Choices and kernels in bipolar valued digraphs

Raymond Bisdor

Fa ulty of Law, E onomi s andFinan e, University of Luxembourg, Avenue dela

Faïen erie162a, L-1511 LUXEMBOURG.

Mar Pirlot

Fa ulté Polyte hnique de Mons,9, rue de Houdain,B-7000 Mons, Belgium.

Mar Roubens

Fa ulté Polyte hnique de Mons,9, rue de Houdain,B-7000 Mons, Belgium.

Abstra t

We explore the extension of the notion of kernel (independent, dominant or

ab-sorbent,non-empty subset)ofa digraphto valuedgraphs (orvaluedrelations). We

denevariousnaturalextensionsandshowtherelationshipbetweenthem.Thiswork

haspotential interestfor appli ationsin hoi e de isionproblems.

Keywords:De isionMaking,Outrankingmethods,Choi e,Kernel,Digraphs,

Valuedrelations, Fuzzy sets.

1 Introdu tion

Dominant kernels have been initially proposed as solutions of games by von

Neumann and Morgenstern (1944). Roy (1968)has suggested to use them as

a set of potentially good andidates among whi h to hoose when fa ing a

de ision problem. In the latter ontext, the preferen e relation on the set of

alternatives is usually built using some sort of a majority rule with vetoes

(Roy and Bouyssou, 1993), whi h results in a relation that is not ne essarily

transitive or omplete and may have y les.

Email addresses: raymond.bisdorffuni.lu(Raymond Bisdor),

(or fuzzy) one. The value atta hed to an ar

(a, b)

may express for instan e the degree with whi h

a

is preferred to

b

, a redibility index assigned to the preferen eof

a

over

b

,et .Intheperspe tiveofgeneralizingtovaluedrelations themethodsusedfor hoosingamonganitesetofalternativesonthebasisof

risp (

{0, 1}

-valued) relations,one an think of a generalizationof the notion of kernel ofvalued graphsor relations.

Asisusualwhenstudyingvalued(fuzzy) ounterpartsofnotionsthathaverst

beenintrodu edforordinary( risp)sets,thereareseveralwaysofgeneralizing

thenotionofkernel.The maingoalofthis paperistoexplorethe relationship

between various natural denitions of kernels for valued graphs and see also

how they relate to kernels of risp graphs.Our analysis is restri ted to nite

graphs.

Inse tion2,were allthedenitionsofkernelsinthe risp ase,relatingmainly

the set-theoreti denition (interms of independent, dominant or absorbent,

non-empty subset) and the algebrai one by means of the so- alled kernel

equationsystem.Notethat we onsidertwosorts ofkernels,the dominantand

the absorbent one, that are just dual of one another in the sense that the

absorbent kernels of a graph are the dominant kernels of the graph obtained

by reversing the orientation of the ar s of the originalone.

Se tion3introdu esavaluationofthear sbydegreesbelongingtoanordered

set

L

;this set isequipped withanegation (anantitoneone-to-oneoperation). If

L

ontains anodd numberof elements, there isan elementthat is equal to itsnegation;thisplaysthe roleofanundeterminedlevel

0

 thatmaybeused forinstan e for odingla kofinformation,likeamissingar inthegraph.One

distinguishesthe levels of

L

that are above

0

 (positive)and those below

0

 (negative).Wethen assume agraph with ar s assigned avalue in

L

. If

L

has a`

0

,we maysuppose,withoutlossofgenerality,that thegraphis omplete, assigningthe value

0

to the missing ar s. This graph isalsoa valued relation and we use indierentlyboth terminologies.

The rst idea for generalizing kernels, is to ut the relation above

0

, yielding a risp relation orgraph, and to make prot of the set-theoreti denition of

the kernels of the obtained risp graph. We all su h kernels

L

-independent, dominant (or absorbent) hoi es, reservingthe term kernel forthe solutions

ofthekernelequationsystem.Avarietyofdegreesofquali ationaredened

andasso iatedtothesesubsets;theymaybeinterpretedas hara terizingtheir

quality asa set of good alternatives(for dominantsubsets) oras aset of bad

alternatives (for absorbent subsets). We show that

L

-independent, dominant (resp.absorbent) hoi es anbedenedalternativelybymeansoftheirdegree

that are valued (fuzzy) sets. We identify a subset of solutions,the maximally

sharp ones, and we show in se tion 4 that they orrespond to the set of

L

-independent dominant (or absorbent) hoi es that were initiallydened. The

proofusingxpointargumentis onstru tive,enablingtondthe

L

-valued kernels starting from the independent dominant (or absorbent) hoi es. The

degreeofsharpness oftheformerisshowntobeequaltothedegreeofgoodness

(or badness) of the latter. From a theoreti al point of view, this is rather

satisfa tory sin e the various generalizations of the kernel lead to essentially

the same notion, though in a non-trivial way, as the reader will see. One

should add that there are perspe tives, using

L

-valued kernels, to ontribute to methods for hoosing among a set of alternatives, but su h developments

are left for another publi ation.

2 Kernels of risp binary relations

2.1 Independent dominant and absorbent hoi es

In this se tion, we onsider a nite set

X

= {a, b, c, d, . . .}

of alternatives endowed with abinaryrelation

R

.Thisrelationisinterpretedasapreferen e relation; we do not assume any parti ular property, su h as transitivity, of

R

; the preferen e maythusresult fromthe aggregation of multipleattribute informationthroughsomekindofamajorityruleora ordingwithsomeform

ofa on ordan e-dis ordan e prin iplelikethey are implemented forinstan e

inthe Ele tremethods(Roy, 1968; Roy and Bouyssou, 1993).

We denoteby

a R b

the fa tthat the pair

(a, b)

belongsto the relation

R

; we phrase

a R b

as 

a

is preferred to

b

 (this not ex luding that

b

may also be preferred to

a

;

R

is not supposed tobean asymmetri relation).

The set

X

endowed with the relation

R

an also be viewed as a dire ted graph (digraph, for short)

G

= (X, R)

where

X

represents the set of verti es and

(a, b)

is anar of the graphif and onlyif

(a, b) ∈ R

.

It has been suggested in the Ele tre I method (Roy, 1969) to use

inde-pendent dominant subsets of X as potential good hoi e if the relation

R

is a y li .

Denition 1 (Independent dominantand absorbent hoi es).

that

∀a 6= b ∈ K, (a, b) 6∈ R

.

A dominant hoi e or dominant subset in

G

is a hoi e set

K

⊂ X

su h that

∀a 6∈ K, ∃ b ∈ K, (b, a) ∈ R

.

Anabsorbent hoi e orabsorbentsubset in

G

isa hoi eset

K

⊂ X

su hthat

∀a 6∈ K, ∃ b ∈ K, (a, b) ∈ R

.

Example 1. The graph

G

1

= (X

1

, R

1

)

, where

X

1

= {a, b, c, d}

and

R

1

=

{(a, b), (a, c), (a, d), (b, c), (b, d)}

is illustrated in gure 1. There is a unique independent dominant hoi e (good hoi e) that is the singleton

{a}

; the unique independentabsorbent hoi e (bad hoi e)is

{c, d}

.

a

b

c

d

Figure1.Graph ofthe preferen erelation in example 1

Example 2. The graph

G

2

= (X

2

, R

2

)

, where

X

2

= {a, b, c, d}

and

R

2

=

{(a, b), (b, c), (c, d), (d, a)}

is illustratedingure2.In thisexample,

{a, c}

and

{b, d}

are two independent hoi es that are both dominantand absorbent.

a

b

c

d

Figure2.Graph ofthe preferen erelation in example 2

If a relation

R

is a y li , then it has a unique dominant (resp. absorbent) hoi e. If, in addition, the relation is transitive, it is a partial order and the

setofalternativesthathavenoprede essor(resp.su essor)inthegraphforms

an independent dominant (resp. absorbent) hoi e. The above existen e and

uniqueness property is amotivationfor hoosing among the alternativesthat

belong to the independent dominant hoi e. In ase one aims at eliminating

bad alternatives, asymmetri argumentwould lead to hoosing theminthe

independent absorbent hoi e set, again for a y li digraphs. We loose su h

We now introdu e an algebrai denition of a independent dominant (resp.

absorbent) hoi e. Ea h subset

K

⊆ X

is asso iated its hara teristi (row) ve tor

Y

: X → {0, 1}

with

Y

(a) =







1

if

a

∈ K

0

otherwise

.

(2.1) for all

a

∈ X

.

Denition 2 (Dominant and absorbent kernels).

Adominant kernel (Bisdorand Roubens,1996a)isa hara teristi ve tor

Y

that isa solution of the following system of Boolean equations:

(Y ◦ R )(a) = ¬Y (a),

for all

a

∈ X,

(2.2)

where

Y

◦ R

is the max min produ t of the Boolean matri es

Y

and

R

,i.e.

(Y ◦ R )(a) = max

b

6=a

min (Y (b), R (b, a)),

for all

a

∈ X

and

¬

istheordinarynegation,i.e.

¬Y (a) = 1−Y (a)

.Thesystemofequations (2.2) reads more expli itlyas:

max

b

6=a

min (Y (b), R (b, a)) = 1 − Y (a),

for all

a

∈ X.

(2.3)

We all absorbent kernel a hara teristi ve tor

Y

, solution of the system of Booleanequations: for all

a

∈ X

,

(R ◦ Y

t

)(a) = max

b6=a

min (R(a, b), Y (b)) = ¬Y (a) = 1 − Y (a),

(2.4)

where

Y

t

denotes the transposed ( olumn) hara teristi ve tor.

We denote

K

dom

_(G)

(

K

abs

_(G)

) the possibly empty set of dominant (resp.

ab-sorbent)kernels in

G

,i.e. the solutionsof thesystem of equations(2.2) (resp. (2.4)).

Proposition 1.

Thesetofindependentdominant(resp.absorbent) hoi esof

X

endowed with therelation

R

)

isthesetofdominant(resp.absorbent)kernelsof

G

= (X, R )

.

R

representing the relation

R

into four parts

R

K K

, R

K K

, R

K K

, R

KK

; for in-stan e

R

K K

orresponds tothe ases

R(a, b)

inwhi hboth

a

and

b

belong to

K

;the three otherpartsare dened similarly.The system ofequation(2.2) is rewritten as

[Y

K

Y

_K

] ◦







R

K K

R

K K

R

KK

R

KK







= [Y

K

Y

K

].

(2.5)

One veries easily that ve tor

Y

exa tly hara terizes a subset

K

that is independent and dominant if and only if the above system of equations is

satised. A similar argument applies to independent absorbent hoi es and

absorbent kernels.

Inthesequelweshalluseindierentlythetermsindependentdominant(resp.

absorbent) hoi es and dominant (resp. absorbent)kernels.

2.3 Histori al note

Followingtheobservationthatthe independentabsorbent hoi einana y li

graph orresponds to the kernel of the asso iated Grundy fun tion, Riguet

(1948)introdu edthenamenoyau (kernel)forsu ha hoi eset.(Absorbent)

kernelswereinthesequelextensivelystudiedbyBerge(1958,1970)inthe

on-textofmodelingtheNimgame.More resultson(absorbent)kernels,

on ern-ingsolutionsof dierentgames,havebeen reportedby S hmidtandStröhlein

(1985, 1989). Re ently, Bang-Jensen and Gutin (2001) reviewed the link

be-tween kernel-solvability and perfe t graphs.

Internally stableand dominating hoi es,i.e.dominantkernelswere originally

introdu ed by J. von Neumann and O. Morgenstern under the name game

solution in the ontext of game theory (von Neumann and Morgenstern,

1944). B.Roy proposed the same on ept in the ontext of the multi riteria

Ele tre de ision aid methods (Roy, 1968, 1985; Roy and Bouyssou, 1993).

Ambiguous hoi es, i.e. both dominant and absorbent hoi es at the same

time,were proposedby Bisdor(2002a)aspotential luster andidatesinthe

ontextof multi riteria lustering.

Inthegeneral(non-oriented)theoryofgraphs,kernelsappear underthename

ofindependentdominatingsetsinHaynesetal.(1998).Ourorienteddominant

(resp. absorbent)version isthere identiedas aninside (resp.outside)

semi-dominatingset orsometimesmoresimplyasaninkernel (resp.outkernel).

Theabsorbentversionofthekernelequationsystem (2.4)wasrstintrodu ed

by S hmidtandStröhlein (1985,1989)inthe ontextof theirthorough

(1996a,b); Bisdor(1997).

3 Kernels in valued digraphs

3.1

L

-valued binary relations

Let

X

be a nite set of alternatives and

R

˜

, a valued binary relation on

X

. We assume that the values assigned by

R

˜

to ea h pair of alternativesbelong toa nite ordered set

L

: {c

0

, c

1

, . . . , c

M

}

with

c

0

< c

1

< . . . < c

M

. The value

˜

R(a, b) ∈ L

is often interpreted as the degree of redibility of the assertion 

a

is preferred to

b

(or 

a

is at least as good as

b

). We allsu h a relation a

L

-valued binary relation(or shortly

L

-vbr).

The set

L

isendowed with a negation operator

¬

that maps

c

i

onto

c

M

−i

for all

i

= 0, . . . , M

. This operator has a xpoint

c

m

i

M

is equal to

2m

, with

m

a positive integer. Bisdor (2002a) has for efully argued that one should take

M

an even number in pra ti e; the median redibility level

c

m

is then interpreted asalogi allyundetermined value; itmayen ode amissingar . In

ase

M

isodd thenegationhasnoxpointatallandthereisthusnologi ally undeterminedvalue. Fromnowon,fornotationalsimpli ity,wemaptheset

L

monotoni ally ontothe following sets of integers, endowed with their natural

order:

−m, −(m − 1), . . . , −1, 0, 1, . . . , m − 1, m

if

M

= 2m

−m, −(m − 1), . . . , −1, 1, . . . , m − 1, m

if

M

= 2m − 1;

(3.1)

weuseornotthevalue

0

dependingonwhether

|L|

isoddoreven,respe tively; in ase

|L|

is odd (i.e.

M

= 2m = |L| − 1

), the median redibility level

c

m

is mapped onto

0

. Sin e this mappingisan orderisomorphism, the operators

max

and

min

an be seen as a ting indierently on

L

or its image on the integers;moreover, the negationoperator

¬

on

L

orresponds with takingthe oppositeofanintegerintheimageset. Wehen eforthidentify

L

withthesets ofintegersdened in(3.1),endowed withtheirnaturalorderandthenegation

operator,whi h isjust takingthe opposite.Wedene the subset ofpositive

levels

L

>0

(resp. negative levels

L

<0

) as being the set of positive integers

{1, 2, . . . , m}

(resp. negative integers

{−m, . . . , −1}

); the set of nonnegative levels

L

≥0

(resp. nonpositive levels

L

≤0

)is dened as the omplement of

L

<0

(resp.

L

>0

).Of ourse,thereisnodieren ebetween

L

>0

and

L

≥0

(orbetween

L

<0

and

L

≤0

)unless

|L|

isodd(i.e.

M

iseven);inthelatter ase,thedieren e onsists of the median redibility level 

0

. In the sequel, for

x

∈ L

, we shall write

x >

0

(resp.

<

0, ≥ 0, ≤ 0

) for

x

∈ L

Interpreting theelementsof

L

astruthvaluesorlogi allevels,assuggestedby Bisdor (2002a), leads to all the elements of

L

>0

, true levels and those of

L

<0

,falselevels. If

R(a, b) > 0

we ouldsay thattheproposition

(a, b) ∈ R

 is

L

-true. If, on the ontrary,

R

(a, b) < 0

, the proposition 

(a, b) ∈ R

 is

L

-false. If

R(a, b) = 0

, i.e. the median level, we say that the proposition 

(a, b) ∈ R

 is

L

-undetermined. The redibility level of a onjun tion (resp. a disjun tion) of

L

-valued propositions is obtained by using the 

min

 (resp. the 

max

) operatordened on

L

.

Twosimplespe ial asesof

L

are thefollowing.If

L

hasonlytwoelements,we getba ktothe lassi alBooleanlatti e

L

2

= {c

0

, c

1

} = {−1, 1}

;

c

0

or

−1

isthe symbol for false, while

c

1

or

+1

is the symbol for true. The three-valued latti e

L

3

(

M

= 2m = 2

) orresponds to the ase in whi h

−1

represents false,

0

represents a logi allyundetermined level and

+1

represents true.

3.2 Independent dominant or absorbent hoi es in a

L

-valued graph

Wedenoteby

G

L

= (X, R)

a omplete

L

-valueddigraphwithvertexset

X

;an ar

(a, b)

of

G

L

isassignedthe value

R(a, b)

.Thesimplestway ofgeneralizing thenotionsofindependentdominant(resp.absorbent) hoi ein

G

L

isthrough

using a ut. Dene the appli ation

τ

mappingthe latti e

L

onto the Boolean latti e

L

2

asfollows:thesetoflevels

L

>0

areallmappedonto

+1

(true)while thelevelsof

L

≤0

are allmapped onto

−1

(false).This appli ationextendsto a transformationof the

L

-valued graph

G

L

= (X, R)

into the ordinary graph

τ G

= (X, τ R)

;

(a, b)

is an ar of the relation

τ R

i

R(a, b) > 0

; otherwise, there isno ar

(a, b)

in the graph

τ G

.

Denition 3 (Independent dominant or absorbent hoi es in

G

L

). The set

K

⊆ X

is anindependent dominant (resp. absorbent) hoi e in

G

L

= (X, R)

if it isanindependent dominant(resp. absorbent) hoi e in

τ G

= (X, τ R)

. Note that with this denition an independent dominant (resp. absorbent)

hoi e in

G

L

= (X, R)

isan ordinary subsetof

X

and not a fuzzy subsetof

X

.

Using the values in

L

, we an dene various

L

-valued quali ation degrees atta hed toany non-empty subset

K

of

X

.

Denition 4 (Degrees of quali ation of hoi es). Let

K

be a non-empty subset of

X

.The degree of independen e of the hoi e

K

is dened as

∆

ind

_{(K) =}







m

if

|K| = 1

,

min

b

6=a

b

∈K

min

a∈K

{¬ R(a, b)}

otherwise

.

K

is onsidered tobe

L

-independent if

∆

ind

_{(K) > 0}

.

The degree of dominan e of a hoi e

K

orresponds to

∆

dom

(K) =







m

if K =X

,

min

a

6∈K

max

b

∈K

{R(b, a)}

otherwise

.

(3.3)

K

is onsidered tobe

L

-dominant if

∆

dom

_{(K) > 0}

.

The degree of absorben e of a hoi e

K

is equal to

∆

abs

(K) =







m

if K = X

,

min

a

6∈K

max

b

∈K

{R(a, b)}

otherwise

.

(3.4)

K

is onsidered tobe

L

-absorbent if

∆

abs

_{(K) > 0}

.

The quali ation of

K

as being an independent and dominant hoi e orre-sponds to

Q

i−dom

(K) = min(∆

ind

_{(K), ∆}

dom

_(K)).

(3.5)

The quali ation of

K

as being an independent and absorbent hoi e, orre-sponds to

Q

i−abs

(K) = min



∆

ind

(K), ∆

abs

(K)



(3.6)

Denition 5 (Potentially good and bad hoi es). The set

K

⊆ X

is a po-tentially good hoi e if

Q

i−dom

_{(K) ∈ L}

>0

, i.e. if

K

is

L

-independent and

L

-dominant.The set

K

⊆ X

is a potentially bad hoi e if

Q

i−abs

_{(K) ∈ L}

>0

, i.e.

if

K

is

L

-independent and

L

-absorbent. We denote by

C

p−good

_(G

L

)

(respe tively

C

p−bad

_(G

L

)

) the (possibly empty) set of potentiallygood(respe tively potentially bad) hoi es in

G

L

. The

or-responden e between potentially good hoi es (resp. potentially bad hoi es)

andindependentdominant(resp.absorbent) hoi esof

G

L

(denition3)isnot

entirely straightforward due to the possible existen e of the undetermined

medianlevel

0

.

Proposition 2 (L.Kitainik, 1993).

Let

G

L

= (X, R)

bean

L

-valued digraphand let

τ G

= (X, τ R)

be its asso i-ated risp digraph.Wehave:

C

i−dom

_(G

L

) ⊆ K

dom

_{(τ G)}

and

C

i−abs

_(G

L

) ⊆ K

abs

_{(τ G).}

Proof. Consider

K

6= ∅

su hthat

∆

ind

_{(K) ∈ L}

>0

∆

ind

(K) ∈ L

>0

⇔ min

b6=a

b

∈K

min

a∈K

(¬ R(a, b)) > 0

⇔ ∀a 6= b ∈ K : R(a, b) < 0

⇒ ∀a 6= b ∈ K : (a, b) 6∈ τ R

⇔ K

is anindependent hoi e in

τ G

= (X, τ R).

Consider now

K

6= ∅

su h that

∆

dom

_{(K) > 0}

.

∆

dom

_{(K) ∈ L}

>0

_{⇔ min}

a

6∈K

max

b

∈K

R(b, a) > 0

⇔ ∀a 6∈ K, ∃ b ∈ K : R(b, a) > 0

⇔ ∀a 6∈ K, ∃ b ∈ K : (b, a) ∈ τ R

⇔ K

is adominant set in

τ G.

Usingthe same arguments,

∆

abs

_{(K) > 0 ⇔ K}

isan absorbent set in

τ G.

The propositionis animmediate onsequen e of the previous results.

From the rst part of the previous proof (independen e), it learly appears

thatthe sets

C

p−good

_(G

L

)

and

K

dom

_{(τ G)}

maybedierentonlyif

R

(a, b)

takes thelogi allyundeterminedvalue

0

;theremayindeedexistsets

K

⊆ X

thatare independent dominant hoi es of

τ G

but have aquali ation

Q

i-dom

(K)

equal to

0

. A similar remark holds for independent absorbent hoi es. The follow-ing orollary establishes the pre ise orresponden e between potentially good

(resp. potentially bad) hoi es and independent dominant (resp.absorbent)

hoi es in

τ G

(whi h are also the dominant (resp. absorbent) kernels in

τ G

, asestablished in proposition 1).

Corollary 1.

The set of potentially good(resp. potentially bad) hoi es in

G

L

= (X, R)

is theset ofindependentdominant(resp.absorbent) hoi es

K

in

τ G

= (X, τ R)

for whi h

Q

i-dom

Consider the rst part of the proof of proposition 2.We have

∆

ind

(K) ∈ L

>m

⇔ K

is anindependent set in

τ G

i

R(a, b) 6= 0

forall

a, b

∈ K

.Therestoftheprooffollowsthatofproposition 2.

Example 3. The following example (see gure 3) has been suggested by

BernardRoy(Lausanne, 1995;see Bisdor(2000)).Let

G

L

besu h that

X

=

{a, b, c}

,

L

= {−5, . . . , 0, . . . , 5}

and

R

: {R(a, b) = −5

,

R(b, a) = R(c, a) =

R(c, b) = 1

,

R(b, c) = R(a, c) = 5}

.

a

b

c

1

5

1

1

5

Figure3.Graph ofthe preferen erelation in example 3

Table 1

Degreesof quali ation ofall hoi es in example 3

hoi e

∆

ind

_∆

dom

_∆

abs

_Q

i−dom

_Q

i−abs

{a}

5

−5

1

−5

1

{b}

5

1

−5

1

−5

{c}

5

1

5

1

5

{a, b}

−1

5

1

−1

−1

{a, c}

−5

1

5

−5

−5

{b, c}

−5

1

5

−5

−5

{a, b, c}

−5

5

5

−5

−5

Table 3 shows the degrees of quali ation for all hoi es in

G

L

. All

single-tons are

L

-independent. All hoi es, ex ept

{a}

, are

L

-dominant and all are

L

-absorbent, ex ept

{b}

. Therefore, the singletons

{b}

and

{c}

give poten-tially good single hoi es whereas

{a}

and

{c}

give potentially bad single hoi es. Choi es onsisting of pairs, as well asthe hoi e onsisting of

X

, are neither potentially good nor potentially bad, as they all la k the required

One may argue that the indi es

Q

i−dom

and

Q

i−abs

an be used to sele t a

good hoi e amongseveral possibleones;forinstan e,inexample3,onesees

from table 3 that

{b}

is a more onvin ing potentially good hoi e than

{c}

sin e

{c}

is alsoa potentiallybad hoi e. We referthe readerto Bisdorand Roubens (2004) forfurther developments of this idea.

Proposition2 suggests a way of omputing the good hoi es (and the bad

ones) of

L

-valued graphs: ompute the independent dominant hoi es of

τ G

; then verify, by omputing their degree of independen e

∆

ind

and he king

whether

∆

ind

_>

₀

,that they are potentiallygood hoi es.

3.3

L

-valued dominant and absorbent kernels

We now address the generalization of the kernel equation systems (2.2) and

(2.4). Choi es and kernels of

L

-valued relations an be dened as fuzzy sets; the degree of membership of an element

a

of

X

to su h a set belongs to the set

L

.

A hara teristi ve tor

Y

˜

ofafuzzysubsetof

X

isanappli ationfrom

X

onto

L

, i.e. a row ve tor

[ ˜

Y

(x), x ∈ X]

. If

Y

˜

is to be interpreted as a hoi e and depending on the ontext,

Y

˜

(a)

, for any

a

∈ X

, may be interpreted as the degree of truth or the redibility of the assertion element

a

belongs to the hoi e

Y

˜

. Values of

Y

˜

(a)

above

0

are interpreted as meaning that

a

tends moretobelongto

Y

˜

than not;the onverse interpretationholdswhen

Y

˜

(a)

is below

0

;

Y

˜

(a) = 0

represents anundetermined situation.

It is rather straightforward to extend the kernel equation systems (2.2) and

(2.4) to the ase in whi h

R

is

L

-valued and we a ept solutions that are

L

-valued. Theseequations be ome: for all

a

∈ X

:

( ˜

Y

◦ R)(a) = max

b

6=a

h

min



Y

˜

(b), R(b, a)

i

= ¬ ˜

Y

(a) = − ˜

Y

(a),

(3.7)

(R ◦ ˜

Y

t

)(a) = max

b

6=a

h

min



R(a, b), ˜

Y

(b)

i

= ¬ ˜

Y

(a) = − ˜

Y

(a).

(3.8)

Applying the negation operator to the above equations yields the following

formulation that willbe used inthe sequel:

˜

Y

(a) = min

b

6=a

[max(− ˜

Y

(b), −R(b, a))],

∀a ∈ X

(3.9)

˜

Y

(a) = min

b

6=a

[max(−R(a, b), − ˜

Y

(b))],

∀a ∈ X.

We denote by

Y

dom

_(G

L

)

(resp.

Y

abs

_(G

L

)

) the set of solutions of equation system 3.7 (resp. 3.8).

Example4. Consider

G

L

= (X, R)

with

X

= {a, b, c}

,

L

= {−5, . . . , 0, . . . , 5}

(

M

= 2m = 10

)and

R

= {R(a, b) = −3}

,

R

(a, c) = 4

,

R

(b, a) = 1

,

R

(b, c) =

5

,

R(c, a) = 2

,

R(c, b) = 3}

(see gure 4).

a

b

c

1

−3

5

3

2

4

Figure4.Graph ofthe preferen erelation in example 4

The orresponding dominantkernel equation system is:

h

˜

Y

(a) ˜

Y

(b) ˜

Y

(c)

i

◦















-

−3

4

1

-

5

2

3

-













=

h

− ˜

Y

(a) − ˜

Y

(b) − ˜

Y

(c)

i

(3.11)

Its solutionsare shown intable 2.Solution

Y

˜

0

is ompletely

L

-undetermined; Table 2

Solutionsof EquationSystem (3.11).The starredsolutions aremaximally sharp.

solution

Y

˜

(a)

Y

˜

(b)

Y

˜

(c)

˜

Y

0

0

0

0

˜

Y

1

−1

1

−1

˜

Y

2

−1

2

−2

˜

Y

₃

∗

−1

3

−3

˜

Y

4

−1

−1

1

˜

Y

₅

∗

−2

−2

2

it doesn't hara terize any hoi e at all and we may ignore it. If we apply

the operator

τ

to the fuzzy hara teristi ve tors, solutions

Y

˜

1

,

Y

˜

2

and

Y

˜

∗

3

yield the same risp hoi e

{b}

, whereas solutions

Y

˜

4

and

Y

˜

∗

5

yield the same risp hoi e

{c}

. Note that both subsets of solutions are organized as hains ofmoreandmorelogi allydeterminedsolutions,i.e. omingin reasingly lose

to the values

−5

and

+5

that represent the maximal degree of falseness and truth, respe tively. In ea hgroup of solutions,we shall fo us on the maximal

solutions,

Y

˜

∗

3

and

Y

˜

∗

all an

L

-fun tion any

L

-valued fun tion dened on

X

. We mainly onsider

L

-fun tions

Y

˜

su h that

Y

˜

(a) 6= 0

, for all

a

∈ X

; the set of su h

L

-fun tions is denoted by

L

0

.

Denition 6 (Sharpness). Let

Y , ˜

˜

Z

be two fun tions in

L

0

. We say that

Y

˜

is atleast as sharp as

Z

˜

,whi h wedenote by

Z 4 ˜

˜

Y

,i, for all

a

∈ X

, either

˜

Y

(a) ≤ ˜

Z(a) < 0

or

0 < ˜

Z(a) ≤ ˜

Y

(a)

. The asymmetri part

≺

of

4

is dened as follows:

a

≺ b

i

a 4 b

and not

b 4 a

.

The sharpness relation

≺

isapartial order(asymmetri and transitive rela-tion) onthe set of

L

-fun tionsin

L

0

.

To ea h fun tion

Y

˜

in

L

, we asso iate a risp subset of X, whi h we denote by

K

Y

˜

and dene by:

K

Y

˜

= τ ˜

Y

= {a ∈ X

su h that

Y

˜

(a) > 0}.

(3.12)

We all

K

˜

Y

the stri t median ut hoi e asso iated with

K

˜

Y

.

Forany

L

-fun tion

Y

˜

∈ L

0

, thedegree of sharpness

Q

sharp

_{( ˜}

_Y

₎

of

Y

˜

isdened as:

Q

sharp

( ˜

Y

) = min( min

a

∈K

Y

_˜

˜

Y

(a), min

a

6∈K

Y

_˜

− ˜

Y

(a)).

(3.13)

This index interprets asa degree of sharpness of

Y

˜

sin e we have

˜

Y 4 ˜

Z

⇒

Q

sharp

( ˜

Y

) ≤ Q

sharp

_{( ˜}

_Z).

(3.14)

Of ourse,wemayhave

Q

sharp

_{( ˜}

_Y

_{) ≤ Q}

sharp

_{( ˜}

_Z)

andneither

Y 4 ˜

˜

Z

nor

Z 4 ˜

˜

Y

. Denition 7 (

L

-valueddominantand absorbent kernels).

We all

L

-dominant (resp.

L

-absorbent)kernel a

L

-valued hara teristi ve -tor

Y

˜

that satises the followingthree onditions:

• ˜

Y

is a solutionof equation system (3.7) (resp. (3.8));

•

itismaximal with respe t tothe sharpness relation

4

inthe set

Y

dom

_(G

L

)

(resp.

Y

abs

_(G

L

)

)ofadmissiblesolutionsof(3.7) (resp.(3.8)),i.e.thereisno othersolution

Z

˜

su h that

Y

˜

≺ ˜

Z

;

•

itbelongs to

L

0

, i.e.

Y

˜

(a) 6= 0

, for all

a

∈ X

We denote by

F

dom

_(G

L

)

(resp.

F

abs

_(G

L

)

) the (possibly empty) set of

L

-dominant(

L

-absorbent) kernels of

G

L

.

Revisitingexample4,wehave

Y

˜

∗

3

and

Y

˜

∗

5

(seeTable 2)as

L

-dominantkernels of

G

L

; defuzzyfying

Y

˜

∗

3

using the operator

τ

yields

τ ˜

Y

∗

3

= {b}

. In a similar way, we get

τ ˜

Y

∗

systems yields the solutions

[Y (a) = 2, Y (b) = −1, Y (c) = −2]

and

[Y (a) =

−4, Y (b) = −4, Y (c) = 4]

as the

L

-absorbent kernels in

G

L

.

In example 3, the digraph

G

L

admits the set

F

i−dom

_(G

L

) = { ˜

Y

₁

∗

, ˜

Y

₂

∗

}

of

L

-dominant kernels and the set

F

bad

_(G

L

) = { ˜

Y

∗

3

, ˜

Y

4

∗

}

of

L

-absorbent kernels (see table 3). Applying the operator

τ

to these

L

-valued kernels yields the hoi e sets that were already obtained for example 3in se tion3.2.

Table 3

L

-valued kernelsin the digraphof Example 3

kernel

˜

Y

(a)

Y

˜

(b)

Y

˜

(c)

L

-dominant

˜

Y

₁

∗

−1

5

−5

˜

Y

₂

∗

−1

−1

1

L

-absorbent

˜

Y

₃

∗

1

−1

−1

˜

Y

₄

∗

−5

−5

5

Fromthelatterexample,itappearsthat

L

-valuedkernelsandpotentiallygood or potentially bad hoi es of a valued graph are strongly related.In the next

se tion,we show the formal orresponden e between those notions.

4 Relating

L

-valued kernels and potentially good and bad hoi es

Inthisse tion,weshallonlydealwiththe dominant ase;our resultsextend

mutatismutandis totheabsorbent ase.Itiseasytoestablishalinkbetween

asubsetofsolutionsofthedominantkernelequationsystem(3.7)andasubset

of dominantkernels of

τ G

that we dene below.

Denition8(Determinedsolutionsofthedominantkernelequationsystem).

A solution

Y

˜

of the dominantkernel equationsystem (3.7)is said determined if it belongs to

L

0

i.e.

Y

˜

(a) 6= 0

, for all

a

∈ X

. We denote by

Y

dom

0

(G

L

)

the set

Y

dom

_(G

L

) ∩ L

0

of determined solutionsof (3.7).

Denition 9 (

R

-determined subsets). A subset

K

of

X

inthe valued graph

G

L

= (X, R)

is

R

-determined if

R(a, b) 6= 0

for all

a, b

∈ K

.

Lemma 1. If

Y

˜

belongs to

L

0

and is a solution of the dominant kernel equation system (3.7), then

τ ˜

Y

= K

Y

˜

is a

R

-determined dominant kernel of

Proof. Using the fa t that

Y

˜

is a solution of the dominant kernel equation system, we apply

τ

toboth sides of (3.7),yielding

−τ ˜

Y

(a) = max

b6=a

[min(τ ˜

Y

(b), τ R(b, a))],

whi h is exa tly (2.2) with

Y

= τ ˜

Y

. It remains to be proven that

K

˜

Y

is

R

-determined. Assume that there are

a, x

∈ K

Y

˜

with

R(a, x) = 0

. Sin e

Y

˜

belongs to

L

0

by hypothesis,

Y

˜

(a), ˜

Y

(x) > 0

. Usingthe negative form (3.9) of the kernel equationsystem, wehave for that parti ular

a

∈ K

˜

Y

:

0 < ˜

Y

(a) ≤ max(− ˜

Y

(x), −R(x, a)) = 0,

a ontradi tion. 4.1 A useful transformation Theelementsof

Y

dom

0

(G

L

)

i.e.thedeterminedsolutionsofthedominantkernel equation system an be viewed as the xpoints of a transformation

T

that operates on

L

0

.

Denition 10 (The transformation

T

). The transformation

T

maps a fun -tion

U

˜

∈ L

0

, onto

T ˜

U

dened by:

T ˜

U

(a) = min

x

6=a

max(− ˜

U

(x), −R(x, a)),

∀a ∈ X.

The xpoints of

T

in

L

0

are the solutions of (3.9) or, equivalently, of (3.7), sin e

T ˜

U

= ˜

U

⇔

U

˜

(a) = min

x

6=a

max(− ˜

U

(x), −R(x, a)),

∀a ∈ X.

Lemma 2. If

K

is a

R

-determined dominant kernel of

τ G

and

U

˜

∈ L

0

is su h that

K

U

˜

= K

, then

T ˜

U

∈ L

0

and

K

T ˜

U

= K

. Moreover, for all

a

∈ K

U

˜

, we may ompute

U

˜

(a) as follows:

T ˜

U

(a) =

min

x

6=a; x∈K

U

˜

−R(x, a)

!

∧

min

x

6∈K

U

˜

max(− ˜

U

(x), −R(x, a))

!

(4.1)

and for all

a

6∈ K

U

˜

, wehave:

T ˜

U

(a) = min

x

∈K

U

_˜

max(− ˜

U

(x), −R(x, a))

(4.2)

•

for

x

∈ K

,

R(x, a) < 0

and thus, for

x

∈ K

,

max(¬ ˜

U

(x), −R(x, a)) =

−R(x, a)

(this alsoproves that(4.1) is equivalentto (3.9),for

a

∈ K

);

•

forall

x

6∈ K

,

max(− ˜

U

(x), −R(x, a)) ≥ − ˜

U

(x) > 0

.

If

a

6∈ K

,

T ˜

U

(a) < 0

be ause there is

x

∈ K

su h that

R(x, a) > 0

; for su h an

x

,

max(− ˜

U

(x), −R(x, a)) < 0

and hen e

U

˜

(a) < 0

.Moreover, for all

x

not in

K

,

max(− ˜

U

(x), −R(x, a)) > 0

, hen e (4.2) isvalid.

Corollary 2. Let

Y

˜

∈ L

0

. The fun tion

Y

˜

is a solution of equation system (3.9) (or equivalently of (3.7)),i

Y

˜

satises

˜

Y

(a) =

min

x

6=a; x∈K

Y

_˜

−R(x, a)

!

∧

min

x

6∈K

Y

_˜

max(− ˜

Y

(x), −R(x, a))

!

,

(4.3) for all

a

∈ K

˜

Y

, and

˜

Y

(a) = min

x

∈K

Y

˜

max(− ˜

Y

(x), −R(x, a)),

(4.4) for all

a

6∈ K

˜

Y

. Proof.

[⇒]

Lemma1 indi ates that

τ ˜

Y

= K

˜

Y

is a

R

-determineddominant kernel of

τ G

; therefore

Y

˜

fulllsthe hypotheses of lemma2 yielding the result.

[⇐]

Weprovethat

K

˜

Y

isa

R

-determineddominantkernelof

τ G

.Inviewof (4.3)andthefa tthat

Y

˜

(a) > 0

whenever

a

∈ K

˜

Y

,wededu ethat

R(x, a) < 0

for all

a, x

∈ K

˜

Y

. Similarly, for all

a

6∈ K

˜

Y

, we dedu e from

Y

˜

(a) < 0

and equation (4.4) that there exists at least one

x

in

K

˜

Y

su h that

R(x, a) > 0

. Hen eforth,

K

˜

Y

isa

R

-determineddominantkernelof

τ G

.Usingthefa tthat

K

Y

˜

is a kernel of

τ G

, lemma 2 yields that

T ˜

Y

an be omputed using (4.1) and (4.2); by hypothesis,

Y

˜

satises (4.3) and (4.4); therefore

Y

˜

is a xpoint of

T

,hen e

Y

˜

belongs to

Y

dom

0

(G

L

)

.

4.2 Obtaining a

L

-dominant kernel from a

R

-determined dominant kernel of

τ G

Weusetransformation

T

toobtaina

L

-dominantkernelfroma

R

-determined dominant kernel of

τ G

. The main result we need is the fa t that

T

respe ts the relation at least as sharp as on any subset of

L

-fun tions

Y

˜

su h that

τ ˜

Y

is a

R

-determined dominantkernel.

Lemma 3. Let

K

bea

R

-determineddominantkernel of

τ G

and

U , ˜

˜

V

∈ L

0

be su hthat

K

˜

Proof. Forany

a

∈ K

,

T ˜

V

(a) =

min

x

6=a; x∈K

V

_˜

−R(x, a)

!

∧

min

x

6∈K

V

_˜

max(− ˜

V

(x), −R(x, a))

!

.

Sin e the fun tions

min

and

max

are nonde reasing and for all

x

6∈ K

,

− ˜

V

(x) ≥ − ˜

U

(x)

, weget

T ˜

V

(a) ≥ T ˜

U

(a)

.

Forany

a

6∈ K

,

T ˜

V

(a) = min

x∈K

˜

V

max(− ˜

V

(x), −R(x, a))

.Sin ethe fun tions

min

and

max

are nonde reasing and for all

x

∈ K

,

− ˜

V

(x) ≤ − ˜

U

(x)

, we get

T ˜

V

(a) ≤ T ˜

U

(a)

.

Startingfromthe hara teristi fun tion of a

R

-determineddominant kernel of

τ G

and applying iteratively

T

to this fun tion, one obtains, after a nite number of iterations, anelement of

Y

dom

0

(G

L

)

that is at least as sharp asany other element

Y

˜

of

Y

dom

0

(G

L

)

su h that

K

˜

Y

= K

.

Proposition3.Let

K

bea

R

-determineddominantkernelof

τ G

and

U

˜

∈ L

0

be dened by

U

˜

(a) = m

if

a

∈ K

and

−m

otherwise. For some nite integer

n

,

T

n

˜

U

= T

n+1

_U

˜

and

Z

˜

= T

n

˜

U

isanelementof

Y

dom

0

(G

L

)

su hthat

K

Z

˜

= K

; we have furthermore

Y 4 ˜

˜

Z

for all element

Y

˜

of

Y

dom

0

(G

L

)

with

K

˜

Y

= K

.

Proof. The fun tion

U

˜

dened in the proposition belongs to

L

0

. Sin e, by lemma 2,

K

T ˜

U

= K

U

˜

, it is obvious from the denition of

U

˜

that

T ˜

U 4 ˜

U

. Applying lemma 3, we get

T

i+1

_{U 4 T}

˜

i

˜

U

, for all

i

. Due to the niteness of

L

and

X

, there is a nite number

n

su h that

T

n+1

_U

˜

is not stri tly less sharp

than

T

n

˜

U

, whi h means that

T

n+1

_U

˜

_{= T}

n

˜

U

. For su h a

n

,

Z

˜

= T

n

˜

U

is an element of

Y

dom

0

(G

L

)

. It is moreover a maximal element w.r.t. the relation

4

in the subset of elements

Y

˜

of

Y

dom

0

(G

L

)

su h that

K

˜

Y

= K

. Let

Y

˜

be su h an element. We have

T ˜

Y

= ˜

Y

and

Y 4 ˜

˜

U

. Applying lemma 3, we get

T

n

˜

Y 4 T

n

˜

U

, whi h means that

Y 4 ˜

˜

Z

.

It was not evident atrst sight that, for all

R

-determined dominant kernels

K

of

τ G

,therearesolutions

Y

˜

of(3.9)with

K

Y

˜

= K

.Itwasnotevidenteither that the set of solutions

Y

˜

of (3.9) with

K

˜

Y

= K

, when non-empty, admits a uniquemaximallysharpelement.Puttingtogetherlemma1andproposition3,

weestablishtheexisten eofabije tive orresponden ebetween

R

-determined dominantkernels of

τ G

and determined

L

-dominantkernels of

G

L

.

Theorem 1.

K

is a

R

-determined dominant kernel of

τ G

i there is a de-termined

L

-dominantkernel

Y

˜

su hthat

K

˜

There is a link between the quali ation

Q

i-dom

(K)

of a dominant kernel

K

of

τ G

and the sharpness quali ation of all elements

Y

˜

of

Y

dom

0

(G

L

)

su h that

K

Y

˜

= K

. We rst prove that the degree of sharpness of an element of

Y

dom

0

(G

L

)

isnotlargerthanthequali ationofthe orresponding

R

-determined dominantkernel.

Lemma 4. If

K

is a

R

-determined dominant kernel of

τ G

, any determined solution

Y

˜

of thekernelequationsystem (3.7)issu hthatthe hoi e

K

˜

Y

= K

in

τ G

veries:

Q

sharp

( ˜

Y

) ≤ Q

i-dom

(K)

(4.5)

Proof. Assumethat

Q

sharp

_{( ˜}

_Y

_{) > Q}

i-dom

(K)

.Let

Q

sharp

_{( ˜}

_Y

_{) = r}

and

Q

i-dom

(K) =

k

,with

r > k >

0

. Thequali ation

Q

i-dom

(K) = k

isequal eitherto

−R(b, a)

for some

a, b

∈ K

or to

max

x∈K

R(x, y)

, for some

y

6∈ K

.

In the former ase, using (4.3), weget

r

≤ ˜

Y

(a) ≤ −R(b, a) = k

,a ontradi -tion.

In the latter ase, there is

a

∈ K

su h that

R(a, y) = max

x

∈K

R(x, y) =

k

. Therefore, for all

x

∈ K

,

R

(x, y) ≤ k

. Using (4.4), we have

Y

˜

(y) =

min

x

∈K

Y

˜

max(− ˜

Y

(x), −R(x, y))

.Forall

x

∈ K

,

− ˜

Y

(x) ≤ −r ≤ −k ≤ −R(x, y)

††

;

therefore

Y

˜

(y) ≥ −k

. But, by denition of

Q

sharp

_{( ˜}

_Y

₎

,

Y

˜

(y) ≤ −r

hen e

−r ≥ ˜

Y

(y) ≥ −k

,a ontradi tion.

Wenallyprovethatthe sharpnessofthe

L

-dominantkernel

Y

˜

orresponding toa

R

-determined dominant kernel

K

is equal to the quali ation of

K

. Theorem 2. For any

R

-determined dominant kernel

K

, the

L

-dominant kernel

Z

˜

orrespondingto

K

is su h that

Q

sharp

_{( ˜}

_Z

_{) = Q}

i-dom

(K)

.

Proof. In view of lemma 4, it su es to prove that

Q

sharp

_{( ˜}

_Y

_{) ≥ Q}

i-dom

(K)

. Let

Q

i-dom

(K) = k

, with

k >

0

.Dene the

L

-fun tion

W

˜

∈ L

0

as

˜

W

(a) =











k

if

a

∈ K

−k

if

a

6∈ K.

Bylemma 2,

T ˜

W

belongsto

L

0

and

K

T ˜

W

= K

W

˜

= K

. Using(4.1) and (4.2),

††

we showthat

W 4 T ˜

˜

W

. Indeed, for

a

∈ K

,

T ˜

W

(a) =

min

x

6=a;x∈K

W

˜

−R(x, a)

!

∧

min

x

6∈K

W

˜

max(− ˜

W

(x), −R(x, a)) ≥ k

!

,

and, for

a

6∈ K

,

T ˜

W

(a) = min

x

∈K

W

˜

max(− ˜

W

(x), −R(x, a))

= max(−k, min

x

∈K

W

˜

−R(x, a))

≤ −k.

Rememberthedenitionof

U

˜

∈ L

0

that wasused inproposition3:

U

˜

(a) = m

if

a

∈ K

and

−m

otherwise.Wehave

W 4 ˜

˜

U

and,usinglemma3,

T

i

˜

W 4 T

i

˜

U

, for all

i

= 1, 2, . . .

. Sin e, for some

n

,

T

n

˜

U

= ˜

Z

, the

L

-dominant kernel orresponding to

K

, we have:

˜

W 4 T

n

W 4 T

˜

n

U

˜

= ˜

Z

and, using (3.14),

Q

sharp

_{( ˜}

_W

_{) = k ≤ Q}

sharp

_{( ˜}

_Z

₎

. 4.4 Remarks

The above results promptthe followingremarks.

(1) Theundetermined value

0

,whensu havalue appearsinthe set

L

,plays aspe ialrole andhas tobe arefullydealtwith.Inlemma2forinstan e,

if we either drop the restri tion that the dominant kernel

K

of

τ G

is a

R

-determined one or that

U

˜

(x) 6= 0

for all

x

∈ X

, we an no longer make sure that

T ˜

U

(a) > 0

for all

a

∈ K

. This means that the trans-formed hara teristi fun tion

T ˜

U

would not satisfy

K

T ˜

U

= K

U

˜

. Sin e we are interested in determined solutions of the kernel equation system,

wedidnotpay attentiontoundetermined solutions,and orrelatively,we

onsidered onlythe

R

-determined dominant hoi es of

τ G

.

(2) Throughoutthis paper,wehaveassumed thatboththeset

X

andtheset

of the real numbers (for instan e the interval

[−1, 1]

), the onstru tion of the xpoint asin proposition 3 stillworks but the onvergen e might

take aninnite number of steps.

5 Con lusion

This paper aimsat exploringthe relationshipbetween independent dominant

(resp. absorbent) hoi es and solutions of the kernel equation system in the

ase of valued relations.Wehaveshown thatthere is aone-to-one

orrespon-den ebetween

L

-dominant(resp.

L

-absorbent)kernels ofa valued graphand the (

R

-determined) independent dominant (resp. absorbent) hoi es of the stri t median ut digraph

τ G

. Furthermore, starting with an independent dominant (resp. absorbent) hoi e of

τ G

, the te hnique of proof provides a way of building the orresponding

L

-dominant (resp.

L

-absorbent) kernel as the xpointofatransformation.This has animportant onsequen efromthe

algorithmi point of view: nding

L

-valued kernels is not more di ult than nding the kernels of a risp graph. Indeed, starting with a (

R

-determined) kernel of

τ G

and applying the transformation (denition 10) at most

m|X|

times yields the xpoint.Of ourse,this does not makethe determinationof

the

L

-valued kernels an easy task sin e we know that nding all kernels of a risp relation is a omputationally hard problem (this was proven by V.

Chvátal; see Garey and Johnson(1979), p.204).

Theorem2alsohasinterestinitsown:thedegreeofsharpnessofa

L

-dominant kernel

Y

˜

isequal tothe degreeof potentialgoodness

Q

i−dom

of the asso iated

risp kernel

K

˜

Y

.The omparison of risp kernels (or equivalently of indepen-dentdominantorabsorbent hoi es of

τ G

)done attheend ofse tion4about examples3 and 4 an thus be transposed to

L

-valuedkernels.

Referen es

BangJensenJ.andGutin,G.,Digraphs:Theory,algorithmsandappli ations.

SpringerVerlag, London, 2001.

Berge, C., Théorie des graphes et ses appli ations.Dunod,Paris, 1958.

Berge, C., Graphes et hypergraphes. Dunod,Paris, 1970.

Bisdor,R.andRoubens,M.,Ondening fuzzykernelsfrom

L

-valuedsimple graphs.In:Pro eedings Information Pro essing and Management of

Un er-tainty,IPMU'96,Granada, July1996, 593599.

Bisdor,R. and Roubens, M.,On dening and omputing fuzzy kernels from

S ienti Publishers, Singapoure,1996, 113123.

Bisdor,R., On omputing kernelsfrom

L

-valuedsimple graphs.In: Pro eed-ings 5th European Congress on Intelligent Te hniques and Soft Computing

EUFIT'97, (Aa hen, September 1997),vol. 1,97103.

Bisdor,R.,Logi alfoundationoffuzzy preferentialsystemswith appli ation

tothe Ele tre de isionaid methods, Computers & Operations Resear h,

27,2000, 673687.

Bisdor, R., Ele tre-like lustering from a pairwise fuzzy proximity index.

European Journal of Operational Resear h, 138/2, 2002,320331.

Bisdor, R. and Roubens, M., Le temps des noyaux .... In: Bair, J. and

Henry, V. (ed.) Regards roisés sur les méthodes quantitatives de gestion.

LesÉditionsde l'Université de Liège, Liège (Belgium),2004, pp. 4154.

Garey,M.R. and Johnson, D.S., Computers and intra tability, A guide to the

theory of NP-Completeness. Freemanand Co., New York, 1979.

Haynes, T.W.,Hedetniemi,S.T.andSlater,P.J.,Fundamentalsofdomination

in graphs.Mar el Dekker In . New-York, 1998.

Kitainik,L.,Fuzzyde ision pro edures withbinary relations:towards aunied

theory. KluwerA ademi Publ.,Boston, 1993.

Riguet, J., Relations binaires, fermetures, orrespondan es de Galois. Bull.

So . Math. Fran e, 76, 1948,114155.

Roy,B.,Classementet hoixen présen ede pointsde vuemultiples.RIRO 8,

5775.

Roy, B., Algèbre moderne et théorie des graphes. Dunod, Paris, 1969.

Roy, B., Méthodologie Multi ritère d'aide à la dé ision. E onomi a, Paris,

1985.

Roy, B. and Bouyssou, D., Aide multi ritère à la dé ision: Méhodes et as.

E onomi a,Paris, 1993.

S hmidt,G. and Ströhlein,Th., On kernels ofgraphs and solutionsof games:

a synopsis based on relations and xpoints. SIAM, J. Algebrai Dis rete

Methods, 6, 1985, 5465.

S hmidt, G. and Ströhlein, Th., Relationen und Graphen. Springer-Verlag,

Berlin,1989.Englishversion:Relationsandgraphs,SpringerVerlag,Berlin,

1993.

vonNeumann, J.and Morgenstern,O., Theory of gamesand e onomi