan property if and only if it has the followingform:

f⁻¹ ( _n

)

i=¹

f(x_i)−ⁿ−¹ 2

*

. (9)

Inparticular,fortwovariables:

f⁻¹ +

f(x)+ ^f(y)−¹₂ -

. (10)

Proposition9.Letf : [⁰,1]→[⁰,1]^beânîncreasing^bijection,ν_∈_[0,1]ând^x=(x1,. . . ,xn),wherexi∈[⁰,1]^,^w=(w1,. . . ,wn), wi>0,n(x)= ^f⁻¹(1−^f(x)).Theweightedgeneraloperatoraν,w(x)satisfiestheself-DeMorganproperty,ifandonlyif,ⁿ

i=¹wi=^1, orν₌ f⁻¹0₁

2

1=ν^∗.

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Proof. Theself-DeMorganpropertymeansthat n(aν,w(x))=^a^ν^,^w(n(x))

holdsforallx; i.e.

f⁻¹ .

1− ( _n

)

i=¹

wi(f(xi)−^f(

ν

))+ ^f(

ν

)

*/

= ^f⁻¹ ( _n

)

i=¹

wi(1− ^f(xi)− ^f(

ν

))+ ^f(

ν

)

* .

Let A:=,ⁿ

i=¹

wif(xi) and B:=,ⁿ

i=¹

wi.Since f(x) isstrictlyincreasing,wehavetoshowthat 1−^[^A−^f(

ν

) (B−¹)]= [^B−^A−^{B f}(

ν

)+ ^f(

ν

)].

1. First,weshowthatthisconditionissufficient.If B=^1,^then^we^get¹−[Â]= [¹−Â]^,^whichâlways^holds.Îf ^f(ν)= ¹₂^, thenweget1−2

A− ^B⁻2¹

3=2

B−^A− ^B2+¹2

3.Using thefactthat 1−[^x]= [^x]^always^holds, ^we^can^readily ^see^that thetwosidesareequal.

2. Second,weshowthatthisconditionisalsonecessary.

(a) First, letusassume that ν_̸=0;^1.^For ^x=(ν, . . . ,ν), A= ^f(ν)B,so on theleft hand sideweget1−[^f(ν)]^,^which meansthatthecuttingfunctioncanbeomitted.Thus2f(ν)(B−¹)=^B−^1,^from^which^B=,ⁿ

i=¹

w_i=^1,ôr ^f(ν)= ¹2. (b) For ν₌0,we get1−[Â]= [^B−Â]^. ^For^x0=(x0, . . . ,x0), where0<x0<1, A= ^f(x0)B;i.e. 1−[^f(x0)B]= [(1−

f(x0))B]^,^where^the^cutting^function ^can^be^omitted,^since ^f(x0)B>0 and(1− ^f(x0))B>0.Thus B=,ⁿ

i=¹ w_i=^1.

(c) Forν₌1,weget1−[^A−^B+¹]= [−^A+¹]^.^For^x0=(x0, . . . ,x0),0<x0<1, A= ^f(x0)B,so1−[^f(x0)B−^B+¹]= [−^f(x0)B+¹]^{; i.e.} [^B− ^f(x0)B]= [¹− ^f(x0)B]^must^hold.

• ^If ^B=,ⁿ

i=¹

w_i≤^1,^then^the^cutting^function^can^be^omitted,^and^we^get ^B=,ⁿ

i=¹ w_i=^1.

• ^If ^B=,ⁿ

i=¹

w_i≥^1,^then^let ^f(x₀):= ,n¹ i=1

wi

= ¹_B^.^So^we^get ^B≤^1,^and ^B=,ⁿ

i=¹

w_i=^{1 must}^hold. ✷

Proposition10.Theweightedgeneraloperatora_ν_,w(x)iscommutative,ifandonlyifw1=^w²= · · · =^wn. Proof. Trivial. ✷

Corollary1.Acommutativeweightedgeneraloperatorfulfillstheself-DeMorganpropertyifandonlyif w=_n¹ ôrν₌ν^∗,where f(ν^∗)=¹₂^;î.e.ît^hasôneôf^the^following^forms:

f⁻¹ (1

n )n i=¹

f(xi)

*

(11) or

f⁻¹ (

w . _n

)

i=¹

f(x_i)−ⁿ₂ /

+¹₂

*

. (12)

Remark8.Note that(11)isaspecialcaseof(12)for w=¹n.

Remark9.If a_ν_,w is commutative and satisfies the self-De Morgan operator, then it is independent of the parameter ν. Thereforethelowerindex ν canbeomitted,andwewillrefertothiscasesimplyasa_w.

As wehaveseen,theweightedgeneral operator oftheform f⁻¹ +

w 4 _n

,

i=¹

f(xi)−ⁿ2

5 +¹2

-

,iscommutativeandsatisfies theself-DeMorganproperty.Withsuchnicepropertiesitisagoodideatogiveitadistinctivename.

Definition13.Theoperator aw(x)= ^f⁻¹

( w

. _n )

i=¹

f(xi)−ⁿ₂ /

+¹₂

*

, (13)

where w>0,iscalledtheweightedaggregativeoperator.

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Proposition11.Theweightedgeneraloperatora_ν,w(x)satisfies 1. Theboundaryconditionaν,w(0)=^0,îfândônlyîfν₌0or,ⁿ

i=¹wi≥¹^(for^acommutativeoperator:w≥¹_n^);

2. Theboundaryconditionaν,w(1,. . . ,1)=^1,îfândônlyîfν₌1or,ⁿ

i=1

wi≥¹^(for^acommutativeoperator:w≥¹_n^);

3. Bothoftheabove-mentionedboundaryconditionsifandonlyif,ⁿ

i=¹

wi≥¹^(for^acommutativeoperator:w≥¹n);

4. a_ν_,w(ν,. . . ,ν)=ν.

Proof. Let B:=,ⁿ

i=¹ w_i.

1. aν,w(0)= ^f⁻¹ +4_n

,

i=1

wi(−^f(ν)) 5

+^f(ν) -

=^0,îf ândônlyîf ^f(ν)(1−^B)≤^{0; i.e.}ν₌0 or ,ⁿ

i=1

wi≥^1.

2. a_ν,w(1,. . . ,1)= ^f⁻¹ +4 _n

,

i=¹

wi(1− ^f(ν)) 5

+ ^f(ν) -

=^1,îfândônlyîf(1−^f(ν))B+ ^f(ν)≥^{1; i.e.}(1−^B)(f(ν)−¹)≥^0, soν₌1 or ,ⁿ

i=¹ w_i≥^1.

3. Itfollowsfromtheprevioustwostatements.

4. a_ν,w(ν,. . . ,ν)= ^f⁻¹^[^f(ν)]=ν. ✷

Remark10.Notethat forcommutativeoperators,thecondition ,ⁿ

i=¹

w_i≥^{1 is}^equivalent^to ^w≥¹n. 5.2. Bisymmetry

Animportant property ofaggregation functions concernsthe grouping character; i.e. whether itis possible to build a partial aggregation for subgroups of input values, and then to get the overall value by combining these partial results.

A strongformofsuchacondition isassociativity,whichallowsusto startwiththeaggregationprocessbeforeknowing all inputstobeaggregated.However,associativityisaratherrestrictiveproperty.Associativityandidempotencytogethercancel theeffectofrepeatingargumentsintheaggregationprocedure,soitisnotpossibletosimulatethepresenceofweightsby repeating arguments.A weaker condition isbisymmetry, whichexpresses thefact that theaggregation oftheelements of anymatrixcanbeperformedfirstontherows,thenonthecolumns,orconversely.Thisnaturalpropertymeansthatinthe caseofnjudgesand mcandidates,say,theoverallscoreofthecandidatescanbecalculatedbyfirstaggregatingthescores ofeachcandidate,and thenaggregatingthese overallvalues;oranalternativewayistofirst aggregatethescoresgiven by eachjudge andthenaggregate these values.Thefollowingpropositions characterize bisymmetricand associativefunctions (seeAczél[1,2]).

Proposition12.Anoperatoro: [⁰,1]ⁿ→R^iscontinuous,strictlyincreasing,idempotent,andbisymmetricifandonlyifitrepresents aquasi-linearmean;i.e.thereisacontinuousandstrictlymonotonicfunction f: [⁰,1]→R^such^that

o(x)= ^f⁻¹ . _n

)

i=¹

w_if(x_i) /

,

wherewi>0, ,n i=¹

wi=^1.

Proposition13.Anoperatoro: [⁰,1]ⁿ→R^iscontinuous,strictlyincreasingandbisymmetricifandonlyifitrepresentsaquasi-linear function;i.e.thereisacontinuousandstrictlymonotonicfunction f: [⁰,1]→R^such^that

o(x)= ^f⁻¹ . _n

)

i=¹

w_if(x_i)+^b /

,

wherewi>0,b∈R^.

Ifinsteadofbisymmetrythefunction satisfiesthestrongerconditionsofcommutativityand associativity,thenwehave thefollowingcorollary whenw_i=^1.

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Proposition14.Anoperatoro: [⁰,1]ⁿ→Rîscontinuous,strictlyincreasing,commutativeandassociativeifandonlyifitrepresents aquasi-linearfunctionwithw_i=^1;î.e.^thereîsâ^continuousând^strictly^monotonic^function ^f: [⁰,1]→R^such^that

o(x)= ^f⁻¹ ( _n

)

i=¹

f(x_i)+^b

* ,

b∈R^.

Proposition15.Theweightedaggregativeoperatorwithweightsw≤¹nisbisymmetric.

Proof. 1. Since0≤^f(x)≤^1,⁰≤,ⁿ

i=¹

f(xi)≤^n.^Therefore,⁰≤^w 4 _n

,

i=¹

f(xi)−ⁿ2

5

+¹2≤^1,^soⁱⁿ^(13),^the^cutting^function ^can beomitted,andso theoperator hastheform ofthefunctioninProposition 13,whichmeansitisbisymmetric. ✷

6. Thetwo-variablegeneralandweightedaggregativeoperator

Now, weexamine theweightedaggregativeoperator oftwo variables.

Corollary2.Acommutativeweightedgeneraloperatora_ν,wfulfillstheself-DeMorganproperty(seeDefinition 2)ifandonlyifw=¹2

orν₌ν^∗,where f(ν^∗)=¹2;i.e.theweightedaggregativeoperatoroftwovariableshasthefollowingform:

f⁻¹ +

w(f(x)+ ^f(y)−¹)+¹₂ -

. (14)

Proof. ItfollowsdirectlyfromProposition 9. ✷

Remark11.Notethat for w= ¹₂^,⁽¹⁴⁾^has ^the^following^form:

f⁻¹ +_f

(x)+ ^f(y) 2

-

. (15)

Thisistheso-calledgeneralarithmeticmean,wherethecuttingfunctioncanbeomitted.

Corollary3.Atwo-variableweightedaggregativeoperatoraw, n(a_w(n(x),x))=^aw(n(x),x)=

ν

^∗,

and,inparticular,aw(0,1)=^aw(1,0)=ν^∗,whereν^∗₌ f⁻¹0

12

1.

Corollary4.Atwo-variablecommutativeweightedgeneraloperatora_ν_,wsatisfiestheboundaryconditions 1. a_ν,w(0,0)=^0,îfândônlyîf^w≥¹2orν₌0;

2. a_ν,w(1,1)=^1,îfândônlyîf^w≥¹2orν₌1.

Proof. ItfollowsdirectlyfromProposition 11. ✷

Corollary 5.Atwo-variablecommutative weightedaggregativeoperatoraw satisfies theboundary conditionsaw(0,0)=⁰ ând a_w(1,1)=^1,îfândônlyîf^w≥¹₂^.

Corollary6.Aweightedaggregativeoperatoraw(x,y)satisfiestheboundaryconditionsaw(0,0)=⁰ândâ^w(1,1)=^1,îfândônlyîf ithasthefollowingform:

a_w(x,y)= ^f⁻¹ +

w(f(x)+^f(y)−¹)+¹₂ -

, (16)

wherew≥¹2.

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Fig. 2.The weighted aggregative operatoraw for f(x)= ¹ 1+1−^ν^dν_d ¹⁻x^x

,ν_d₌0.8.

Proposition16.Aweightedaggregativeoperatoraw(x,y),whichsatisfiestheboundaryconditions,hasthefollowingproperty(see Fig. 2):

1. Ifx,y≤ν,thenaw(x,y)≤ν, 2. Ifx,y≥ν,thena_w(x,y)≥ν.

Proof. Aweightedaggregativeoperatoraw(x,y),whichsatisfiestheboundaryconditionshas thefollowingform:

aw(x,y)= ^f⁻¹ +

w(f(x)+^f(y)−¹)+¹₂ -

,

where w≥¹2.

1. First,weconsiderthecasewhereνisthefixpointofthenegation;i.e.ν₌ν^∗, f(ν)=¹2.

(a) If x,y≤ν,thenfromtheincreasingpropertyof f(x),wefindthat f(x),f(y)≤¹2; i.e. w(f(x)+ ^f(y)−¹)+¹2≤¹2, soaw(x,y)≤ν.

(b) Ifx,y≥ν,thenfromtheincreasingpropertyof f(x),wefindthat f(x),f(y)≥¹₂^{; i.e.} ^w(f(x)+ ^f(y)−¹)+¹₂≥¹₂^, soaw(x,y)≥ν.

2. Second,weconsider thecasewhere w= ¹2.From x,y≤ν followsthat f(x),f(y)≤ ^f(ν).Therefore, f⁻¹2_f₍_x₎₊_f₍_y₎

2

3≤

f⁻¹[f(ν)]=ν. ✷

Proposition17.Aweightedaggregativeoperator,withw1=^w2=^1,^has^the^following^properties^(see^{Fig. 3):}

1. Ifx,y≤ν^∗,thena1(x,y)isconjunctive;i.e.∀^x,y a1(x,y)≤^min(x,y). 2. Ifx,y≥ν^∗,thena1(x,y)isdisjunctive;i.e.∀^x,y a1(x,y)≥^max(x,y).

3. Ifx≤ν^∗_≤y,ory≤ν^∗_≤x thena1(x,y)isaveraging;i.e.∀^x,y min(x,y)≤^a¹(x,y)≤^max(x,y),whereν^∗₌ f⁻¹0₁

2

1.

Proof. Theoperator hasthefollowingform:

a1(x,y)= ^f⁻¹ +

(f(x)+ ^f(y)−¹)+¹₂ -

= ^f⁻¹ +

f(x)+^f(y)−¹₂ -

.

1. Let us assume that x≤^y≤ν^∗. From the increasing property of f(x), we see that f(x)≤ ^f(y)≤ ^f(ν^∗)= ¹2; i.e.

a1(x,y)= ^f⁻¹2

(f(x)+ ^f(y)−¹)+¹₂3

≤^x=^min(x,y).

2. Let us assume that ν^∗_≤x≤^y. ^From ^the ^increasing ^property ^of ^f(x), we see that ¹₂ = ^f(ν^∗)≤ ^f(x)≤ ^f(y); i.e.

a1(x,y)= ^f⁻¹2

(f(x)+ ^f(y)−¹)+¹₂3

≥^y=^max(x,y).

3. Let us assume that x ≤ν^∗ _≤ y. If x ≤ν^∗ _≤ y, then f(x) ≤ ¹2 ≤ ^f(y), so min(x,y) = ^x≤^a¹(x,y) = ^f⁻¹6 (f(x) + ^f(y)−¹)+¹2

7≤^y=^max(x,y). ✷

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Fig. 3.Uninorm-like property of the weighted aggregative operatora1(x,y).

Remark12.Theabove-mentionedpropertyholdsifandonlyif w=^1.^For ^w>1,theconjunctiveanddisjunctiveproperties hold,buttheaveragingpropertydoesnot.

Remark13.As wehaveseen,a1(x,y) has auninorm-likeproperty(seeProposition 17)and itsatisfiestheself-DeMorgan property as well. However, it is not associative (since a1(0,1)=^a¹(1,0)= ^f⁻¹0₁

2

1=ν^∗) and therefore it cannot be a uninorm.Note that aggregative operators in the strictcase (see Dombi [7])are always associative and therefore theyare specialuninorms.

Note that by substituting n(x) and y in thecommutative self-De Morgan weighted aggregative operator, the operator a(n(x),y)hascertainpropertiesthataresimilartothoseexpectedofapreferenceoperator.Preferencemodeling isafunda- mental partofseveralapplied fieldsofdecision-making [15].Intheclassical theory,preference isabinaryrelationclosely relatedtoimplications,withthemeaning

xR y ⇐⇒ ^“yis not worse thanx”.

Preferences between alternatives can also be described by a valued preference relation p, such that the value p(x,y) is normalized,anditisunderstoodasthedegreetowhichthestatement“yisnotworsethanx”istrue:p(x,y)=^{truth of}(y≥ x). Here, p is acontinuous function,which is strictlydecreasing in thefirst variable and strictly increasingin thesecond one,and p(x,y)=ⁿ(p(y,x))mustalsohold.Thereforeitissensibletodefinepreferenceinthefollowingway:

Definition14.Let w>0 be arealparameter and f : [⁰,1]→[⁰,1] ^beânîncreasing ^bijection.^Moreover, ^letûs ^define^the preferenceoperatoras pw(x,y)=â^w(n(x),y)= ^f⁻¹2

w(f(y)− ^f(x))+ ¹2

3.

Remark14.Notethatthenegationoperatorgeneratedby f(x): [⁰,1]→[⁰,1]^can^be^expressedⁱⁿ^the^following^way:

n(x)= ^f⁻¹8

(f(

ν

^∗)− ^f(x))+ ^f(

ν

^∗)9

, (17)

whereν^∗₌ f⁻¹0₁

2.1

Corollary7.Wehaveshownthatthegeneralformula

aν,w(x):= ^f⁻¹ ( _n

)

i=¹

wi(f(xi)−^f(

ν

))+ ^f(

ν

)

*

(18)

fortheweightedgeneraloperatorincludesthefollowingspecialcases:

1. For f(ν)=¹ând^wⁱ=¹∀î,îtîsâconjunctionwithgenerator1− ^f(x). 2. For f(ν)=⁰ând^wi=¹∀î,îtîsâdisjunctionwithgenerator f(x). 3. For f(ν)=¹₂ôr,ⁿ

i=¹wi=^1,^it^satisfies^the^self-De^Morgan^property.

4. For f(ν)= ¹2andw1= · · · =^wⁿ^,^or^for^wi=¹n,itisaweightedaggregativeoperator(acommutativeself-DeMorganoperator).