• Practical additions to the axioms:c3. c is a continuous functionc4. c is involutive

Usually,

1,01,0: c

1,0xA 1,0xA

xcx AA

0110 cc bcacbaba 1,0,

xPxPC ~~: ACA xc

4c3c

2,1 cc

and

• Axioms (skeleton):

c1.

c2.

(boundary conditions)

(monotonicity)

• A family of functions C satisfy c1,c2

• Complement

1,0, aaacc

tafortafor

ac01

1,01,0

ta

0

1

1 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1

.1

.2

.3

.4

.5

.6

.7

.8

.9

0

1

ac

aa

ac

aac cos121

1)

2)

satisfies c1,c2

t=threshold

satisfies c1,c2,c3

• Some examples:

aaac

11 ,1

www aac

1

1 ,0w

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1

.1

.2

.3

.4

.5

.6

.7

.8

.9

0

1

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1

.1

.2

.3

.4

.5

.6

.7

.8

.9

0

1

0

2

10

5.9.

5.w

1w

2w

5w

Classical complement

3) (M. Sugeno)

4) (R. Yager)

• More examples:

(e is equilibrium,

eec 5.01 eee

Not increasing!

Increasing!

ae

e=c(e)

c(a)

c

c

eaacaeaaca

e!

1,0e

c ce

c

Every fuzzy complement hasat most one equilibrium.Illustration:

If

If

has an equilibrium then

satisfies c3 (is continuous) then

)

• Equilibrium of complement:for efor classical fuzzy complement

011

021

if

ifec

0

1

1-1 2 3 4 5 6w

0

w

cwe

1

21

wewc

ce

“ Yager “:

• Equilibrium of Sugeno-complement

(with respect toismembership ad

c a acaaac dd ac

c ad! 1,0a If

If ce ccd ee

For 1,0a aaccacad Example: Yager-c

2

Involutive!

ia00

2ce

1

1a

aac di

iac

acaacc dii

acw

2ce 2

satisfies c3 then

(e is equilibrium) then

• Dual point:If given a complement and any membership degree then the

for which ‘s dual point)

• Axiomatic skeleton:

1,01,01,0: t xxtx BABA ,

]1,0[1, aaatt1. (boundary conditions)t2. ]1,0[,,, baabtbat (commutativity)t3.t4.

]1,0[,,,, cbacatbatcb (monotonicity)(associativity) ]1,0[,,,,,, cbacbtatcbatt

• Some usual restrictions (practical motivation)t5.t6a.t6b.t7.

t is a continuous function

aaat ,(idempotence)(subidempotence)

• Intersection (t-norm)

aaat , ]1,0[',',,',','' bababatbatbbaa

1,01,01,0: s xxsx BABA ,

• Axiomatic skeleton:

s1.s2.s3.s4.

(boundary conditions)

]1,0[,,, baabsbas ]1,0[,,,, cbacasbascb

]1,0[,,,,,, cbacbsascbass

(commutativity)(monotonicity)(associativity)

• Some usual restrictions:

s5.s6a.s6b.s7.

s is a continuous function

aaas , (idempotence)(superidempotence)

• Union (t-conorm, s-norm)

]1,0[0, aaas

aaas , ]1,0[',',,',','' bababasbasbbaa

• Intersection:1.2.3.4.

• Union:

• Some examples:

),min(, babat abbat , )1,0max(, babat

otherwiseaifbbifa

bat,0

1,1,

,min

1.2.3.4. ),max(, babas

abbabas , ),1min(, babas

otherwiseaifbbifa

bas,1

0,0,

,max

R e f e r e n c e F u z z y U n i o n ss - n o r m

F u z z y I n t e r s e c t i o n st - n o r m

R a n g e o fP a r a m e t e r

S c h w e i z e r &S k l a r [ 1 9 6 1 ] ppp ba

1

111,0max1 ppp ba1

1,0max ,p

H a m a c h e r[ 1 9 7 8 ]

ab

abba

11

2 abba

ab 1

,0

F r a n k [ 1 9 7 9 ]

1111log1

11

sss ba

s

1111log

sss ba

s ,0s

Y a g e r [ 1 9 8 0 ]

www ba

1

,1min

www ba

1

11,1min1 ,0w

D u b o i s &P r a d e [ 1 9 8 0 ]

,1,1max

1,,minbabaabba

,,max baab 1,0

D o m b i [ 1 9 8 2 ] 1

11111

1

ba

1

11111

1

ba

,0

Include algebraic norms: abba aband

• Some classes of fuzzy set unions and intersections

s t

h1.

1,01,0: nh 2n xxxhx

nAAAA ,...,,21

00,...,0,0 h

Very general axiomatic requirements:

11,...,1,1 h(boundary conditions)

h2. for arbitrary andia ib nNi niniii NibhNiahbai (monotonicity)

• Practical additions

h3. h is a continuous functionh4. h is symmetric for all the arguments

hiphi NiahNiah

ip : arbitrary permutation

• Aggregation operations

... dcbadcbadcba

:,...,, 21 naaaFor given

? maxs

naaa ,...,,max 21

Unions

naaa ,...,,min 21

Intersections

mint

? is the area of averaging operations

nnn aaaahaa ,...,max,...,,...,min 111 Generalized means:

1

211

...,...,

naaaaah n

n 0R

satisfies h1-h4h

• Union & intersection can be extended to n-ary operations because of associativity:

naaah ,...,,min 21

naaah ,...,,max 21

nnaaah ...210

naaah n

...21

1

naaa

nh1...11

21

1

• If h4 is not necessary (diff. importance of arguments)

1

1

n

iii

w awh so that 11

n

iiw

• Generalized means:

min max maxumini

Averaging operations

Union operations

Intersection operations

0

0

w

p

0

0

w

p

Generalized means

Yager Yager

Schweizer/SklarSchweizer/Sklar

Dombi Dombi

• Various classes of aggregation operations